Destructive Testing Methods:

1) Acid-Soluble (Total) Chloride (ASTM C1152/AASHTO T 260): Measures all chloride ions, including those bound in the cement paste.

2) Water-Soluble Chloride (ASTM C1218/AASHTO T 260): Measures only free chloride ions, which are primarily responsible for steel rebars' corrosion.

3) Titration Method: A common, highly accurate laboratory method (potentiometric or silver nitrate) to determine the precise chloride concentration.

4) X-Ray Fluorescence: It is used for high-precision, detailed imaging of chloride profiles and concentration levels.

Non-Destructive/In-Situ Testing Methods:

These methods allow for rapid, field-based assessment without damaging the structure.

1) Electrochemical Sensors: Use ion-selective electrodes or embedded sensors to monitor the potential difference, which corresponds to the free chloride ion concentration in the pore solution.

2) Electrical Resistivity: Measures the corrosion rates due to the penetration of chloride ions into the concrete microstructure (ASTM C1202).

3) Silver Nitrate: A qualitative method that changes color to show the depth of chloride penetration.

4) Ground Penetrating Radar, microwave spectroscopy, and Nuclear Magnetic Resonance are used for quick, non-destructive, and in-situ monitoring of chloride penetration.

1) Phenolphthalein Indicator: The most common method; a 1% solution is sprayed on a freshly broken concrete surface. Uncarbonated (alkaline) concrete turns bright pink, while carbonated (low pH) concrete remains colorless

2) Carbonation Depth Measurement: Measures the distance from the concrete surface to the pink boundary, indicating how far CO₂ has penetrated

3) Half-Cell Potential Measurements: A non-destructive method utilized to estimate the probability of corrosion for steel rebars.

4) Linear Polarization Resistance (LPR): Utilized to determine the instantaneous corrosion rate of the steel rebars.

5) Electrochemical Impedance Spectroscopy: A technique that monitors the rise in concrete resistivity as pores fill with CaCO₃.

6) Gravimetric Mass Loss: A highly accurate method used in laboratory settings to calculate the precise amount of metal lost to corrosion.

Type I (Gumbel)

ξ = 0

$\mathit{exp}\left[-\mathit{exp}\left(-\frac{X-\mu }{\sigma }\right)\right]$

$\frac{1}{\sigma }\times \mathit{exp}\left[-\frac{X-\mu }{\sigma }-\mathit{exp}\left(-\frac{X-\mu }{\sigma }\right)\right]$

Type II (Fréchet)

ξ ˃ 0, X > µ

$\exp \left(-{\left(\frac{\sigma }{X-\mu }\right)}^{\xi }\right)$

$\left(\frac{\xi }{\sigma }\right){\left(\frac{\sigma }{\mathrm{ }X-\mu }\right)}^{(\xi +1)}\exp \left(-{\left(\frac{\sigma }{X-\mu }\right)}^{\xi }\right)$

Type III (Weibull)

ξ ˂ 0, X < µ

$1-\exp \left(-{\left(\frac{X-\mu }{\sigma }\right)}^{\xi }\right)$

$\left(\frac{\xi }{\sigma }\right){\left(\frac{\mathrm{ }X-\mu }{\mathrm{ }\sigma }\right)}^{(\xi -1)}\exp \left(-{\left(\frac{X-\mu }{\mathrm{ }\sigma }\right)}^{\xi }\right)$

G=CV $-\sqrt{\frac{2\times D_{e,\mathrm{ }\mathit{CO}2}\times \left(\frac{{\mathit{CO}}_2}{100}\right)\times t}{0.218\times (C+\mathit{kP})}}$ $f\mathrm{ }\left(T\right)=\exp \left[\frac{U_c}{R}\times \left(\frac{1}{T_r}-\frac{1}{T}\right)\right]$ f(RH) = (1 − (RH/100))^2.2

Un-cracked Concretes

$\mathrm{G}=\mathrm{CV}-\sqrt{\frac{2\mathrm{ }{\mathrm{f}}_{\mathrm{T}}\left(\mathrm{t}\right)\times {\mathrm{D}}_{\mathrm{co}2}\left(\mathrm{t}\right)}{\mathrm{a}}{\mathrm{K}}_{\mathrm{urban}}\underset{2000}{\overset{\mathrm{t}}{\int }}{\mathrm{C}}_{\mathrm{co}2}\left(\mathrm{t}\right)\mathrm{ dt}}$$\mathrm{G}=\mathrm{CV}-\sqrt{\frac{2{\mathrm{f}}_{\mathrm{T}}\left(\mathrm{t}\right)\times {\mathrm{D}}_{\mathrm{co}2}\left(\mathrm{t}\right)}{\mathrm{a}}{\mathrm{K}}_{\mathrm{urban}}\underset{2000}{\overset{\mathrm{t}}{\int }}{\mathrm{C}}_{\mathrm{co}2}\left(\mathrm{t}\right)\mathrm{dt}}$$G=\mathit{CV}-\sqrt{\frac{2\mathrm{ }{f}_T\left(t\right)\times D_{\mathit{co}2}\left(t\right)}{a}{K}_{\mathit{urban}}\underset{2000}{\overset{t}{\int }}{C}_{\mathit{co}2}\left(t\right)\mathrm{ }\mathit{dt}}$ $f_t\left(t\right)=e^{\frac{E}{R}\times \left(\frac{1}{293}-\frac{1}{273+T_{\mathit{average}\mathrm{ }}\left(t\right)}\right)}$ $T_{\mathit{average}\mathrm{ }}\left(t\right)=\frac{\sum _{i=2000}^{t}T\left(t\right)}{t-1999}$ ${\mathrm{f}}_{\mathrm{t}}\left(\mathrm{t}\right)={\mathrm{e}}^{\frac{\mathrm{E}}{\mathrm{R}}\times \left(\frac{1}{293}-\frac{1}{273+{\mathrm{T}}_{\mathrm{average }}\left(\mathrm{t}\right)}\right)}$${\mathrm{f}}_{\mathrm{t}}\left(\mathrm{t}\right)={\mathrm{e}}^{\frac{\mathrm{E}}{\mathrm{R}}\times \left(\frac{1}{293}-\frac{1}{273+{\mathrm{T}}_{\mathrm{average}}\left(\mathrm{t}\right)}\right)}$

$G=\mathit{CV}-X_c\left(t\right)=\sqrt{\frac{2\mathrm{ }{D}_{\mathit{co}2}\left(t\right)\mathrm{ }}{a}{K}_{\mathit{urban}}{C}_{\mathit{co}2}\mathrm{ }(t-1999)}$

G= CV$-\left(\sqrt{\frac{2\times D_{e,\mathrm{ }\mathit{CO}2}\times \left(\frac{{\mathit{CO}}_2}{100}\right)\times t}{0.218\times (C+\mathit{kP})}}\times (2.816\times \sqrt{W_c}+1)\right)$

Cracked Concrete

G=CV$-\left(\sqrt{\frac{2\times D_{e,\mathrm{ }\mathit{CO}2}\times \left(\frac{{\mathit{CO}}_2}{100}\right)\times t}{0.218\times \left(C+\mathit{kP}\right)}}\times \left(11.4\times \sqrt{W_c}+1\right)\right)$ $f\mathrm{ }\left(T\right)=\exp \left[\frac{U_c}{R}\times b\times \left(\frac{1}{T_r}-\frac{1}{T}\right)\right]$ f(RH) = (1 − (RH/100))^2.2

$g\left(C_s,\mathrm{ }{\mathrm{ }x,C}_{\mathit{th}},\mathrm{ }{D}_c\right)=C_{\mathit{th}}-C_s\left[1-\mathrm{erf}\left(\frac{x}{2\sqrt{D_c\times t}}\right)\right]$ $D_{\mathrm{c}}=D\times f_1\left(\mathrm{T}\right)\times f_2\left(\mathrm{t}\mathrm{e}\right)\times \mathrm{ }{f}_3\left(\mathrm{RH}\right)$ $f_1\left(T\right)=\mathrm{Exp}\left[\frac{E_a}{R}\left(\frac{1}{T_1}-\frac{1}{T_2}\right)\right]$ $f_2\left(\mathit{te}\right)=\xi +\left(1-\xi \right){\left(\frac{28}{t_e}\right)}^{\frac{1}{2}}$ $f_3\left(\mathit{RH}\right)={\left[1+\frac{{\left(1-\mathit{RH}\right)}^4}{{\left(1-{\mathit{RH}}_c\right)}^4}\right]}^{-1}$

Uncracked concrete

$g\left(C_s,\mathrm{ }{\mathrm{ }x,C}_{\mathit{th}},\mathrm{ }{D}_c\right)=C_{\mathit{th}}-C_s\left[1-\mathrm{erf}\left(\frac{x}{2\sqrt{D_{\mathit{cc}}\times t}}\right)\right]$ $D_{\mathit{cc}}=\frac{\left({A\times D}_a\right)+{(A}_{\mathit{cr}}\times D_{\mathit{cr}})}{(A+A_{\mathit{cr}})}$ $\left\{\begin{array}{c}{D}_{\mathit{cr}}\approx 2\times {10}^{-11}w-4\times {10}^{-10}\mathrm{ }30\mathrm{ }\mathit{\mu m}\le w\le 80\mathrm{ }\mathit{\mu m}\\ D_{\mathit{cr}}\approx 14\times {10}^{-10}w>80\mathrm{ }\mathit{\mu m}\end{array}\right.$ $D_a={10}^{-6.77\times {\left(\frac{W}{C}\right)}^2+10.10\times \left(\frac{W}{C}\right)-14.64}\left(\mathit{withoutSCM}\right)$ $D_a={10}^{-0.79\times {\left(\frac{W}{C}\right)}^2+3.40\times \left(\frac{W}{C}\right)-13.10}\left(\mathit{withSCM}\right)$ Another Performance Function Method: $g\left(C_s,\mathrm{ }{\mathrm{ }x,C}_{\mathit{th}},\mathrm{ }{D}_c\right)=C_{\mathit{th}}-C_s\left[1-\mathrm{erf}\left(\frac{x}{2\sqrt{D_c\times t}}\right)\right]$ $D_{\mathrm{c}}=D\times f_1\left(\mathrm{T}\right)\times f_2\left(\mathrm{t}\mathrm{e}\right)\times \mathrm{ }{f}_3\left(\mathrm{RH}\right)\times \mathrm{f}\left(\mathrm{w}\right)$ f(w)=1+347.85w-1642.57w²+4189w³ (w ranges from 0.1 to 0.4 mm) according to Park et al.

Cracked concrete

Assumptions

Diffusion-Dominant Transport: It is assumed that chloride ingress is governed primarily by Fickian diffusion in saturated or partially saturated pores, ignoring convective transport mechanisms (e.g., absorption during wet-dry cycles). Homogeneous Matrix: The model assumes a homogeneous concrete matrix and uniformly distributed concrete cover, often ignoring micro-cracks and localized defects.

Sharp pH Front: The model assumes the carbonation front represents a sharp, distinct drop in the pore solution's pH from above 12 down to approximately 9, at which point the passive film on the steel becomes unstable. Diffusion-Based Progress: CO₂ transport is treated as a diffusion-driven process described generally by Fick’s laws.

Sources of Uncertainty

Critical Chloride Threshold: It is highly variable. Depending on the steel type, cement chemistry, and the local pH of the pore solution, the threshold fluctuates widely, making it one of the largest sources of epistemic uncertainty in service-life modeling. Environmental / Exposure Conditions: Surface chloride concentration (C_s) and the aging factor rely heavily on microclimates, wetting-drying cycles, and temperature, all of which exhibit intrinsic statistical randomness (aleatoric uncertainty). Model Accuracy: Analytical equations, including the error-function solution, contain simplification errors. Using deterministic constants for naturally convoluted physicochemical processes necessitates the use of partial safety factors or full probabilistic frameworks (like Monte Carlo simulations) to account for scatter.

Model Uncertainty: Standard equations rely on empirical simplifications (such as the square-root-of-time law) that cannot perfectly map non-linear microclimatic shifts and unpredictable weather changes. Material Variability: The permeability and alkalinity of concrete can vary significantly batch-to-batch, and cast-in-place concrete properties rarely match idealized laboratory conditions. Climate Change: Projected increases in atmospheric (CO₂) and shifting global temperatures directly alter carbonation rates, making historic baseline averages unreliable.

Practical Applications

Probabilistic Design: Used in modern structural design codes (such as the fib Model Code for Concrete Structures) and ISO standards to perform full-probabilistic service life design. Engineers set a target reliability index (often targeting a failure probability of 10% to 7% for serviceability to determine the required concrete cover thickness and maximum allowable diffusion coefficients.

Practically, carbonation-induced corrosion is considered a Serviceability Limit State (SLS). Since cross-sectional loss is generally negligible in the absence of chlorides, the primary structural threat is expansive cracking and concrete spalling resulting from rust accumulation. Thus, the function is applied during the structural design phase to specify adequate concrete cover and concrete mix qualities, or during the assessment of existing structures to plan preventative maintenance and surface treatments. Because of the high uncertainties, engineers rarely use this limit state function with single deterministic values. Instead, they apply probabilistic methods (like Monte Carlo Simulations within the fib Model Code framework) to calculate the Probability of Corrosion Initiation (usually aiming for a target reliability index (β= 1.3) for depassivation).

A1: Raw Material Supply

Cement/Binder

82% - 89%

Calcination of limestone comprises ~50% of CO₂, and fossil fuel combustion for rotary kilns consumes about ~40% of CO₂.

Steel Rebar

7% - 11%

Primary steel manufacturing from iron ore and scrap in blast furnaces.

Aggregates

2% - 4%

Quarrying, crushing, and washing coarse and fine aggregates.

A2: Transport

Material Delivery

2% - 4%

Diesel is consumed in transporting raw materials (aggregates, cement, steel) to mixing plants.

A3: Manufacturing

Mixing & Processing

1% - 2%

Electricity and fossil fuels are utilized at the ready-mix or precast concrete plant.

50/50 FA/SG

2.38

40/60 FA/SG

1.01

OPC

6.70

Mechanism

Sacrificial zinc coating (cathodic protection).

Forms a protective passive oxide film

Corrosion Resistance

Less Superior compared to SS.

Superior

Cost

(1000-1300 USD)/ ton in the USA

(2500-4000 USD)/ ton in the USA

Durability

Less Robust compared to SS.

More Robust

Canadian Code (CSA A23.3-19)

Based on CSA A23.3-19, the acceptable crack widths in RC are generally designed based on exposure conditions to ensure durability and prevent corrosion (CSA A23.3):

1) Interior Environment: 0.40 mm.

2) Exterior Environment: 0.30 mm to 0.33 mm.

3) Corrosive Environment: 0.15 mm to 0.20 mm.

4) Water Retaining Structures: (<0.2 mm).

The Canadian climate necessitates a focus on Freeze-Thaw (F-classes) and Chloride/De-icer exposure (C-classes) for RC structures having a fixed service life (e.g., 50 or 75 years).

Egyptian Code of Practice (ECP 203)

Under the Egyptian Code of Practice (ECP 203), crack width limits are not explicitly determined by broad geographic climate regions. Instead, the code defines limits based on the specific environmental exposure conditions (humidity, presence of aggressive chemicals) and the structure's vulnerability. The Standard target service life is conventionally assumed to be 50 to 75 years for typical concrete buildings.

According to the Egyptian Code of Practice (ECP 203), the allowable crack width in RC structures typically varies from 0.1 mm to 0.3 mm, depending on environmental exposure conditions and structural type.

Environmental Exposure:

1) Aggressive Environments: Often limited to 0.1 mm.

2) Normal Environments: Generally limited to 0.2 mm–0.3 mm.

3) Water tanks: the allowable crack width is often 0.1 mm or 0.2 mm.

American Concrete Institute (ACI-224R-01)

ACI-224R-01 establishes tolerable crack widths based purely on environmental exposure and aesthetic needs, rather than strict service life. Standard design service life for permanent structures is typically 50 to 120 years, but ACI's crack limits are defined by the severity of the surrounding climate.

Acceptable crack widths in concrete vary from 0.1 mm to 0.41 mm, depending on the exposure condition (ACI 224R-01/ACI 350-06).

1) Protective membrane: 0.41 mm.

2) Humidity, moist air: 0.30 mm.

3) Deicing chemicals: 0.18 mm.

4) Seawater, wetting/drying: 0.15 mm.

5) Water-retaining structures: 0.10 mm.

American Association of State Highway and Transportation Officials (AASHTO)

AASHTO sets a default design life of 75 years for concrete bridges. While the specifications establish limits based on exposure classes, they do not directly classify crack widths by regional climate. Instead, maximum crack width (w) is tied to specific environmental exposure conditions.

AASHTO specified crack width limitations to 0.4 mm for Class 1 (severe exposure/deicing chemicals) and 0.3 mm for Class 2 (moderate exposure) to ensure durability. These limits help control reinforcement corrosion, particularly in bridge decks and superstructure components.

Saudi Building Code (SBC 304)

The Saudi Building Code (SBC 304) relies on American Concrete Institute (ACI 318) methodologies but features customized durability requirements suited for Saudi Arabia’s severe climate. Crack width limits primarily depend on the environmental exposure class rather than just the geographic region or target service life. Under the Saudi Building Code, standard structures are typically designed for a target service life of 50 years, while major infrastructure projects (bridges, high-rises) are typically designed for 100 years.

The acceptable crack width in concrete according to SBC 304 depends on the environmental exposure conditions, with a common maximum limit of 0.30 mm to 0.40 mm for general RC members under service loads.

1) Moderate Exposure: Maximum crack width is often taken as 0.40 mm.

2) Severe Exposure: Maximum crack width is restricted to 0.30 mm.

3) Water Retaining Structures: For enhanced watertightness, more restrictive limits are applied, often requiring crack widths to be less than 0.2 mm.

British Standard European Norm (BS EN 1992-1-1/Eurocode 2) and BS 8110

Under both British and European structural concrete standards, allowable crack width limits (Wmax) are primarily governed by exposure severity, durability, and the likelihood of steel corrosion

The generally accepted maximum crack width for RC members is 0.3 mm for most structural applications. This limit is primarily for aesthetics and serviceability; tighter limits of 0.1 mm to 0.2 mm are required for severe, aggressive, or watertight environments.

A

Area of Uncracked Concrete (mm²)

Acr

Area of the Crack (mm²)

Cth

Chloride Threshold Level (kg/m³)

Cs

Surface Chloride Concentration (kg/m³)

CV

Concrete Cover for the rc Deck

Dc

Corrected Chloride Diffusion Coefficient (m²/s)

D

Chloride Diffusion Coefficient (m²/s)

Da

Apparent Chloride Diffusion Coefficient (m²/s)

Dcr

Chloride Diffusion Coefficient Inside the Crack (m²/s)

Dcc

Total Diffusion Coefficient for Cracked Concrete (m²/s)

E

Activation Energy of the Diffusion Process (40 kj/mol)

f1(T)

A factor that Represents the Maximum Temperature

f2(te)

A factor that Represents the Equivalent Maturation Time of Concrete

f3(RH)

A factor that Represents the Influence of Relative Humidity

G

Performance Function for the Carbonation-Induced Corrosion Initiation Stage

R

Gas Constant (8.314 × 10−3 kj/mol k)

Ti

Time of Corrosion Initiation due to Chlorides (years)

T(t)

Temperature at time t (°c)

w

Crack Width (mm)

x

Concrete Cover Depth Surrounding the Steel Rebar (mm)

X

Climate Variable that will be Considered

μ

Location Parameter in gevd

μ(t)

Time-varying Location Parameter

σ

Scale Parameter

ξ

Shape Parameter

ψ

Scale Parameter