A stress magnification factor for plates with welding-induced curvatures

The fatigue strength of thin-walled structures can be reduced significantly by non-linear secondary bending effects resulting from geometrical imperfections such as axial and angular misalignments. The welding-induced distortions can cause a critical increase of the structural hot-spot stress in the vicinity of the weld. Traditionally, the classification society rules for the fatigue strength assessment of welded ship structures suggest an analytical formula for a stress magnification factor 𝑘 𝑚 for axial and angular misalignment under axial loading condition. Recently, the well-known analytical solution for the angular misalignment has been extended to account for the curvature effect. The present paper analyses the effect of non-ideal, intermediate boundary conditions between fixed and pinned ends. In this regard, the fixity factors ρ (with 0 ≤ 𝜌 ≤ 1 from ideally pinned to clamped conditions) are introduced in order to model the actual constraint on the rotation close to the ends. Under tension, a non-negligible decrease of the 𝑘 𝑚 factor is observed in relation to the reduction of the fixity factor at the welded end, while the fixity factor related to the loaded end has a minor effect on the 𝑘 𝑚 factor. Under compression, the reduction of the beam end fixity factors results into lower buckling resistance.


INTRODUCTION
Over the last decades, in the scientific research there has been a push towards sustainable metal structures.Aiming at reducing the economic and environmental impact, whilst maintaining highly performing mechanical properties, the 1 Contact author: federica.mancini@aalto.fimanufacturing of thin-walled steel structures currently stands as one of the most targeted engineering lightweight solution, especially for large structures such as cruise ships.
Thin-walled structures are particularly prone to structural stability and durability problems, as they show considerable susceptibility to manufacturing-induced imperfections.Depending on the engineering field, there are different thresholds to quantify what is a thin component.In shipbuilding industry, given that the code specifications (i.e.[1]) state a minimum allowed thickness  0 = 5 mm for ship deck components, a wall is defined thin when its thickness is less than 5 mm [2], [3].The fatigue stress assessment of these structures is a major challenge for current research.This mainly relates to the detrimental effect on the global fatigue strength caused by welding processes, which are inevitable in the assembly of the block-based, modular superstructure [3].Part of the challenge also comes from the interaction between the local mechanical behavior of the welds and the global response of the ship [3], [4].For this reason, recognizing the influencing factors in the ship performance is of fundamental importance.In this regard, even though low-heat-input welding methods such as laser-hybrid welding can guarantee a less severe impact (see [2][5] [6]), geometric and material imperfections generate a non-negligible stress concentration in the weld region.Since this may eventually cause fatigue failures, attention has been devoted to the presence of such imperfections.
The currently preferred methodology for the fatigue strength assessment of large structures such as ships is the structural stress approach [7].The analysis identifies a stress magnification factor as the ratio between the hot-spot structural stress and the nominal stress, i.e.   =   /  .This factor is referred to as   , and its analytical definition is provided by shipbuilding code specifications and recommendations (see [1][7] [8]).
The computation of the   factor for plates with transverse butt joints currently accounts for the axial, , and the angular, , misalignment (see Figure 1).The assessment is limited to the presence of a global angular misalignment, i.e. to flat distortions.The analytical models provided by the design rules (see Eqs ( 1) and ( 2)) consider a one-dimensional, linearly distorted Euler-Bernoulli beam under axial tension.The   factors derived from the analytical models for fixed and pinned boundary conditions (BCs) at the plate supports were introduced in [9] as: respectively, where  = (2/)√(3  /).Parameters ,  and  are the length, the thickness and the angular misalignment, respectively;   is the tensile membrane stress (i.e. the nominal stress) applied to the structure, and  is the modulus of elasticity.These solutions reveal a non-linear relationship between the   factor and the remotely applied load (see Figure 2).This nonlinearity is analytically derived by including the developing deflection into the equilibrium equation of the structure as a secondary bending moment.
Figure 2. Non-linear relationship between the stress magnification factor and the applied nominal stress.[2] Resent research has clearly demonstrated that the simple models (in Eqs ( 1) and ( 2)) do not accurately predict the stress concentration for welded thin plates used in ship deck stiffened panels.In fact, the plates may take very irregular shapes (see e.g. Figure 3), including local angular misalignment at the weld location [2][3] [5].Therefore, design codes nowadays, do not recommend the utilization of plates thinner than 5 mm.
Figure 3. Measured welding-induced distortion over the length of butt-welded plates in a ship deck stiffened panel.[5] Non-linear Finite Element Analyses (FEA) is successfully used to calculate structural stress for welded thin plates (see e.g.[5]).
Compared to a time-consuming FEA, an analytical model for the stress magnification factor prediction considerably eases the concept design phase, which is limited in time.On this matter, Shen et al. [10] have introduced a modified   factor to include the effect of a curvature defined as a quadratic function, while Zhou et al. [11] defined equivalent dummy loads applied to a straight beam in order to account for a non-linear distortion.These studies have shown valid geometrically non-linear analysis approaches for the structural stress problem, but their engineering use is challenged by the complexity of the analytical formulations.Thus, Mancini et al. [12] have recently developed a more compact solution.The study considers a non-linear beam theory, which utilizes the von Kármán kinematic assumption.
This assumption allows the analysis to consider moderate rotations of the beam model, thus including the secondary bending moments caused by the welding-induced curvature.This analytical solution applies to a pre-deformed Euler-Bernoulli beam, modelled with ideally pinned or clamped boundary conditions (BCs) and a linear lateral deflection superimposed to a half-sine curvature.This paper extends the analytical approach introduced in [12] to non-ideal slope BCs at the beam ends.This novel study utilizes the definition of fixity factors to include the rotational stiffness of the joints, see also [13] and [14].The analytical study considers a half-sine and a quarter-sine curvature and aims at understanding the impact of non-ideal BCs on the   factor computation.

2.1
The assumptions of the analytical model The three-dimensional continuum problem of the distorted plate in a stiffened panel is reduced to a one-dimensional beam problem, given that thin plates with a geometric slenderness / >= 10 (i.e. the ratio between the beam length and thickness) are considered; see [15], [16] and [17].Small elastic Copyright © 2020 ASME displacements and moderate rotations are assumed, meaning that the Euler-Bernoulli Theory is considered along with the von Kármán assumption in the definition of the axial strain   .As shown below, the axial strain becomes: where  and  are the axial and vertical coordinates, while  and  are the axial displacement and the deflection of the beam.
Based on the Principle of Virtual Displacement (see e.g.[15][16]), the equilibrium of the beam of length  and virtual displacements  and , considering the external (  ) and internal (  ) works, is: where   and   are the internal normal force and moment.The parameter  is the external axial load.Including an initial distortion  0 , a double integration of the -dependent terms in Equation ( 4) gives: where  and  are two integration constants, which depend on specific beam configurations, as shown afterwards (see Sec.3).
Figure 4 shows the used sign convention for the internal normal and shear (  ) forces and moments.The analytical derivation assumes the constitutive relation described by the Hooke's law (  =   ) for linear-elastic materials.Furthermore, the curvature of the deflected beam axis, , is linearized as (see, e.g.[17]): According to [17], considering the equation   = −/, the bending moment distribution can be approximated as: In the equation,   is the principal moment of inertia of the cross-section of area , while ′′ indicates the second derivative of the deflection  with respect to the -coordinate.The substitution of Equation (7) in Equation ( 5) leads to the derivation of an Ordinary Differential Equation (ODE), which is solved by imposing the deflection and the slope BCs of the beam.
The ideal slope BCs of the end  and  (indicated as θ a,b ) are extended to non-ideal conditions as a function of the beam ends rotational stiffnesses  , .That is: The parameters  , ranges between 0, i.e. a pinned end configuration, to ∞, i.e. a fully fixed end configuration.
According to the relation the fixity factors  , will substitute the rotational stiffnesses, as they limit their variability range between 0 (for a pinned end) and 1 (for a fixed end).A fixity factor represents the intermediate BC as the amount of allowed rotation.For instance,  , = 0.5 means that 50% of the rotation hindered by an ideally clamped condition is free to occur.This factor can be estimated from the numerical analysis of a stiffened panel as an alteration of the curvature, i.e. of the bending moment, along the centerline.By using the Central Difference method formulation [18], the effect of the fixity factor ( , ) is related to the structure deflection.Thereby, the fixity factor relates to the departure from the final deflection expected for an ideally fixed condition.

2.2
The stress magnification factor definition The   factor is defined as: i.e. the ratio between the hot-spot structural stress and the nominal applied stress.For a beam element subjected to an axial load , the nominal applied stress is the membrane stress: The sum of the membrane and the bending stress, i.e.

3
Copyright © 2020 ASME defines the structural stress.Thus, the hot-spot structural stress is determined as the maximum structural stress (see Eq. ( 14)).
Considering the weld location (i.e. = 0), whether the hot-spot is on the top or bottom surface depends on the through-thethickness, linear distribution of the structural stress.

Modelling of the initial curvature
As in [12], the welding-induced distortion geometry is modelled by the superimposition of a half-sine curvature (HSC) and a linear lateral sway, thus including both a local and a global angular misalignment.This geometry, shown in Figure 5, is indicated as HSC model and expressed as: An additional geometry includes a linear lateral sway and a quarter-sine curvature (QSC).It is shown in Figure 6 and described as:

Analysis of HSC model under axial compression
In a step-by-step procedure, the deflection BCs are firstly applied.Given that the half-sine curvature shape fulfils the zerodeflection BC at the beam ends (required for both pinned and fixed ends), it is possible to decouple the analysis of the nonlinear and linear geometric distortions; see Step 1 and 2, respectively.Once the Principle of Superimposition is applied (see Step 3), the problem is solved through the Slope Deflection Method development [13] in Step 4. The study is completed with the computation of the related stress magnification factor for specific cases explained in Section 3.5.
Step 1 -Equation ( 5) is adapted to a half-sine curved beam under axial compression, i.e. to the internal actions and total deflection BCs written as: with  1 indicating the beam deflection in Step 1, and  0, 0 the imposed initial curvature.By applying the approximation in Equation ( 7), the resulting ODE is where  = √/  .The homogeneous solution to this ODE can be assumed as  ℎ1 () =  1 cos() +  2 sin(), while the particular solution takes the form  1 () =  sin(/).
The constants  1 ,  2 and  are found by imposing as deflection BCs.Thus, the deflection and slope-deflection equations are derived.
Step 2 -Equations ( 5) and ( 7) are now adapted to the BCs: where  is the final position reached by the right end of the beam.Thereby, the ODE for the laterally deflected beam under compression becomes: The equations for the beam deflection and slope-deflection are computed according to the particular solution  2, () =  3 +  4  and to the deflection BCs: Step 3 -The total deflection and slope deflection equations related to the HSC beam model are derived by applying the Principle of Linear Superposition, that is: Based on Equation ( 7), an -dependent equation for the bending moment follows.The solution is undetermined, as the two moments   and   are unknown.
Step 4 -A system of two equations is obtained by imposing the slope BCs in Equation ( 8).The fixity factors act as an alteration of the fully fixed ends, which implies  , = 0.By isolating the two unknown bending moments   and   , the system can be written in matrix form as: where  is the quantity also defined in the IIW recommendations (see Sec. 1).Notice that, for the beam equilibrium: Moreover, the terms   ,   ,      are not shown in his paper, as their explicit form can be found in the analytical study conducted by Aristizabal-Ochoa in [13].

Analysis of QSC model under axial compression
In the procedure for the QSC model, the linear and nonlinear distortions are studied together in Step 1*, as the quarter of sine does not satisfy the zero-deflection boundary conditions of the HSC model.
Step 1* -The resulting ODE in Equation ( 28 Step 2* -The same slope BCs applied to the HSC model are utilized to reach the system of two equations in the two unknowns   and   , which in matrix form becomes:

Adaptation of the solution to tensile axial load
The procedures shown in Sections 3.2 and 3.3 equally apply to tensile axial load (i.e. > 0).The final solutions for tensile loading condition are obtained by changing the sign of the load .This implies that:  → .Thus, for the properties of the trigonometric functions: (sin  =  sinh ) and (cos  = cosh ).

3.5
Stress magnification factor evaluation by varying one fixity factor at a time The impact of intermediate slope BCs at the beam ends is studied by varying one fixity factor at a time.So, two case studies are defined for both the HSC and QSC models: • Case 1.The fixity factor related to the loaded end,   , of the beam is varied between 0 and 1, while the weld is modelled as an ideal clamp (i.e.  = 1).• Case 2. The weld is modelled as a non-ideal BC, meaning that   varies between 0.5 and 1, while the loaded end is ideally clamped (i.e.  = 1).In Case 2, the weld is never treated as an ideally pinned BC (i.e.  > 0), as such a case would not be physically meaningful.The analytical formulations for the stress magnification factor   derived in Case 1 (  = 1,   = ) and Case 2 (  = 1,   = ) for the two models are shown in Table 1.These formulations have been verified with less than 2% error by comparison with the FEA for ideally clamped and pinned beam configurations.
Notice that, for the given beam distortions (see Sec. 3.1), the hot-spot structural stress arises at the top surface.So that the formulations in the table are determined as:

3.6
Parameter selection for the case studies Two beam configurations are considered: one is based on common small-scale specimen dimensions (see Table 2), while 5 Copyright © 2020 ASME the other presents a high geometric slenderness (see

RESULTS AND DISCUSSION
In this section, the results from the case studies are presented and discussed in order to describe the influence of non-ideal BCs on the stress magnification factor for curved thin plates.Firstly, the bending stress on the top and bottom surfaces in the weld location are compared to the applied membrane stress.This reveals the hot-spot location and whether the beam deformation is bending-dominated or not.Then, the stress magnification factor is shown as a function of the applied membrane stress to study the non-linear relationship between the   factor and the applied load.
The stress state of the slender HSC model under tensile load in Figure 7 is strongly dominated by the bending stress.In Case 1 (see Figure 7(a)), the variation of the fixity factor at the loaded end does not affect the bending stress felt at the weld location.This suggests that remote BCs do not affect the stress state, and so the deformation mechanism, in the weld region.A different trend is shown for Case 2 in Figure 7(b), where the rotation of the loaded end remains fully fixed (  = 1) .In fact, for   ≠ 1, the bending stress values become constant at low applied membrane stress.For instance, at about   = 100 for   = 0.5, as well as at about 200 MPa for   = 0.75, the bending stress is not the dominant action anymore.
The same characteristics are observed for the stress state of the QSC model with high geometrical slenderness ( / = 150).Only the Case 2 is shown in Figure 8, where, compared to the HSC model, half of the / ratio results in much lower bending stress.
For lower geometric slenderness ratio (/ = 42), the HSC model undergoes the state of stress described in Figure 9.The bending stress becomes comparable to the membrane stress as a result of the increased beam bending stiffness.In Case 1 (with   = 1), it is always slightly higher than the tensile membrane stress.Under compression, the bending behavior remains the dominant one whenever the constraints react to the rotation to some extent.In fact, only in case of ideally pinned condition at the loaded end, the membrane stress temporarily dominates the stress field.This happens during the transition of the stress state from compression to tension on the top surface (see dot-dashed lines in Figure 9(a)).Differently, in Case 2, with intermediate BC at the weld location (i.e.0.5 ≤   ≤ 1), the bending stress, Case 1 ( ρ a = 1, ρ b = ρ ) HSC With γ = 3ρsinβ + (1 − ρ)βcosβ.
If P > 0, →.In Figure 10, the plots illustrate the state of stress felt by the QSC model with low slenderness ratio.In Case 2, the bending stress slightly dominates the tensile membrane stress for the ideally clamped configuration, while it becomes smaller than the other for intermediate BCs at the weld location (i.e.smaller   , see Figure 10(b)).Under compression, the bending action leads the beam mechanical response.This also occurs in Case 1 in Figure 10(a).However, in this case, to smaller bending stiffness at the loaded end (i.e.smaller   ) corresponds a higher bending stress.9(a) Case 1, rotation is fixed at the welded end.In general, the stress magnification factor   is always computed on the top surface, as it is the most excited one for the chosen beam configurations.In fact, the top bending stress is always consistent in sign with the applied membrane stress, so that their summation is higher than the one at the bottom surface.The reason behind this phenomenon is in the definition of the initial curvatures of the HSC and QSC models.Indeed, the resulting lever arm for the axial tensile load determines a bending moment that generates a tension on the top surface.While, under compression, the initial curvature sign leads the top surface to be the one under compression at the weld location (i.e. the curvature sign tends to stay negative).Furthermore, the analysis of the models with reduced geometrical slenderness (/ = 42 for HSC model and / = 21 for QSC model) shows that the definition of the beam end BCs can also change the stress state of the component from bending-to membrane-dominated.
In Figures 11 and 12, the stress magnification factor as a function of the applied membrane stress is shown for those 7 Copyright © 2020 ASME conditions in which the variation of the fixity factors have a significant impact.According to the results, having intermediate BCs at the loaded end matters mostly under compression.Under tension, anything that happens at a relatively large distance from the weld does not affect the stress concentration at the hot-spot.On the contrary, if the joint cannot be assumed as an ideal clamp (i.e. in Case 2) , the prediction of the   factor must consider the fixity factor   , especially in case of very slender structures.In fact, a reduced bending stiffness implies higher sensitivity to secondary bending moments and to the related influencing factors.
Case 2, rotation is fixed at the loaded end.

Figure 6 . 1 -
Figure 6.1-D beam distorted by a linear lateral sway and a quarter-sine curvature (QSC model).

Figure 7 .
Figure 7. Bending stress compared to the membrane stress on the top and bottom surfaces of the beam at the weld location.HSC model, / = 300.

Figure 8 .
Figure 8. Bending stress compared to the membrane stress on the top and bottom surfaces of the beam at the weld location.QSC model, / = 150.

9(b) Case 2 ,
rotation is fixed at the loaded end.

Figure 9 .
Figure 9. Bending stress compared to the membrane stress on the top and bottom surfaces of the beam at the weld location.HSC model, / = 42.

10(a) Case 1 ,
rotation is fixed at the welded end.10(b) Case 2, rotation is fixed at the loaded end.

Figure 10 .
Figure 10.Bending stress compared to the membrane stress on the top and bottom surfaces of the beam at the weld location.QSC model, / = 21.The (  .  ) trend for a very slender configuration is shown in Figure11in relation to Case 2 and under tensile load only.The figure refers to the HSC model, which is comparable to the QSC model in the same conditions.If the loaded end is ideally clamped against rotation and the weld is modelled as a non-ideal connection, the stress magnification factor significantly changes.For instance, when   = 0.5, the   factor is more than halved if compared to the fully rigid configuration.At high tensile load, the   factor stabilizes around 5 for ideally clamped ends (i.e.  ≈ 4  as in Figure7(b)) and around 2 (i.e.  ≈   ) for the other cases.Figure12highlights the impact of non-ideal BCs on the HSC model with low slenderness ratio.Under compression, the buckling resistance of the structure is equally altered in Case 1 and Case 2. The only difference relates to the ideally pinned configuration at the loaded end, which undergoes a change in the dynamic of deformation of the beam (see dot-dashed line in Figure12(a)).According to the results, having intermediate BCs at the loaded end matters mostly under compression.Under tension, anything that happens at a relatively large distance from the weld

Figure 12 highlights
the impact of non-ideal BCs on the HSC model with low slenderness ratio.Under compression, the buckling resistance of the structure is equally altered in Case 1 and Case 2. The only difference relates to the ideally pinned configuration at the loaded end, which undergoes a change in the dynamic of deformation of the beam (see dot-dashed line in Figure12(a)).

Table 3 )
that is typical for ship deck structures.The tables refer to the HSC model.In the study of the QSC model, the geometric slenderness ratios are halved to / = 21 and / = 150, respectively.The beams are made of structural steel S355, which has a module of elasticity  of 206.8 GPa, Poisson's ratio  = 0.3 and a yielding strength of 355 MPa.

Table 1 .
Analytical formulation for the stress magnification factor including one non-ideal slope BC