On the modelling of distorted thin-walled stiffened panels via a scale reduction approach for a simplified structural stress analysis

To reduce the weight of cruise ships, the shipbuilding industry is interested in thin-walled superstructures, where the thickness of butt-welded plates in stiffened panels is under 5 mm. Compared to thick plates, thin ones can develop severe welding-induced distortions, which limit the validity of recommended early-design structural stress assessment methods. Therefore, computationally costly 3D non-linear numerical analysis must be used. For a quick and effective early design, this paper investigates on simplified computational models for thin-walled panels under uni-axial tension, considering the distortions measured on 4-mm thick, full-scale ship-deck panels. A 2D simply-supported analytical plate is shown to be sufficiently accurate (i.e., error < 10% ) at 150MPa for a distortion with maximum slope within 0.02rad and amplitude smaller than the thickness ( 𝑡 ). Moreover, a 1D beam numerical analysis efficiently predicts local stresses around the butt weld when the maximum distortion amplitude is below 0 . 6 × 𝑡 mm. The distortion needs to be considered at least up to a length of half of the plate width from the weld location and can represent longitudinal profiles within 60% of the plate width. In conclusion, the early structural stress assessment of thin-walled panels can be significantly simplified, thus helping bridge the gap between complex numerical analysis and simplified analytical solutions.


Introduction
Passenger ships are thin-walled structures, in which the buoyancy needs to compensate for the weights of the ship structure, the cargo, and the passengers.Typically, on cruise ships, the structural weight can be up to 80% of the overall weight, which is four times more than on generic cargo ships, thus justifying the great interest in reducing the economic and environmental impact of these large metallic structures.The structural weight reduction must comply with safety, strength, and durability requirements, which leads to new design and manufacturing challenges in developing lightweight solutions.Therefore, structural assessment methods and quality criteria (such as serviceability, accidental, fatigue, or ultimate strength) must be revised accordingly [1].
Among others, plate thickness () reduction is an effective weightsaving strategy in structures like ships, where the decks are modular structures made of hundreds of welded stiffened panel units, consisting of a plate attached to a single-sided stiffener with a flat bar or bulbflat profile.However, the welding process introduces macro-geometric distortions that affect the global mechanical response of the panels [2,3].Given that the plate bending stiffness is reduced proportionally to ( 3 ), these distortions are more severe on thin and slender plates, thus significantly changing their performance, especially in terms of fatigue and ultimate strength when acting as part of the hull-girder [4,5].This is because the distortions generate nonlinear secondary bending moments on the plates under membrane forces.
Several studies have been carried out on the impact of initial geometric distortions on buckling and ultimate strength of stiffened panels, generally regarded as their most critical design aspects [2,3].These studies focus on the change in the global stiffness of the panel under increasing compression load based on observed load-end-shortening behaviour.For this purpose, simplified closed-form analytical solutions have been available since the second half of the 20th century (see, e.g., [6]), and since then, various expansions have been derived to combine loads and more complex framing solutions [7][8][9][10][11][12].The general approach has been to idealise the real initial distortions with trigonometric series that can describe typical shapes known as 'hungry horse', 'mountain mode', 'spoon mode', and 'sinusoidal mode' [13].However, if the interest is on the local response of the panel in critical sections (e.g., welds or other hot spots of stress localisation), as is, for instance, in fatigue analysis, investigations should assess the difference between considering idealised and actual distortions and their impact on the panel under tensile loads.
Generally, plane element-based models are utilised to assess the influence of macro-geometry, including distortions (see, e.g., [14,15]) https://doi.org/10.1016/j.tws.2024.111637Received 14 September 2023; Received in revised form 8 December 2023; Accepted 25 January 2024 and weld profiles (see, e.g., [16][17][18][19]).Further complexity arises when the welding procedure is considered to study both residual stresses and distortions, which requires coupled three-dimensional transient thermal and nonlinear structural analyses (see, e.g., [20,21]).Focusing on the assessment of welding-induced distortions, especially with complex geometries (see, e.g., [15,22,23]), the analysis of thin-plated stiffened panels relies on the Geometric Nonlinear Finite Element Analysis (GNL-FEA) since early design stages.A comprehensive benchmark on modelling simplifications of a 4 -mm thick, full-scale stiffened panel has been carried out by Lillemäe et al. [15], where five participants compared different modelling strategies.The most accurate numerical simulations in the study showed within 5% of inaccuracy compared to experimental data.The authors limited their study to a single distortion, the modelling of which was not a fast and straightforward procedure to be extended to a large number of different samples.Therefore, despite the mentioned simplifications, GNL-FEA of 3D shell element-based models with accurate initial distortions is not computationally efficient for the early design of large structures with hundreds of repeating units like ships [24].This explains the interest in using design standards and recommendations, such as IIW [14,25], to estimate geometric stress concentrations with stress magnification factors based on highly simplified models or experimental benchmarks.This is a common approach in the context of fatigue assessment of welded structures under tension.However, current standards are based on relatively thick plates (i.e., over 5-mm thickness).As a result, for example, shipbuilding design codes (e.g., [26]) limit the thickness of the deck plate to a minimum of 5 mm.Nevertheless, geometry measurements conducted in [22,27] showed that thin-walled panels have sufficient fatigue strength based on the design S-N curves given in the IIW recommendations, despite that lateral distortion and axial misalignment between the welded plates can exceed the current tolerance limit by 35% and 50%, respectively.
To map the limitations of current IIW recommendations for the structural hot-spot stress assessment of welded thin plates, Lillemäe et al. [15] also compared the use of GNL-FEA and linear extrapolation near the welded area to the use of the analytical stress magnification factor for butt-welded plates.Such a stress magnification factor (  factor), multiplied by the applied nominal stress, defines the hot-spot (or peak) structural stress near the weld location by accounting for axial () and global angular () misalignment (or linear lateral sway, ) [14].The study showed that the current formulations for the   factor do not apply for thin plates with curved shapes, for which the definition of a support length has not been defined and significantly affects the calculations.
Therefore, it is relevant to assess the feasibility of performing further simplification on the FE modelling of the distorted panels and extending the existing standards to include the impact of initial distortions to moderately deformed structures, as needed to develop innovative lightweight solutions.In this regard, curvature-based corrections have been recently proposed for the analytical formulation of the   factor in the case of butt-welded thin plates with a simple initial curvature, as in Fig. 1(); see [19,[28][29][30][31].These corrections use a second-order beam theory that models the secondary bending effects by assuming von Kaŕmań kinematics (see, e.g., [6]) and are verified against 1D GNL-FEA.They differ in the analytical description of the initial distortion, which, nevertheless, does not seem to result in a significant difference in terms of stress magnification factors; see comparisons in [30,32].Although the proposed formulations seem to provide a valuable improvement of the current IIW recommendations, different viewpoints are exposed about which one has more engineering relevance as a quick and effective evaluation tool.This is because there is no standard protocol for the definition of the local angular misalignment due to the weldinginduced lateral buckling of the welded plates.Therefore, the accuracy of these formulations is dependent on the geometry measurement procedure and the following assumptions.As an example of proposed improvement to the IIW recommendation, the latter is compared to the Fig. 1.Geometric parameters describing () axial and () angular misalignment between butt-welded plates with a simple initial curvature.Source: Modified from the IIW recommendations [14] and based on the proposed model by Mancini et al. [30].
factor formulations in [30], as shown in Table 1.Other formulations are not included, given that this study does not aim to present further comparisons and evaluation of beam analytical models.The geometric parameters in the table refer to Fig. 1.
Although 1D analysis applies to thin and slender butt-welded plates, nothing has been demonstrated about the validity of 1D analytical formulations for plates in stiffened panels.In such a case, the 2D development of the initial distortion and the presence of a stiffening frame cause an in-plane redistribution of the stresses that may shift the critical location from the centre line and affect the consequent deformation mechanism of the welded plates [4,5,22].In light of this, the present study helps bridge the gap between the commonly accepted numerical approaches and the application of simplified 1D analytical formulations to the structural stress assessment of 4-mm-thick plates in stiffened panels with complex distortion shapes.The geometric distortions are measured through 3D laser scanning on panels cut from a full-scale ship-deck demonstrator block [23]; see Section 3 .In performing a gradual computational scale reduction from 3D to 1D models, the experimentally measured distortions are implemented into the numerical models or approximated using high-order trigonometric series.The present study draws its conclusions based on the reliability of a shell element-based FE 3D model of the distorted panel similar to the ones presented in [15] and validated within ≈10% error against experimental results in [23].The effects of stiffening frame, welds, shape and magnitude of initial distortions under varying uni-axial tension on the mechanics of the panels are studied.To assess simplified analysis strategies, both a well-known analytical plate model by Timoshenko [6] and GNL-FEA of 2D and 1D models are considered.

Theoretical framework
The stiffened panel considered in the study is typical of thin-deck ship structures [22].It is modelled in an (,  , ) coordinate system, as in Fig. 2 , where the main dimensions and the panel elements are indicated.The origin of the coordinate system is arbitrarily located in correspondence with the transverse weld and on the centre line for convenience in measurement data extrapolation, modelling, and postprocessing of the numerical results.The welded plates in the 4 -mm thick panel are considered in plane stress condition, given that the thickness dimension is significantly smaller than the in-plane dimensions [33].The computational models assume von Kaŕmań kinematics, which allows including geometric nonlinear second-order terms related to moderately large rotations of the plate.In addition, for aspect ratios higher than 3, a 1D analytical model representing the longitudinal (and main) dimension of the plates could estimate structural stresses with less than 2%-3% of inaccuracy when the initial distortion is flat along the Y-direction [30].These assumptions justify a scale reduction in the structural assessment of welded plates.Nevertheless, in the case of full-scale stiffened panels, the role of the stiffening frame cannot be neglected straightforwardly for any combination of loads and boundary conditions.Under uni-axial tension, the panel frame is expected to Table 1   factor formulations, as in the current IIW standard [25] and as proposed by Mancini et al. [30], including both axial (ax) and angular (ang) contributions.In the equations,  = 2∕ √ 3  ∕, where   is the tensile membrane stress and  is the elastic modulus of the material.

Pinned BCs Fixed BCs
(IIW [14]) )    [30] 1 + ( follow the global deformation of the welded plates.Thus, its contribution is implied by imposing the equivalent uni-axial displacement to the simplified computational models; see Section 2.2.3.Although a 2D GNL-FEA of distorted plates without the stiffening frame was already presented in [34], there are significant differences in the distortion shapes (see Section 3), thus requiring additional observations before considering further simplifications towards the 1D analysis.

Von Kaŕmań kinematic assumptions
The derivation of von Kaŕmań plate equations is shortly reviewed in this section; extensive explanations can be found, e.g., in [33].The Green-Lagrange strain tensor (  ) is defined as: where  represents partial derivatives and  ,, the displacement in the  ,, direction in the space.In the following explanation, (  ,   ,   ) are replaced by (, , ).For a thin plate in the (, ) plane, plane-stress condition and small deformation are assumed; therefore components in the thickness (i.e., ) direction and higher-order terms of the form in Eq. ( 2) are neglected.
The non-zero strain tensor components become: ) . ( The displacement field for an initially distorted plate is defined as: where (, , ) are the pure axial displacements in the (, , ) directions, and  0 represents an initial distortion.This leads to the von Kaŕmań strain definition in Eqs. ( 9) to (11), where the initial distortion is assumed to be comparable to the thickness dimension and does not generate curvature (i.e., the  1 strain components neglect  0 -related terms).
Considering Hooke's law for a linear-elastic and isotropic material with Young's () and shear moduli (), and Poisson's ratio (), the normal () and shear () stresses are defined as:

Solution strategies
Three different strategies have been used: (i) high-fidelity 3D GNL-FEA, based on shell elements, which has been validated by Mancini et al. [23] with experiments; (ii) 2D analytical plate solution as derived by Timoshenko [6]; and (iii) 1D beam GNL-FEA.

3D: GNL-FEA of the panel model
Based on the observations in [15], the 3D FE model of the stiffened panel is generated by using experimental measurements from the scanned panel using an optical measurement technique and assuming an ideally straight stiffening frame.These measurements are thereafter converted into an FE mesh that is used to carry out the GNL-FEA simulations, as shown in Fig. 3.The scanning procedure and the measured geometries are described in Section 3. The data are postprocessed to make an orphan mesh from the cubic interpolation of three longitudinal profiles extracted from the scans at  = [−120, −5, 120] mm from the centre line and within the transverse frames that are located at  = [−1740, 818.5] mm; see Fig. 3(b,c).Three profiles have been chosen as the minimum number necessary to ensure a reliable representation of the distortions [23], considering that a simple half-wave along the transverse direction may introduce inaccuracy in the case of more irregular shapes [15].The left and right ends of the profiles are aligned to  = 0 mm.The FE mesh consists of square 4 -noded shell elements of 5 mm with reduced integration and with hourglass control (S4R element in Abaqus FE package library).The mesh size at the vicinity of the weld is refined towards meeting the IIW recommendations for the hot-spot structural stress assessment, as in Fig. 3(d).The weld shape is simplified by a rectangular cross-section that is 2 -mm thicker than the plates.Tie constraints are used for the assembly to model uniform deformations along the edges.A coupling constraint is imposed between the right edge of the panel and the point of application of a concentrated, uni-axial tension ( ) to simulate a uniform load on the structure.The boundary conditions selected for the plate edges let the panel slide in the horizontal direction only from the right end, that is, only the U1 degree of freedom is allowed to the lateral and right transverse edges.The material is linear-elastic, with a Young's Modulus of  = 209 GPa and Poisson's ratio of  = 0.3.The described FE model is validated by Mancini et al. [23] and is used in this study as a reference solution to measure the accuracy of the simplified 2D and 1D models presented next.

2D: Analytical solution for the plate model
The generalised governing differential equations for a plate can be derived using the Principle of Virtual Displacement (PVD, or the minimum total potential energy); see, e.g., [33].That is, derived from the minimisation of the total potential energy ((, )).The latter is given in Eq. ( 15), where   (, ) is the external work of loads acting on the structure (in this case, only the uni-axial load ( ) is producing the work ( )), and  (, ) is the energy density function in Eq. ( 16).

𝛱(𝑥, 𝑦)
The differential equations describe a deformation mechanism in which membrane and bending effects are generally interdependent, and reaction forces depend on both external and internal actions in the (, ) plane.They are shown in Eqs.(19), (20), and (21) and are derived by using the PVD by considering: (i) von Kaŕmań strain definitions in Eqs. ( 9) to ( 11); (ii) the stresses in Eqs. ( 12) to (14); and (iii) the definitions of the membrane forces and bending moments as thickness () integrated stress resultants, as in Eqs. ( 17) and (18), respectively.The step-wise derivation of the plate governing differential equations can be found in the available literature (see, e.g., [6,33]).
In these equations,  indicates a lateral load, and the term in square brackets in Eq. ( 21) represents the coupling between membrane and bending mechanisms.In general, not all combinations of initial distortions, loads and boundary conditions allow for a closed-form solution of these equations.Nevertheless, this study considers the case of a simply-supported (SS) plate of length  and width , with small initial curvature, and loading only in the x -direction (i.e., ,   and   are neglected), as shown in Fig. 4.
Notice that the analytical model refers to the (, ) coordinate system in Fig. 4, which is different from the one presented for the panel plate field in Fig. 2 and given in capital letters.For an isotropic material: where . With an initial deformation ( 0 ), the total deflection is the sum of  0 and the deflection ( 1 ) due to axial loading, i.e.,  =  1 +  0 .Using the relation in Eq. ( 22) and   =  , Eq. ( 21) becomes: which means that the plate under membrane actions is subjected to a fictitious load, which is equivalent to . The related solution is relatively simple and given by Timoshenko [6] by using a double trigonometric series that satisfies the plate boundary conditions to describe both the initial distortion and the deflection as: where (, )  are the series coefficients associated with the sinusoidal components of wavelength (∕) and (∕) in -and -direction, respectively.By substitution of Eqs. ( 24) and (25) in Eq. ( 23): from which the plate stress field is derived based on the constitutive equations in Section 2.1.1.
Neglecting   and   is not accurate in modelling a plate that belongs to a larger structure, that is, surrounded by, e.g., web-frames or bulkheads.However, they can be sufficiently smaller than   .Moreover, this analytical solution can be adapted to any initial distortion shape that can be described by a double trigonometric series, which makes it a reasonably good simplification strategy for the structural analysis of the plate field.The plate analytical model is applied to the panels without a longitudinal weld in their plate field, i.e., where geometric symmetry about the centre line is fulfilled; these panels are indicated as P1.1, P1.2, and P1.3 (see Section 3).In such cases, the distortion geometry in the transverse direction is best described by  2 (∕), which however cannot be used to obtain a closed-form solution to this problem.Therefore, (∕) is selected as the most suitable approximation (i.e.,  is always equal to 1 in Eq. ( 24)); see Fig. 5.For comparison purposes, a parabolic approximation is included in Fig. 5.

1D: GNL-FEA of the beam model
Given the rigid constraints imposed by the stiffening frame on the panel plate field, the strain over the Y -direction for the plate is negligible and only due to the Poisson's effect of in-plane stress redistribution.Therefore, a representative beam model (where    = 0) of Young's Modulus  * is typically defined by accounting for the Poisson's effect (i.e.,  * = ∕1 −  2 ), based on which the transverse stress (   ) becomes as in Eq. (27).A pure beam solution (i.e.,  * = ) would apply to rigid rotational constraints over lateral edges free to move in the Y -direction, when the plate reactions in such direction are negligible.This case is not included in the present study.
(27) In the numerical model, however,  * =  because the equivalence in the stress and strain field through the dimensional reduction from the 3D panel to the 1D beam is obtained when the axial displacement of the panel under tension is imposed on the beam model.Such displacement is the relative displacement between the locations of the transverse frame on both the right and left sides of the panel; see Fig. 2. Using an imposed displacement for beam model is in line with the common sub-modelling approach in ship design, where the local analysis of structures is typically conducted under displacement control [24].
In the scale reduction process, it is interesting to understand how much freedom there is in the consideration of boundary conditions and distortion shape.Thereby, a beam translation over the plate width and a spatial truncation of the distortion are performed on the 1D beam model as described next.
• Beam translation over the plate widthregarding the boundary conditions, based on previous studies by Mancini et al. [35], the rotation constraint over the short edges of the plate away from the weld location has a relatively small influence on the structural stress distribution around the weld.However, the influence of the rotational constraints over the long edges of the plate has not been considered in previous studies.Thereby, this paper also investigates to what extent the 1D beam model can be translated over the plate width from the centre line towards the long edges.The translation can be beneficial when the initial distortion is critical in locations away from the plate centre line, in case slightly asymmetric configurations form during manufacturing; see, e.g., the benchmark case presented in [15].To investigate the freedom in choosing the location represented by the beam model, analyses are carried out by modelling a beam using distortion profiles at   = [50, 100, 150, 175] mm from the weld centre line; see Fig. 6.In Section 3, the reader can see that geometric symmetry about the panel centre line is valid for the distortions of three panels, while larger amplitudes on the positive side of the  -axis are observed on two panels.Therefore, considering translations over the positive side is enough for the study.• Spatial truncation of the distortionfurther analyses are dedicated to defining the minimum possible spatial observation window needed for an accurate estimation of the structural stress around the welded area.This means that instead of considering the distortion over the entire length of the plate, a spatial truncation is performed on the beam model.In the present study, the truncation is done at   = ±[100, 400] mm, as shown in Fig. 7.This analysis is about defining a minimum support length to consider in analytical formulations for the structural hot-spot stress.

Panel description
The case study is based on five large-scale panels cut from a shipdeck demonstrator block.The block is shown in Fig. 8, where the panel types P1 and P2 are identified.Accordingly, three panels of type  The panel geometries are described in detail by Mancini et al. [23], where the experimental validation of the panel GNL-FEA described in Section 2.2.1 is presented.The measurements were performed using the ATOS Compact Scan optical measurement system by GOM GmbH, which consists of two 12 -megapixel stereo cameras with a limit length measurement error (i.e., measurement resolution) of 0.126 mm based on a measuring volume of 1200 × 900 × 880 mm 3 [36].The 3D scans were carried out before and after the panels were cut out of the demonstrator block, or only after with and without clamping plates used in the experimental setup.The weld width in the panels ranges between 3.5 and 5 mm.Considering the zero position in the X direction located at the peak of the weld toe, the X -coordinate of the right and left side of the weld seam ( , and  , , respectively) are listed in Table 2.The initial distortions over the plate field vary between  0 ∕ = −1.2 and 0.4 ; as see Fig. 9.The axial misalignment and the lateral sway at the transverse weld location, as defined in the IIW recommendations [14], varies between 0.01 and 0.20 mm and 0.54 and 2.8 mm, respectively, as also shown in Table 2.The definition of the geometric parameters listed in the Table are shown in Fig. 10 Although for the panel analysis under uni-axial tension a load range above 100 MPa and within the elastic range of the material is more relevant from a limit state design perspective, this study investigates the validity of the simplification strategies well below the lower limit, i.e., down to 25 MPa, to assess the applicability limits of the simplified models.
Due to the dimension reduction scheme, the results firstly show the influence of modelling the weld cross-section and the stiffening frame through GNL-FEA of panel and plate field models; Then, the accuracy of the simplified 2D analytical plate solution is investigated.This includes some additional considerations about typical distortion shapes for welded plates in stiffened panels.The description of these shapes is given in Section 4.1.2.Afterwards, the accuracy of the 1D beam GNL-FEA is assessed over the panel centre line; lastly, the beam spatial translation over the Y -direction of the plate field and the distortion truncation are considered, as previously explained in Section 2.2.3.

Influence of stiffening frame and weld modelling
To reduce the problem dimension from a 3D panel to a 2D plate analysis, the influence of including the stiffening frame and the transverse weld on the stress distribution of the panel plate field is investigated through comparison among the GNL-FEA for three models:      Fig. 11 shows that the modelling of both weld and stiffening frame does not affect the plate stress field when they undergo the same axial displacement.Moreover, the accuracy of the stress prediction increases when more terms are included in the Fourier series approximation.Hence, a comparison over a wider load range (  = [25, 250] MPa) is shown in Fig. 12 by considering higher-order Fourier series approximation with  = 19 for panels P1.1 and P1.3.The figure shows the top ( = ∕2) and bottom ( = −∕2) structural-to-nominal stress ratio over the plate centre line from the GNL-FEA of the three models.Fig. 12 indicates that the accuracy of the analytical solution (light blue construction line) increases when at higher loads.Nonetheless, having a close look around the weld, a full agreement is not reached locally.In the figure, the black dots indicate the locations where the error of the analytical model is evaluated.The error percentages at these locations are shown in Fig. 13 for panel P1.1 ( * ), as well as P1.2 (•), and P1.3 (▵).Fig. 13 indicates that the error is fairly low (i.e., < 10%), being less than 5% for   ≥ 150 MPa.Larger errors occur for P1.2 and P1.3 under 25 MPa.

Validity range of the plate analytical solution
Based on the results above, it is possible to predict the stress distribution over the distorted plate field of a stiffened panel by neglecting the presence of the stiffening frame and the weld.This opens a possibility to utilise a non-complex analytical solution that provides fairly accurate results at early design stages.Thereby, this section presents the validity range of the plate analytical solution given in Section 2.2.2.Fig. 14(, ) shows the error percentage as a function of the initial absolute value of the maximum deflection ( 0, ) and slope (|  0 ∕ |  ), respectively, for distortion shapes from panels P1.1 ( * ), P1.2 (•), and P1.3 (▵) under 150 MPa.To have more data points, the analysis is done for the same initial distortions reduced by factors of [0.1, 0.2, … , 0.9].The figure shows that the maximum initial slope is a better control parameter among different shapes to measure the range of validity of the analytical solution.In fact, the slope values follow a unique trend curve regardless of the shape.This curve indicates an accuracy of < 10% for a slope within 0.02 rad when the structure undergoes a tension of 150 MPa.This slope threshold fluctuates with the load applied.The straightening effect at higher loads allows the formulation to cover cases with larger slopes.However, at lower load     levels, e.g., 25 and 50 MPa, the slope threshold reduces, as shown in Fig. 15 The same observations done for the panel shapes can be done for different types of distortion.As an example, Fig. 16 shows two shapes typically used to model welding-induced deformations of plates in stiffened which are a constant amplitude (CA) and a hungry horse (HH) shape [2,3].These distortions are described in Table 3, and related results in terms of error percentage as a function of the initial absolute value of the maximum slope are shown in Fig. 17(, ).

Feasibility of 3D to 1D scale reduction
In this section, the normal stresses over the top and bottom surfaces of the panel, panel plate field, and beam models are shown when GNL-FEA is used for all cases.These include the weld geometry to assess the possibility of predicting the hot-spot structural stress around the weld.As before, the nominal stress ranges between 25 and 250 MPa.Figs.18(, ) and 19(, ) show the results for the bottom and top surfaces, respectively, for panels P1.1 and P2.1.It can be observed that there is good consistency among the models when the panels undergo a stress of at least 150 MPa.Below such value, the beam model becomes inaccurate.This demonstrates that in the range of low stresses, the inplane stresses are not negligible due to the curvature in the transverse direction.The low peak at  = 0 mm is due to the discontinuity between plates and weld geometries.The maximum error reached over the panel centre line within ±10 mm ( 10 ) and ±500 mm ( 500 ) from the weld location is shown in Fig. 20(, ).These distances are used to check both local stress prediction around the weld and the first peak stress location from the weld.Fig. 21 shows the straightening effect measured by the change in the maximum amplitude range over the plate thickness, i.e., ( 1 +  0 )  ∕, where ( 1 +  0 ) is the deformed configuration of the plates.Considering that an error below 10% is acceptable, it is observed that the beam model is suitable when the nominal applied load is ≥ 150 MPa, which corresponds to an imposed axial displacement of about 1.5 mm.At this load level, the panel centre line has straightened enough so that its maximum amplitude is below 0.6 × , corresponding to a maximum slope value below 0.015 rad.

Beam translation over the plate width
This section clarifies whether it is possible to utilise the 1D beam model to represent longitudinal profiles over the plate width that are at   = [50, 100, 150, 175] mm away from the centre line (see Fig. 6).Fig. 22 shows the error in stress evaluation within ±500 mm away from the weld location for the plates P1.1 and P2.1.The former is symmetric about the centre line, thus, the same error range applies to negative values of   .For P2.1, the positive side of the Y axis has slightly larger amplitudes, thus being more interesting for the present study given that the distortions are more severe and thus more challenging to simulate.The errors are shown for stress levels of [100, 150, 250] MPa.

Spatial truncation of the distortion
To further reduce the dimension of the model, the accuracy of the 1D beam model is assessed when the distortion over the panel centre line is truncated to certain spans in the vicinity of the weld; see Fig. 7.An example of stress comparison between panel and beam model is shown in Fig. 23() for panel P1.1, where the truncation occurs at   = ±400 mm away from the weld.In the same manner, the error in the stress evaluation within 50 mm away from the weld for   = 100-400 mm is evaluated for all the panel specimens; see Fig. 23().The figure suggests that the initial distortion should be modelled at least within ±200 mm away from the weld location over the longitudinal direction.

Discussion
The results presented here indicate that simplifications in the analysis of stress localisation of distorted panels are possible, given the loading and boundary conditions of the present study.Specifically, it is observed that the amplitude and slope of the initial distortions define the severity of the nonlinearity of the problem and the feasibility of a scale reduction from 3D to 2D and further 1D computational models.
In structural analysis, a 10% error can be considered acceptable in stress predictions.This tolerance becomes critical in fatigue life prediction, as such an overestimation of the stresses would result in overly conservative underestimations of 25% and 37% for S-N curve slopes equal to  = 3 and  = 5, respectively.Therefore, a 10% error is considered a tolerance limit in this study.
With this consideration and based on the results in Section 4.1.1,it can be said that the stiffening frame modelling can be neglected in the structural stress assessment of the welded plates in stiffened panels under uni-axial tension.If the weld modelling is also ignored, an estimation of the global stress distribution can be done through a simplified 2D analytical model.On the contrary, for a hot-spot structural stress extrapolation at the weld location (as described in the IIW recommendations [14]), the weld cross-section must be considered.Nevertheless, in such a case, a 1D GNL-FEA adapted to the critical stress locations can be used.The presented 2D analytical plate solution is accurate ( ≤ 5%) above 50 MPa and if the maximum initial slope of the distortion does not overcome 0.02 rad; see Section 4.1.2.The given initial distortions are not sufficiently smaller than the thickness for the model assumptions to hold at lower loads.The related error decreases as the load increases, i.e., as the panels straighten.To avoid such an error, the shape approximation in the transverse direction should follow more accurately the gradient in the vicinity of the plate edges (as, e.g., the second-order cosine function shown in Fig. 5).Nevertheless, this, and a change in the BCs, would not allow for a closed-form solution of the plate equations.Furthermore, the smallness of  0 is explored in the linear superposition of initial distortion and deflection.This means that  0 enters the equation as an additional fictitious load (i.e.,  ( 2 ( 1 +  0 )∕ 2 )), but does not affect the considered plate field dimensions.In addition, there is no curvature effect given by the initial distortion in the strain definition, i.e., the terms  1 do not include  0 ; see Eqs. ( 9)-( 11).This leads to missing higher-order non-linear terms related to the initial distortion; see, e.g., the treatment of initial distortion by Steen et al. [37].This is not a major source of error when the applied tension causes a significant straightening of the distortion.However, for severe distortions under low load (i.e., low    and related secondary bending), such simplifications may not apply.Based on Figs. 13 and  21, when (( 1 +  0 ) < ) the plate solution results in less than 10% error.Classifications commonly used to define the severity of an initial distortion come from buckling and ultimate strength analyses [38,39].In this context, the key parameter is the plate slenderness, which for the present case equals: indicating that the plate is extremely slender (sturdy plate  < 2.4 and slender plate  > 2.4).The slenderness value is used to assess the level of severity in terms of the maximum amplitude of the initial distortion Alternatively, the IACS guidelines [39] set the maximum allowable amplitude to  0, = 6-9 mm.The measured distortions showed maximum values around 4.75 mm, thus being from average to severe quality based on Eq. ( 30), although within the maximum allowed limits based on IACS.In buckling, this means that large-deflection plate theory would be needed to model out-of-plane distortions that become gradually larger than the thickness [38].Under low tension, the deflection is not necessarily large, but the simplifying assumptions on the treatment of the initial distortion are not valid, as explained above.Therefore, these classifications help recognise critical cases for the analysis presented in this study.However, they are limited to the amplitude of the distortion, thus missing information on the slope.This means that the ranges may be too conservative or too permissive, depending on the shape of the distortion.The 1D beam GNL-FEA solution is acceptable (i.e.,  < 10%) when the deformed configuration of the 4 -mm thick plates has straightened to a maximum amplitude of 0.6 × , see Figs. 20 and 21.In this regard, neglecting the transverse direction of the distortion is the major source of error at low loads.Based on this study, the initial distortion should be modelled at least within ±200 mm away from the weld location over the plate longitudinal direction.Nevertheless, to capture the first stress peak from weld location, a span of at least 400 mm is needed.This agrees with the general assumption used in plate buckling analysis, where the wavelength of the plate approaches its width, which is the minimum necessary span to consider in plate analyses.Thereby, as a general recommendation, a numerical model for the estimation of the hot-spot structural stress of a distorted plate should include the distortion profile up to a length at least as large as the plate width.Moreover, the beam model can represent distortion profiles away from the centre line within about 60% of the total plate width, which represents up to ±150 mm away from the centre line of the panels under concern.
For the simplified models, the error in the stress estimation decreased visibly at around 150 MPa.However, this load threshold is strictly related to the initial distortion and its gradual straightening under tension, thus not being an additional limitation to define the validity ranges of the proposed models.In general, when the straightening is such that the panel approaches an ideally straight shape, the stress distribution over the transverse direction becomes more uniform, as assumed in a column-beam model.For clarity purposes, the top and bottom contour plots in Fig. 24 show the transverse to normal stress ratio field (22∕11) when the structure is under 25 and 250 MPa, respectively.Since the distortion shape defines the distribution of the secondary bending moment acting on the structure, such distribution does not change with increasing nominal stress and consequent straightening, but only the level of secondary bending stress decreases.This study does not consider analytical shell models of the distorted plates, as they are unsuitable for concise closed-form solutions; and the 1D analysis is limited to numerical simulations to investigate an equivalent support length, as well as the applicability range of beam formulations, which are still unclear.Given that it is understood that the effect of distortions is not interdependent on other stress-rising factors, weld shape imperfections, residual stresses, and material non-linearity are excluded from the study.

Conclusion
To speed up and simplify the assessment of structural stress during the early design of distorted thin-walled panels under uni-axial tension, the study investigated the feasibility of simplified computational models.To bridge the gap between the commonly accepted numerical approaches and simplified methods, a dimension reduction across scales was performed, from computationally costly GNL-FEA of 3D shell-element models, through moderately expensive 2D models, to inexpensive 1D beam ones.The accuracy of a 2D analytical plate solution was also investigated as an alternative to the FE models.The welding-induced distortions of the models were measured through 3D optical scanning on 4 -mm thick panels cut from a full-scale ship-deck demonstrator block.Based on this study, it is concluded that: • Explicit modelling of the panel stiffening frame can be neglected by using rotational constraints over the plate edges and an imposed displacement equivalent to the uni-axial tension applied to the panel structure.• Weld modelling is unnecessary in assessing the global stress distribution of the panel plate field.A 2D analytical von Kármán plate solution yields less than 5% error, if the maximum slope of the initial distortion is within 0.02 rad and the stress level is ≈150 MPa.Considering the given panel geometries, modelling the transverse distortion as a simple half-wave introduces an inaccuracy, which is significant when the distorted configuration is larger than the plate thickness (i.e., ( 1 +  0 )  ∕ > 1).• A 1D beam model is sufficiently accurate ( < 10%) when the plate distortion has straightened to a maximum amplitude of less than 0.6 ×  mm.In other words, when the panel behaviour in the in-plane transverse direction is negligible compared to the longitudinal direction.• A 1D beam model can represent panel longitudinal profiles up to ±150 mm from the centre line for a 456 mm-wide plate field, that is, within ≈60% of the total width.This is promising concerning the application of 1D analytical methods in the assessment of hot spot stress concentrations away from the panel centre line.• The welding-induced initial distortion over the panel length should be considered at least up to ±200 mm (i.e., ± ≈ 0.5 × ) in the length direction to predict the weld local stresses accurately.However, a span that approaches the width of the plate field (i.e.,  = 456 mm in this case) is needed to include the global maximum stresses due to the first buckles of the distortion from the weld location.Such a simplification opens the possibility for the future development of standard protocols for the definition of distortion parameters in simplified analytical formulations.
In conclusion, the presented simplified computational models do not have general validity in the design of stiffened panels.However, for an early-design, quick, and effective structural stress assessment under uni-axial tension, such models are sufficiently accurate in the absence of very sharp gradients of multi-buckled shapes.The estimated stress concentrations due to distortion can be added to those arising from additional factors not addressed in this study, such as residual stresses and material non-linearity.Nevertheless, these conclusions are not extended to different loading conditions.For instance, the stiffening frame could play a major role in the load-carrying capacity of the panel, and modelling of its distortions may be required, as commonly observed in buckling-driven mechanisms.

2 )Fig. 2 .
Fig. 2. Top and side views of the panel model with dimensions in [mm].The plate field (blue area), the centre line (construction line), the transverse and longitudinal welds (dashed lines), and frame details are shown.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .
Fig. 3. FE panel modelling procedure: from () optical 3D scanning of the specimen, through () scan post-processing and distortion profile extrapolation, () orphan mesh generation for the welded plates, and to () FE modelling of the panel under uniform tension ( ).

Fig. 4 .
Fig. 4. Simply-supported plate of length  and width  under uni-axial tension   =  over the -direction.

Fig. 5 .
Fig. 5. Comparison between measured distortions and analytical approximations along the  direction.The absolute values of the measured data (black lines) are normalised with their maximum at relative peak amplitude locations (  ) along the  direction; peak amplitude locations are indicated with blue dots in Fig. 9().

Fig. 6 .Fig. 7 .
Fig. 6.Beam profile modelling the plate field distortion (blue line) at distance   from the centre line ( = 0).Dimensions in [mm].(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 8 .
Fig. 8. Model in global coordinate system (  ,   ,   ) of the ship-deck demonstrator block from which the panel types P1 and P2 were cut.
(i) the full panel model with clamped constraints over the stiffening frame (i.e., simulating in-service condition); (ii) the panel plate field with clamped lateral edges (called Plate (w)); and (iii) a plate without the weld (called Plate (nw)) and zero deflection ( 3) over the lateral edges targeting to approach the plate analytical solution.With the same target, the plate field distortion is approximated by a Fourier series, as shown in Eq. (28), where  is equal to 8 or 19  = 456 mm, and  =  1 +  2 , with  1 = 1740 mm equal to the length of the plate on the left side of the weld and  2 = 818.5 mm equal to the one on the right.The number of terms changes to compare the accuracy of the distortion approximation in terms of stress prediction.Results are shown for panel P1.1 (see Fig. 9) for the plate centre line in Fig. 11 under a load of 150 MPa.

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Fig. 14 .
Fig. 14.Error percentage plotted as a function () of the absolute value of the maximum total deflection and () of the absolute value of the maximum slope of the analytical solution compared to the numerical solution of a simply-supported, distorted plate under 150 MPa.Distortions refer to the panel type P1 in Fig. 9.

Fig. 15 .
Fig. 15.Trend of the maximum initial slope at different loads considering model Plate (nw) with distortions from panel type P1 under 25, 50, and 150 MPa.Exponential trendline equations are shown.

Fig. 16 .
Fig. 16.Examples of CA and HH initial distortion types over the plate centre line.

Fig. 17 .
Fig. 17.Error percentage of the analytical solution compared to the numerical one for a simply-supported distorted plate under 25 and 150 MPa with () CA and () HH distortion types.The error is plotted against the absolute value of the maximum initial slope.

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Fig. 20 .
Fig. 20.Error percentage in the numerical stress estimation of the 1D beam model against the panel model () within 10 mm ( 10 ) and () 500 mm ( 500 ) from the weld location.Missing data beyond  10,500 = 35% are not shown.

Fig. 21 .
Fig. 21.Maximum span to thickness ratio of the distorted configuration ( 1 +  0 ) as a function of the nominal applied load for all panel specimens over their centre line.

Fig. 23 .
Fig. 23.() Stress comparison between panel model and beam with spatial truncation of the distortion at   = 400 mm for panels P1.1; and () error percentage within 50 mm away from the weld of the beam with truncation at   = [400, 200, 100] mm for all panels.

Fig. 24 .
Fig. 24.Contour of the transverse to normal stress ratio 22∕11 of the bottom surface for the panel P1.1 under 25 and 250 MPa.

Table 3
Analytical equations and geometric parameters used in the study for CA and HH distortion types.