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Model-based durability design of concrete structures in Hong Kong–Zhuhai–Macau sea link project
Structural Safety 53 (2015) 1–12
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Structural Safety journal homepage: www.elsevier.com/locate/strusafe
Model-based durability design of concrete structures in Hong Kong–Zhuhai–Macau sea link project Quanwang Li a, Kefei Li a,⇑, Xingang Zhou b, Qinming Zhang a, Zhihong Fan c a
Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Civil Engineering Department, Tsinghua University, Beijing 100084, China Civil Engineering College, Yan-tai University, Yantai 264005, China c CCCC Fourth Harbor Engineering Institute, Guangzhou 510230, China b
a r t i c l e
i n f o
Article history: Received 23 April 2013 Received in revised form 19 June 2014 Accepted 13 November 2014
Keywords: Concrete structures Durability design Chloride ingress Probability Reliability Structural engineering
a b s t r a c t This paper presents the durability design of concrete structures in the Hong Kong–Zhuhai–Macau (HZM) sea link project for a design service life of 120 years. Among all the durability issues, this paper focuses on the model-based durability design of concrete structures in the project subject to the action of marine chlorides. The mechanism and modeling of chloride penetration into concrete are reviewed in depth and the applicability of models for durability design is evaluated. On the basis of long-term exposure tests and in-place structural investigations during the past 30 years in South-eastern China, the design model for concrete structures in the HZM project is established. The statistical properties of model parameters are analyzed in details and compared to other chloride ingress models. Using the established HZM model for chloride ingress and specified durability limit state (DLS), the design parameters are first evaluated through a fully probabilistic analysis in terms of service lives (50 years, 120 years) and target reliability levels (b = 1.3, 1.5, 1.8). The partial factor format for durability design is then established and the partial factors are calibrated from the first order reliability method (FORM) from specified service life and specified reliability level. The design results from the partial factor design equation are given for different exposure zones, and the achieved reliability index is verified for each design through the full probabilistic method. Finally remarks are given on the HZM model and its application in durability design of RC elements. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction When elements of concrete structures are exposed to a marine environment, chloride ions, either air-borne or in sea water, can penetrate the concrete cover and induce the corrosion of reinforcement steel bars [1]. In the recent decades, considerable efforts have been dedicated to the mechanisms and the modeling of chloride ingress and the subsequent steel corrosion processes [2]. The chloride ions can penetrate into concrete elements through different mechanisms in marine environment: pore diffusion, capillary suction, permeation under pressure as well as surface deposit of airborne salts [3,4]. The available modeling on chloride ingress can be divided into two families: the simplified Fick’s diffusion model and more sophisticate physical models. The Fick’s model uses the analytical solution of second Fick’s law with constant boundary conditions to represent the chloride ingress [5]. This model presents a very simplified image of the real transport processes of ⇑ Corresponding author. Tel./fax: +86 10 6278 1408. E-mail address: [email protected] (K. Li). http://dx.doi.org/10.1016/j.strusafe.2014.11.002 0167-4730/Ó 2014 Elsevier Ltd. All rights reserved.
chloride ingress, and envelops all these processes into the wellknown ‘‘apparent’’ chloride diffusion coefficient [6]. Nowadays, more mechanism-based physical models are also available: multi-species models were proposed for chloride transport in pore solution as a concentrated electrolytic solution [7,8]; the thermal effect was considered through thermal-ionic couplings [9]; the non-stationary boundary was modeled by Uji et al. [10]; Li and Li [11] developed a hydro-ionic model to consider specially the drying-wetting actions; the sorption behavior was investigated for chloride transport in unsaturated concretes [12]; the material microstructure aspects and cement hydration processes were incorporated into the chloride diffusivity modeling through upscaling techniques [13,14] and percolation consideration [15]. In principle, these models can all be used in the durability design for chloride ingress of concrete elements provided that they yield reliable results. The reliability of such models comes from two fundamental aspects: firstly the model should be correct or basically correct in a physical sense, and secondly, the model should be robust or tolerant enough to incorporate the randomness of durability processes in design [16]. From the point of view of
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Q. Li et al. / Structural Safety 53 (2015) 1–12
design, a model is acceptable as long as the associated uncertainty is reasonable and can be quantified. By its simplicity and the capacity of being adapted to different exposure cases, the Fick’s model is often preferred in model-based durability design [17,18] and recommended by codes and standards [19,20]. Moreover, long-term monitoring of the apparent chloride diffusion coefficient in real structures [21,22] and the standardization of its determination through chloride profiling method [23] make Fick’s model appropriate for service life design and prediction [24,25]. This approach has been shown to be capable for achieving service life design in marine chloride environments for a target reliability level [26,27]. However, using the Fick’s model in durability design for an expected safety margin (reliability level) necessitates determining the parameter uncertainties since these parameters are both strongly environment and material dependent. The Hong Kong–Zhuhai–Macau (HZM) sea link project is currently under construction. The HZM project includes sea bridges of 28.8 km (three navigable spans), two artificial islands and an immerged tube tunnel of 6.8 km, with a total investment of nearly 12 billion US dollars. The project is situated on the South-eastern coast of the China mainland. One of the technical challenges of HZM project was to achieve the service life of 120 years for the concrete structures in an aggressive marine environment. The concrete structures in the project include the piers, bearing platforms and piles for sea bridges, the tunnel tube segments for the immersed tunnel, and retaining walls and breakwaters for the artificial islands. The annual average temperature at the project site is between 22.3– 23.1 °C, the annual average humidity is 77–80% and the annual average wind speed is between 3.1–6.6 m/s. The analysis of the sea water chemistry show the chloride (Cl) concentration is within 10,700–17,020 mg/L, the sulfate (SO2 4 ) concentration is within 1140–2260 mg/L, and the total salinity is between 25.4 and 32.9 (sea bottom). Under this marine environment the possible durability processes for reinforced concrete (RC)/prestressed concrete (PC) elements were identified as the carbonation-induced corrosion of steel bars, chloride-induced corrosion of steel bars, salt attack on structural concrete as well as internal expansion reactions. From a criticality analysis of these processes in the preliminary study phase, the chloride-induced steel corrosion of RC/PC elements was identified as the controlling process of durability design. Thus, a model-based approach was taken for the durability design, and the uncertainty of model parameters was taken into account through analysis of their statistical properties. Their statistical properties were regressed on the basis of the exposure test results under the same marine environment conditions as the HZM project and structural investigations conducted in the past 30 years. This paper presents the statistical analysis of parameters in the design model, i.e., Fick’s second law. Then, both fully probabilistic and partial factor design formats are given for different reliability levels for durability. The results form the basis for the choice of structural and material parameters for durability of concrete structures in HZM project. This paper is organized as follows. Section 2 is dedicated to the analysis of statistical properties of model parameters on the basis of long term real structure investigation as well as exposure tests. The durability design through the fully probabilistic approach is presented in Section 3, while the durability design in a partial factor format is detailed in Section 4. The concluding remarks are given in the end.
2. HZM model for chloride penetration 2.1. Basic model The empirical Fick’s model assumes a pure diffusion process for chloride ingress into concrete [5]. With a constant chloride concen-
tration Cs imposed on the surface, the profile of chloride concentration C(x,t) along the depth x and with time t is,
x Cðx; tÞ ¼ C s 1 erf pffiffiffiffiffiffiffiffiffiffi 2 DC1 t
ð1Þ
where DCl is the apparent chloride diffusion coefficient in concrete and erf the error function. This apparent diffusion coefficient is found to be time-dependent [28], and a power law is recommended for its aging behavior [29],
DC1 ðtÞ ¼ D0C1
n t0 ¼ D0C1 gðt0 ; t; nÞ t
ð2Þ
with D0Cl = diffusion coefficient at concrete age t0, n = exponent, and g = aging factor of diffusion coefficient DCl. Physically, the decrease of DCl with time is related to the long-term evolution of the microstructure of concrete materials. Since DCl also includes other transport processes than diffusion, the aging coefficients, n or g, are different for different exposure zones in marine environments [30]. Although the power law in Eq. (2) is well supported by in-field measurements, it is not rational to assume that DCl will decrease infinitely with time. Thus, for durability design in the HZM project, the power law is truncated at the end of 30 years and it is assumed that the DCl will be stable after 30 years’ exposure, i.e.,
gðt0 ; t; nÞjt>tD ¼ gðt0 ; tD ; nÞ with tD ¼ 30 years
ð3Þ
This assumption leads to a faster predicted rate for chloride ingress and thus a conservative durability design. For service life design in the HZM project, the durability limit state (DLS) is specified as the corrosion initiation state, usually denoted by a critical (threshold) chloride concentration on the steel surface Ccr [31]. Thus the design equation, for a target service life tSL, for chloride ingress becomes,
2
0
13
xd B C7 6 G ¼ C cr C s 41 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5 ¼ 0 0 2 DC1 gðt0 ; t SL ; nÞt SL
ð4Þ
where xd is the thickness of concrete cover for reinforcement steel. Note that the aging coefficient g in Eq. (4) should equal Eq. (3). Thus this design model contains five parameters Ccr, Cs, xd, D0Cl, g (or n) to be defined for a given target service life tSL. Except for xd, all parameters depend on both environmental exposure conditions and concrete materials. However, in durability design it is assumed that the three parameters Ccr, Cs, g (or n) can be determined through parameter analysis while xd, D0Cl are specified for a given service life and exposure condition. 2.2. Statistical properties of parameters The statistical properties of model parameters are analyzed on the basis of the long-term in-place structural investigations and exposure tests during the last 30 years in the South-eastern coastal regions of China. The exposure tests have been performed at the Zhanjiang Exposure Station where the marine environment is very similar to that at the site of HZM project. 2.2.1. Concrete cover x The concrete cover is the only structural parameter in the design equation, Eq. (4). The thicker the concrete cover, the later the chloride ingress will initiate the corrosion in the steel. For this reason, correct specification of concrete cover thickness as well as its chloride ingress resistance (DCl) is the central issue for durability design under chloride ingress [32–34]. The statistical properties of concrete cover thickness are important for correct estimation of reliability with respect to Eq. (4), and are related closely to the construction methods and practice: stricter control results in smaller
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deviation, while poor construction practice yields a larger deviation. For the durability design of the HZM project, statistical analysis is performed on the data of achieved cover thickness of marine concrete structures constructed after 1996. A total of 24 concrete structures were investigated and 1451 concrete cover thicknesses were measured. The design values of cover thickness are from 40 to 80 mm, and the mean values and standard deviations are evaluated for the achieved concrete covers. Statistical analysis showed that the cover thickness can be described by a normal distribution. Fig. 1 presents histograms for cover thickness for two marine docks constructed in 1997 and 1999. It can be seen that the histogram of ‘measured’ cover thickness is in good agreement with the ‘modeled’ histogram obtained by normal distribution modeling with the measured mean values (M) and standard deviations (SD). For all the achieved thickness data, the average SD is 5.3 mm. 2.2.2. Surface chloride concentration Cs The surface chloride concentration Cs is noted as the mass ratio of chloride with respect to concrete binder. Actually Cs in the design model Eq. (4) refers to the apparent surface concentration, i.e., the value is not from a direct measurement on the concrete surface but from the regression of the chloride concentration profile at a certain exposure age using Fick’s law. This regression provides, for exposure age t, the surface concentration Cs and the apparent diffusion coefficient DCl(t) simultaneously. The surface concentration Cs depends on the sea water salinity and its chloride concentration, but also on the chloride absorption capacity of concrete materials [35]. This concentration is usually expressed as Cs = A(w/b), where the coefficient A can be regarded as environment-dependent and is regressed for different concretes [17–19]. Since the accumulation of surface chloride needs some time, in scale of years, to reach a stable value, Cs is also found to be timedependent [19,36]. To the best of the authors’ knowledge, these two aspects have never been combined together in the literature, but for the HZM project this aspect has to be considered, because the support data of Cs values come from measurements during last 10 years and a correct estimation of stable values for Cs is important to guarantee the safe design for durability. This time-dependent Cs can be expressed as,
C s ðtÞ ¼ Aðw=bÞ
at 1 þ at
ð5Þ
where the coefficient a describes the influence of time t on Cs. This law for surface chloride build-up takes into account both the kinetics and the asymptotic behavior of chloride build-up. Considering the entire service life tSL, the surface chloride concentration can be averaged using Eq. (5) as,
0.30
0.35
Measurement Modeling
M=50.3 mm SD=4.7 mm
0.20 0.15 0.10
1 t SL
0.30
Measurement Modeling
M=70.5 mm SD=5.7 mm
0.25 0.20 0.15 0.10
0.05
0.05
0.00
0.00 <42 42-45 45-48 48-51 51-54 54-57 57-60 >60
Cover thickness (mm)
(a)
ð6Þ
where A0 depends on the environment coefficient A, time coefficient a and service life tSL. This sub-model is used to analyze the data on Cs from 351 specimens selected from the Zhenjiang Exposure Station. The Cs values for the specimens of w/b = 0.35 incorporating fly ash (FA) and slag (SG) as supplementary cementitious materials (SCM) are presented in Fig. 2 for tidal and splashing zones. Both the mean values and dispersions of Cs are illustrated on the figure: the mean values are represented by the regressed pointed lines and the dispersion is illustrated by the scattering points in the same exposure age. It can be found that the stabilization of Cs value is longer in splashing zones, i.e., 7 years, compared to the tidal zones, i.e., within 2 years. The asymptotic mean values for chloride surface concentration Cs0 for tSL are then calculated via Eq. (6) and presented in Fig. 2 by horizontal lines. These values will be used as characteristic value of Cs for durability design of concrete structures in HZM project. For comparison, the Cs values suggested from DuraCrete [18] are also illustrated by horizontal lines, showing that the DuraCtete values underestimate the Cs values for the HZM project. The preliminary proportioning of structural concrete for the HZM project has w/b = 0.35, and its binder has a ternary composition: ordinary Portland cement (50%), FA (20%) and SG (30%). For the exposure site, only a limited number of such concrete specimens are available and their values are marked as ‘‘+’’ in Fig. 2. It can be seen that the Cs values of FA concrete are systematically higher than for SG concrete, while the Cs values of the ternary-binder concrete are in between. The apparently higher Cs values from FA concretes result from the densification effect of secondary hydration between fly-ash particles and the cement hydrates, creating more effective adsorbents for surface chloride ions. Thus for the HZM project, the higher Cs values of FA concrete are conservatively retained for durability design for tidal and splashing zones. For the immerged zone, the analysis of Cs values utilizes a limited number of ternary binder concrete specimens. The available values are presented in Fig. 3, where they are compared with the values from DuraCrete [18]. Again, the DuraCrete values tend to underestimate the Cs values for the HZM project. For the atmospheric zone, no exposure data are available; thus, the recommended values from DuraCrete are used. Note that the Cs mean value suggested by DuraCrete is increased by 30% to account for the possible higher surface concentration of ternary binder concretes suggested from the comparison in Figs. 2 and 3. The statistical properties of Cs (Cs0 ) values are summarized in Table 1.
Frequency (-)
Frequency (-)
0.25
C 0s ¼
tSL w w lnð1 þ at SL Þ ¼ A0 C s ðtÞdt ¼ A 1 b at SL b 0
Z
<60
60-65 65-70 70-75 75-80 Cover thickness (mm)
>80
(b)
Fig. 1. Histograms of concrete cover thickness: (a) Dock in 1997 (280 points), (b) Dock in 1999 (90 points).
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Q. Li et al. / Structural Safety 53 (2015) 1–12
(a)
(b)
(c)
(d)
Fig. 2. Surface chloride concentrations: (a) FA concrete in splashing zone, (b) FA concrete in tidal zone, (c) SG concrete in splashing zone, (d) SG concrete in tidal zone.
Table 1 Statistical properties of C’s values for HZM durability design (w/b = 0.35). Exposure zone
Atmospheric Splashing Tidal Immerged
Fig. 3. Surface chloride concentration in immerged zone.
2.2.3. Critical chloride content Ccr The physical meaning of Ccr is the chloride concentration accumulated at the steel surface capable to initiate the electrochemical reactions of corrosion, i.e., depassivation. This concentration depends on factors from steel material (corrosion resistance), concrete material (moisture content, pH value of pore solution, oxygen availability, electro-resistivity) and external environments (ambi-
Distribution
Log-Normal Log-Normal Log-Normal Log-Normal
Parameters Mean value (% binder)
Deviation (% binder)
2.0 5.4 3.8 4.5
0.31 0.82 0.58 0.68
ent temperature) [31]. Thus it is not surprising that Ccr values reported or recommended in literature vary in wide limits [17– 19,36]. This concentration should be calibrated for HZM project for reliable durability design. For HZM project the Ccr values are regressed from 68 exposure concrete specimens. The concrete materials cover several binder types and w/b ratios from 0.45 to 0.55. The Ccr values were measured on these specimens on broken tests continuously with time. For a broken specimen, the chloride concentration at steel surface was measured, and if the steel corroded the concentration was taken as Ccr. Thus, the measured Ccr values correspond to an already-corroded state of steel and the exact depassivation was surely between the two subsequent broken observations. These Ccr values obtained for splashing zones are presented in Fig. 4 and beta-distribution is used to describe the statistical properties [19]. The mathematical form of beta-distribution writes,
f ðxÞ ¼
a1 b1 Ux UL
Cða þ bÞ xL CðaÞ CðbÞ ðU LÞ U L
ð7Þ
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Q. Li et al. / Structural Safety 53 (2015) 1–12
Four parameters are needed for beta-distribution: two shape factors
a, b, and two distribution limits L (lower limit), U (upper limit). Due to the approximate nature of Ccr determination for a given broken test, the statistical parameters of Ccr are to be determined through optimizing a likelihood function. A likelihood function, L, is employed to determine these parameters,
L ¼
m Y i¼1
P½C cr < xi
n Y
P½C cr > yj
ð8Þ
j¼1
This function calculates the probability associated with the fact that steel bars in m specimens are corroded when chloride concentration reaches xi (i = 1, 2,. . .m) and steel bars in n specimens remain passivated when chloride concentration attains yj (j = 1, 2,. . .n). Maximizing the likelihood function L can obtain the parameters of betadistribution. The obtained distributions are presented in Fig. 4 for specimens exposed in splashing zones. It can be seen that Ccr values tend to augment with the decrease of w/b ratio. The w/b ratio of HZM structural concrete is 0.35 but no direct Ccr data are available for w/b = 0.35. Accordingly a synthetic analysis is performed on the basis of available Ccr statistical properties, and the beta-distribution for HZM structural concrete (w/b = 0.35) calibrated as: a = 0.22, b = 0.36, L = 0.45%, U = 1.25%. This beta-distribution is compared to the DuraCrete model [18] and fib model [19] in
(a)
Fig. 5, showing DuraCrete model values are systemically higher than fib model and the HZM Ccr values lie between. Compared to fib model, the calibrated HZM model for Ccr has similar distribution shape, larger value and has smaller variance. The Ccr values for tidal zone are conservatively taken as the same for splashing zone for HZM project. For atmospheric zone, the values recommended by DuraCrete [18] are retained. For immerged zone, the beta-distribution similar to splashing zone is retained and the mean value recommended by DuraCrete is adopted. Table 2 summarizes the statistical properties for Ccr values used for steel corrosion. 2.2.4. Chloride diffusivity D0Cl and aging exponent n The chloride diffusivity DCl in the design model is time-dependent and determined by diffusivity D0Cl at a given age t0 and the aging exponent n. These two parameters are closely related to material composition and environmental exposure conditions, Table 2 Statistical properties of Ccr values for HZM durability design. Exposure zone
Distribution
Parameters
Atmospheric Splashing/tidal Immerged
Log-Normal Beta Beta
M = 0.85%, SD = 0.13% a = 0.22, b = 0.36, U = 1.25%, L = 0.45% a = 0.23, b = 0.33, U = 3.50%, L = 1.00%
(b)
Fig. 4. Critical chloride concentrations in splashing zone: (a) w/b = 0.55 (b) w/b = 0.40.
(a)
(b)
Fig. 5. Comparison of Ccr distributions in splashing zone between HZM, DuraCrete and fib models for structural concrete with w/b = 0.35: (a) probability density distribution, (b) cumulative probability function.
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Q. Li et al. / Structural Safety 53 (2015) 1–12
both having important influence on the service life. During design process, D0Cl value is usually taken as a design unknown while the aging exponent n is specified as known value for different material compositions and exposure conditions. The aging law of DCl is retained as Eq. (2), and the exposure data from selected specimens of w/b = 0.35 are used for statistical analysis. For regression analysis the aging law is written into logarithm form,
t0 þ ln D0C1 ln ½DC1 ðtÞ ¼ n ln t
ð9Þ
Knowing that this aging law is sensitive to concrete binder composition, the binder of selected specimens includes three families: cement–FA binder with FA content of 30%, 35% and 40% (225 specimens totally), cement–SG binder with SG content of 60%, 70%, 80% (150 specimens totally) and ternary binder (cement–FA–SG) with 20% FA and 30% SG (20 specimens totally). The apparent chloride diffusivity was measured on these specimens after exposure time of 90 d, 180 d, 1 year, 2 years, 4 years and 7 years, and the average values are presented in Fig. 6 for cement–FA concrete and Fig. 7 for cement–SG and ternary binder concretes. Using Eq. (9), the aging exponent n can be interpreted as the slope of ln(DCl) ln(t0/t) curves. The number of ternary binder specimens is limited, about 20 specimens, and the data are just available for exposure ages of 90 d, 180 d and 1 year, resulting in larger regression dispersion
(a)
compared to other exposure data, cf. Fig. 7(c). For comparison the aging exponent n is also calculated according to the Life-365 model [17,37],
n ¼ 0:2 þ 0:4
f FA f SG þ 40 70
ð10Þ
with fFA,SG standing for mass ratio, in percentage, of FA and SG in concrete binder. Table 3 summarizes the obtained n values. It is found that Life-365 model agrees well with the exposure data for cement–FA and ternary binder concrete but overestimates the n values for cement–SG concrete. For the design value of aging exponent n, a synthetic analysis is necessary since the number of ternary specimens is not sufficient to support the statistical analysis. Following the relative contribution of FA and SG to the aging exponent, it is judged that cement– FA binder with 30% FA or cement–SG binder with 60% SG can both be considered representative for the ternary binder concrete in the HZM project. Accordingly, the mean values of aging exponent n are conservatively determined after the cement–SG binder (30% FA) for splashing and tidal zones. For atmospheric zone, no exposure data are available and the n value for ternary binder from Eq. (10) is used. The statistical properties of n and ln(D0Cl) (t0 = 28 d) are determined from Figs. 6 and 7 using respectively the standard deviations for the slop and intercept in linear regression,
(b)
(c)
Fig. 6. Chloride diffusivity of cement–FA concrete in splashing zone (a), tidal zone (b) and immerged zone (c) (DCl in 1012 m2/s and t in day).
(a)
(b)
(c)
Fig. 7. Chloride diffusivity of cement–SG concrete in splashing zone (a), tidal zone (b) and ternary binder concrete in splashing, tidal and immerged zone (c) (DCl in 1012 m2/s and t in day).
Table 3 Aging exponent n from exposure data and Life-365 model. Data source
Life-365 Model Exposure (splashing zone) Exposure (tidal zone) Exposure (immerged zone)
Cement–FA binder
Cement–SG binder
30%
35%
40%
60%
70%
80%
0.44 0.47 0.46 0.44
0.48 0.49 0.49 0.47
0.52 0.51 0.51 0.52
0.54 0.45 0.45 –
0.60 0.52 0.51 –
0.66 0.61 0.58 –
Ternary binder (FA 20%, SG 30%) 0.53 0.54 0.56 0.55
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Q. Li et al. / Structural Safety 53 (2015) 1–12 Table 4 Statistical properties of n values and chloride diffusivity for HZM durability design. Parameter n
Mean value () Deviation () Mean value (1012 m2/s) Deviation (1012 m2/s)
ln(D0Cl)
SD½n ¼ r½lnðDC1 Þ
Atmospheric
Splashing
Tidal
Immerged
0.53 0.079 – –
0.47 0.029 1.413 0.076
0.46 0.029 0.842 0.064
0.44 0.028 0.598 0.097
" #12 m X ðxi xÞ2 ; i¼1
h i SD ln D0C1 ¼ r½ln ðDC1 Þ
3.1. Chloride diffusivity D0Cl and concrete cover thickness xd
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1 þ Pm m 2 i¼1 ðxi xÞ
ð11Þ
where r[ln(DCl)] is the SD between the measured and regressed values of ln(DCl), xi, x are measured and mean values of ln(t0/t), and m is the number of specimens. The standard deviations of n and ln(D0Cl) values are presented in Table 4. For atmospheric zone, the recommendation of 0.15 time mean value is adopted for SD of n values after [18]. The statistical properties of n values are to be used in the subsequent design process. But the mean values and SD for ln(D0Cl) are just for analysis purpose and the values are not to be used in design since because the uncertainty of D0Cl associated with field practice is not included in Table 4 from exposure data and this uncertainty is usually more significant. Taking D0Cl as design unknown, its variation coefficient is set to be 20%, same as fib model code [19]. 3. Design by fully probabilistic method Through the above statistical analysis, the durability design can be performed through Eq. (4) following a fully probabilistic approach. The concrete cover thickness xd and the chloride diffusion coefficient D0Cl are usually retained as main design unknowns for an expected service life tSL and reliability level b. The reliability level b is actually the reliability index corresponding to the failure probability pf with respect to the design equation in Eq. (4), i.e., b = U1(1 pf), where U1() is the inverse CDF of the standard normal distribution. The other parameters and their statistical properties are regarded as known values, dependent on exposure conditions and material compositions. In Section 2 all parameters are analyzed with respect to the target proportioning of structural concrete of the HZM project, thus these parameters and properties are considered dependent only on exposure conditions. In Table 5 are summarized the design parameters (unknown) and known parameters on the basis of statistical analysis in Section 2.
The full probabilistic design through Eq. (4) is performed by Monte-Carlo simulation [38] for the joint occurrence of 5 random variables Cs, Ccr, D0Cl, n, xd. A computer-based program is specifically developed for the full probabilistic solution of Eq. (4). The sampling number for each simulation was 100,000 to obtain ‘‘exact’’ probability of failure. Two service lives and three levels of reliability index are considered: tSL = 50 years, 120 years, and b = 1.3, 1.5 and 1.8. Moreover, two exposure ages are considered for quality control purpose: t0 = 28 d, 56 d. From Figs. 8–10 are presented the design results for relationship between D0Cl (mean value) and xd (mean value) for target reliability index b = 1.3, 1.5 and 1.8. The above figures can serve as design charts to choose concrete cover thickness in terms of required durability quality (D0Cl) for different exposure conditions, reliability levels and service lives. For b = 1.3 in Fig. 8, if D0Cl(t0 = 28 d) is controlled within 3.0 1012 m2/s, the mean value of xd should be not less than 49 mm (50 years) and 73 mm (120 years) for splashing zone; in tidal zone the xd mean value becomes 46 mm (50 years) and 69 mm (120 years); in immerged zone the mean value of cover thickness becomes 36 mm (50 years) and 55 mm (120 years); in atmospheric zone these values are 27 mm (50 years) and 40 mm (120 years). The design values for D0Cl(28 d, 56 d) and xd can be read from Figs. 9 and 10 for b = 1.5 and b = 1.8, respectively.
3.2. Sensitivity analysis for reliability index An important issue in full probabilistic approach is the sensitivity of reliability index with respect to the model parameters and their statistical properties. Focus always on the mean values of D0Cl and xd. Their impact on the reliability index for service life of 120 years is evaluated in the following. The reference design points, in terms of mean values, are selected as D0Cl(28 d) = 3.0 1012 m2/s, D0Cl(56 d) = 2.2 1012 m2/s, xd = 40 mm (atmospheric zone), 73 mm (splashing zone), 69 mm (tidal zone), 55 mm (immerged zone). During the sensitivity analysis, one design parameter varies with the others kept fixed, and the corresponding reliability index is evaluated through HZM model using
Table 5 Statistical properties for model parameters of chloride ingress. Parameter (distribution)
Statistical properties
Immerged zone
Tidal zone
Splashing zone
Atmospheric zone
Surface concentration Cs (Lognormal)
Average (%) Deviation (%) Lower bound L (%) Upper bound U (%) Coefficient a () Coefficient b () Average (%) Deviation (%) Average (1012 m2/s) Coefficient of variance Average () Deviation () Average (mm) Deviation (mm)
4.5 0.68 1.0 3.5 0.23 0.33 – – Design 0.2 0.44 0.028 Design 5.3
3.8 0.58 0.45 1.25 0.22 0.36 – – Design 0.2 0.46 0.029
5.4 0.82 0.45 1.25 0.22 0.36 – – Design 0.2 0.47 0.029
2.0 0.31 – – – – 0.85 0.13 Design 0.2 0.53 0.079
Critical concentration Ccr (beta)
Critical concentration Ccr (Lognormal) Diffusion coefficient D0Cl (Lognormal) Exponent coefficient n (Normal) Concrete cover xd (Normal)
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(a)
(b)
Fig. 8. Concrete cover thickness and chloride diffusion coefficients for b = 1.3: (a) tSL = 50 years (b) tSL = 120 years (solid line for t0 = 28 d, dash-line for t0 = 56 d).
(a)
(b)
Fig. 9. Concrete cover thickness and chloride diffusion coefficients for b = 1.5: (a) tSL = 50 years (b) tSL = 120 years (solid line for t0 = 28 d, dash-line for t0 = 56 d).
(a)
(b)
Fig. 10. Concrete cover thickness and chloride diffusion coefficients for b = 1.8: (a) tSL = 50 years (b) tSL = 120 years (solid line for t0 = 28 d, dash-line for t0 = 56 d).
Monte-Carlo simulation. The results are presented as the achieved reliability index in terms of mean value of xd, D0Cl(28 d) and D0Cl(56 d) in Figs. 11 and 12. It can be seen that the reliability index b is sensitive to the mean values of both concrete cover thickness xd and the apparent chloride diffusivity D0Cl(t0 = 28 d, 56 d). The sensitivity of b to xd is similar for all exposure zones, and a reduction of 5 mm in xd results in roughly a drop of 0.4 for b. The reliability index b is less sensitive to D0Cl for atmospheric zone than for the other exposure zones. In atmospheric zone, a reduction of 0.4 1012 m2/s in D0Cl(t0 = 28 d) or 0.3 1012 m2/s in D0Cl(t0 = 56 d) results in roughly an increase of 0.2 for b while for other exposure conditions (splashing, tidal and immerged zones) D0Cl needs to decrease 0.22 1012 m2/s (t0 = 28 d) or 0.17 1012 m2/s (t0 = 56 d) to increase b by 0.2. The reliability index b is also influenced by the dispersion (SD) of
these design parameters. However, compared with their mean values the impact of SD is less significant, so the sensitivity with respect to their SD is not discussed here.
4. Design by partial factor method 4.1. Design equation Although the full probabilistic method established in the previous section can perform durability design for chloride ingress for the HZM project, a design method in partial factor format is still expected because the partial factor method is much easier to use and much accessible to design engineers. Thus, this part is dedicated to establishing the design method in partial factor format
9
Q. Li et al. / Structural Safety 53 (2015) 1–12 cðnomÞ
xdd ¼ xd
Dx
ð14Þ
On the basis of the statistical analysis of achieved concrete cover thickness in Section 2.2.1, the standard deviation for thickness larger than 50 mm is 5.26 mm, and the fractile of 5% in the lognormal distribution corresponds roughly to 10 mm. Thus, for cast-in-place concrete elements Dx is taken as 10 mm. For pre-fabricated concrete elements, the concrete cover is subject to stricter control, thus this tolerance is taken as 5 mm, same as recommendations from DuraCrete [19] and fib model [20]. In terms of characteristic values and partial factors, Eq. (12) becomes,
2 G¼
Fig. 11. Sensitivity of reliability index with respect to cover thickness.
2 G¼
6 C ds 41
0
13
xdd B C7 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 0;d d 2 DC1 g ðt0 Þ tSL
1
ccr
ð12Þ
0;c d c C ccr ; C ds ¼ cCS C cs ; D0;d C1 ¼ cD DC1 ; g ðt 0 Þ ¼ cg g ðt 0 Þ
0
c6 Cs C s 41
c
13
xcd Dx B C7 ffiA5 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0;c c 2 cD DC1 cg g t SL
ð15Þ
In this partial factor format, all parameters, except concrete cover thickness, adopt directly the mean values in statistical analysis as characteristic values. For concrete cover thickness, the tolerance expression in Eq. (14) is used to calculate the characteristic value. The design parameters are summarized in Table 7 Partial factors of design parameters of HZM model for chloride ingress.
The superscript ‘d’ stands for the design value for parameters. Introducing partial factor, the design values for Ccr, Cs and D0Cl write,
C dcr ¼
ccr
C ccr
4.2. Characteristic values and partial factors
on the basis of the first order reliability method (FORM), determining the characteristic values and the partial factors for design parameters. In partial factor format, the design equation, G, taking always the corrosion initiation as DLS, writes,
C dcr
1
Target reliability b⁄
Exposure zone
ccr
cCs
cg
cD
1.3
Atmospheric Splashing Tidal Immerged
1.07 1.66 1.66 2.00
1.08 1.04 1.05 1.04
1.32 1.04 1.03 1.03
1.07 1.07 1.07 1.01
1.5
Atmospheric Splashing Tidal Immerged
1.09 1.67 1.67 2.00
1.10 1.05 1.06 1.04
1.38 1.05 1.05 1.04
1.08 1.08 1.09 1.02
1.8
Atmospheric Splashing Tidal Immerged
1.10 1.67 1.67 2.00
1.12 1.06 1.07 1.05
1.51 1.06 1.06 1.05
1.11 1.10 1.10 1.02
ð13Þ
where the superscript ‘c’ stands for the characteristic values and c for the partial factors. The aging coefficient g has the expression in Eq. (3). Since the aging exponent n observes normal distribution g follows lognormal distribution. This partial factor format, according to structural design customs, is not used for concrete cover thickness xd. Instead, a tolerance is attributed to concrete cover thickness, linking its design and characteristic (nominal) values,
(a)
(b)
Fig. 12. Sensitivity of reliability index with respect to chloride diffusivity at: (a) 28 d (b) 56 d.
Table 6 Characteristic values of design parameters for partial factor format. Exposure zone
Cccr (% binder)
Ccs (% binder)
gc (28 d) [gc (56 d)]
12 D0,c m2/s) Cl (28 d, 56 d) (10
xcd (mm)
Atmospheric Splashing Tidal Immerged
0.85 0.75 0.75 2.0
1.98 5.44 3.82 4.50
0.047 0.061 0.067 0.074
3.0 3.0 3.0 3.0
50 80 80 60
[0.067] [0.084] [0.091] [0.100]
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Q. Li et al. / Structural Safety 53 (2015) 1–12
Table 8 Design results by partial factor method and achieved reliability indices. Target reliability b⁄
Exposure zone
tSL = 50 years D0,c Cl
(28 d) (10
tSL = 120 years 12
2
m /s)
xcd
(mm)
b ()
12 D0,c m2/s) Cl (28 d) (10
xcd (mm)
b ()
1.3
Atmospheric Splashing Tidal Immerged
3.5 3.0 3.0 3.5
32 53 51 43
1.75 1.81 1.99 1.90
3.5 3.0 3.0 3.5
45 78 75 63
1.56 1.68 1.80 1.81
1.5
Atmospheric Splashing Tidal Immerged
3.5 3.5 3.0 3.0
33 54 51 43
1.86 1.93 1.99 1.90
3.5 3.0 3.0 3.5
47 80 77 64
1.72 1.84 1.96 1.88
1.8
Atmospheric Splashing Tidal Immerged
3.5 3.5 3.0 3.0
34 55 52 44
1.96 2.04 2.11 2.01
3.5 3.0 3.0 3.5
49 82 79 65
1.91 2.00 2.12 1.96
Table 6 for all exposure conditions. The characteristic values of D0Cl and xd in Table 6 are the reference points for partial factor calibration and will be taken as design unknowns in partial factor format design. The partial factors are calibrated using first-order reliability method (FORM) with specified statistical properties for parameters and specified reliability level. The partial factors are evaluated by minimizing the distance between the origin and the failure surface in the normalized independent variable space [16]. Table 7 provides the partial factors for b = 1.3, 1.5 and 1.8, and the calibration procedure is detailed in Appendix. It is noted that the partial factors have only slight change (less than 0.02) as tSL changes from 50 to 120 years, and the factors are nearly identical for taking t0 = 28 d or 56 d for D0Cl. Therefore, the partial factors in Table 7 apply for service lives of 50–120 years and for both exposure ages (28 d, 56 d) of D0Cl. 4.3. Design results by partial factor method Using the design equation Eq. (15) and the characteristic values in Table 6 and the partial factors in Table 7, one is capable to make the durability design for chloride ingress taking the apparent chloride diffusivity D0Cl and concrete cover thickness xd as design unknowns for expected reliability levels. Table 8 presents the results for b = 1.3, 1.5 and 1.8 and different exposure zones, and the results of b = 1.3 serve as the basis for durability requirements in the preliminary design of concrete structures in the HZM project. The achieved reliability index is checked for these design results via the full probabilistic method using Monte-Carlo simulation. The achieved reliability indices are also presented in Table 8. It can be seen that the achieved reliability index b is generally higher than the target one b⁄, and for a target reliability of 1.3 the design results via partial factor format achieve actually a reliability index b = 1.75–1.99 for tSL = 50 years and b = 1.56–1.80 for tSL = 120 years. This is mainly due to the conservative adoption of the concrete cover tolerance as 10 mm in Eq. (15), see Appendix for details. 5. Concluding remarks 1. For the durability design of concrete structures in the HZM project, the empirical Fick’s law is retained and adapted for the project site environmental conditions and the concrete material proportioning. The results on long-term structural investigation and exposure tests during the last 30 years in South-eastern costal region in China are used to determine the values and statistical properties of concrete cover thickness, concrete surface
chloride concentration, critical chloride concentration for steel corrosion initiation and the time-dependent chloride diffusivity of concrete. On the basis of these data, a HZM model is established for durability design for chloride ingress of concrete structures in the HZM project. 2. On the basis of available statistical properties of model parameters, a full probabilistic method is set up for the durability design of concrete structure for chloride ingress for intended service lives and expected reliability levels. The design adopts the corrosion initiation state as DLS and takes chloride diffusivity at given age D0Cl and concrete cover thickness xd as main design unknowns, and design is performed for service lives of 50 years and 120 years and for reliability index b = 1.3, 1.5 and 1.8. The sensitivity analysis shows that the achieved reliability of design is rather sensitive to the mean values of diffusivity and cover thickness. 3. For easier engineering use, a partial factor format of durability design for chloride ingress is calibrated from first-order reliability method, and the characteristic values and the partial factors are determined for different levels and reliability index and different exposure zones. Taking chloride diffusivity and cover thickness as design unknowns, their values are given for tSL = 50 years, 120 years and target reliability index b⁄ = 1.3, 1.5 and 1.8. The verification through full probabilistic method gives achieved reliability index higher than the target one, mainly because the conservative value of concrete cover tolerance. 4. The design results through partial factor method with b⁄ = 1.3 are retained for preliminary design of concrete structure in the HZM project. This apparently low reliability level is justified by the conservative considerations during the design: first DLS is taken as corrosion initiation state, a rather early state thus conservative for service life; second, the achieved reliability index through partial factor method for b⁄ = 1.3 is actually above 1.5 by full probabilistic verification; finally the design is just for preliminary phase and additional protection measures are to be implemented and the reliability level can be increased. The updating of reliability index of durability levels by implemented protection measures is to be treated in later publications.
Acknowledgments The research is supported by the China National Science and Technology Support Planning Project No. 2011BAG07B04. The support of Management Bureau of HZM project (Zhuhai, China) is also acknowledged. Special thankfulness is extended to Prof. Bruce
11
Q. Li et al. / Structural Safety 53 (2015) 1–12 Table A.1 Calculation procedure of partial factors for splashing zone design (Target b⁄ = 1.3). Values Characteristic value () Sensitivity factor a () Normalized point Z Design point xi⁄ Partial factor ci (or Dx)
Ccr 0.75% 0.855 1.11 0.45% 1.66
Cs
xd
5.44% 0.165 0.22 5.67% 1.04
80 mm 0.261 0.34 73.2 mm 1.8 mm
D0Cl (28 d) 12
g 2
3.0 10 m /s 0.279 0.36 2.36 1012 m2/s 1.07
0.061 0.311 0.40 0.064 1.04
Ellingwood for his constructive suggestions and linguistic help in the revision of this paper.
exposure conditions can be calculated following the same procedure.
Appendix A. Calibration algorithm for partial factors
References
The limit state function is given in Eq. (4), containing 5 random variables. The partial factors are determined by first-order reliability method (FORM). The FORM involves the mapping of the set of to a set of independent standard normal basic random variables X variables Z. This results in the mapping of the limit state failure surface given by Eq. (4) to a failure surface in standard normal space,
[1] Tuutti K. Corrosion of steel in concrete. CBI research report 4: 82. Stockholm: Swedish Cement and Concrete Research Institute; 1982. [2] Bohni H. Corrosion in reinforced concrete structures. Cambridge: Woodhead Publishing; 2005. [3] Weyers RE, Prowell BD, Sprinkel MM. Concrete bridge protection, repair and rehabilitation relative to reinforcement corrosion: a methods application manual. SHRP-S-360; 1993. [4] Kropp J. Chlorides in concrete. In: Performance criteria for concrete durability, RILEM Report 12, London: E&FN SPON; 1995, p. 138–64. [5] Crank J. The mathematics of diffusion. 2nd ed. Oxford: Clarendon Press; 1975. [6] Collepardi M, Marcialis A, Turriziani R. Penetration of chloride ions into cement pastes and concrete. J Am Ceram Soc 1972;55:534–5. [7] Olivier T, Ollivier JP, Nilsson LO. Numerical simulation of multi-species transport through saturated concrete during a migration test – MsDiff code. Cem Concr Res 2000;30:1581–92. [8] Hosokawa Y, Yamada K, Johannesson B, Nilsson LO. Development of a multispecies mass transport model for concrete with account to thermodynamic phase equilibriums. Mater Struct 2011;44:1577–92. [9] Samson E, Marchand J. Modeling the effect of temperature on ionic transport in cementitious materials. Cem Concr Res 2007;37:455–68. [10] Uji K, Matsuoka Y, Maruya T. Formulation of an equation for surface chloride concrete of concrete due to permeation if chloride. In: Page CL, Treadaway KWJ, Bamforth PB, editors. Corrosion of reinforcement in concrete. London: Taylor & Francis; 1990. p. 258–68. [11] Li KF, Li CQ. Modeling hydroionic transport in cement-based porous materials under drying-wetting actions. J Appl Mech 2013;80:020904 (1-9). [12] Nguyen T, Baroghel-Bouny V, Dangla P. Prediction of chloride ingress into saturated concrete on the basis of a multi-species model by numerical calculations. Comp Concr 2006;3:401–21. [13] Pivonka P, Hellmich C, Smith D. Microscopic effects on chloride diffusivity of cement pastes – a scale-transition analysis. Cem Concr Res 2004;34:2251–60. [14] Oh BH, Jang SY. Prediction of diffusivity of concrete based on simple analytic equations. Cem Concr Res 2004;34:463–80. [15] Bentz DP. Influence of silica fume on diffusivity in cement based materials II, multi-scale modeling of concrete diffusivity. Cem Concr Res 2000;30:1121–9. [16] Palle TC, Baker MJ. Structural reliability theory and its application. Heidelberg, New York: Springer-Verlag; 1982. [17] Violetta B. Life-365 service life prediction model. Concr Int 2002;24:53–7. [18] DuraCrete. Probabilistic performance based durability design: modeling of degradation. DuraCrete Project Document BE95-1347/R4-5. The Netherlands; 1998. [19] Fédération International du Béton. Model code for service life design. Bulletin 34. Lausanne: fib; 2006. [20] Chinese Civil Engineering Society. Guide for durability design and construction of concrete structures (CCES01-2004). Beijing: China Building Industry Press; 2005. [21] Bamforth PB, Price WF. An international review of chloride ingress into structural concrete. Transport Research Laboratory Report 359. Scotland; 1997. [22] Tang LP, Gulikers JP. On the mathematics of time-dependent apparent chloride diffusion coefficient in concrete. Cem Concr Res 2007;37:589–95. [23] Nordtest Method. Concrete, hardened: accelerated chloride penetration (NT Build 443). Espoo: NORDTEST; 1995. [24] Andrade C, Tavares F, Castellote M, Petre-Lazars I, Climent MA, Vera G. Comparison of chloride models: the importance of surface condition. In: Proc. 2nd international symposium on advances in concrete through science and engineering, September 2006, Quebec City, Canada. [25] Nilsson LO. Present limitations of models for predicting chloride ingress into reinforced concrete structures. J Phys IV France 2006;136:123–36. [26] Sarja A, Vesikari E. Durability design of concrete structures. In: RILEM Report 14, London: E&FN SPON; 1996. [27] Lounis Z. Risk-based maintenance optimization of aging highway bridge decks. In: Advances in engineering structures, mechanics and construction. The Netherlands: Springer; 2007. p. 723–34. [28] Bamforth PB. The derivation of input data for modeling chloride ingress from eight-year UK coastal exposure trials. Mag Concr Res 1999;51:87–96.
f ðZ 1 ; Z 2 ; . . . ; Z n Þ ¼ 0
ðA-1Þ
The reliability index b is defined in Z space as the shortest distance from the origin to the failure surface. Meanwhile, the sensitivity fac , is also obtained. If the target reliability is b⁄, then the design tor, a point in the normalized Z space is,
Z ¼ b a
ðA-2Þ
By using the inverse mapping,
xi ¼ F 1 X i U zi
ðA-3Þ
corx for the original basic variables X one obtains the set of values . In Eq. (A-3), F 1 ðÞ stands for the responding to the design point Z Xi inversed probability distribution function of random variable Xi, UðÞ is the standard normal distribution function. If the values x were to be used as the design values in a deterministic design, the resulting design would have a reliability index b⁄. Thus a satisfactory set of partial factors is given by:
ci ¼
xci xi
ðA-4Þ
where xci is the characteristic value of the resistance variable Xi, and by:
ci ¼
xi xci
ðA-5Þ
where xic is the characteristic value of the loading variable Xi. Take the design in splashing zone as example. The initial characteristic value of xd is assumed 80 mm, D0Cl(28 d) is assumed 3.0 1012 m2/s and the service life tSL is 120 years. The detailed calculations are shown in Table A.1, and the concrete cover is noted by a tolerance Dx. The initial reliability index is b = 2.26 and the target index is b = 1.3. From the table, one can see that the required tolerance in concrete cover thickness is significantly less than the specified value 10 mm. This provides conservativeness to the partial factor design using 10 mm as tolerance and explains why the ‘actual’ reliability indices achieved by partial factor format are larger than the target ones. If the design value of xd is changed to 50 mm, the design value of D0Cl(28 d) changed to 2.0 1012 m2/s, or the service life is changed to 50 years, the partial factors have only slight change, indicating that these factors are not sensitive to the pre-assumed design values. Partial factors for the other
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[29] Mangat PS, Molloy BT. Model for long term chloride penetration in concrete. Mater Struct 1994;25:404–11. [30] DuraCrete. General guidelines for durability design and redesign. DuraCrete Project Document BE95-1347/R17. The Netherlands; 2000. [31] Angst U, Elsener B, Larsen CK, Vennesland O. Critical chloride content in reinforced concrete – a review. Cem Concr Res 2009;39:1122–38. [32] Hobbs DW, Matthews JD, Marsh BK. Minimum requirements of durable concrete: carbonation- and chloride-induced corrosion, freeze-thaw attack and chemical attack. Crowthorne: British Cement Association; 1998. [33] EuroCode2 : design of concrete structures (prEN 1992-1-1). Brussels: European Committee for Standardization; 2002. [34] American Concrete Institute Committee 318. Building code requirements for structural concrete and commentary (ACI318-05, ACI318R-05). Farmington Hills: ACI; 2005.
[35] Ann KY, Ahn JH, Ryou JS. The importance of chloride content at the concrete surface in assessing the time to corrosion of steel in concrete structures. Constr Build Mater 2009;23:239–45. [36] Bouteiller V, Cremona C, Baroghel-Bouny V, Maloula A. Corrosion initiation of reinforced concretes based on Portland or GGBS cements: chloride contents and electrochemical characterizations versus time. Cem Concr Res 2012;42:1456–67. [37] Ehlen MA, Thomas MDA, Bentz EC. Life-365 service life prediction model™ version 2.0. Concr Int 2009;31:41–6. [38] Landau DP, Binder K. A guide to Monte Carlo simulations in statistical physics. Cambridge: Cambridge University Press; 2005. p. 432.
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Model-based durability design of concrete structures in Hong Kong–Zhuhai–Macau sea link project Quanwang Li a, Kefei Li a,⇑, Xingang Zhou b, Qinming Zhang a, Zhihong Fan c a
Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Civil Engineering Department, Tsinghua University, Beijing 100084, China Civil Engineering College, Yan-tai University, Yantai 264005, China c CCCC Fourth Harbor Engineering Institute, Guangzhou 510230, China b
a r t i c l e
i n f o
Article history: Received 23 April 2013 Received in revised form 19 June 2014 Accepted 13 November 2014
Keywords: Concrete structures Durability design Chloride ingress Probability Reliability Structural engineering
a b s t r a c t This paper presents the durability design of concrete structures in the Hong Kong–Zhuhai–Macau (HZM) sea link project for a design service life of 120 years. Among all the durability issues, this paper focuses on the model-based durability design of concrete structures in the project subject to the action of marine chlorides. The mechanism and modeling of chloride penetration into concrete are reviewed in depth and the applicability of models for durability design is evaluated. On the basis of long-term exposure tests and in-place structural investigations during the past 30 years in South-eastern China, the design model for concrete structures in the HZM project is established. The statistical properties of model parameters are analyzed in details and compared to other chloride ingress models. Using the established HZM model for chloride ingress and specified durability limit state (DLS), the design parameters are first evaluated through a fully probabilistic analysis in terms of service lives (50 years, 120 years) and target reliability levels (b = 1.3, 1.5, 1.8). The partial factor format for durability design is then established and the partial factors are calibrated from the first order reliability method (FORM) from specified service life and specified reliability level. The design results from the partial factor design equation are given for different exposure zones, and the achieved reliability index is verified for each design through the full probabilistic method. Finally remarks are given on the HZM model and its application in durability design of RC elements. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction When elements of concrete structures are exposed to a marine environment, chloride ions, either air-borne or in sea water, can penetrate the concrete cover and induce the corrosion of reinforcement steel bars [1]. In the recent decades, considerable efforts have been dedicated to the mechanisms and the modeling of chloride ingress and the subsequent steel corrosion processes [2]. The chloride ions can penetrate into concrete elements through different mechanisms in marine environment: pore diffusion, capillary suction, permeation under pressure as well as surface deposit of airborne salts [3,4]. The available modeling on chloride ingress can be divided into two families: the simplified Fick’s diffusion model and more sophisticate physical models. The Fick’s model uses the analytical solution of second Fick’s law with constant boundary conditions to represent the chloride ingress [5]. This model presents a very simplified image of the real transport processes of ⇑ Corresponding author. Tel./fax: +86 10 6278 1408. E-mail address: [email protected] (K. Li). http://dx.doi.org/10.1016/j.strusafe.2014.11.002 0167-4730/Ó 2014 Elsevier Ltd. All rights reserved.
chloride ingress, and envelops all these processes into the wellknown ‘‘apparent’’ chloride diffusion coefficient [6]. Nowadays, more mechanism-based physical models are also available: multi-species models were proposed for chloride transport in pore solution as a concentrated electrolytic solution [7,8]; the thermal effect was considered through thermal-ionic couplings [9]; the non-stationary boundary was modeled by Uji et al. [10]; Li and Li [11] developed a hydro-ionic model to consider specially the drying-wetting actions; the sorption behavior was investigated for chloride transport in unsaturated concretes [12]; the material microstructure aspects and cement hydration processes were incorporated into the chloride diffusivity modeling through upscaling techniques [13,14] and percolation consideration [15]. In principle, these models can all be used in the durability design for chloride ingress of concrete elements provided that they yield reliable results. The reliability of such models comes from two fundamental aspects: firstly the model should be correct or basically correct in a physical sense, and secondly, the model should be robust or tolerant enough to incorporate the randomness of durability processes in design [16]. From the point of view of
2
Q. Li et al. / Structural Safety 53 (2015) 1–12
design, a model is acceptable as long as the associated uncertainty is reasonable and can be quantified. By its simplicity and the capacity of being adapted to different exposure cases, the Fick’s model is often preferred in model-based durability design [17,18] and recommended by codes and standards [19,20]. Moreover, long-term monitoring of the apparent chloride diffusion coefficient in real structures [21,22] and the standardization of its determination through chloride profiling method [23] make Fick’s model appropriate for service life design and prediction [24,25]. This approach has been shown to be capable for achieving service life design in marine chloride environments for a target reliability level [26,27]. However, using the Fick’s model in durability design for an expected safety margin (reliability level) necessitates determining the parameter uncertainties since these parameters are both strongly environment and material dependent. The Hong Kong–Zhuhai–Macau (HZM) sea link project is currently under construction. The HZM project includes sea bridges of 28.8 km (three navigable spans), two artificial islands and an immerged tube tunnel of 6.8 km, with a total investment of nearly 12 billion US dollars. The project is situated on the South-eastern coast of the China mainland. One of the technical challenges of HZM project was to achieve the service life of 120 years for the concrete structures in an aggressive marine environment. The concrete structures in the project include the piers, bearing platforms and piles for sea bridges, the tunnel tube segments for the immersed tunnel, and retaining walls and breakwaters for the artificial islands. The annual average temperature at the project site is between 22.3– 23.1 °C, the annual average humidity is 77–80% and the annual average wind speed is between 3.1–6.6 m/s. The analysis of the sea water chemistry show the chloride (Cl) concentration is within 10,700–17,020 mg/L, the sulfate (SO2 4 ) concentration is within 1140–2260 mg/L, and the total salinity is between 25.4 and 32.9 (sea bottom). Under this marine environment the possible durability processes for reinforced concrete (RC)/prestressed concrete (PC) elements were identified as the carbonation-induced corrosion of steel bars, chloride-induced corrosion of steel bars, salt attack on structural concrete as well as internal expansion reactions. From a criticality analysis of these processes in the preliminary study phase, the chloride-induced steel corrosion of RC/PC elements was identified as the controlling process of durability design. Thus, a model-based approach was taken for the durability design, and the uncertainty of model parameters was taken into account through analysis of their statistical properties. Their statistical properties were regressed on the basis of the exposure test results under the same marine environment conditions as the HZM project and structural investigations conducted in the past 30 years. This paper presents the statistical analysis of parameters in the design model, i.e., Fick’s second law. Then, both fully probabilistic and partial factor design formats are given for different reliability levels for durability. The results form the basis for the choice of structural and material parameters for durability of concrete structures in HZM project. This paper is organized as follows. Section 2 is dedicated to the analysis of statistical properties of model parameters on the basis of long term real structure investigation as well as exposure tests. The durability design through the fully probabilistic approach is presented in Section 3, while the durability design in a partial factor format is detailed in Section 4. The concluding remarks are given in the end.
2. HZM model for chloride penetration 2.1. Basic model The empirical Fick’s model assumes a pure diffusion process for chloride ingress into concrete [5]. With a constant chloride concen-
tration Cs imposed on the surface, the profile of chloride concentration C(x,t) along the depth x and with time t is,
x Cðx; tÞ ¼ C s 1 erf pffiffiffiffiffiffiffiffiffiffi 2 DC1 t
ð1Þ
where DCl is the apparent chloride diffusion coefficient in concrete and erf the error function. This apparent diffusion coefficient is found to be time-dependent [28], and a power law is recommended for its aging behavior [29],
DC1 ðtÞ ¼ D0C1
n t0 ¼ D0C1 gðt0 ; t; nÞ t
ð2Þ
with D0Cl = diffusion coefficient at concrete age t0, n = exponent, and g = aging factor of diffusion coefficient DCl. Physically, the decrease of DCl with time is related to the long-term evolution of the microstructure of concrete materials. Since DCl also includes other transport processes than diffusion, the aging coefficients, n or g, are different for different exposure zones in marine environments [30]. Although the power law in Eq. (2) is well supported by in-field measurements, it is not rational to assume that DCl will decrease infinitely with time. Thus, for durability design in the HZM project, the power law is truncated at the end of 30 years and it is assumed that the DCl will be stable after 30 years’ exposure, i.e.,
gðt0 ; t; nÞjt>tD ¼ gðt0 ; tD ; nÞ with tD ¼ 30 years
ð3Þ
This assumption leads to a faster predicted rate for chloride ingress and thus a conservative durability design. For service life design in the HZM project, the durability limit state (DLS) is specified as the corrosion initiation state, usually denoted by a critical (threshold) chloride concentration on the steel surface Ccr [31]. Thus the design equation, for a target service life tSL, for chloride ingress becomes,
2
0
13
xd B C7 6 G ¼ C cr C s 41 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5 ¼ 0 0 2 DC1 gðt0 ; t SL ; nÞt SL
ð4Þ
where xd is the thickness of concrete cover for reinforcement steel. Note that the aging coefficient g in Eq. (4) should equal Eq. (3). Thus this design model contains five parameters Ccr, Cs, xd, D0Cl, g (or n) to be defined for a given target service life tSL. Except for xd, all parameters depend on both environmental exposure conditions and concrete materials. However, in durability design it is assumed that the three parameters Ccr, Cs, g (or n) can be determined through parameter analysis while xd, D0Cl are specified for a given service life and exposure condition. 2.2. Statistical properties of parameters The statistical properties of model parameters are analyzed on the basis of the long-term in-place structural investigations and exposure tests during the last 30 years in the South-eastern coastal regions of China. The exposure tests have been performed at the Zhanjiang Exposure Station where the marine environment is very similar to that at the site of HZM project. 2.2.1. Concrete cover x The concrete cover is the only structural parameter in the design equation, Eq. (4). The thicker the concrete cover, the later the chloride ingress will initiate the corrosion in the steel. For this reason, correct specification of concrete cover thickness as well as its chloride ingress resistance (DCl) is the central issue for durability design under chloride ingress [32–34]. The statistical properties of concrete cover thickness are important for correct estimation of reliability with respect to Eq. (4), and are related closely to the construction methods and practice: stricter control results in smaller
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Q. Li et al. / Structural Safety 53 (2015) 1–12
deviation, while poor construction practice yields a larger deviation. For the durability design of the HZM project, statistical analysis is performed on the data of achieved cover thickness of marine concrete structures constructed after 1996. A total of 24 concrete structures were investigated and 1451 concrete cover thicknesses were measured. The design values of cover thickness are from 40 to 80 mm, and the mean values and standard deviations are evaluated for the achieved concrete covers. Statistical analysis showed that the cover thickness can be described by a normal distribution. Fig. 1 presents histograms for cover thickness for two marine docks constructed in 1997 and 1999. It can be seen that the histogram of ‘measured’ cover thickness is in good agreement with the ‘modeled’ histogram obtained by normal distribution modeling with the measured mean values (M) and standard deviations (SD). For all the achieved thickness data, the average SD is 5.3 mm. 2.2.2. Surface chloride concentration Cs The surface chloride concentration Cs is noted as the mass ratio of chloride with respect to concrete binder. Actually Cs in the design model Eq. (4) refers to the apparent surface concentration, i.e., the value is not from a direct measurement on the concrete surface but from the regression of the chloride concentration profile at a certain exposure age using Fick’s law. This regression provides, for exposure age t, the surface concentration Cs and the apparent diffusion coefficient DCl(t) simultaneously. The surface concentration Cs depends on the sea water salinity and its chloride concentration, but also on the chloride absorption capacity of concrete materials [35]. This concentration is usually expressed as Cs = A(w/b), where the coefficient A can be regarded as environment-dependent and is regressed for different concretes [17–19]. Since the accumulation of surface chloride needs some time, in scale of years, to reach a stable value, Cs is also found to be timedependent [19,36]. To the best of the authors’ knowledge, these two aspects have never been combined together in the literature, but for the HZM project this aspect has to be considered, because the support data of Cs values come from measurements during last 10 years and a correct estimation of stable values for Cs is important to guarantee the safe design for durability. This time-dependent Cs can be expressed as,
C s ðtÞ ¼ Aðw=bÞ
at 1 þ at
ð5Þ
where the coefficient a describes the influence of time t on Cs. This law for surface chloride build-up takes into account both the kinetics and the asymptotic behavior of chloride build-up. Considering the entire service life tSL, the surface chloride concentration can be averaged using Eq. (5) as,
0.30
0.35
Measurement Modeling
M=50.3 mm SD=4.7 mm
0.20 0.15 0.10
1 t SL
0.30
Measurement Modeling
M=70.5 mm SD=5.7 mm
0.25 0.20 0.15 0.10
0.05
0.05
0.00
0.00 <42 42-45 45-48 48-51 51-54 54-57 57-60 >60
Cover thickness (mm)
(a)
ð6Þ
where A0 depends on the environment coefficient A, time coefficient a and service life tSL. This sub-model is used to analyze the data on Cs from 351 specimens selected from the Zhenjiang Exposure Station. The Cs values for the specimens of w/b = 0.35 incorporating fly ash (FA) and slag (SG) as supplementary cementitious materials (SCM) are presented in Fig. 2 for tidal and splashing zones. Both the mean values and dispersions of Cs are illustrated on the figure: the mean values are represented by the regressed pointed lines and the dispersion is illustrated by the scattering points in the same exposure age. It can be found that the stabilization of Cs value is longer in splashing zones, i.e., 7 years, compared to the tidal zones, i.e., within 2 years. The asymptotic mean values for chloride surface concentration Cs0 for tSL are then calculated via Eq. (6) and presented in Fig. 2 by horizontal lines. These values will be used as characteristic value of Cs for durability design of concrete structures in HZM project. For comparison, the Cs values suggested from DuraCrete [18] are also illustrated by horizontal lines, showing that the DuraCtete values underestimate the Cs values for the HZM project. The preliminary proportioning of structural concrete for the HZM project has w/b = 0.35, and its binder has a ternary composition: ordinary Portland cement (50%), FA (20%) and SG (30%). For the exposure site, only a limited number of such concrete specimens are available and their values are marked as ‘‘+’’ in Fig. 2. It can be seen that the Cs values of FA concrete are systematically higher than for SG concrete, while the Cs values of the ternary-binder concrete are in between. The apparently higher Cs values from FA concretes result from the densification effect of secondary hydration between fly-ash particles and the cement hydrates, creating more effective adsorbents for surface chloride ions. Thus for the HZM project, the higher Cs values of FA concrete are conservatively retained for durability design for tidal and splashing zones. For the immerged zone, the analysis of Cs values utilizes a limited number of ternary binder concrete specimens. The available values are presented in Fig. 3, where they are compared with the values from DuraCrete [18]. Again, the DuraCrete values tend to underestimate the Cs values for the HZM project. For the atmospheric zone, no exposure data are available; thus, the recommended values from DuraCrete are used. Note that the Cs mean value suggested by DuraCrete is increased by 30% to account for the possible higher surface concentration of ternary binder concretes suggested from the comparison in Figs. 2 and 3. The statistical properties of Cs (Cs0 ) values are summarized in Table 1.
Frequency (-)
Frequency (-)
0.25
C 0s ¼
tSL w w lnð1 þ at SL Þ ¼ A0 C s ðtÞdt ¼ A 1 b at SL b 0
Z
<60
60-65 65-70 70-75 75-80 Cover thickness (mm)
>80
(b)
Fig. 1. Histograms of concrete cover thickness: (a) Dock in 1997 (280 points), (b) Dock in 1999 (90 points).
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Q. Li et al. / Structural Safety 53 (2015) 1–12
(a)
(b)
(c)
(d)
Fig. 2. Surface chloride concentrations: (a) FA concrete in splashing zone, (b) FA concrete in tidal zone, (c) SG concrete in splashing zone, (d) SG concrete in tidal zone.
Table 1 Statistical properties of C’s values for HZM durability design (w/b = 0.35). Exposure zone
Atmospheric Splashing Tidal Immerged
Fig. 3. Surface chloride concentration in immerged zone.
2.2.3. Critical chloride content Ccr The physical meaning of Ccr is the chloride concentration accumulated at the steel surface capable to initiate the electrochemical reactions of corrosion, i.e., depassivation. This concentration depends on factors from steel material (corrosion resistance), concrete material (moisture content, pH value of pore solution, oxygen availability, electro-resistivity) and external environments (ambi-
Distribution
Log-Normal Log-Normal Log-Normal Log-Normal
Parameters Mean value (% binder)
Deviation (% binder)
2.0 5.4 3.8 4.5
0.31 0.82 0.58 0.68
ent temperature) [31]. Thus it is not surprising that Ccr values reported or recommended in literature vary in wide limits [17– 19,36]. This concentration should be calibrated for HZM project for reliable durability design. For HZM project the Ccr values are regressed from 68 exposure concrete specimens. The concrete materials cover several binder types and w/b ratios from 0.45 to 0.55. The Ccr values were measured on these specimens on broken tests continuously with time. For a broken specimen, the chloride concentration at steel surface was measured, and if the steel corroded the concentration was taken as Ccr. Thus, the measured Ccr values correspond to an already-corroded state of steel and the exact depassivation was surely between the two subsequent broken observations. These Ccr values obtained for splashing zones are presented in Fig. 4 and beta-distribution is used to describe the statistical properties [19]. The mathematical form of beta-distribution writes,
f ðxÞ ¼
a1 b1 Ux UL
Cða þ bÞ xL CðaÞ CðbÞ ðU LÞ U L
ð7Þ
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Q. Li et al. / Structural Safety 53 (2015) 1–12
Four parameters are needed for beta-distribution: two shape factors
a, b, and two distribution limits L (lower limit), U (upper limit). Due to the approximate nature of Ccr determination for a given broken test, the statistical parameters of Ccr are to be determined through optimizing a likelihood function. A likelihood function, L, is employed to determine these parameters,
L ¼
m Y i¼1
P½C cr < xi
n Y
P½C cr > yj
ð8Þ
j¼1
This function calculates the probability associated with the fact that steel bars in m specimens are corroded when chloride concentration reaches xi (i = 1, 2,. . .m) and steel bars in n specimens remain passivated when chloride concentration attains yj (j = 1, 2,. . .n). Maximizing the likelihood function L can obtain the parameters of betadistribution. The obtained distributions are presented in Fig. 4 for specimens exposed in splashing zones. It can be seen that Ccr values tend to augment with the decrease of w/b ratio. The w/b ratio of HZM structural concrete is 0.35 but no direct Ccr data are available for w/b = 0.35. Accordingly a synthetic analysis is performed on the basis of available Ccr statistical properties, and the beta-distribution for HZM structural concrete (w/b = 0.35) calibrated as: a = 0.22, b = 0.36, L = 0.45%, U = 1.25%. This beta-distribution is compared to the DuraCrete model [18] and fib model [19] in
(a)
Fig. 5, showing DuraCrete model values are systemically higher than fib model and the HZM Ccr values lie between. Compared to fib model, the calibrated HZM model for Ccr has similar distribution shape, larger value and has smaller variance. The Ccr values for tidal zone are conservatively taken as the same for splashing zone for HZM project. For atmospheric zone, the values recommended by DuraCrete [18] are retained. For immerged zone, the beta-distribution similar to splashing zone is retained and the mean value recommended by DuraCrete is adopted. Table 2 summarizes the statistical properties for Ccr values used for steel corrosion. 2.2.4. Chloride diffusivity D0Cl and aging exponent n The chloride diffusivity DCl in the design model is time-dependent and determined by diffusivity D0Cl at a given age t0 and the aging exponent n. These two parameters are closely related to material composition and environmental exposure conditions, Table 2 Statistical properties of Ccr values for HZM durability design. Exposure zone
Distribution
Parameters
Atmospheric Splashing/tidal Immerged
Log-Normal Beta Beta
M = 0.85%, SD = 0.13% a = 0.22, b = 0.36, U = 1.25%, L = 0.45% a = 0.23, b = 0.33, U = 3.50%, L = 1.00%
(b)
Fig. 4. Critical chloride concentrations in splashing zone: (a) w/b = 0.55 (b) w/b = 0.40.
(a)
(b)
Fig. 5. Comparison of Ccr distributions in splashing zone between HZM, DuraCrete and fib models for structural concrete with w/b = 0.35: (a) probability density distribution, (b) cumulative probability function.
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Q. Li et al. / Structural Safety 53 (2015) 1–12
both having important influence on the service life. During design process, D0Cl value is usually taken as a design unknown while the aging exponent n is specified as known value for different material compositions and exposure conditions. The aging law of DCl is retained as Eq. (2), and the exposure data from selected specimens of w/b = 0.35 are used for statistical analysis. For regression analysis the aging law is written into logarithm form,
t0 þ ln D0C1 ln ½DC1 ðtÞ ¼ n ln t
ð9Þ
Knowing that this aging law is sensitive to concrete binder composition, the binder of selected specimens includes three families: cement–FA binder with FA content of 30%, 35% and 40% (225 specimens totally), cement–SG binder with SG content of 60%, 70%, 80% (150 specimens totally) and ternary binder (cement–FA–SG) with 20% FA and 30% SG (20 specimens totally). The apparent chloride diffusivity was measured on these specimens after exposure time of 90 d, 180 d, 1 year, 2 years, 4 years and 7 years, and the average values are presented in Fig. 6 for cement–FA concrete and Fig. 7 for cement–SG and ternary binder concretes. Using Eq. (9), the aging exponent n can be interpreted as the slope of ln(DCl) ln(t0/t) curves. The number of ternary binder specimens is limited, about 20 specimens, and the data are just available for exposure ages of 90 d, 180 d and 1 year, resulting in larger regression dispersion
(a)
compared to other exposure data, cf. Fig. 7(c). For comparison the aging exponent n is also calculated according to the Life-365 model [17,37],
n ¼ 0:2 þ 0:4
f FA f SG þ 40 70
ð10Þ
with fFA,SG standing for mass ratio, in percentage, of FA and SG in concrete binder. Table 3 summarizes the obtained n values. It is found that Life-365 model agrees well with the exposure data for cement–FA and ternary binder concrete but overestimates the n values for cement–SG concrete. For the design value of aging exponent n, a synthetic analysis is necessary since the number of ternary specimens is not sufficient to support the statistical analysis. Following the relative contribution of FA and SG to the aging exponent, it is judged that cement– FA binder with 30% FA or cement–SG binder with 60% SG can both be considered representative for the ternary binder concrete in the HZM project. Accordingly, the mean values of aging exponent n are conservatively determined after the cement–SG binder (30% FA) for splashing and tidal zones. For atmospheric zone, no exposure data are available and the n value for ternary binder from Eq. (10) is used. The statistical properties of n and ln(D0Cl) (t0 = 28 d) are determined from Figs. 6 and 7 using respectively the standard deviations for the slop and intercept in linear regression,
(b)
(c)
Fig. 6. Chloride diffusivity of cement–FA concrete in splashing zone (a), tidal zone (b) and immerged zone (c) (DCl in 1012 m2/s and t in day).
(a)
(b)
(c)
Fig. 7. Chloride diffusivity of cement–SG concrete in splashing zone (a), tidal zone (b) and ternary binder concrete in splashing, tidal and immerged zone (c) (DCl in 1012 m2/s and t in day).
Table 3 Aging exponent n from exposure data and Life-365 model. Data source
Life-365 Model Exposure (splashing zone) Exposure (tidal zone) Exposure (immerged zone)
Cement–FA binder
Cement–SG binder
30%
35%
40%
60%
70%
80%
0.44 0.47 0.46 0.44
0.48 0.49 0.49 0.47
0.52 0.51 0.51 0.52
0.54 0.45 0.45 –
0.60 0.52 0.51 –
0.66 0.61 0.58 –
Ternary binder (FA 20%, SG 30%) 0.53 0.54 0.56 0.55
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Q. Li et al. / Structural Safety 53 (2015) 1–12 Table 4 Statistical properties of n values and chloride diffusivity for HZM durability design. Parameter n
Mean value () Deviation () Mean value (1012 m2/s) Deviation (1012 m2/s)
ln(D0Cl)
SD½n ¼ r½lnðDC1 Þ
Atmospheric
Splashing
Tidal
Immerged
0.53 0.079 – –
0.47 0.029 1.413 0.076
0.46 0.029 0.842 0.064
0.44 0.028 0.598 0.097
" #12 m X ðxi xÞ2 ; i¼1
h i SD ln D0C1 ¼ r½ln ðDC1 Þ
3.1. Chloride diffusivity D0Cl and concrete cover thickness xd
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 1 þ Pm m 2 i¼1 ðxi xÞ
ð11Þ
where r[ln(DCl)] is the SD between the measured and regressed values of ln(DCl), xi, x are measured and mean values of ln(t0/t), and m is the number of specimens. The standard deviations of n and ln(D0Cl) values are presented in Table 4. For atmospheric zone, the recommendation of 0.15 time mean value is adopted for SD of n values after [18]. The statistical properties of n values are to be used in the subsequent design process. But the mean values and SD for ln(D0Cl) are just for analysis purpose and the values are not to be used in design since because the uncertainty of D0Cl associated with field practice is not included in Table 4 from exposure data and this uncertainty is usually more significant. Taking D0Cl as design unknown, its variation coefficient is set to be 20%, same as fib model code [19]. 3. Design by fully probabilistic method Through the above statistical analysis, the durability design can be performed through Eq. (4) following a fully probabilistic approach. The concrete cover thickness xd and the chloride diffusion coefficient D0Cl are usually retained as main design unknowns for an expected service life tSL and reliability level b. The reliability level b is actually the reliability index corresponding to the failure probability pf with respect to the design equation in Eq. (4), i.e., b = U1(1 pf), where U1() is the inverse CDF of the standard normal distribution. The other parameters and their statistical properties are regarded as known values, dependent on exposure conditions and material compositions. In Section 2 all parameters are analyzed with respect to the target proportioning of structural concrete of the HZM project, thus these parameters and properties are considered dependent only on exposure conditions. In Table 5 are summarized the design parameters (unknown) and known parameters on the basis of statistical analysis in Section 2.
The full probabilistic design through Eq. (4) is performed by Monte-Carlo simulation [38] for the joint occurrence of 5 random variables Cs, Ccr, D0Cl, n, xd. A computer-based program is specifically developed for the full probabilistic solution of Eq. (4). The sampling number for each simulation was 100,000 to obtain ‘‘exact’’ probability of failure. Two service lives and three levels of reliability index are considered: tSL = 50 years, 120 years, and b = 1.3, 1.5 and 1.8. Moreover, two exposure ages are considered for quality control purpose: t0 = 28 d, 56 d. From Figs. 8–10 are presented the design results for relationship between D0Cl (mean value) and xd (mean value) for target reliability index b = 1.3, 1.5 and 1.8. The above figures can serve as design charts to choose concrete cover thickness in terms of required durability quality (D0Cl) for different exposure conditions, reliability levels and service lives. For b = 1.3 in Fig. 8, if D0Cl(t0 = 28 d) is controlled within 3.0 1012 m2/s, the mean value of xd should be not less than 49 mm (50 years) and 73 mm (120 years) for splashing zone; in tidal zone the xd mean value becomes 46 mm (50 years) and 69 mm (120 years); in immerged zone the mean value of cover thickness becomes 36 mm (50 years) and 55 mm (120 years); in atmospheric zone these values are 27 mm (50 years) and 40 mm (120 years). The design values for D0Cl(28 d, 56 d) and xd can be read from Figs. 9 and 10 for b = 1.5 and b = 1.8, respectively.
3.2. Sensitivity analysis for reliability index An important issue in full probabilistic approach is the sensitivity of reliability index with respect to the model parameters and their statistical properties. Focus always on the mean values of D0Cl and xd. Their impact on the reliability index for service life of 120 years is evaluated in the following. The reference design points, in terms of mean values, are selected as D0Cl(28 d) = 3.0 1012 m2/s, D0Cl(56 d) = 2.2 1012 m2/s, xd = 40 mm (atmospheric zone), 73 mm (splashing zone), 69 mm (tidal zone), 55 mm (immerged zone). During the sensitivity analysis, one design parameter varies with the others kept fixed, and the corresponding reliability index is evaluated through HZM model using
Table 5 Statistical properties for model parameters of chloride ingress. Parameter (distribution)
Statistical properties
Immerged zone
Tidal zone
Splashing zone
Atmospheric zone
Surface concentration Cs (Lognormal)
Average (%) Deviation (%) Lower bound L (%) Upper bound U (%) Coefficient a () Coefficient b () Average (%) Deviation (%) Average (1012 m2/s) Coefficient of variance Average () Deviation () Average (mm) Deviation (mm)
4.5 0.68 1.0 3.5 0.23 0.33 – – Design 0.2 0.44 0.028 Design 5.3
3.8 0.58 0.45 1.25 0.22 0.36 – – Design 0.2 0.46 0.029
5.4 0.82 0.45 1.25 0.22 0.36 – – Design 0.2 0.47 0.029
2.0 0.31 – – – – 0.85 0.13 Design 0.2 0.53 0.079
Critical concentration Ccr (beta)
Critical concentration Ccr (Lognormal) Diffusion coefficient D0Cl (Lognormal) Exponent coefficient n (Normal) Concrete cover xd (Normal)
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Q. Li et al. / Structural Safety 53 (2015) 1–12
(a)
(b)
Fig. 8. Concrete cover thickness and chloride diffusion coefficients for b = 1.3: (a) tSL = 50 years (b) tSL = 120 years (solid line for t0 = 28 d, dash-line for t0 = 56 d).
(a)
(b)
Fig. 9. Concrete cover thickness and chloride diffusion coefficients for b = 1.5: (a) tSL = 50 years (b) tSL = 120 years (solid line for t0 = 28 d, dash-line for t0 = 56 d).
(a)
(b)
Fig. 10. Concrete cover thickness and chloride diffusion coefficients for b = 1.8: (a) tSL = 50 years (b) tSL = 120 years (solid line for t0 = 28 d, dash-line for t0 = 56 d).
Monte-Carlo simulation. The results are presented as the achieved reliability index in terms of mean value of xd, D0Cl(28 d) and D0Cl(56 d) in Figs. 11 and 12. It can be seen that the reliability index b is sensitive to the mean values of both concrete cover thickness xd and the apparent chloride diffusivity D0Cl(t0 = 28 d, 56 d). The sensitivity of b to xd is similar for all exposure zones, and a reduction of 5 mm in xd results in roughly a drop of 0.4 for b. The reliability index b is less sensitive to D0Cl for atmospheric zone than for the other exposure zones. In atmospheric zone, a reduction of 0.4 1012 m2/s in D0Cl(t0 = 28 d) or 0.3 1012 m2/s in D0Cl(t0 = 56 d) results in roughly an increase of 0.2 for b while for other exposure conditions (splashing, tidal and immerged zones) D0Cl needs to decrease 0.22 1012 m2/s (t0 = 28 d) or 0.17 1012 m2/s (t0 = 56 d) to increase b by 0.2. The reliability index b is also influenced by the dispersion (SD) of
these design parameters. However, compared with their mean values the impact of SD is less significant, so the sensitivity with respect to their SD is not discussed here.
4. Design by partial factor method 4.1. Design equation Although the full probabilistic method established in the previous section can perform durability design for chloride ingress for the HZM project, a design method in partial factor format is still expected because the partial factor method is much easier to use and much accessible to design engineers. Thus, this part is dedicated to establishing the design method in partial factor format
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Q. Li et al. / Structural Safety 53 (2015) 1–12 cðnomÞ
xdd ¼ xd
Dx
ð14Þ
On the basis of the statistical analysis of achieved concrete cover thickness in Section 2.2.1, the standard deviation for thickness larger than 50 mm is 5.26 mm, and the fractile of 5% in the lognormal distribution corresponds roughly to 10 mm. Thus, for cast-in-place concrete elements Dx is taken as 10 mm. For pre-fabricated concrete elements, the concrete cover is subject to stricter control, thus this tolerance is taken as 5 mm, same as recommendations from DuraCrete [19] and fib model [20]. In terms of characteristic values and partial factors, Eq. (12) becomes,
2 G¼
Fig. 11. Sensitivity of reliability index with respect to cover thickness.
2 G¼
6 C ds 41
0
13
xdd B C7 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A5 0;d d 2 DC1 g ðt0 Þ tSL
1
ccr
ð12Þ
0;c d c C ccr ; C ds ¼ cCS C cs ; D0;d C1 ¼ cD DC1 ; g ðt 0 Þ ¼ cg g ðt 0 Þ
0
c6 Cs C s 41
c
13
xcd Dx B C7 ffiA5 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0;c c 2 cD DC1 cg g t SL
ð15Þ
In this partial factor format, all parameters, except concrete cover thickness, adopt directly the mean values in statistical analysis as characteristic values. For concrete cover thickness, the tolerance expression in Eq. (14) is used to calculate the characteristic value. The design parameters are summarized in Table 7 Partial factors of design parameters of HZM model for chloride ingress.
The superscript ‘d’ stands for the design value for parameters. Introducing partial factor, the design values for Ccr, Cs and D0Cl write,
C dcr ¼
ccr
C ccr
4.2. Characteristic values and partial factors
on the basis of the first order reliability method (FORM), determining the characteristic values and the partial factors for design parameters. In partial factor format, the design equation, G, taking always the corrosion initiation as DLS, writes,
C dcr
1
Target reliability b⁄
Exposure zone
ccr
cCs
cg
cD
1.3
Atmospheric Splashing Tidal Immerged
1.07 1.66 1.66 2.00
1.08 1.04 1.05 1.04
1.32 1.04 1.03 1.03
1.07 1.07 1.07 1.01
1.5
Atmospheric Splashing Tidal Immerged
1.09 1.67 1.67 2.00
1.10 1.05 1.06 1.04
1.38 1.05 1.05 1.04
1.08 1.08 1.09 1.02
1.8
Atmospheric Splashing Tidal Immerged
1.10 1.67 1.67 2.00
1.12 1.06 1.07 1.05
1.51 1.06 1.06 1.05
1.11 1.10 1.10 1.02
ð13Þ
where the superscript ‘c’ stands for the characteristic values and c for the partial factors. The aging coefficient g has the expression in Eq. (3). Since the aging exponent n observes normal distribution g follows lognormal distribution. This partial factor format, according to structural design customs, is not used for concrete cover thickness xd. Instead, a tolerance is attributed to concrete cover thickness, linking its design and characteristic (nominal) values,
(a)
(b)
Fig. 12. Sensitivity of reliability index with respect to chloride diffusivity at: (a) 28 d (b) 56 d.
Table 6 Characteristic values of design parameters for partial factor format. Exposure zone
Cccr (% binder)
Ccs (% binder)
gc (28 d) [gc (56 d)]
12 D0,c m2/s) Cl (28 d, 56 d) (10
xcd (mm)
Atmospheric Splashing Tidal Immerged
0.85 0.75 0.75 2.0
1.98 5.44 3.82 4.50
0.047 0.061 0.067 0.074
3.0 3.0 3.0 3.0
50 80 80 60
[0.067] [0.084] [0.091] [0.100]
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Q. Li et al. / Structural Safety 53 (2015) 1–12
Table 8 Design results by partial factor method and achieved reliability indices. Target reliability b⁄
Exposure zone
tSL = 50 years D0,c Cl
(28 d) (10
tSL = 120 years 12
2
m /s)
xcd
(mm)
b ()
12 D0,c m2/s) Cl (28 d) (10
xcd (mm)
b ()
1.3
Atmospheric Splashing Tidal Immerged
3.5 3.0 3.0 3.5
32 53 51 43
1.75 1.81 1.99 1.90
3.5 3.0 3.0 3.5
45 78 75 63
1.56 1.68 1.80 1.81
1.5
Atmospheric Splashing Tidal Immerged
3.5 3.5 3.0 3.0
33 54 51 43
1.86 1.93 1.99 1.90
3.5 3.0 3.0 3.5
47 80 77 64
1.72 1.84 1.96 1.88
1.8
Atmospheric Splashing Tidal Immerged
3.5 3.5 3.0 3.0
34 55 52 44
1.96 2.04 2.11 2.01
3.5 3.0 3.0 3.5
49 82 79 65
1.91 2.00 2.12 1.96
Table 6 for all exposure conditions. The characteristic values of D0Cl and xd in Table 6 are the reference points for partial factor calibration and will be taken as design unknowns in partial factor format design. The partial factors are calibrated using first-order reliability method (FORM) with specified statistical properties for parameters and specified reliability level. The partial factors are evaluated by minimizing the distance between the origin and the failure surface in the normalized independent variable space [16]. Table 7 provides the partial factors for b = 1.3, 1.5 and 1.8, and the calibration procedure is detailed in Appendix. It is noted that the partial factors have only slight change (less than 0.02) as tSL changes from 50 to 120 years, and the factors are nearly identical for taking t0 = 28 d or 56 d for D0Cl. Therefore, the partial factors in Table 7 apply for service lives of 50–120 years and for both exposure ages (28 d, 56 d) of D0Cl. 4.3. Design results by partial factor method Using the design equation Eq. (15) and the characteristic values in Table 6 and the partial factors in Table 7, one is capable to make the durability design for chloride ingress taking the apparent chloride diffusivity D0Cl and concrete cover thickness xd as design unknowns for expected reliability levels. Table 8 presents the results for b = 1.3, 1.5 and 1.8 and different exposure zones, and the results of b = 1.3 serve as the basis for durability requirements in the preliminary design of concrete structures in the HZM project. The achieved reliability index is checked for these design results via the full probabilistic method using Monte-Carlo simulation. The achieved reliability indices are also presented in Table 8. It can be seen that the achieved reliability index b is generally higher than the target one b⁄, and for a target reliability of 1.3 the design results via partial factor format achieve actually a reliability index b = 1.75–1.99 for tSL = 50 years and b = 1.56–1.80 for tSL = 120 years. This is mainly due to the conservative adoption of the concrete cover tolerance as 10 mm in Eq. (15), see Appendix for details. 5. Concluding remarks 1. For the durability design of concrete structures in the HZM project, the empirical Fick’s law is retained and adapted for the project site environmental conditions and the concrete material proportioning. The results on long-term structural investigation and exposure tests during the last 30 years in South-eastern costal region in China are used to determine the values and statistical properties of concrete cover thickness, concrete surface
chloride concentration, critical chloride concentration for steel corrosion initiation and the time-dependent chloride diffusivity of concrete. On the basis of these data, a HZM model is established for durability design for chloride ingress of concrete structures in the HZM project. 2. On the basis of available statistical properties of model parameters, a full probabilistic method is set up for the durability design of concrete structure for chloride ingress for intended service lives and expected reliability levels. The design adopts the corrosion initiation state as DLS and takes chloride diffusivity at given age D0Cl and concrete cover thickness xd as main design unknowns, and design is performed for service lives of 50 years and 120 years and for reliability index b = 1.3, 1.5 and 1.8. The sensitivity analysis shows that the achieved reliability of design is rather sensitive to the mean values of diffusivity and cover thickness. 3. For easier engineering use, a partial factor format of durability design for chloride ingress is calibrated from first-order reliability method, and the characteristic values and the partial factors are determined for different levels and reliability index and different exposure zones. Taking chloride diffusivity and cover thickness as design unknowns, their values are given for tSL = 50 years, 120 years and target reliability index b⁄ = 1.3, 1.5 and 1.8. The verification through full probabilistic method gives achieved reliability index higher than the target one, mainly because the conservative value of concrete cover tolerance. 4. The design results through partial factor method with b⁄ = 1.3 are retained for preliminary design of concrete structure in the HZM project. This apparently low reliability level is justified by the conservative considerations during the design: first DLS is taken as corrosion initiation state, a rather early state thus conservative for service life; second, the achieved reliability index through partial factor method for b⁄ = 1.3 is actually above 1.5 by full probabilistic verification; finally the design is just for preliminary phase and additional protection measures are to be implemented and the reliability level can be increased. The updating of reliability index of durability levels by implemented protection measures is to be treated in later publications.
Acknowledgments The research is supported by the China National Science and Technology Support Planning Project No. 2011BAG07B04. The support of Management Bureau of HZM project (Zhuhai, China) is also acknowledged. Special thankfulness is extended to Prof. Bruce
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Q. Li et al. / Structural Safety 53 (2015) 1–12 Table A.1 Calculation procedure of partial factors for splashing zone design (Target b⁄ = 1.3). Values Characteristic value () Sensitivity factor a () Normalized point Z Design point xi⁄ Partial factor ci (or Dx)
Ccr 0.75% 0.855 1.11 0.45% 1.66
Cs
xd
5.44% 0.165 0.22 5.67% 1.04
80 mm 0.261 0.34 73.2 mm 1.8 mm
D0Cl (28 d) 12
g 2
3.0 10 m /s 0.279 0.36 2.36 1012 m2/s 1.07
0.061 0.311 0.40 0.064 1.04
Ellingwood for his constructive suggestions and linguistic help in the revision of this paper.
exposure conditions can be calculated following the same procedure.
Appendix A. Calibration algorithm for partial factors
References
The limit state function is given in Eq. (4), containing 5 random variables. The partial factors are determined by first-order reliability method (FORM). The FORM involves the mapping of the set of to a set of independent standard normal basic random variables X variables Z. This results in the mapping of the limit state failure surface given by Eq. (4) to a failure surface in standard normal space,
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f ðZ 1 ; Z 2 ; . . . ; Z n Þ ¼ 0
ðA-1Þ
The reliability index b is defined in Z space as the shortest distance from the origin to the failure surface. Meanwhile, the sensitivity fac , is also obtained. If the target reliability is b⁄, then the design tor, a point in the normalized Z space is,
Z ¼ b a
ðA-2Þ
By using the inverse mapping,
xi ¼ F 1 X i U zi
ðA-3Þ
corx for the original basic variables X one obtains the set of values . In Eq. (A-3), F 1 ðÞ stands for the responding to the design point Z Xi inversed probability distribution function of random variable Xi, UðÞ is the standard normal distribution function. If the values x were to be used as the design values in a deterministic design, the resulting design would have a reliability index b⁄. Thus a satisfactory set of partial factors is given by:
ci ¼
xci xi
ðA-4Þ
where xci is the characteristic value of the resistance variable Xi, and by:
ci ¼
xi xci
ðA-5Þ
where xic is the characteristic value of the loading variable Xi. Take the design in splashing zone as example. The initial characteristic value of xd is assumed 80 mm, D0Cl(28 d) is assumed 3.0 1012 m2/s and the service life tSL is 120 years. The detailed calculations are shown in Table A.1, and the concrete cover is noted by a tolerance Dx. The initial reliability index is b = 2.26 and the target index is b = 1.3. From the table, one can see that the required tolerance in concrete cover thickness is significantly less than the specified value 10 mm. This provides conservativeness to the partial factor design using 10 mm as tolerance and explains why the ‘actual’ reliability indices achieved by partial factor format are larger than the target ones. If the design value of xd is changed to 50 mm, the design value of D0Cl(28 d) changed to 2.0 1012 m2/s, or the service life is changed to 50 years, the partial factors have only slight change, indicating that these factors are not sensitive to the pre-assumed design values. Partial factors for the other
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