In modern engineering, particularly for critical structures such as offshore platforms and long-span bridges, the application of steel castings is indispensable. These components offer unparalleled design flexibility and excellent mechanical properties, allowing for the creation of complex nodal connections that are both efficient and reliable. However, when these structures are deployed in aggressive environments, especially marine settings, they are subjected to a synergistic and damaging combination of cyclic loading and corrosive media. This phenomenon, known as corrosion fatigue, is a primary degradation mechanism that can lead to catastrophic failure if not properly accounted for in the design and maintenance phases. The corrosion process, often accelerated by chlorides in seawater, interacts with the cyclic stresses to initiate and propagate cracks at a rate far exceeding that of fatigue in an inert environment. This necessitates a robust and accurate methodology for predicting the service life of steel castings under such demanding conditions.
The cornerstone of fatigue life assessment for metallic components is the stress-life (S-N) curve, which defines the relationship between the applied stress amplitude (S) and the number of cycles to failure (N). For design against corrosion fatigue, deriving an accurate S-N curve from experimental data is paramount. Traditional approaches often rely on deterministic models like the Basquin equation or its modifications (e.g., Stromeyer model), which describe the S-N relationship using piecewise linear segments or simple power laws with a fatigue limit. These methods, while codified in various standards like DNV, IIW, and Eurocode, possess inherent limitations. Their piecewise nature introduces artificial拐点, and they often struggle to provide a uniformly good fit across the entire spectrum of stress levels, from high-cycle to low-cycle fatigue regions. Furthermore, these deterministic curves represent only a central tendency, typically the median life (50% failure probability), and do not inherently account for the statistical scatter that is intrinsic to fatigue data, especially in corrosive environments. For reliable design, especially for safety-critical applications involving high-performance steel castings, it is essential to model this scatter and derive S-N curves corresponding to different probabilities of failure (e.g., 5% for conservative design).
This leads to the core challenge: finding a versatile and statistically sound distribution model to characterize the fatigue life data of steel castings. Commonly used distributions like the Normal or Log-Normal, fitted via least squares or maximum likelihood methods, can be unstable with different datasets and computationally intensive for models with multiple parameters. A powerful alternative is the Weibull distribution. Originally developed for reliability engineering, the Weibull distribution has proven exceptionally suitable for modeling time-to-failure data, including fatigue life. Its flexibility stems from its shape parameter, which allows it to model a wide variety of failure rate behaviors. In this analysis, I employ the three-parameter Weibull distribution to model the corrosion fatigue life of steel castings and their welded connections. This model offers a more refined fit by introducing a location parameter, providing greater flexibility to capture the curvature of the S-N relationship accurately. The primary objective is to establish a probabilistic S-N curve model based on the three-parameter Weibull distribution that can reliably predict the corrosion fatigue life of steel castings under simulated marine conditions, thereby offering a superior tool for the assessment and design of durable marine structures.
The Three-Parameter Weibull Distribution Model for S-N Curves
The foundation of the probabilistic model is the three-parameter Weibull cumulative distribution function (CDF). The standard form is given by:
$$F_3(x; \beta, \delta, \lambda) = 1 – \exp\left[ -\left( \frac{x – \lambda}{\delta} \right)^\beta \right]$$
for \(x \geq \lambda\), where \(-\infty < \lambda < +\infty\), \(\delta > 0\), and \(\beta > 0\). Here, \(x\) is the random variable, \(\beta\) is the shape parameter, \(\delta\) is the scale parameter, and \(\lambda\) is the location parameter. The function \(F_3(x)\) represents the probability that the random variable \(X\) is less than or equal to \(x\).
To adapt this distribution to fatigue life modeling, we define the random variable \(x\) as a function of both the fatigue life \(N\) and the stress amplitude \(S\). A highly effective transformation for this purpose is:
$$x = (\ln N – B)(\ln S – C)$$
where \(B\) and \(C\) are correction parameters for the logarithmic fatigue life and logarithmic stress amplitude, respectively. This formulation effectively linearizes the relationship in the log-log domain while allowing for adjustment through \(B\) and \(C\). Substituting this into the Weibull CDF yields the core probabilistic S-N model:
$$Q(N, S) = 1 – \exp\left\{ -\left[ \frac{(\ln N – B)(\ln S – C) – \lambda}{\delta} \right]^\beta \right\}$$
subject to the condition \((\ln N – B)(\ln S – C) \geq \lambda\). In this equation, \(Q(N, S)\) denotes the failure probability at a given stress amplitude \(S\) and cycle count \(N\). This model can be solved to express \(N\) as a function of \(S\) and \(Q\), providing explicit probabilistic S-N curves.
Parameter Estimation Methodology
Estimating the five parameters (\(B, C, \beta, \delta, \lambda\)) robustly and efficiently is crucial. The process is conducted in two main stages.
Stage 1: Estimating Correction Parameters B and C
The parameters \(B\) and \(C\), along with an auxiliary parameter \(\mu\), are estimated by minimizing the sum of squared errors based on the conditional expectation. For a given stress level, the expected value of \(\ln N\) is:
$$E[\ln N | (\ln S – C)] = B + \frac{\mu}{\ln S – C}$$
Therefore, we seek to minimize the function \(R\):
$$R = \sum_{i=1}^{n} \left[ \ln N_i – B – \frac{\mu}{\ln S_i – C} \right]^2$$
where \(n\) is the total number of specimens, and \((\ln S_i, \ln N_i)\) are the data points. As this is a non-linear minimization problem, obtaining good initial guesses is important to ensure convergence. Initial values can be found by selecting three distinct stress levels (e.g., the three highest), calculating the mean log-life \(\bar{\ln N}_j\) for each, and solving the resulting system of three equations from the conditional expectation formula for \(B\), \(C\), and \(\mu\). This initial estimate is then refined using a standard non-linear optimization algorithm.
Stage 2: Estimating Weibull Parameters \(\beta\), \(\delta\), and \(\lambda\) using PWM
Once \(B\) and \(C\) are determined, the sample values \(x_i\) are computed as \(x_i = (\ln N_i – B)(\ln S_i – C)\). These \(x_i\) values are then sorted in ascending order. Instead of resorting to complex iterative methods like Maximum Likelihood Estimation for the three Weibull parameters, I employ the robust and straightforward Probability Weighted Moments (PWM) method. The PWM of orders \(p, r, s\) for a distribution is defined generally. For our purpose, we use the first three PWMs where \(p=1\) and \(s=0\). The theoretical PWMs \(M_{1,r,0}\) for the three-parameter Weibull distribution are:
$$M_{1,0,0} = \lambda + \delta \cdot \Gamma\left(1 + \frac{1}{\beta}\right) = \mu_x$$
$$M_{1,1,0} = \frac{\lambda}{2} + \left(1 – \frac{1}{2^{1+1/\beta}}\right) \delta \cdot \Gamma\left(1 + \frac{1}{\beta}\right)$$
$$M_{1,2,0} = \frac{\lambda}{3} + \left(1 – \frac{1}{2^{1+1/\beta}} + \frac{1}{3^{1+1/\beta}}\right) \delta \cdot \Gamma\left(1 + \frac{1}{\beta}\right)$$
where \(\Gamma(\cdot)\) is the Gamma function. The corresponding unbiased sample estimates \(\hat{M}_{1,r,0}\) are calculated from the ordered sample \(x_{(1)} \leq x_{(2)} \leq … \leq x_{(n)}\):
$$\hat{M}_{1,0,0} = \frac{1}{n} \sum_{i=1}^{n} x_i$$
$$\hat{M}_{1,1,0} = \frac{1}{n(n-1)} \sum_{i=1}^{n} (i-1) x_i$$
$$\hat{M}_{1,2,0} = \frac{1}{n(n-1)(n-2)} \sum_{i=1}^{n} (i-1)(i-2) x_i$$
By equating the sample estimates to their theoretical counterparts, we can solve for the Weibull parameters. I present two effective solution paths derived from these equations.
Solution Path 1 (Direct): The shape parameter \(\beta\) is obtained by numerically solving the following equation derived from the ratios of the PWMs:
$$\frac{3\hat{M}_{1,2,0} – \hat{M}_{1,0,0}}{2\hat{M}_{1,1,0} – \hat{M}_{1,0,0}} = \frac{2 – 3\cdot 2^{-1/\beta} + 3^{-1/\beta}}{1 – 2^{-1/\beta}}$$
Subsequently, the scale and location parameters are calculated as:
$$\delta = \frac{2\hat{M}_{1,1,0} – \hat{M}_{1,0,0}}{(1 – 2^{-1/\beta}) \cdot \Gamma(1 + 1/\beta)}$$
$$\lambda = \hat{M}_{1,0,0} – \delta \cdot \Gamma(1 + 1/\beta)$$
Solution Path 2 (Approximate): An alternative, often very accurate, formula for the shape parameter is:
$$\beta = \left( 7.859c + 2.9554c^2 \right)^{-1}$$
$$\text{where } c = \frac{2\hat{M}_{1,1,0} – \hat{M}_{1,0,0}}{3\hat{M}_{1,2,0} – \hat{M}_{1,0,0}} – \frac{\ln 2}{\ln 3}$$
With \(\beta\) from this formula, \(\delta\) and \(\lambda\) are again computed using the equations from Solution Path 1. Both paths yield valid solutions, demonstrating the existence of multiple valid parameter sets for the Weibull model that fit the data closely.
Analysis and Validation on Corrosion Fatigue Data for Steel Castings
To validate the proposed methodology, I apply it to experimental corrosion fatigue data for a high-strength cast steel grade (similar to G20Mn5QT) and its butt welds to structural steel. The tests were conducted in a 3.5 wt.% NaCl solution under fully reversed axial loading (stress ratio R = -1), simulating a marine splash/atmospheric zone environment.
Case 1: Corrosion Fatigue of Cast Steel Base Metal
The experimental data for the cast steel base metal consists of 21 specimens tested at six stress amplitudes. The measured cycles to failure \(N_{c,cor}\) are listed in the table below.
| Stress Amplitude \(S_{c,cor}\) (MPa) | Specimen | Experimental \(N_{c,cor}\) | Weibull Model Fit \(N_{c,cor}\) (Path 1) | Weibull Model Fit \(N_{c,cor}\) (Path 2) |
|---|---|---|---|---|
| 280 | 1 | 10,053 | 18,500 | 19,586 |
| 2 | 14,191 | |||
| 3 | 23,721 | |||
| 4 | 35,710 | |||
| 240 | 1 | 49,973 | 77,961 | 81,836 |
| 2 | 94,134 | |||
| 3 | 170,940 | |||
| 200 | 1 | 171,771 | 267,094 | 278,344 |
| 2 | 241,446 | |||
| 3 | 243,001 | |||
| 4 | 256,708 | |||
| 180 | 1 | 332,565 | 466,167 | 484,209 |
| 2 | 482,769 | |||
| 3 | 541,841 | |||
| 4 | 675,013 | |||
| 140 | 1 | 1,397,554 | 1,302,650 | 1,344,890 |
| 2 | 1,411,627 | |||
| 3 | 1,864,879 | |||
| 100 | 1 | 2,842,079 | 3,377,490 | 3,467,480 |
| 2 | 2,869,171 | |||
| 3 | 4,068,438 |
Following the two-stage procedure, the estimated parameters for the cast steel are:
Stage 1: \(B = 19.4923\), \(C = 6.5166\), \(\mu = 8.4978\).
Stage 2 (Path 1): \(\beta = 5.5486\), \(\delta = 2.1263\), \(\lambda = 6.5337\).
Stage 2 (Path 2): \(\beta = 2.5404\), \(\delta = 1.0881\), \(\lambda = 7.5319\).
The predicted median (50% failure probability) lives from both Weibull solution paths are shown in the table. They are in very close agreement, with a relative difference of less than 5% across all stress levels. This confirms the existence of multiple, equally valid parameter sets for the Weibull model. The S-N curves generated are smooth and show a characteristic curvature, effectively capturing the data trend without artificial拐点.
For comparison, a traditional Stromeyer model (\(S = A \cdot N^b + D\)) was also fitted to the same data. The performance comparison is stark:
| Stress Amplitude \(S_{c,cor}\) (MPa) | Experimental Mean \(N_{c,cor}\) | Weibull Model (Path 1) \(N_{c,cor}\) | Error (%) | Stromeyer Model \(N_{c,cor}\) | Error (%) |
|---|---|---|---|---|---|
| 280 | 20,919 | 18,500 | -11.6 | 34,461 | 64.7 |
| 240 | 105,016 | 77,961 | -25.8 | 74,405 | -29.1 |
| 200 | 228,232 | 267,094 | 17.0 | 184,908 | -19.0 |
| 180 | 508,047 | 466,167 | -8.2 | 312,913 | -38.4 |
| 140 | 1,558,020 | 1,302,650 | -16.4 | 1,097,440 | -29.6 |
| 100 | 3,259,896 | 3,377,490 | 3.6 | 5,888,440 | 80.6 |
The average absolute error for the Weibull model is 13.8%, significantly lower than the 43.5% average error for the Stromeyer model. The Weibull model provides a consistently better fit, particularly at the high and low stress extremes where the Stromeyer model deviates substantially.
A key advantage of the probabilistic Weibull model is the ability to generate S-N curves for any desired failure probability \(Q\). For design, curves for low probabilities (e.g., \(Q = 0.05\) or 5%) are critical. The full probabilistic model for the cast steel (using Path 1 parameters) is:
$$Q(N_{c,cor}, S_{c,cor}) = 1 – \exp\left\{ -\left[ \frac{(\ln N_{c,cor} – 19.4923)(\ln S_{c,cor} – 6.5166) – 6.5337}{2.1263} \right]^{5.5486} \right\}$$
Solving this equation for \(N\) at specific \(Q\) values yields a family of curves. All experimental data points fall logically between the 5% and 95% probability curves, validating the model’s ability to describe the statistical scatter in the fatigue life of these steel castings.
Case 2: Corrosion Fatigue of Butt Welds on Steel Castings
The integrity of connections involving steel castings is often governed by the welded joints. I applied the same methodology to corrosion fatigue data from butt welds joining the cast steel to a structural steel plate. The experimental results are summarized below.
| Stress Amplitude \(S_{w,cor}\) (MPa) | Specimen | Experimental \(N_{w,cor}\) | Weibull Model Fit \(N_{w,cor}\) (Path 1) | Weibull Model Fit \(N_{w,cor}\) (Path 2) |
|---|---|---|---|---|
| 280 | 1 | 83,970 | 41,295 | 39,717 |
| 2 | 21,184 | |||
| 3 | 29,125 | |||
| 240 | 1 | 148,556 | 104,959 | 101,363 |
| 2 | 134,219 | |||
| 3 | 92,367 | |||
| 200 | 1 | 300,641 | 252,421 | 244,718 |
| 2 | 175,043 | |||
| 3 | 225,480 | |||
| 180 | 1 | 444,517 | 385,611 | 374,541 |
| 2 | 397,349 | |||
| 3 | 416,672 | |||
| 140 | 1 | 891,107 | 884,816 | 862,566 |
| 2 | 590,007 | |||
| 3 | 695,508 | |||
| 100 | 1 | 2,103,315 | 2,033,950 | 1,990,090 |
| 2 | 2,538,303 | |||
| 3 | 2,065,954 |
The parameter estimation for the weld data yielded:
Stage 1: \(B = 19.4754\), \(C = 6.9426\), \(\mu = 11.5940\).
Stage 2 (Path 1): \(\beta = 2.6670\), \(\delta = 1.3780\), \(\lambda = 10.3690\).
Stage 2 (Path 2): \(\beta = 5.0636\), \(\delta = 2.3850\), \(\lambda = 9.4025\).
Again, both Weibull solutions provide nearly identical fits (within ~3%). The comparison with the Stromeyer model for the welds further demonstrates the superiority of the Weibull approach.
| Stress Amplitude \(S_{w,cor}\) (MPa) | Experimental Mean \(N_{w,cor}\) | Weibull Model (Path 1) \(N_{w,cor}\) | Error (%) | Stromeyer Model \(N_{w,cor}\) | Error (%) |
|---|---|---|---|---|---|
| 280 | 44,760 | 41,295 | -7.7 | 57,175 | 27.7 |
| 240 | 125,048 | 104,959 | -16.1 | 101,636 | -18.7 |
| 200 | 233,722 | 252,421 | 8.0 | 200,704 | -14.1 |
| 180 | 419,513 | 385,611 | -8.1 | 297,387 | -29.1 |
| 140 | 725,540 | 884,816 | 22.0 | 759,714 | 4.7 |
| 100 | 2,235,857 | 2,033,950 | -9.0 | 2,668,600 | 19.4 |
The average absolute error for the Weibull model is 11.8%, compared to 19.0% for the Stromeyer model. The Weibull model’s performance is more consistent across the stress range. The full probabilistic model for the butt welds (Path 1) is:
$$Q(N_{w,cor}, S_{w,cor}) = 1 – \exp\left\{ -\left[ \frac{(\ln N_{w,cor} – 19.4754)(\ln S_{w,cor} – 6.9426) – 10.3690}{1.3780} \right]^{2.6670} \right\}$$
This model also successfully bounds the experimental data with its 5% and 95% probability curves, providing a complete statistical description of the weld’s corrosion fatigue performance.
Conclusion and Engineering Implications
The analysis presented establishes a robust and effective framework for the probabilistic assessment of corrosion fatigue life in steel castings and their critical welded connections. The adoption of the three-parameter Weibull distribution, coupled with the Probability Weighted Moment estimation method, addresses key shortcomings of traditional S-N curve fitting techniques.
The primary conclusions are as follows. Firstly, the three-parameter Weibull model produces S-N curves with a natural, smooth curvature that provides a superior fit to experimental corrosion fatigue data compared to conventional piecewise-linear or simple power-law models like the Stromeyer equation. This is quantitatively demonstrated by the significantly lower average fitting errors for both cast steel base metal (13.8% vs. 43.5%) and butt welds (11.8% vs. 19.0%). Secondly, the PWM method provides a straightforward, non-iterative pathway for estimating the Weibull parameters, avoiding the convergence issues and complexity of methods like Maximum Likelihood Estimation. The existence of two valid solution paths yielding nearly identical results reinforces the robustness and flexibility of the Weibull model for this application.
Most importantly, this methodology transcends deterministic life prediction. By its very nature, it quantifies the statistical scatter in fatigue life, enabling the derivation of S-N curves for any specified probability of failure. This is a fundamental requirement for modern reliability-based design and maintenance planning for structures employing steel castings in corrosive environments. Engineers can use the 5% failure probability curve for conservative design or use the full distribution to perform probabilistic life assessments and risk analyses. Therefore, the proposed model offers a powerful, practical, and statistically rigorous tool for enhancing the safety, durability, and economy of marine and other corrosion-prone structures that rely on the performance of high-quality steel castings.