Comparative study between analytical and FE analysis for the ultimate strength assessment of pitted platings

Abstract

The ultimate strength assessment of platings affected by pitting corrosion wastage is a basic issue for the scantling and design of ship structures. In the past, several nonlinear FE analyses were performed to investigate the incidence of pitting degree and corrosion depth on the ultimate strength of plate panels, with the main aim of providing some approximate formulations, useful at least in the preliminary project phase. Based on actual state of art, the main aim of current research is to provide an analytical formulation for the ultimate strength assessment of platings with random pitting corrosion wastage, by solving the Marguerre non-linear governing differential equations for large deflection analysis of platings in the post-buckling regime. In this respect, a comparative analysis between the analytical solution and a series of FE results is preliminarily performed for uncorroded platings, combining different levels of initial geometrical imperfections and welding residual stresses. Subsequently, the comparative analysis is extended to platings affected by pitting corrosion wastage. Hence, different levels of pitting and corrosion intensity degrees are properly combined in order to investigate the goodness of the proposed analytical formulation.

Keywords

Plating ultimate strength Marguerre equations pitting corrosion wastage FE analysis

1. Introduction

In the last century, the ultimate strength analysis of steel-plated structures was studied by a variety of researchers throughout the world [4,8,9,11,12], to correctly assess the ultimate strength of large sea-going ships [1–3], even if only recently research activities investigated the incidence of random pitting corrosion wastage on plating ultimate strength, following the growing interest of Classification Societies on the risk and reliability assessment of ageing structures [10]. In this respect, Khedmati et al. [6], Jiang and Soares [5], Zhang et al. [15] among others provided some practical design formulas to predict the plate strength degradation, based on a series of non-linear finite element analyses.

Based on actual state-of-art, there is still need to investigate the ultimate strength of platings affected by pitting corrosion wastage under uniaxial compression. In this respect, the analytical formulation based on the well-known Marguerre equations is extended to pitted platings and a comparative analysis is performed with the FE results derived by a series of elasto-plastic large-deflection simulations performed by Ansys Mechanical APDL. Current results show a very good agreement between the FE values and the analytical formulation that reveals to be reliable for the practical assessment of the ultimate strength of platings affected by pitting corrosion wastage.

2. Ultimate strength of uncorroded platings

2.1. Solution of Marguerre equations

The post-buckling behavior of platings under uniaxial compression can be assessed by solving the coupled nonlinear governing differential equations of large-deflection plate theory, namely the equilibrium equation and the compatibility condition that, in absence of pressure loads, can be resembled as follows [7]: $\begin{array}{l} D \nabla^{4} w - t [\frac{\partial^{2} F}{\partial x^{2}} \frac{\partial^{2} (w + w_{0})}{\partial x^{2}} - 2 \frac{\partial^{2} F}{\partial x \partial y} \frac{\partial^{2} (w + w_{0})}{\partial x \partial y} \\ (1) & + \frac{\partial^{2} F}{\partial y^{2}} \frac{\partial^{2} (w + w_{0})}{\partial y^{2}}] = 0 \\ \nabla^{4} F - E [{(\frac{\partial^{2} w}{\partial x \partial y})}^{2} - \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} + 2 \frac{\partial^{2} w_{0}}{\partial x \partial y} \frac{\partial^{2} w}{\partial x \partial y} - \frac{\partial^{2} w_{0}}{\partial x^{2}} \frac{\partial^{2} w}{\partial y^{2}} \\ (2) & - \frac{\partial^{2} w}{\partial x^{2}} \frac{\partial^{2} w_{0}}{\partial y^{2}}] = 0 \end{array}$ having denoted by $D (t)$ the plating flexural rigidity (thickness), by F the Airy stress function, by $w_{0}$ (w) the initial (added) deflection field and by E the material Young modulus. After determining the Airy stress function and the added deflection field, the membrane stresses inside the plating can be easily calculated as follows: $\begin{matrix} (3) & σ_{x} = \frac{\partial^{2} F}{\partial y^{2}}; σ_{y} = \frac{\partial^{2} F}{\partial x^{2}}; τ_{x y} = \frac{\partial^{2} F}{\partial x \partial y} \end{matrix}$

Equations (1) and (2) can be efficiently solved by the energy-principle or Galerkin method, after expanding the initial deflection field into a suitable double sine trigonometric series, satisfying the simple support boundary conditions at plate edges: $\begin{matrix} (4) & w_{0} (x, y) = A_{0 m} sin \frac{m π x}{a} sin \frac{π y}{b} \end{matrix}$ where $A_{0 m}$ is the initial deflection amplitude, a (b) is the plating length (breadth) and m is the half wave number in the longitudinal direction which is equal to the minimum integer satisfying the following inequality, as a function of the plate aspect ratio α: $\begin{matrix} (5) & α ⩽ \sqrt{m (m + 1)} \end{matrix}$ Similarly, the added deflection field is assumed to include the only buckling mode: $\begin{matrix} (6) & w (x, y) = A_{m} sin \frac{m π x}{a} sin \frac{π y}{b} \end{matrix}$ having denoted by $A_{m}$ the unknown added deflection amplitude. After replacing Eq. (4) and (6) into Eq. (2), the Airy stress function is derived: $\begin{array}{rcl} F (x, y) & = & {(σ_{x, a v} + σ_{r x})}^{2} \frac{y^{2}}{2} \\ (7) & + \frac{E A_{m} (A_{m} + 2 A_{0 m})}{32} (\frac{α^{2}}{m^{2}} cos \frac{2 π m x}{a} + \frac{m^{2}}{α^{2}} sin \frac{2 π y}{b}) \end{array}$ where $σ_{x, a v}$ is the average compressive stress and $σ_{r x}$ is the welding stress that depends on the yield strength $σ_{Y}$ , the tensile block width $b_{t}$ and the welding tension block parameter η, as depicted in Fig. 1: $\begin{matrix} (8) & σ_{r x} = \{\begin{matrix} σ_{Y} & for 0 ⩽ y < b_{t} \\ - η σ_{Y} & for b_{t} ⩽ y < b - b_{t} \\ σ_{Y} & for b - b_{t} ⩽ y ⩽ b \end{matrix} \end{matrix}$ Hence, the total deflection field $\overline{w} = w + w_{0}$ is determined by the energy principle: $\begin{array}{l} \int_{0}^{a} \int_{0}^{b} {D \nabla^{4} w - t [\frac{\partial^{2} F}{\partial x^{2}} \frac{\partial^{2} \overline{w}}{\partial x^{2}} - 2 \frac{\partial^{2} F}{\partial x \partial y} \frac{\partial^{2} \overline{w}}{\partial x \partial y} + \frac{\partial^{2} F}{\partial y^{2}} \frac{\partial^{2} \overline{w}}{\partial y^{2}}]} \\ (9) & \times sin \frac{m π x}{a} sin \frac{π y}{b} d x d y = 0 \end{array}$ that is resembled by the following equation, with respect to the variable $X_{m} = A_{m} / β^{2} t$ , having denoted by $β = b / t \sqrt{σ_{Y} / E}$ the plating slenderness parameter: $\begin{matrix} (10) & C_{1} X_{m}^{3} + C_{2} X_{m}^{2} + C_{3} X_{m} + C_{4} = 0 \end{matrix}$ The non-dimensional coefficients of Eq. (10) can be easily determined as follows: $\begin{array}{l} (11.1) & C_{1} = \frac{π^{2}}{16} β^{4} (\frac{m^{4}}{α^{3}} + α) \\ (11.2) & C_{2} = 3 X_{0 m} C_{1} \\ C_{3} = 2 X_{0 m}^{2} C_{1} - \frac{m^{4}}{α} β^{2} (φ + \frac{1 + η}{π} sin \frac{π η}{1 + η}) \\ (11.3) & + \frac{m^{2} π^{2}}{12 α (1 - ν^{2})} {(\frac{m}{α} + \frac{α}{m})}^{2} \\ (11.4) & C_{4} = - X_{0 m} β^{2} \frac{m^{2}}{α} (φ + \frac{1 + η}{π} sin \frac{π η}{1 + η}) \end{array}$ In all cases $X_{0 m} = A_{0 m} / β^{2} t$ is the non-dimensional initial deflection amplitude and $φ = σ_{x, a v} / σ_{Y}$ is the edge function, namely the average to yield strength ratio.

Fig. 1.

Welding residual stress field.

2.2. Yielding criterion

After determining the unknown variable $X_{m}$ , maximum and minimum values of longitudinal and transverses stresses are determined by Eq. (3) as follows: $\begin{array}{l} (12.1) & σ_{x, min} = - σ_{x, a v} - \frac{m^{2} π^{2} E A_{m} (A_{m} + 2 A_{0 m})}{8 a^{2}} cos \frac{2 π b_{t}}{b} \\ (12.2) & σ_{x, max} = - σ_{x, a v} + \frac{m^{2} π^{2} E A_{m} (A_{m} + 2 A_{0 m})}{8 a^{2}} \\ (12.3) & σ_{y, min} = - \frac{π^{2} E A_{m} (A_{m} + 2 A_{0 m})}{8 b^{2}} \\ (12.4) & σ_{y, max} = \frac{π^{2} E A_{m} (A_{m} + 2 A_{0 m})}{8 b^{2}} \end{array}$

Hence, the plating ultimate strength $ϕ_{u}$ , namely the ultimate $σ_{u}$ to the yield strength $σ_{Y}$ ratio, is reached when first occurrence of yielding at plate corners, longitudinal or transverse edges, occurs: $\begin{array}{l} (13.1) & σ_{x, min}^{2} + σ_{y, min}^{2} - σ_{x, min} σ_{y, min} = σ_{Y}^{2} \\ (13.2) & σ_{x, min}^{2} + σ_{y, max}^{2} - σ_{x, min} σ_{y, max} = σ_{Y}^{2} \\ (13.3) & σ_{x, max}^{2} + σ_{y, min}^{2} - σ_{x, max} σ_{y, min} = σ_{Y}^{2} \end{array}$

2.3. Geometrical imperfections

The initial deflection amplitude can be assessed on the basis of onboard deflection measurements or by some empirical formulations proposed in the past by several researchers. In this respect, it must be pointed out that the geometrical pattern of the initial deflection field is quite complex and, for platings simply supported at all edges with multi-wave shape in the longitudinal direction, it is generally modelled by the so-called thin-horse mode, based on a double sine trigonometric series, whose coefficients depend on the panel aspect ratio. Nevertheless, as stressed by Paik et al. [9], this deflection mode can be efficiently replaced by the buckling mode shape, provided by Eq. (4). Hence, in current analysis, the buckling mode shape is embodied together with average initial geometrical imperfections, namely $X_{0 m} = 0.100$ , in conjunction with different levels of welding residual stresses [14]: $\begin{matrix} (14) & η = \{\begin{matrix} 0.05 & slight level \\ 0.15 & average level \\ 0.30 & severe level \end{matrix} \end{matrix}$ In this respect, it must be pointed out that the levels of geometrical imperfections provided by Smith et al. [14] were developed for the thin-horse deflection mode. Anyway, they were subsequently extended to the buckling mode applied in current analysis, as carried out by Paik et al. [7], among others

3. Ultimate strength of pitted platings

Based on the main outcomes of past research activities, the ultimate strength of pitted platings is influenced by the pitting intensity degree ( $DOP$ ) and the pitting corrosion degree ( $DOC$ ). The former is the percentage area of the plate panel affected by corrosion wastage and is generally assessed, for practical purposes, by means of the pitting intensity diagrams reported in Fig. 2. The latter, instead, refers to the thickness loss in the pitted area and generally ranges from 0 up to 50%.

Fig. 2.

Pitting intensity diagrams.

In current analysis, the ultimate strength of pitted platings $φ_{u, pit}$ is assessed after determining the ultimate strength $φ_{u}$ of the plate panel by the analytical procedure detailed in Section 2. In this respect, the slenderness parameter of the uncorroded plating β has to be preliminarily replaced by the equivalent slenderness parameter $β_{e q}$ that, in turn, refers to the mean thickness of the corroded plating [13]: $\begin{matrix} (15) & β_{e q} = \frac{β}{1 - DOP \cdot DOC} \end{matrix}$ Subsequently, the ultimate strength of platings affected by pitting corrosion wastage is determined by the following design formula [13]: $\begin{matrix} (16) & φ_{u, pit} = φ_{u} (1 - 1.5 DOP \cdot DOC) \end{matrix}$ where a further reduction of the ultimate capacity occurs, to account for the strength loss at the edges between the uncorroded and pitted areas. This formulation accounts for the reduction of the transverse cross-section of randomly pitted platings, as regards the corresponding uniformly corroded plating, so yielding to an increase of the stress field. In this respect, the applied procedure is based on two subsequent steps: (i) evaluation of the ultimate strength of a uniformly corroded plating, based on the equivalent slenderness parameter provided by Eq. (15), depending on DOP and DOC degrees; (ii) assessment of the ultimate strength drop-off of pitted platings by Eq. (16). The reliability of the embodied formulation will be verified in Section 4 by a comparative analysis with a series of non-linear large-deflection FE simulations, carried out by Ansys Mechanical APDL, where the random distribution of pitted areas was simulated based on different combinations of DOP and DOC degrees. From this point of view, it was preliminarily verified that the ultimate strength drop-off is slightly dependent on the random distribution of pitted areas, but it mainly depends on DOP and DOC degrees that are the key parameters affecting the ultimate strength of pitted platings.

Fig. 3.

Modelling of pitted areas by FE analysis.

Hence, the 4-node SHELL181 element was applied to mesh the plate panel, as this element is suitable for thin and moderately thick shell structures, in conjunction with the elastic-perfectly plastic material model. Furthermore, the APDL code allows to account for: (i) welding residual stresses, that are superimposed to the FE model as a pre-stress field, and (ii) initial imperfections, by properly varying the vertical coordinate of each node on the basis of a given initial displacement function. Finally, random corrosion wastage is included in the FE model by reducing the thickness of all shell elements belonging to the random pitted areas on the basis of the assumed DOC degree, as depicted in Fig. 3.

4. Results

4.1. Platings without pitting corrosion wastage

The effectiveness of the analytical method is preliminarily investigated by a comparative analysis with some FE results, obtained by Ansys Mechanical APDL. In this respect, Fig. 4 reports the comparative analysis, with reference to no, slight, average and severe levels of welding residual stresses. Particularly, the plating slenderness parameter ranges from 1.0 up to 2.0, with 0.1 step, and then up to 4.0, with 0.5 step. Current results show a very good agreement between the analytical formulation and the FE results, which implies that the solution obtained by the Marguerre equations can be applied for the ultimate strength assessment of platings affected by pitting corrosion wastage.

Fig. 4.

Comparative analysis for uncorroded platings.

4.2. Platings with pitting corrosion wastage

The ultimate strength assessment of pitted platings is performed by systematically increasing the DOP degree from 5% up to 50%, while the DOC degree is set equal to 25%. Particularly, the ultimate strength analysis is performed for platings without and with severe welding residual stresses, respectively. Besides, the nonlinear FE analysis is performed with reference to a plating slenderness parameter equal to 2.0 and 3.0. Current results are reported in Figs 5–8 for pitted platings with 5%, 10%, 25% and 50% DOP degrees, respectively. In all cases, a very good agreement between the analytical values and the FE results is recognized, so proving the goodness of the approximate formulation for pitted platings reported in Section 3.

Fig. 5.

Comparative analysis for pitted platings – DOP = 5%, DOC = 25%.

Fig. 6.

Comparative analysis for pitted platings – DOP = 10%, DOC = 25%.

Fig. 7.

Comparative analysis for pitted platings – DOP = 25%, DOC = 25%.

Fig. 8.

Comparative analysis for pitted platings – DOP = 50%, DOC = 25%.

The effectiveness of the analytical method for the ultimate strength assessment of platings affected by pitting corrosion wastage is further investigated by the comparative analysis with the FE results reported in Table 1. As previously said, two β-values are considered, each one coupled with no and average welding residual stresses. Hence, different combinations of $DOP$ and $DOC$ degrees are analysed up to 50%. Based on current results, reported in Figs 9 and 10, a very good agreement between the analytical formulation and the FE results is achieved, so proving that Eq. (15) and (16) can be applied with confidence for practical design purposes. Besides, it is confirmed that pitting corrosion wastage significantly affects the ultimate strength of platings under uniaxial compression. In this respect, the ultimate strength reduction mainly depends on the product between DOP and DOC degrees, based on Eq. (15) and (16). Hence, platings with different pitting and corrosion degrees behave almost identically if the product between these two variables is the same. Finally, by Figs 9 and 10 it is gathered that the ultimate strength reduction is an almost linear function of the pitting degree.

Table 1

Ultimate strength of pitted platings

DOP %	DOC %	$η = 0.00$				$η = 0.15$

		$β = 2$		$β = 3$		$β = 2$		$β = 3$

		FE	Analytical	FE	Analytical	FE	Analytical	FE	Analytical
5	10.0	0.674	0.684	0.492	0.496	0.631	0.623	0.428	0.424
	25.0	0.652	0.672	0.478	0.489	0.610	0.610	0.418	0.416
	50.0	0.630	0.652	0.454	0.475	0.574	0.591	0.393	0.404
10	10.0	0.669	0.676	0.486	0.492	0.625	0.615	0.421	0.419
	25.0	0.638	0.652	0.463	0.475	0.595	0.591	0.398	0.404
	50.0	0.586	0.611	0.427	0.450	0.535	0.552	0.371	0.381
25	10.0	0.650	0.652	0.469	0.475	0.601	0.591	0.408	0.404
	25.0	0.582	0.592	0.428	0.438	0.539	0.534	0.365	0.370
	50.0	0.483	0.500	0.354	0.378	0.422	0.445	0.305	0.317
50	10.0	0.617	0.611	0.451	0.450	0.571	0.552	0.387	0.381
	25.0	0.518	0.500	0.380	0.378	0.458	0.445	0.311	0.317
	50.0	0.353	0.340	0.256	0.272	0.292	0.295	0.207	0.224

Fig. 9.

Comparative analysis between analytical formulation and FE results for pitted platings – $η = 0.00$ .

Fig. 10.

Comparative analysis between analytical formulation and FE results for pitted platings – $η = 0.15$ .

5. Conclusions

An analytical formulation, based on the solution of the large-deflection Marguerre equations, and devoted to the strength assessment of platings under uniaxial compression, was developed and applied to investigate the incidence of pitting corrosion wastage on the ultimate strength of platings under uniaxial compression. Particularly, the analytical solution for uncorroded platings was extended to plate panels affected by pitting corrosion wastage, by properly modifying the plate slenderness parameter and the edge function for the ultimate strength assessment. Hence, a comparative analysis with the FE results obtained by a series of elasto-plastic large-deflection simulations, was carried out by Ansys Mechanical APDL, in order to verify the effectiveness of the proposed formulation.

Based on current results, the analytical solution of Marguerre equations allows efficiently determining the ultimate strength of uncorroded platings for different combinations of initial geometrical imperfections and welding residual stresses. Furthermore, the proposed formula allows efficiently estimating the ultimate strength reduction of platings affected by pitting corrosion wastage. In this respect, several combinations of pitting and corrosion degrees were analysed, in conjunction with different levels of geometrical imperfections and welding residual stresses. Current results clearly show that the approximate formulation for pitted platings is effective for practical design purposes. Furthermore, it was verified that the ultimate strength drop-off mainly depends on the product between pitting and corrosion intensity degrees, which implies that platings with different DOP and DOC parameters behave almost identically, if relevant product is the same.

Current outcomes need to be further investigated and extended to platings with different levels of initial geometrical imperfections and different boundary conditions. In this respect, current results are quite promising for further research activities, devoted to the ultimate strength assessment of pitted platings by analytical formulations.

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