Development of a Design Formula for Estimating the Residual Strength of Corroded Stiffened Cylindrical Structures

Abstract

This paper develops a novel design formula to estimate the residual strength of corroded stiffened cylindrical structures. It extends a previously established ultimate strength formulation for intact cylinders by introducing a corrosion-induced strength reduction factor. The foundational formula considers failure mode interactions like yielding, local buckling, overall buckling, and stiffener tripping. This research utilizes recent experimental and numerical investigations on corroded ring-stiffened cylinder models. Experimental results validate the numerical analysis method, showing good agreement in collapse pressures (2–4% difference) and shapes. The validated numerical method is then subject to an extensive parametric study, systematically varying corrosion characteristics. Results indicate a clear relationship between corrosion volume and strength reduction, with overall buckling being more sensitive. Based on these comprehensive results, a new empirical strength reduction factor (${\rho }_c$) is derived as a function of the corrosion volume ratio ($V_{non}$). This factor is integrated into the existing ultimate strength formula, allowing direct residual strength estimation for corroded structures. The proposed formula is rigorously verified against experimental and numerical data, showing excellent agreement (mean 1.00, COV 5.86%). This research provides a practical, accurate design tool for assessing the integrity and service life of corroded stiffened cylindrical structures.

1. Introduction

As principal structural elements in a wide array of marine and offshore engineering, stiffened cylindrical shells are integral to submarine pressure hulls, the legs of tension leg platforms, spars, and the foundations of floating offshore wind turbines. Throughout their operational lifetime, these structures face harsh marine environments and potential accidental loads. Significant damage can be inflicted by incidents such as collisions with support vessels, dropped objects, grounding, or slamming. Collision damage is particularly critical, as it has the potential to severely compromise structural integrity and may lead to catastrophic failure. Moreover, continuous exposure to corrosive media like seawater results in material degradation, typically manifesting as a reduction in shell thickness, which consequently diminishes the hull’s structural strength. A precise assessment and prediction of the residual strength of these structures after they have sustained corrosion or collision damage is therefore paramount for ensuring operational safety, maintaining structural integrity, and estimating the remaining service life. Because immediate repair of such damage is frequently impractical due to economic or technical constraints, it is essential to develop robust and practical methodologies for evaluating the residual strength of compromised structures to support informed engineering decisions.

Although significant progress has been made in understanding the behavior of stiffened cylinders, prior research has largely focused on the ultimate strength of intact structures or depended on experimental and numerical analyses of damaged sections without establishing a direct design methodology for residual strength. A discernible gap exists in the availability of practical design formulae capable of directly estimating the residual strength of stiffened cylindrical structures affected by corrosion. This study, therefore, seeks to address this critical gap by putting forth a novel design formula. The demand for such predictive tools is high in critical applications like submarine hull maintenance and offshore platform integrity management, where timely and accurate assessment of damaged structures is paramount. Historically, the absence of robust, easily applicable design formulae for damaged structures has necessitated conservative operational limits or costly detailed analyses. This study aims to fill that void by providing a practical methodology. The formula is intended to function as a practical tool for engineers, enabling them to assess the residual strength of corroded stiffened cylindrical structures and thereby facilitating more accurate integrity assessments and life cycle management.

The structural integrity of cylindrical shells under external pressure has been a central focus of research for many decades. Early inquiries were concentrated on the ultimate strength of both unstiffened and stiffened intact structures. An understanding of elastic buckling was first established through foundational theoretical work on uniform thickness cylinders by researchers such as von Mises [1], Windenburg and Trilling [2], and Southwell [3].

Crucial data was provided by experimental validations from Slankard and Nash [4] and Kirstein and Slankard [5], with the latter study noting a significant strength reduction in welded models attributable to imperfections and residual stresses. Reynolds [6] provided additional experimental data and formulae concerning inelastic buckling. The use of ring stiffeners to improve buckling resistance spurred extensive experimental programs, particularly by Kendrick [7,8,9,10], who examined various scantlings and failure modes, including local, overall, and interactive buckling in welded steel ring-stiffened cylinders. While Lunchick [11] focused on axisymmetric yield failure, Miller and Kinra [12] performed hydrostatic tests on models representing offshore structures, highlighting the effect of residual stresses. The impact of manufacturing imperfections on overall buckling modes was emphasized by Yamamoto et al. [13], and Frieze [14] later quantified the reduction in strength caused by welding residual stresses in flat-bar ring-stiffened cylinders. In recent years, the work of Cho et al. [15,16,17] has produced comprehensive ultimate strength formulations for ring-stiffened cylinders and conical shells through extensive experimental and numerical studies; their formulations account for all relevant failure modes, including shell yielding, local buckling, overall buckling, and stiffener tripping. Zhang et al. [18] investigated Titanium-alloy ring-stiffened cylinders, identifying shell yielding and fracture as prominent failure modes. For intact ring-stiffened cylinders, design rules are available in the standards from BSI (PD 5500), DNV GL, and ABS, and the accuracy of these formulae and FEA predictions has been evaluated by MacKay et al. [19,20].

However, real-world structures are inevitably affected by initial imperfections and damage accrued during service, such as corrosion and dents. The impact of geometric imperfections and residual stresses on the ultimate strength of ring-stiffened cylinders has been addressed by researchers like Krenzke [21], Bushnell [22], Faulkner [23], and Smith and Kirkwood [24], with Kendrick [10] also emphasizing their importance. Studies on corrosion damage have gained prominence, with MacKay et al. [25,26] conducting significant experimental and numerical work on cylinders with artificial corrosion damage (shell thinning). Their findings indicated that shell thinning promotes early yielding and inelastic buckling, thereby reducing collapse strength, particularly affecting overall collapse more than interframe collapse, with one-sided thinning eccentricity being a critical factor. Viljoen et al. [27] correlated experimentally derived corrosion rates for HY-80 steel with operational depth reduction in a submarine, using FEA on ring-stiffened sections with varying hull thinning. Zhang et al. [28] numerically and experimentally investigated resin egg-shaped shells with corrosion thinning, observing that ultimate strength diminishes with increased corrosion depth and extent. Mendoza and Marín-López [29] numerically analyzed HY-80 steel cylinders with localized thickness reductions, finding that even a 30% thickness reduction over a significant area resulted in a relatively small decrease in collapse load, with buckling occurring elastically. Yu and Qiao [30] introduced an efficient non-uniform spline finite strip method (N-u SFSM) for buckling analysis of structures with local abnormalities like corrosion. Regarding dent damage, Harding and Onoufriou [31] reported on axial compression tests of ring-stiffened cylinders with static lateral load-induced dents. Cerik [32] numerically studied the ultimate strength of locally dented steel stiffened cylinders under axial compression, while Cho et al. [33] investigated the residual strength of cylinders damaged by dynamic mass impact under hydrostatic pressure. Do et al. [34] proposed formulations for predicting permanent damage from lateral mass impacts and the subsequent residual strength under hydrostatic pressure.

Despite this body of research, a direct and practical design formula for estimating the residual strength of corroded stiffened cylindrical structures, suitable for general engineering application, is notably absent. Many existing studies depend on detailed FEA or specific experimental setups, which are not always practical for initial design phases or for rapid assessments of in-service structures. Prevailing design codes, such as DNV-RP-C203 [35], ABS [36], PD 5500 [37], DNV GL [38], and API [39], primarily focus on intact structures or offer general guidance that requires substantial interpretation or supplementary analysis for corroded conditions. While MacKay and van Keulen [40] explored a partial safety factor approach using nonlinear FEA, highlighting its accuracy, they did not propose a direct residual strength formula for corrosion. Similarly, studies by Freitas et al. [41] on innovative sliding stiffeners and Zhang et al. [42] on inner corrugated pressure shells focused on enhancing the buckling performance of novel intact designs rather than addressing the residual strength of conventional corroded structures. Therefore, this study seeks to build upon the foundational understanding of intact strength [15] and insights from experimental/numerical work on damaged structures [34] to develop a much-needed practical design tool for assessing the residual strength of corroded stiffened cylinders.

Drawing upon comprehensive prior research, the primary objective of this paper is to develop a novel and practical design formula for estimating the residual strength of corroded ring-stiffened cylindrical structures. This study aims to bridge the current gap in direct design methodologies for such corroded structures by integrating a corrosion-induced strength reduction factor into the robust ultimate strength formulation for intact structures proposed by Cho et al. [15]. The research methodology involves a critical validation of a numerical analysis method using experimental data from corroded stiffened cylinders. Subsequently, an extensive parametric study, leveraging the validated numerical method and building upon results from Park et al. [43], will be conducted to systematically analyze the relationship between various corrosion damage characteristics and the observed strength reduction. This analysis will lead to the derivation of an empirical strength reduction factor that quantifies the impact of corrosion volume on structural capacity, which will then be directly applied to the existing ultimate strength formula. The newly developed residual strength design formula will be rigorously verified against both existing experimental data and the extensive numerical data from the parametric study, assessing its accuracy and reliability for practical engineering applications. This research aims to provide a practical and reliable design tool for evaluating the structural integrity and remaining operational life of corroded stiffened cylindrical structures, contributing significantly to their safe and efficient management.

2. Ultimate Strength Design Formula for Intact Stiffened Cylinders

2.1. Overview of the Base Design Formula

The ultimate strength of ring-stiffened cylinders under hydrostatic pressure is a critical design consideration for submarine pressure hulls and other offshore structures. The foundational design formula employed in this study, proposed by Cho et al. [15], is a robust method for estimating the collapse pressure of intact ring-stiffened cylinders. This formula is rooted in the quadratic Merchant–Rankine formulation, specifically designed to account for the complex interactions between various dominant failure modes: local buckling ($P_L$), overall buckling ($P_{OA}$), stiffener tripping ($P_T$), and shell yielding ($P_Y$). The general form of this interaction formula is expressed as follows:

$${\left({\displaystyle \frac{P_c}{{\rho }_L{P}_L}}+{\displaystyle \frac{P_c}{{\rho }_{OA}{P}_{OA}}}+{\displaystyle \frac{P_c}{{\rho }_T{P}_T}}\right)}^2+{\left({\displaystyle \frac{P_c}{P_Y}}\right)}^2=1$$

(1)

where $P_L$, $P_{OA}$, and $P_T$ are the characteristic pressures for local buckling, overall buckling, and stiffener tripping, respectively. $P_Y$ is the yield pressure of the cylinder. The terms ${\rho }_L, {\rho }_{OA}$ and ${\rho }_T$ are knockdown factors that account for imperfections and residual stresses in predicting the actual collapse for each buckling mode.

This foundational formula was extensively validated against a comprehensive database of 107 published test results, predominantly from steel structures, as detailed in Cho et al. [15].

2.2. Constituent Buckling and Yield Pressures

The ultimate strength formula relies on the accurate calculation of several characteristic pressures, each representing a distinct failure mode.

2.2.1. Local Buckling Pressure $\left(P_L\right)$

This mode involves the loss of stability of the cylindrical shell between adjacent ring stiffeners, often forming circumferential wave lobes. The local buckling pressure is based on a von Mises [1] solution for radially loaded, simply supported cylindrical shells:

$$P_L={\displaystyle \frac{\left(E{t}_s/R_m\right)}{\left\{n_L^2-1+\frac{1}{2}{\left(\frac{\pi R_m}{L_s}\right)}^2\right\}}}\left[{\displaystyle \frac{1}{{\left\{n_L^2{\left(\frac{L_s}{\pi R_m}\right)}^2+1\right\}}^2}}+{\displaystyle \frac{t_s^2}{12R_m^2(1-{\nu }^2)}}{\left\{n_L^2-1+{\left({\displaystyle \frac{\pi R_m}{L_s}}\right)}^2\right\}}^2\right]$$

(2)

where $E$ is Young’s modulus, $t_s$ is the shell thickness, $R_m$ is the mean radius, $n_L$ is the circumferential wave number, $L_s$ is the stiffener spacing, and $\nu$ is Poisson’s ratio.

2.2.2. Overall Buckling Pressure $\left(P_{OA}\right)$

This failure mode encompasses the deformation of both the shell and the ring stiffeners over an entire compartment length. It is characterized by long wavelengths in both circumferential and axial directions and typically occurs when the stiffeners are relatively small compared to the shell thickness and the cylinder is relatively long. The overall buckling pressure is calculated using Bryant’s two-term approximation [44], which combines von Mises and Bresse’s relations [45]:

$$P_{OA}={\displaystyle \frac{E{I}_{fe}(n_{OA}^2-1)}{L_s{R}_m^3}}\left({\displaystyle \frac{E{t}_s}{R_m}}\right)\left[{\displaystyle \frac{1}{\left\{n_{OA}^2-1+0.5{\left(\frac{\pi R_m}{L_c}\right)}^2\right\}{\left\{n_{OA}^2{\left(\frac{\pi R_m}{L_c}\right)}^2+1\right\}}^2}}\right]$$

(3)

where $I_{fe}$ represents the second moment area of the combined cross-section of the ring-frame and the shell, considering the effective breadth of the shell plating, $n_{OA}$ is the circumferential wave number for overall buckling, and $L_c$ is the compartment length.

2.2.3. Stiffener Tripping Pressure $\left(P_T\right)$

Stiffener tripping is characterized by the rotation of the stiffener about its connection to the cylindrical shell. This mode can significantly influence the overall buckling behavior, especially as the load increases. The elastic tripping stress (${\sigma }_t$) is proposed by Faulkner [46] and is combined with the yield strength to estimate the equivalent external pressure:

$$P_T={\displaystyle \frac{{\sigma }_T{P}_{yf}{R}_s}{{\sigma }_Y{R}_f}}$$

(4)

where ${\sigma }_T$ is the tripping stress, $P_{yf}$ is the circumferential yield pressure, $R_s$ is the measured radius from the centroid of ring stiffener cross-section, and $R_f$ is the measured radius from the standing flange.

2.2.4. Yield Pressure $\left(P_Y\right)$

Shell yielding occurs when the mean stress in the plating at the mid-bay between stiffeners reaches the material’s yield point. The yield pressure of the combined shell and ring stiffener, considering the von Mises yield criterion, is defined as follows:

$$P_Y={\displaystyle \frac{{\sigma }_Y{t}_s}{R_m}}{\displaystyle \frac{1}{\sqrt{0.25 - 0.5\left(1-\gamma G\right)+{\left(1-\gamma G\right)}^2}}}$$

(5)

where $\gamma$ is the stiffener parameter, and $G$ is the shear modulus.

2.3. Knockdown Factors for Intact Structures

Knockdown factors (${\rho }_L$, ${\rho }_{OA}$, ${\rho }_T$) are empirical reduction coefficients applied to the characteristic pressures to account for manufacturing imperfections, residual stresses, and other real-world discrepancies that reduce the actual collapse strength from theoretical predictions. These factors were derived through a rigorous regression analysis utilizing a comprehensive dataset of 107 published test results for ring-stiffened cylinders. The derivation process involved iteratively determining the best-fit relationships between the knockdown factors and non-dimensional geometrical and material parameters, such as stiffener height to thickness ($h_w/t_w$), stiffener spacing to radius ($L_f/R_m$), radius to shell thickness ($R_m/t_s$), and the ratio of Young’s modulus to yield strength ($E/{\sigma }_Y$). The derived knockdown factors are given by the following:

$${\rho }_L=0.674\mathit{exp}\left\{0.0006\left({\displaystyle \frac{h_w}{t_w}}\right)\left({\displaystyle \frac{\sqrt{R_m{L}_s}}{t_s}}\right)\left({\displaystyle \frac{E}{1000{\sigma }_Y}}\right)\right\}$$

(6)

$${\rho }_{OA}=1.055\mathit{exp}\left\{0.167\left({\displaystyle \frac{h_w}{t_w}}\right)\left({\displaystyle \frac{E}{1000{\sigma }_Y}}\right)\right\}$$

(7)

$${\rho }_T=207374\mathit{exp}\left\{0.0088\left({\displaystyle \frac{L_s}{R_m}}\right)\left(\sqrt{{\displaystyle \frac{L_s}{t_s}}}\right)\right\}$$

(8)

2.4. Development and Verification of the Intact Strength Formula

The knockdown factors were derived through a rigorous regression analysis using a comprehensive database of 107 published test results for ring-stiffened cylinders. This process aimed to achieve the best fit between theoretical predictions and actual collapse pressures, minimizing bias and scatter. The derived knockdown factors are specific empirical equations.

The accuracy of the developed ultimate strength formula for intact cylinders was extensively verified against this test data, demonstrating superior accuracy and consistency compared to other existing design codes [35,36,37,38,39]. The formula showed a mean ratio of actual to predicted collapse pressure ($X_m$) of 1.00 and a Coefficient of Variation (COV) of 10.39% for 102 models. This high reliability confirms its suitability as the fundamental basis for developing a residual strength model for corroded structures.

3. Experimental and Numerical Investigation of Corroded Stiffened Cylinders

3.1. Experimental Program on Corroded Models

To investigate the residual strength of corroded ring-stiffened cylinders, an experimental program was conducted [43]. Four steel ring-stiffened cylinder models were fabricated, as illustrated in Figure 1: two intact (RSC-CD-1 and RSC-CD-3) and two artificially corroded (RSC-CD-2 and RSC-CD-4) by precise machining on the internal shell surface. The corrosion mimicked uniformly thinned circular areas with varying diameters and depths, representing up to a 25% reduction in local shell thickness. More detailed information regarding the experimental models and test results, including specimen dimensions and specific test procedures, can be found in Park et al. [43]. The chosen diameters (60 mm, 100 mm) represent a range of localized damage sizes typically observed or considered in previous studies on submarine structures, aiming to cover various extents of pitting corrosion effects. Initial shape imperfections, critical for buckling behavior, were meticulously measured using a 3D coordinate measuring machine and considered for analysis. Mild steel was used as the model material, and its material properties were determined through tensile tests.

Hydrostatic collapse tests were performed, revealing significant strength reductions in the corroded models compared to intact ones. RSC-CD-2 showed a 6.3% strength reduction, while RSC-CD-4 exhibited a 12.1% reduction. The failure modes observed, as shown in Figure 2, included local/overall interactive buckling for RSC-CD-1 and RSC-CD-2 (with local buckling initiating at the corroded area for RSC-CD-2), and overall buckling for RSC-CD-3 and RSC-CD-4.

3.2. Numerical Analysis and Validation

Nonlinear finite element (FE) analyses were conducted using Abaqus 6.23 (Simulia, 2023) to simulate the collapse behavior of the stiffened cylinders. The FE models incorporated the measured initial shape imperfections to accurately reflect real-world conditions.

The material properties used in the FE models were those determined from tensile tests on the mild steel specimens, as detailed in Park et al. [43]. The boundary conditions applied simulated the experimental setup, as also described in Park et al. [43]. The models utilized four-node shell elements (S4R) with hourglass control and reduced integration. Mesh convergence tests optimized the element sizes, using a 10 × 10 mm element size for most areas and a finer 5 × 5 mm size in the corroded regions, as detailed in Park et al. [43].

The numerical collapse pressures showed excellent agreement with the experimental results, with differences of only 2–4%. Crucially, the simulated collapse shapes from the FE analyses remarkably matched the failure patterns observed in the physical experiments. This close correlation validates the numerical methodology as a reliable tool for estimating the residual strength of corroded ring-stiffened cylindrical structures.

4. Parametric Study on Corrosion Damage Characteristics

4.1. Design of the Parametric Study

Following the rigorous validation of the numerical analysis method against experimental results, an extensive parametric study was conducted to systematically investigate the effects of various corrosion damage characteristics on the residual strength of stiffened cylindrical structures. This study utilized the validated FE models based on the geometries of RSC-CD-2 and RSC-CD-4, as described in Section 3. The primary objective was to generate a comprehensive dataset that elucidates the relationship between the extent of corrosion damage and the resulting strength reduction.

The corrosion damage was systematically varied by modifying both the diameter and depth of the artificially corroded areas. For the RSC-CD-2 and RSC-CD-4 series, the dimensions of the corroded areas were defined as follows: outer corrosion diameter ($d_{co1}$) and depth ($h_{co1}$), and inner corrosion diameter ($d_{co2}$) and depth ($h_{co2}$).

Specifically, for the RSC-CD-2-60, 80, and 100 series, the $d_{co1}$ values were set to 60 mm, 80 mm, and 100 mm, respectively, while $d_{co2}$ was maintained at 100 mm. For each corrosion case, the depths ($h_{co1}$ and $h_{co2}$) were varied from 1.0 mm to 5.0 mm at 1.0 mm intervals. The same corrosion damage characteristics were applied to the RSC-CD-4-60, -80, and -100 series. This systematic variation resulted in a total of 32 numerical computation cases, encompassing a wide range of corrosion scenarios. The original models (RSC-CD-2 and RSC-CD-4) maintained their identical specifications of shell thickness (6.1 mm), mean diameter (1155 mm), and stiffener spacing (200 mm). Figure 3 provides a detailed description of the corrosion damages for the parametric study, and

Table 1. Summary of corrosion damages in numerical models for parametric studies (adapted from [43]).

Model	${\mathit{d}}_{\mathit{c}\mathit{o}1}$ (mm)	${\mathit{h}}_{\mathit{c}\mathit{o}1}$ * (mm)	${\mathit{d}}_{\mathit{c}\mathit{o}2}$ (mm)	${\mathit{h}}_{\mathit{c}\mathit{o}2}$ * (mm)
RSC-CD-2	60	1.0	100	1.5
RSC-CD-2-60 series	60	1.0~5.0	100	1.0~5.0
RSC-CD-2-80 series	80	1.0~5.0	100	1.0~5.0
RSC-CD-2-100 series	100	1.0~5.0	100	1.0~5.0
RSC-CD-4	60	1.0	100	1.5
RSC-CD-4-60 series	60	1.0~5.0	100	1.0~5.0
RSC-CD-4-80 series	80	1.0~5.0	100	1.0~5.0
RSC-CD-4-100 series	100	1.0~5.0	100	1.0~5.0

* Note: corrosion depths $h_{co1}$ and $h_{co2}$ vary in 1.0 mm increments (1.0, 2.0, 3.0, 4.0, 5.0 mm) for each series.

summarizes the sizes of these damages in the numerical models.

4.2. Results of Parametric Study: Strength Reduction Analysis

The results of the extensive parametric study are presented in terms of strength reduction against the corrosion volume ratio ($V_{non}$). The corrosion volume ratio is defined as the ratio of the volume of corrosion damage ($V_c$) to the shell volume of one bay of the model ($V_{m, 1bay}$), calculated based on the intact shell thickness. The ordinate in the corresponding figures denotes the strength reduction rate of the corroded model relative to its intact counterpart.

The study revealed a clear and consistent trend, as illustrated in Figure 4: as the volume of corrosion damage increased, the rate of strength reduction also increased. For the RSC-CD-2 series, which primarily exhibited local buckling collapse, the strength reductions gradually decreased with increasing corrosion damage. However, this reduction became less pronounced when $V_{non}$ exceeded approximately 1.5%. In contrast, for the RSC-CD-4 series, characterized by overall buckling collapse, the strength reduction showed an interesting behavior: it was relatively insensitive to $V_{non}$ when the ratio was less than 0.8%, but beyond this point, the strength decreased noticeably and sharply. This observation suggests that the overall buckling failure mode is more significantly influenced by the degree of corrosion damage than the local buckling mode. This comprehensive parametric dataset formed the foundation for the subsequent development of the corrosion strength reduction factor.

5. Development of the Residual Strength Design Formula

5.1. Methodology for Deriving the Strength Reduction Factor

The objective of developing a residual strength design formula for corroded stiffened cylinders is to accurately quantify the strength degradation due to material loss. Rather than developing an entirely new design formula, the approach taken in this study is to extend the validated ultimate strength design formula for intact stiffened cylinders [15] by incorporating a corrosion-induced strength reduction factor $\left({\rho }_c\right)$. This method maintains the robustness and accuracy of the existing intact formula while specifically accounting for the detrimental effects of corrosion.

The proposed residual collapse pressure $\left(P_c\right)$ is thus expressed as the product of the collapse pressure of the non-corroded cylinder $\left(P_{c,intact}\right)$ and the corrosion strength reduction factor. The overall form of the residual strength design formula becomes the following:

$${\left({\displaystyle \frac{P_c}{{\rho }_c{\rho }_L{P}_L}}+{\displaystyle \frac{P_c}{{\rho }_{OA}{P}_{OA}}}+{\displaystyle \frac{P_c}{{\rho }_T{P}_T}}\right)}^2+{\left({\displaystyle \frac{P_c}{P_Y}}\right)}^2=1$$

(9)

The core task was to empirically derive the ${\rho }_c$ factor based on the extensive numerical parametric study results discussed in Section 4. The parameter chosen to represent the extent of corrosion damage is the non-dimensional volume ratio $\left(V_{non}\right)$, defined as the ratio of the corrosion damage volume $\left(V_c\right)$ to the volume of one bay of the model ($V_{m, 1bay}$). This parameter effectively captures the overall material loss due to corrosion.

5.2. Proposed Strength Reduction Factor

Based on the analysis of the parametric study results, a new strength reduction factor was empirically derived. The relationship between the strength reduction and the corrosion volume ratio was analyzed, and a best-fit curve was established. The derived equation for the corrosion strength reduction factor is given by the following:

$${\rho }_c=0.91\times \mathit{exp}\left(-44.62V_{non}\right)$$

(10)

This exponential form effectively captures the nonlinear degradation of strength as the volume of corrosion damage increases. The factor ${\rho }_c$ directly scales the ultimate strength capacity of the intact structure, effectively transforming the intact strength formula into a residual strength prediction tool. When $V_{non}$ is zero (i.e., no corrosion), ${\rho }_c$ approaches 0.91. This initial value is a result of the empirical calibration during the best-fit regression analysis of the parametric study data, aiming to provide the most accurate overall prediction across the entire range of corrosion conditions, including the baseline intact state. It effectively serves as an optimized starting point within the derived exponential function that best fits the collected data.

As $V_{non}$ increases, ${\rho }_c$ decreases, reflecting the diminishing strength.

The development of this ${\rho }_c$ factor is a crucial contribution, allowing for a direct and quantifiable assessment of corrosion’s impact on the overall collapse pressure of stiffened cylindrical structures.

6. Verification of the Proposed Residual Strength Design Formula

6.1. Verification with Experimental and Numerical Data

The reliability and accuracy of the newly proposed residual strength design formula, which incorporates the corrosion strength reduction factor, were rigorously verified against both experimental data and the extensive numerical data obtained from the parametric study. This comprehensive verification process is critical to ensuring the formula’s applicability for practical engineering design.

For verification against experimental data, the collapse pressures predicted by the new design formula were compared with the actual measured collapse pressures from the corroded test models (RSC-CD-2 and RSC-CD-4) described in Section 3. The results, summarized in

Table 2. Comparison of collapse pressure (in the experiment, numerical study, and proposed formula) for corroded models.

Model	Collapse Pressure (MPa)			Xm (Exp./Num.)	Xm (Exp./Eq.)
	Experiment	Numerical	Equation
RSC-CD-1	3.04	2.99	3.11	1.02	0.98
RSC-CD-2	2.85	2.9	2.88	0.98	0.99
RSC-CD-3	2.6	2.62	2.53	0.99	1.03
RSC-CD-4	2.28	2.37	2.35	0.96	0.97
Mean				0.99	0.99
COV				2.30%	2.58%

, showed excellent agreement. For RSC-CD-1 (intact), the difference between the proposed formula and experimental results was 2.3% (3.04 MPa vs. 3.11 MPa predicted). For RSC-CD-2 (corroded), the difference was 1% (2.85 MPa vs. 2.88 MPa predicted). Similarly, for RSC-CD-3 (intact), the difference was 2.7% (2.60 MPa vs. 2.53 MPa predicted). For RSC-CD-4 (corroded), the difference was 1.3% (2.28 MPa vs. 2.35 MPa predicted). Overall, the difference between the predicted collapse pressures and experimental results for corroded models was consistently within 1–3%.

The formula was also extensively verified against the large dataset generated from the parametric numerical study (Section 4). The mean value of the ratio of predicted collapse pressure to numerical collapse pressure was found to be 1.00, with a COV of 5.86%. This low COV indicates high consistency and minimal scatter in the predictions, confirming the formula’s robust performance across a wide range of corrosion scenarios. The bias check, as shown in Figure 5, also confirmed that no significant bias or skewness was observed, indicating that the developed formula is well behaved and does not systematically overestimate or underestimate the strength across different corrosion levels. The comparison of predicted collapse pressure versus numerical collapse pressure is further illustrated in Figure 6.

6.2. Discussion on Accuracy and Reliability

The verification results clearly demonstrate the high accuracy and reliability of the proposed residual strength design formula. The ability to predict collapse pressures within 1–3% of experimental values for corroded models is a significant improvement for practical engineering applications. The low COV of 5.86% against the extensive numerical dataset further confirms the formula’s consistency and its capability to capture the complex behavior of corroded stiffened cylinders under hydrostatic pressure.

This level of accuracy is comparable to, and in many cases superior to, the prediction capabilities of existing design codes for intact structures, while specifically addressing the critical aspect of corrosion damage. The development of this formula provides naval architects and structural engineers with a valuable tool for performing more realistic and reliable assessments of corroded pressure hulls, thereby enhancing safety and optimizing maintenance strategies for aging marine structures. The systematic approach of validating the numerical method, conducting a comprehensive parametric study, and empirically deriving a strength reduction factor based on this data, has resulted in a robust and practical design solution.

6.3. Limitations and Future Work

While the proposed residual strength design formula demonstrates high accuracy and reliability, it is important to acknowledge its current limitations, which also suggest avenues for future research.

Firstly, the derivation of the corrosion strength reduction factor and its subsequent verification heavily relied on a limited number of experimental data points for corroded stiffened cylinders (four models in total). Although these experimental results were crucial for validating the numerical method, the primary dataset for derivation was generated through numerical parametric studies. While the numerical method was thoroughly validated, an expanded experimental database, encompassing a wider range of corrosion damage configurations (e.g., varying patterns, sizes, and locations beyond the mid-bay uniform thinning), would further enhance the robustness and generality of the proposed design formula. Additional hydrostatic collapse experiments, as also suggested in the conclusion of the related conference presentation, are needed to increase the accuracy and reliability of both the numerical analysis method and the design formula.

Secondly, the current study primarily focused on pitting corrosion, specifically represented by uniform circular thinning localized in the mid-bay of the cylinder. In reality, corrosion can manifest in various forms, such as other types of pitting corrosion, grooving corrosion, or more complex, non-uniform patterns. The current design formula might not fully capture the nuanced effects of such diverse corrosion morphologies. Future research should particularly emphasize numerical investigations on models with grooving corrosion damages, in addition to exploring other forms of pitting, to significantly extend the applicability of the design formula. Furthermore, future work should consider chemical analyses of corroded materials to understand how different metals react, as this influences corrosion mechanisms and structural response [47].

Finally, the design formula and its associated knockdown factors are derived from specific ranges of geometrical parameters and material properties, as represented by the experimental and numerical models. Extrapolating the formula significantly beyond these ranges might introduce uncertainties. Further parametric studies covering a broader spectrum of structural dimensions (e.g., shell thickness, diameter-to-thickness ratio, stiffener dimensions) and material types would contribute to enhancing the formula’s universality and reducing potential biases for different design conditions. The eventual aim is to potentially refine the design formula to accommodate more complex interactions between corrosion and various buckling modes.

7. Conclusions

This paper successfully presents the development of a novel design formula for accurately estimating the residual strength of stiffened cylindrical structures subjected to corrosion damage. Building upon a previously validated ultimate strength formulation for intact ring-stiffened cylinders, this study effectively integrates a new corrosion-induced strength reduction factor (${\rho }_c$) to quantify the detrimental effects of material loss due to corrosion.

The research leveraged a comprehensive approach, starting with experimental investigations of corroded ring-stiffened cylinder models. These experiments demonstrated significant residual strength reduction in damaged models compared to intact ones (6.3% to 12.1% reduction), highlighting the critical need for an accurate assessment method. The experimental results were then used to rigorously validate a numerical analysis method, showing excellent agreement in collapse pressures (2–4% difference) and observed collapse shapes.

The validated numerical method was subsequently employed to conduct an extensive parametric study, systematically varying corrosion damage characteristics (diameter and depth) across numerous scenarios. This study revealed a clear relationship between the volume ratio of corrosion damage ($V_{non}$) and strength reduction, with overall buckling failure modes exhibiting greater sensitivity to corrosion compared to local buckling modes.

Based on the comprehensive results from this parametric study, a new empirical strength reduction factor was successfully derived. This factor directly accounts for the influence of corrosion volume on structural capacity. The proposed residual strength design formula, which incorporates this ${\rho }_c$ factor into the existing intact ultimate strength equation, was then rigorously verified. Verification against both experimental data and the extensive numerical parametric study results demonstrated high accuracy and reliability, with mean values of 1.00 and COV of 5.86%. The collapse pressure differences between the proposed formula and experimental results for corroded models were consistently within 1–3%.

Despite its high accuracy, the current formula’s limitations include a reliance on a limited experimental dataset for corroded models, primarily focusing on uniform thinning in the mid-bay. Future work should involve additional hydrostatic collapse experiments and numerical investigations of diverse corrosion morphologies, such as pitting and grooving corrosion, to further expand the formula’s applicability and enhance its robustness.

The proposed formulation is structured for straightforward implementation, and future work could include the development of a user-friendly computational tool to automate structural assessments. This significantly contributes to advancing the safe and efficient design, operation, and maintenance of marine and offshore structures subjected to corrosive environments.