A Computational Investigation of the Group Effect of Pits on the Ultimate Strength of Steel Plates

In this work, we focus on assessing the group effect of localized corrosion on the ultimate strength of the marine structural plates and study the load-deformation behaviors of plates of various slenderness and uniaxial compression. Meanwhile, we investigate different corroded patterns from a single circular pit to 25 circular pits distributed over the plate and carry out hundreds of nonlinear finite element simulations by combining the number, depth, distribution of pits with imperfections and slenderness of plate. The distribution of multiple pits causes scattering of stress concentration on the plate, then the plastic section of plate changes with wider distribution of damage simultaneously. The ultimate strength arises when un-loading zone comprised of the yielding strips and holes extends across the plate. It can be concluded that the corroded condition defined as group effect of pits manipulates the deformation state and the loading capacity of plate at the ultimate strength mode that coincides with the proportion of effective loading area and section in the process of post-buckling. To validate the effect of pits group, we perform the numerical experiments of the post-buckling of steel plates containing pits in a row with different orientation.


Introduction
In the marine structures, metal plates exposed to the seawater environment are at higher risk of corrosion (Canfield, 2007;Erika, 2009). Pits caused by corrosion can reduce the design values of both structural scantlings and strengths. Improperly maintained ageing structure could finally lead to disastrous casualties in rough seas and heavy weather. Thus, it is important to assess the residual strength of ageing structure properly either with pits in order to reflect vessel's inspection and maintenance program.
Failure characteristics for plates, the most common structural elements found on marine structures, are important for structures. The ultimate strength of the plate was defined by the load-shortening curves (Zhou, 2010). For an imperfect plate containing out-of-plane deformations, a moment develops upon loading and this causes a decreasing plastic force for the increasing out-of-plane deformation. The failure load is bounded by the buckling curve and the plasticity curve. Nonlinear finite element analysis is necessary to investigate the ultimate strength according to the post-buckling behavior of plates (Scheperboer et al., 2016). The effects of pits as a function of the pit shape, size and location were studied expressly. Paik et al. (2003) investigated the ultimate strength of plates with different degrees of pit under axial compressive load based on a series of nonlinear finite element analysis. It was concluded that the ultimate strength of plate with scattered pits could be calculated by the smallest cross-section, but the computational error was stochastic. Dunbar et al. (2004) investigated the effects of local pits applied to plates typically found in ship structures by the finite element analyses. Pits were applied at several different volumes and central areas. It was found that a pitted area would descend the post-ultimate response. Ok et al. (2007) assessed the effects of localized pits on the ultimate strength of plates by the finite element analyses. Because the strength degradation due to pits is more difficult and complex than general area-wide degeneration, pits were grouped together with a rectangular shape if fairly close to each other. The results depended on the equivalence of the length, breadth and depth of the rectangle. Paik (2007) investigated the ultimate strength of steel plates with a single circular defect. The plate aspect ratio was not sensitive to the ultimate strength of steel plates under longitudinal axial compressive loads. Then, the shape and size of the equivalent area would be effective to calculate the ultimate strength. Silva et al. (2013) investigated the effects of a corrosion thickness on the ultimate strength of plates with fully distributed corrosion subjected to uniaxial compressive load. A Monte Carlo simulation was conducted in order to account to the randomness associated with deterioration. The result mainly depends on the mean depth, which means that the depth of pits would be a key factor. Rahbar-Ranji (2013, 2014 had also used random corrosion surfaces to study elastic buckling strength of plates and had concluded that one-sided and both-sided pitted plates had almost the same strength reduction. Rahbar-Ranji et al. (2015) utilized the finite element method to analyze the ultimate strength of plate by varying the parameters, including the plate thickness, pit depth and the area of pitting. It was found that the ultimate strength was reduced by increasing the pit depth to thickness ratio. Sultana et al. (2015) performed the finite element analysis to investigate the effect of random corrosion on the compressive strength capacity of marine structural units. Variables included the extent of corrosion, slenderness ratio and aspect ratio. Corrosion-induced volume loss resulted in a great reduction of the ultimate strength for slender plates. Saad-Eldeen et al. (2016) performed experiments to study the effect of corrosion conjugated with cracks on the local and global responses of thin rectangular steel plates. The trend of the ultimate capacity and strain were affected by the combination of the damage degradation levels, which resulted in complex collapse modes. Garbatov et al. (2017) presented numerical and experimental analyses on the effect of the non-uniform corrosion degradation on the ultimate strength of plates. The ultimate strength decreases monotonously as a non-linear function of the estimated degree of corrosion.
As shown above, the localized corrosion has a great influence on the ultimate strength reduction. From the global strength point of view (Zhou, 2010), as the volume of pits increases, the absorbed strain-energy in need to reach the ultimate strength decreases, resulting in less developed strain with the increment of load. On the other hand, the local responses of the pitted plates differ from those of the intact one (Xu, 2016). Then, the differences in the considered parameters and loading conditions could lead to inconsistent findings due to the complex interactions between the parameters. Thus, researchers tend to specify a limited number of global variables aiming to focus on the most important ones such as the slenderness ratio, degree of pit and volume loss. Nevertheless, there is a certain amount of volatility in the calculation of strength degradation due to pits. The global variables are available, but not in sufficient quantities to obtain the ultimate strength of plates. Then, local effect of pits would be detrimental to global strength acting as the potential stress raisers.
The objective of this paper is to investigate the effects of pits, which distributed over the plate, on the ultimate strength of plates under uniaxial compression. Hundreds of nonlinear finite element simulations are carried out to systematically investigate the strength by combinating several factors, including the number, depth and distribution of pits, imperfections and slenderness of plate. The results of these analyses are used to derive new essential factor to predict the ultimate strength and strength reduction of the marine structural plates with pits under uniaxial compression.
2 Details on the finite element model θ The resistance of damaged plates subjected to uniaxial compression were calculated by the finite element method. The study considered the important parameters, as determined by the researches on steel plates with different slenderness, distributions and depths of pits. In the study, the modelled plates were square plates with the ratio b/t ranging between 30 and 100, i.e. the side length of b = 6000 mm and a thickness, t, between 60 and 200 mm. In the nonlinear finite element ultimate strength analysis for the pitted plate, the real pits on the surfaces of structural plates generally appear a conical shape and/or a spherical shape (Nakai et al., 2004). For simplification of the FEM analysis, both the conical and spherical pits are expected to be simplified as the cylindrical one with the same corroded volume as the corresponding pit and the relative errors of the ultimate strength of plates with pits between the FEA results and the experiment results are smaller than 10% (Zhang et al., 2016). Therefore, the assumptions about the plates and pits in the FEA are reasonable and the results and conclusions are reliable in the previous research. The plates had one central pit or 5, 13 or 25 pits distributed over the plate and 5 pits in a row, as shown in Fig. 1. The angle between the alignment of row and lateral direction is defined as . The ratio percentages of the pit depth to the original plate thickness, presented by p, are 25%, 50% and 100% (hole). The ratio percentage of the pit area to the original plate surface area is 4% in all the patterns considered. Steel with a Young's modulus of 200 GPa, no strain hardening, a Poisson' s ratio of 0.3, a density of 7.90 mg/m 3 , a yield stress of 335 MPa was used for the model. In addition, there was an analyzed case without any damage as reference model, which was taken to establish a comparison basis for pitted cases (i.e. to investigate the effect of corrosion on the ultimate strength characteristics). This model was called 'REF' henceforth.
The simulated compressive test setup is presented in Fig. 1, where the plate specimen has been imposed the conditions of constrained displacement and rotation on the upper and bottom boundaries. The uni-axial compressive loading was applied along the vertical direction. An average level of initial deflection was needed by the finite element software to induce instability. The first eigen-mode of the elastic buckling analysis had been considered as the initial imperfection mode, which is a half sine wave in the lateral directions in the plane of the plate. And the amplitude of wave with a maximum of 2 mm was used. In ship structures, the plate members were restricted with varied boundary, and the buckling and post-buckling behaviors would be different. The two-edge supported plate was related to the basic mechanic behavior of the plate (Zhou, 2010), so they were discussed in detail to get subtle influence in this paper.
The plates were analyzed with the finite element programme ABAQUS. For each plate, the critical elastic buckling load and mode were determined with an eigen value analysis and the failure load was obtained by a geometrical and material non-linear analysis on an imperfect plate. Plates were modelled using the element type C3D8R, which is a solid element containing 8 nodes with the reduced integration and hourglass control. Some examples of the adopted mesh are presented in Fig. 2. Each plate has been modelled with 5×10 6 or more elements. The large number of elements was required to allow for an effective prediction of buckling as well as to obtain a proper description of the stress state near the defects, where large stresses and steep stress gradients developed. Fig. 3 shows a typical case in the model 'plate with 5 defects' in this regard.
3 Verification of initial model accuracy σ The critical elastic buckling stress, e , has been determ-σ ined as the first eigenvalue of a buckling analysis with ABAQUS. At the first validation step, e from the analysis is compared with the theoretical value for the unpitted plate: σ where K c is the buckling factor, equal to 4 for the square plate with the fixed up and bottom support. As an example, for a plate with b/t = 30, the critical elastic buckling stress from the eigenvalue analysis, and the theoretical value are e = 786 MPa and 802 MPa, respectively.
σ For an imperfect plate containing out-of-plane deformation, a moment develops upon loading and this causes a decreasing force for increasing the out-of-plane displacement. Fig. 4 shows an example graph of the results. According to the chart, it is clear that the amount of compressive force and the displacement increase initially. After a while, the displacement increases, but the force is nearly constant. The force begins to decrease while the displacement increases. The ultimate strength, fail , can be assumed as the nominal stress corresponding to the maximum force value (F fail ) in this moment, which is depicted in the graph.
As a validation step, the resistance of a steel plate of the grade A36 (Young's modulus of 200 GPa, a Poisson's ratio of 0.3 and a yield stress of 250 MPa) with a single, central defect is determined by using the finite element model with the same sets presented and compared with the results of references (El-Sawy et al., 2004;Shanmugam et al., 1999) for plates with b/t = 30, 40 and 50 and normalized hole dia-  3. The square plates are simply supported around all sides and plate edges and ends were not enforced to remain straight. The ratios between the resistance, fail , and the yield stress, y , from the analyses are provided in Table 1. The results are in agreement basically for all the cases. The current research results in slightly higher resistances. Reasons should be the differences in the size of the imperfection applied. In this section, the imperfection with the maximum value of t/30 is utilized. In addition, minor differences may be present in the boundary conditions applied.
4 Results and discussion σ Fig. 5 provides the ultimate strength, fail , of a plate of σ σ σ σ σ σ σ σ σ steel containing 1, 5, 13 or 25 pits equally distributed over the plate. The resistance is presented as the normalized value " fail / y (left graph) and fail / fail_25 (right graph)", in which fail_25 is the resistance of the plate with 25 pits. For the same depth of corrosion, the resistance decreases with the increasing plate slenderness and the curve for different numbers of pits are similar to each other. The resistance of plates with a single pit is in all cases smaller than that of plates with multiple pits with the same pit depth as shown in the right graph of Fig. 5. The resistance of plate with 5 defects (pits perforate plate, DOC = 100%) is the largest one in each task with different slenderness, but this trend changes when the pits do not perforate (DOC = 50%, DOC = 25%). Fig. 6 provides the ultimate strength and polynomial fitting curve, fail , of a plate of steel containing 1 pit with different depths of pit. The resistance is presented as its normalized value, fail / ref , in which ref is the ultimate strength of that un-pitted plate. The resistance of plate containing hole (DOC = 100%) is the smallest in the cases with the same slenderness. For the same depth of corrosion, the resistance increases with the increasing slenderness. The results of plates with different numbers of pits are similar. The results are consistent with Fig. 5 (left plane). Hence, the number of pits subtlely influences the resistance while multiple pits distribute over the plate, and the depth of the pits influences the values jointly.
The deflection modes and spreads of the yielding on the compressed surface at the ultimate strength state for differσ σ

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FU Qiang et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 665-674 ent patterns (perforated pits, DOC = 100%) are shown in Fig. 7. The deflection mode is consistent with the initial imperfection mode as shown by the up panel on Fig. 7. However, it is noted that the pattern of the defect group causes some changes of the yielding portions. In the plate with one hole, the spread of the yielding focuses more on the middle panel of the plate, which is concurrent with the excessive deflection of the portion of initial imperfection, and causes con-siderable descending of the loading area and section ultimately. The distribution of multiple pits causes scattering of the stress concentration on the plate, and then the effective loading area and the section of plate change with wider distribution of damage simultaneously. In addition, the yielding strips markedly format in the plates with 13 and 25 holes, which is in agreement with the stress concentration fields caused by holes, and then the inefficient portions of the plate occur even though do not yield. For the plate with 5 holes, there is only the hazy outline of the yielding stripes because the stress concentration diminishes rapidly when the distance between the edges of holes (L dh ) is larger than the radius (r) of the hole. The load out of the plane deformation curves of plates with different holes all start with the property of the linear elasticity similarly when there are loading portions on the plates, which means that the bearing capacity depends on the loading area and section, as shown in Fig. 8. The ultimate strength arises when the unloading zone comprised of the yielding strips and holes extends across the plate as shown in Fig. 7, which means the loading areas vanish. Then, the ratio between the distance and radius of pits should be an effective parameter for the  FU Qiang et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 665-674 formation of the yielding strips. On the other hand, the first eigen-mode of the elastic buckling comprises of one half wave that occurs in the plate's longitudinal direction as shown in the right panel of Fig. 7, along which the effective loading would be differentiable because of the local deformation. Then, the holes in different positions would reduce effective loading area and section diversely. In general, the pattern of the defect group manipulates the ultimate strength level of the structure. The spreads of the yielding of the pitted plate with different pit depths on the compressed surface at the ultimate strength state are shown in Fig. 9. The deflection mode is consistent with the first eigen-mode of the elastic buckling analysis, and bends toward the un-pitted surface. The spread of the yielding is concurrent with the deflection. With the increase in the depth of pits, the spread of yielding mainly

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FU Qiang et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 665-674 focuses on the stress concentration parts around the pits and unloading takes place in the pitted parts. The deeper pits make the effective loading section of the plate decrease. But the deflection at the ultimate strength modes is different from that of the perforation patterns, and then influences the ultimate strength level of the structure. As shown in Fig. 9, with the decrease in the depth of pits, the spread of yielding is around the peak line, which is concurrent with the excessive deflection of the portion of initial imperfection. Because of undamaged parts, the stress concentration and yielding strips around the damage are removed and then the effect of the pit group is reduced. Then, the effective loading area and section of the plate with multiple pits increases effectively, which results in the change of trend as shown in Fig.  5 (right graphs). As shown above, the relative position of damage with imperfection shape would be of two effective factors that influence the ultimate strength of the pitted plate. To validate the effect of the factor, the plates of steel containing 5 pits θ θ σ σ σ θ σ θ in a row with different orientations, in which L dh = r and the distribution of the pits is controlled by the angle ( ) between the alignment and lateral direction as shown in Fig.  1. Based on the essential factor discussed above, the ultimate strength of the plate with = 0° would be the minimum one because the pits with limited distance would conveniently format the unloading row in the center of half wave of the imperfection shape. Fig. 10 provides the ultimate strength ( fail ) of these plates, which is presented as its normalized value, fail / y . For the same depth of the corrosion and orientation of the alignment, the resistance decreases with the increasing plate slenderness. As expected, the strength of the plate with the orientation of the alignment = 0° is the smallest. Then, Fig. 11 provides the ratio between the resistance of the plates and un-pitted plate ( ref ) as a function of the angle of the alignment, , for the cases (DOC = 100%) with different slenderness. The loss of the resistance increases with the decreasing angle in cases of the same slenderness. Fig. 12 provides the deflection modes and spreads of the yielding on the compressed surface at the ultimate strength state for the case with different orientations of the alignment. Although the initial imperfection modes (deformation) of different cases consistent with the first mode of the θ θ elastic buckling analysis (Fig. 3a) are similar, the deflection at the ultimate strength modes are very different. With the increase in the angle between the orientation of the alignment and the peak line of initial imperfection mode, the spread of yielding is not around the zone of pits, and occurs at both sides of the plate, as shown in Fig. 12, which is not concurrent with the excessive deflection of the portion of initial imperfection. For the scenario of = 0°, the pits are at the place consistent with the wave crest of initial impaction and the bended mode of the plate, which result in the stress concentration of the local section and unloading (yielding). For the scenario of = 90°, the pits are at the place perpendicular to the wave crest of initial impaction and the bended mode of the plate, which influence the effective section of the plate insignificantly and the load is borne by the un-pitted parts of plate, and then the ultimate strength level of the plate increases monotonously. θ Fig. 13 provides the resistance of plates with different depths of corrosion. With the same orientation of the alignment ( = 0°), the resistance decreases with the increasing  depth of corrosion for the same plate slenderness. The results of the plates with different orientations of the alignment are similar. Fig. 14 provides the spreads of yielding on the compressed surface at the ultimate strength state for the cases with different depths of pits. Although the initial im-perfection modes (Fig. 3a) are similar, the deflections at the ultimate strength modes are different. As shown in Fig. 14a, the unloading area of plates with the pitted row in a small angle row is similar to that of the perforated plates (DOC = 100%). With the increase in the angle, the spread of yielding is more focused on the peak line, which is concurrent with the excessive deflection of the portion of initial imperfection. This would be concluded that the stress concentration around the damage is weakened due to the remnant material. In addition, the pits in different positions would reduce the effective loading area and section diversely along the first eigen-mode of the elastic buckling comprising of half wave. The effective loading section of the portion of initial imperfection increases with bigger angle between the alignment and peak line. σ Fig. 15 provides the ultimate strength, fail , of a plate with different pitted patterns. The resistance is deducted from this graph and represented as its normalized value, σ σ θ θ fail / y . For the same depth of corrosion, the strength of plate with the orientation of the alignment = 0° is the smallest one and = 90° is the biggest one. In all cases, the initial imperfection modes obtained by the elastic buckling analyses are similar, and depend on the boundary and load conditions. However, the pitted condition, defined as pits group, manipulates the yielding pattern of the portion of the plastic zone directly, and then influences the deflection state of the plate and loading capacity at the ultimate strength mode, which coincides with the proportion of the effective loading area and section.

Conclusions
In the work presented herein, the group effect of pits on the resistance of square steel plates subjected to uniaxial compression has been investigated utilizing the nonlinear FEA. The unloading pattern comprised of the yielding strips and defects extends across the plate, and then the inefficient portions of plate occur even though do not yield. The ratio between the distance and radius is an effective parameter for the formation of yielding strips. Then, the distribution of pits causes the effective loading area and section of the plate changes with wider distribution of pits simultaneously, which manipulates the ultimate strength. To validate the effect of the pits distribution, the FEA of plates of steel containing pits in a row with different orientations are carried out. As expected, the pits manipulate the yielding pattern of the portion of the plastic zone directly, and then influence the deflection state of plate and loading capacity at the ultimate strength mode, which coincides with the proportion of the effective loading area and section.