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This paper re-evaluates recently published quasi-static tests on laser-welded thin-walled steel structures in order to discuss the fundamental challenges in collision simulations based on finite element analysis. Clamped square panels were considered, with spherical indenter positioned at the mid-span of the stiffeners and moved along this centerline in order to change the load-carrying mechanism of the panels. Furthermore, the use of panels with single-sided flat bar stiffening and web-core sandwich panels enabled the investigation of the effect of structural topology on structural behavior and strength. The changes in loading position and panel topology resulted in different loading, structural and material gradients. In web-core panels, these three gradients occur at the same locations making the panel global responses sensitive for statistical variations and the failure process time-dependent. In stiffened panel with reduced structural gradient, this sensitivity and time-dependency in failure process is not observed. These observations set challenges to numerical simulations due to spatial and temporal discretization as well as the observed microrotation, which is beyond the currently used assumptions of classical continuum mechanics. Therefore, finally, we discuss the potential of non-classical continuum mechanics as remedy to deal with these phenomena and provide a base for necessary development for future.

openaire: EC/H2020/745770/EU//SANDFECH The strength of laser-welded web-core sandwich plates is often limited by buckling. In design of complex thin-walled structures the combination of possible structural and material combinations is basically infinite. The feasibility of these combinations can be assessed by using analytical, numerical and experimental methods. At the early design stages such as concept design stage, the role of analytical methods is significant due to their capability for parametric description and extremely low computational efforts once the solutions have been established for prevailing differential equations. Over the recent years significant advances have been made on analytical strength prediction of web-core sandwich panels. Therefore, aim of the present paper is to show impact of this development to the design space of web-core sandwich panels in buckling. The paper reviews first, briefly the differential equations of a 2-D micropolar plate theory for web-core sandwich panels and the Navier buckling solution for biaxial compression recently derived by Karttunen et al. (Int J Solids Struct 170(1):82–94, 2019) by exploiting energy methods. By comparing the micropolar and widely-used classical first-order shear deformation plate theory (FSDT) solutions, it is shown that the different equivalent single layer (ESL) formulations and plate aspect ratios have a significant impact on the practical outcomes of the feasible design space and this way motivating further developments for micropolar formulations from practical structural engineering viewpoint.

The paper investigates the post-buckling response of web-core sandwich plates through classical continuum mechanics assumptions. The compressive loading is assumed to be in the direction of the web plates. Equivalent Single Layer (ESL) plate formulation is used with the kinematics of the First order Shear Deformation Theory (FSDT). During the initial, membrane-dominated loading stages, it is observed that the effect of finite size of the periodic microstructure is barely influences the plate responses. At the higher loads, when bending is activated, the finite size of the microstructure activates secondary shear-induced bending moments at the unit cells of the plate. A method to capture the envelope of the maximum values of these bending moments is presented. The findings are validated with the shell element models of the actual 3D-geometry. Finally, the physical limits of the classical continuum mechanics are discussed in the present context.

openaire: EC/H2020/745770/EU//SANDFECH Marine structures are advanced material and structural assemblies that span over different length scales. The classical structural design approach is to separate these length scales. The used structural models are based on classical continuum mechanics. There are multiple situations where the classical theory breaks down. Non-classical effects tend arise when the size of the smallest repeating unit of a periodic structure is of the same order as the full structure itself. The aim of the present paper is to discuss representative problems from different length scales of ship structural design.

The construction of lightweight structures poses new design challenges as a result of the different mechanics of deformation experienced by thinner-plated structures. Because of a reduced bending stiffness, thin plates are particularly sensitive to welding-induced distortions, which include a curvature, in addition to the axial and global angular misalignment observed on thick plates. The curvature shape and amplitude determine a local angular misalignment at the welded joint, which causes non-negligible secondary bending effects. Therefore, the commonly used stress magnification factors km solution for flat plates need a further development to include the curvature effect. This study proposes new analytical formulations, which extend the applicability of the existing solutions to the assessment of the structural stress of a curved thin plate under an axial load. The improved formulations are consistent with the geometrical non-linear finite element analysis under compression (up to $\sim 80$∼80% of the buckling limit) and tension (up to the yield strength). A sensitivity analysis is presented in order to show the dominant role of the curvature effect in the estimation of the km factor. Regardless of the load applied, the presence of the curvature causes a flat plate solution inaccuracy greater than 10% when the local angular misalignment is more than 1.25 times higher than the global angular misalignment in the case of a thin and slender structure.

This paper investigates experimentally the fatigue strength of high-strength steel, which has undergone the normal shipyard production process of plasma cutting, grinding, and sandblasting. The study includes steels with the yield strength of 355 and 690 MPa. The tested specimens are of dog-bone shape and represent the large-scale situation of a cruise ship balcony opening corner, loaded in shear or tension. The influence of surface roughness, internal inclusions, hardness, and residual stress on the fatigue strength are studied and discussed. Compared to the design curve as well as to the untreated surfaces, the results show significantly improved fatigue strength under constant amplitude loading at a load ratio of R = 0.1. However, very flat or even rising slope of the S-N curve indicates variations in the material and surface quality as well as in the residual stress. Surprisingly, internal defects even up to 100 μm in size did not decrease the fatigue strength.

Overload response of the stiffening frames in ship side structure due to ice loading is an important design consideration for ships operating in ice infested waters. By overload, we mean loads that are larger than assumed by the rules. Therefore, the response of ice strengthened grillage frames is investigated under a range of idealized rectangular pressure patches and by assuming different boundary conditions for the structural units. A flat, representative grillage of an ice-strengthened ship is considered and analysed using non-linear Finite Element Method. The response of the grillage frames is compared with the isolated frame response. Two frame types are considered: flatbars and L-frames. Finite element simulations revealed that patch length has strong effect on the frame deformation mode. The key characteristic that differentiates the response under shorter and longer patches is the longitudinal membrane stretching of the shell plating. Longer patches tend to suppress this deformation mode that leads to similar frame behaviour observed in isolated frame analysis. It is further shown that overload capacity of grillage frames reduces with increasing patch length to levels observed in isolated frame analysis. Analysis of plastic strain development in the frames and plating revealed that plastic strain localized faster in frames, but shell plating is more sensitive to patch height variations. This renders frames more susceptible to fracture than plating. Finally, the local failure mechanisms of the L-frames tend to diminish the load sharing capability and so negatively affects the overload behaviour.

Ductile fracture in large structures is often resolved with non-linear finite element (FE) simulations employing structural shell elements which are larger than localization zone. This makes solution element size dependent and calibration of material parameters complex. Therefore, the paper explores the ability of numerical simulations to capture the penetration resistance of stiffened panels after determining steel material fracture ductility at different stress states. The numerical simulations are compared with experiments performed with rigidly fixed 1.2 m square panels penetrated with half-sphere indenter until fracture took place. Response of the panels was measured in terms of indentation force versus indenter displacement. In parallel, tensile tests were performed with four different flat specimens extracted from the face sheet of panels to characterize the material fracture ductility at different stress states. Panel simulations were performed with two fracture criteria: one calibrated based on the test data from dog-bone specimen and other calibrated based on the data from all tensile tests. To evaluate the fracture criteria in terms of their capacity to handle mesh size variations, mesh size was varied from fine to coarse. Results suggest that fracture criterion calibrated based on the range of stress states can handle mesh size variations more effectively as displacement to fracture showed considerably weaker mesh size dependence.

Present paper investigates the factors contributing to the stress response of steel sandwich panel deck joints when applied into the hull girder of modern passenger vessel. The emphasis is put on the fatigue analysis, which is the critical design criterion in application of these structures to ships. Two types of joints are considered, i.e., a symmetrical and a nonsymmetrical prismatic joint with respect to the geometrical midplane of the sandwich panel. Stiffness properties are derived for the joints in order to enable the modeling of sandwich structures as part of the hull girder finite element model. The influence of sandwich joints to hull girder response is studied. Notch stress analysis is used for the fatigue strength estimation. The applicability of notch stress method for steel sandwich panels is validated with experiments. The investigation shows that the fatigue strength of steel sandwich panels does not depend only on local stress concentration factors at details, but also on the global stiffness of the structural hull girder system, the sandwich panels, and thus the joint response must be taken into account.

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 km 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 km 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 km factor. Under compression, the reduction of the beam end fixity factors results into lower buckling resistance.

Equivalent single layer (ESL) elements provide an easy and computationally effective way to model stiffened plates in finite element analysis of ship structures. Secondary stiffeners are incorporated into the plate or shell formulation. In the free vibration analysis, these elements ignore inertia induced local deformation of plating between the secondary stiffeners. Oscillating motion causes inertia induced body load that locally deforms the plate. This local deformation may have a significant effect on the global modal frequencies of a deck structure. This paper presents a method for correcting ESL modal frequencies by modifying generalised mass and stiffness of the modes. The modification is based on the kinetic and strain energies of the local deformations. Energy components are derived from local consideration of plate in cylindrical bending under enforced support vibration. The method is validated in a case study of ship deck structure against shell mesh results, and good agreement is found.

With modern automated welding processes, such as laser-hybrid welding, it is possible to produce high-quality welds with good resistance to fatigue. However, although the overall quality of the weld may be good, the joints can still contain significant geometrical variation and geometrical imperfections at the weld notches. In previous research for thin laser-hybrid welded butt joints, small undercut-type imperfections were observed in both toe and root side. These imperfections are usually short along the weld direction and may not be visible when the geometry of the weld is examined. However, such imperfections work as local stress risers and may influence the fatigue strength significantly based on the weakest link principle. Severity of the local notch geometry is commonly analysed by the two-dimensional analyses, which assumes constant geometry in weld direction. This approach is conservative for short undercuts due to macro-support of the surrounding material. In this study, the difference between two-dimensional and three-dimensional stress analyses for short semi-elliptic undercuts is examined using stress averaging approach. The study utilises parametric notch models, where geometric parameters, such as notch length, depth, radius and opening angle, are varied to examine the difference for different undercut shapes.

The paper presents a synthesis of experimental fatigue tests and theoretical studies of laser stake-welded T-joints in steel sandwich panels. The experiments indicate that the slope of the fatigue strength curves varies significantly depending on the type of loading the joint experiences, but also on the geometry. Therefore, the focus is on the influence of crack tip plasticity when the joints are loaded under tension and bending. This is investigated by two different approaches based on square root of J-integral. With this parameter, the experimental fatigue curves meet at the fatigue limit for all of the loading modes and geometries. To investigate the change in slope in the medium-high cycle fatigue range, the suitability of stress gradient and plastic zone size as defined by Irwin are studied. It is shown that the gradient of principal stress correlates well with the variation of slope, but it does not give a practical tool for fatigue assessment of the joints. Instead, based on the plastic zone approach, a new method is proposed that permits the estimation of number of cycles to failure under bending directly from the tension fatigue curve. The method is verified with experiments.

In this work, we create a framework for linear buckling and free vibration analyses of sandwich beams using a microstructure-dependent Timoshenko beam model founded on the modified couple-stress theory. The stiffness parameters of a structural web-core sandwich panel are determined by unit cell analysis. An extension to homogeneous cores is also carried out. By employing the exact general solution to the governing equations of the beam, an accurate approximate finite element stiffness matrix is formulated. Furthermore, the static shape functions are used to derive consistent linear geometric stiffness and mass matrices. A convergence study shows that the approximate finite element has good accuracy although the hyperbolic terms of the exact general solution have been expanded into only relatively low-order polynomial series. Results from examples show that the microstructure-dependent beam can predict critical buckling loads and natural frequencies with very good accuracy when compared to more sophisticated finite element models. Unlike the classical Timoshenko beam model, the microstructure-dependent model yields accurate results also when the sandwich assembly is transversely flexible and the bending stiffness of the faces non-negligible.

Laser stake-welded steel sandwich panels are widely used in engineering due to their high stiffness-to-weight ratios. The welds are thinner than the plates they join so that there are two crack-like notches on each side of a weld. As a consequence, the welded joints are susceptible to fatigue. In this study, as a remedy to the fatigue problem, low-density H80-grade Divinycell polyvinylchloride foam is bonded adhesively to the voids of stake-welded web-core sandwich beams. The foam reduces shear-induced stresses in the stake-welds. The choice of Divinycell H80 is founded on earlier J-integral-based finite element fatigue assessments of sandwich panels. Empty and the H80-filled sandwich beams are tested in three-point-bending for stiffness, ultimate strength and fatigue (load ratio R = 0.05). The failure modes in the weld joint region are studied using scanning electron microscopy. The experimental results show that the filling increases the stiffness of the sandwich beams by a factor of three while the weight is increased only by 6%. The ultimate strength is increased by 2.7 times. As for the fatigue behavior, the slope increases from m = 4.508 of empty panels to m = 7.321 of filled panels while the load level at 2 million cycles increases by a factor of 8.5. Keywords: Sandwich panels, Ultimate strength, fatigue strength, Hybrid structure, Scanning electron microscopy

This paper investigates the influence of material non-linearity on load carrying mechanism and strain path in stiffened panel. First, clamped stiffened panel with dimensions of 1.2 x 1.2 m was penetrated with rigid indenter until fracture took place. Second, panel material was characterized with standard tensile tests using flat test coupons extracted from the face sheet of the panel. Failure strain for different element lengths was calibrated using iterative state-of-the-art procedure. Numerical finite element simulations were performed using failure strain calibrated with tensile tests. Comparison of numerical and experimental force-displacement curves of panel clearly shows that this widely used approach is not sufficient for reliable element size independent numerical simulations. The reason is that failure strain scaling depends on the element size as well as stress state. The stress state in the structural component however, can considerably vary from that observed in tensile test.

In this paper, Reddy’s third-order shear deformable plate theory is employed for the analysis of centrosymmetric anisotropic plate structures within strain gradient elasticity. The general three-dimensional anisotropic gradient theory is reduced to a two-dimensional formulation for the analysis of thick plate structures. The third-order shear deformation theory (TSDT) takes into account quadratic variation of the transverse shear strains of the plate and does not require shear correction factors. In order to investigate the case of small strains but moderate rotations, the von Karman strains are considered. The TSDT is also simplified to anisotropic Kirchhoff plate theory within gradient elasticity. To study specific material properties in more detail, the (Kirchhoff and TSDT) gradient plate theory of general anisotropy is simplified to the more practical case of orthotropic plates. It is observed that the gradient theory provides the capability to capture the size effects in anisotropic plate structures. As case studies, the bending and buckling behaviors of the simply supported orthotropic (Kirchhoff and TSDT) plates are studied. Variationally consistent boundary conditions are also discussed. Finally, analytical solutions are presented for the bending and buckling of simply supported orthotropic Kirchhoff plates. The effects of internal length scales on deflections and buckling loads are presented.

In this study, we develop an exact microstructure-dependent Timoshenko beam finite element. First, a Timoshenko beam model based on the modified couple-stress theory is reviewed briefly. The general closed-form solution to the equilibrium equations of the beam is derived. The two-dimensional in-plane stress distributions along the beam in terms of load resultants are obtained. In addition, the general solution can be used to model any distributed load which can be expressed as a Maclaurin series. Then, the solution is presented in terms of discrete finite element (FE) degrees of freedom. This representation yields the exact shape functions for the beam element. Finally, the exact beam finite element equations are obtained by writing the generalized forces at the beam ends using the FE degrees of freedom. Three calculation examples which have applications in micron systems and sandwich structures are presented.

Utilization of thin plates together with laser-based welding processes allows manufacturing of large weight efficient steel structures. However, the fatigue strength of welds in thin-plate structures with plate thicknesses below 5 mm is observed to have large variation, which brings challenges to fatigue strength assessment. One possible reason for this variation is the increased influence of actual weld geometry that is neglected in common fatigue strength assessment approaches utilizing geometry idealization. To reveal this influence the fatigue strength of 3 mm thick laser-hybrid welded butt joints were studied using the measured microscale weld geometry and the notch stress approach. Notch stresses were defined using Neuber’s stress averaging approach which allows the determination of the fatigue-effective stress without fictitious geometric modifications. For the studied specimens the large scatter of fatigue strength in the high-cycle region could be explained using this approach with high-resolution weld profile measurements combined with thorough finite-element analysis. It was observed that axial misalignment in narrow laser-hybrid welds causes a significant notch stress increase on the root side reducing the fatigue strength dramatically in terms of structural and nominal stress. In order to capture the increased notch stress it is crucial to use a significantly smaller stress averaging length than commonly assumed for welded joints.

A homogenization method for geometric nonlinear analysis of structural core sandwich panels is proposed. The method provides high computational performance based on an efficient separation of scales. In the macroscale, the sandwich panel is discretized with an equivalent single layer of shell elements. The macroscale shell stiffness matrix is nonlinear, obtained from the analysis of a representative volume element. Prescribed displacement boundary conditions are applied to the representative unit based on the strain definitions of the first-order shear deformation theory. Changes in local wavelength in the post-buckling are considered in the analyses. Manufacturing-induced imperfections are introduced to local and global scales. The method allows for description of buckling in these two scales and is shown to hold good accuracy with respect to equivalent 3D FEM models. Examples include web-core and corrugated-core sandwich panels. The method can be extended to any periodic structure of complex local topology. It can be easily implemented to commercial FE packages.

A web–core steel sandwich panel is a lightweight structure where thin plates are welded together by laser-welding technique. The plates form a T-joint which has in the center a weld thinner than the plates themselves. Thus the rotational stiffness of the joint is not infinite. The paper investigates the influence of T-joint rotational stiffness on the lowest natural vibration frequency of the panel. The methods used in the study have different kinematic assumptions. Equivalent single-layer (ESL) theory is used to obtain the frequency of the global vibration. The local vibrations are predicted using an isolated part of the panel, the I-beam model. In addition, three-dimensional model of a sandwich panel is analyzed. Finite element method (FEM) and analytical solution are used to obtain the frequencies. First-order shear deformation theory (FSDT) is used. The joint is considered through its rotational stiffness whose quantitative values are presented in the literature. Four different cross-sections with industrial relevancy are considered. The rotational stiffness of the T-joint affects the transverse shear stiffness of the panels. The results show up to 22% reduction of the fundamental frequency when compared with the case of the rigid joint for the global vibration mode. The effect on local vibrations is up to 11% in the case of asymmetric rotation in the T-joint and is otherwise insignificant. The study furthermore outlined the limitations of the ESL approach for assessment of natural frequencies in web–core sandwich panels depending on the vibration mode shape. The results show that the rotational stiffness of the T-joint has to be considered in the conceptual design of these structures.

The paper investigates the fatigue strength of laser stake-welded T-joints subjected to reversed bending. The fatigue tests are carried out with the load ratio, R ≈ −0.8. The experimental data is firstly analysed using the nominal stress approach and then by the J-integral as the local fatigue strength parameter in the finite element (FE) assessment. The nominal stress approach demonstrated that the fatigue strength of the investigated T-joints is lower than encountered for any other steel joint under reversed tensile loading. The results also showed that the fatigue strength of this joint under the load ratio R ≈ −0.8 increases with respect to R = 0 bending by 22.6% in the case of the nominal stress approach and 13% in the case of the J-integral approach. However, the slopes of the fatigue resistance curves for different load ratios appear very similar, suggesting that the load ratio has an insignificant influence to the slope. In contrast to the similar slopes, the scatter indexes were different. The nominal stress approach shows that the scatter index is 3.4 times larger for R ≈ −0.8 than R = 0 bending. The J-integral approach showed that the scatter index for R ≈ −0.8 is only 67% larger than in the R = 0 case because the weld geometry is modelled in the FE analysis.

This paper gives an overview of the factors that affect the strength and structural design of advanced thin-walled marine structures with reduced plate thickness or alternative topologies to those used today in the marine industry. Due to production-induced initial deformations and resulting geometrical non-linearity, the classical division between primary, secondary, and tertiary responses becomes strongly coupled. Volume-averaged, non-linear response of structural element can be used to define the structural stress–strain relation that enables analysis at the next, larger, length scale. This, today’s standard homogenization process, needs to be complemented with localization, where the stresses are assessed at the details, such as welds for fatigue analysis. Due to this, the production-induced initial distortions need to be considered with high accuracy. Another key question is the length-scale interaction in terms of continuum description. Non-classical continuum mechanics are needed when consecutive scales are close. Strain-gradients are used to increase the accuracy of the kinematical description of beams, plates, and shells. The paper presents examples of stiffened and sandwich panels covering limit states such as fatigue, non-linear buckling, and fracture.

The 2-D approximation functions based on a general exact 3-D plate solution are used to derive locking-free, rectangular, 4-node Mindlin (i.e., first-order plate theory), Levinson (i.e., a third-order plate theory), and Full Interior plate finite elements. The general plate solution is defined by a biharmonic mid-surface function, which is chosen for the thick plate elements to be the same polynomial as used in the formulation of the well-known nonconforming thin Kirchhoff plate element. The displacement approximation that stems from the biharmonic polynomial satisfies the static equilibrium equations of the 2-D plate theories at hand, the 3-D Navier equations of elasticity, and the Kirchhoff constraints. Weak form Galerkin method is used for the development of the finite element model, and the matrices for linear bending, buckling and dynamic analyses are obtained through analytical integration. In linear buckling problems, the 2-D Full Interior and Levinson plates perform particularly well when compared to 3-D elasticity solutions. Natural frequencies obtained suggest that the optimal value of the shear correction factor of the Mindlin plate theory depends primarily on the boundary conditions imposed on the transverse deflection of the 3-D plate used to calibrate the shear correction factor.

Sandwich panel strips with prismatic cores are modelled using the modified couple stress theory and their elastic size-dependent bending behaviour investigated. Compatibility between the discrete sandwich and continuum beam kinematics is first discussed. A micromechanics-based framework to estimate effective mechanical properties is provided and unit cell models constructed with elementary beam elements to determine the stiffness parameters of various prismatic cores. Numerical studies show that the modified couple stress Timoshenko beam enhances the static deflection predictions of the classical Timoshenko model. A sensitivity measure based on structural ratios is proposed to estimate the influence of size effects in the global beam-level response. The parameters governing size effects in elastic sandwich beams are identified: Face-to-core thickness ratio, core density and topology, vertical corrugation order and set of load and boundary conditions. Size effects are shown more pronounced in low-density coresthat rely on corrugation bending as shear-carrying mechanism. Based on the external load, boundary conditions and sensitivity factor, one can assess whether size effects are non-negligible in a given engineering structure.

Groundings are among the most common and serious maritime accidents. The shape of the sea bottom is one of the important factors that determine the extent of grounding damage on ships, including loss of watertightness. This paper presents a four-step methodology for mathematically analyzing sea bottom shapes, where individual peaks are identified and isolated from larger datasets. To these individual peaks mathematical models are fitted and the goodness of the fit is evaluated. The aim is to develop mathematical rock models that can be used in grounding damage analysis. The method is applied to bottom topography data from the two busiest tanker harbors in Finland. As potential mathematical rock models common assumptions found in literature as well as new suggestions are used. It was observed that rocks vary in shape and size locally as well as between different geographical areas. This highlights the need to have rock models that are flexible enough to model a wide range of sizes and shapes so that the local conditions can be taken into account. This includes modeling asymmetrical shapes, which is not taken into account in the current models. The analysis results suggest using a binormal function to describe the bottom shape: This model gives overall better goodness-of-fit test results than the cone and polynomial models. In the data the binormal model can model well the also the bottom shapes where the polynomial and conical models had a good fit in statistical terms. Although the paper discusses data from the Gulf of Finland, the framework for analyzing bottom shapes is expected to be applicable to other sea areas as well. (C) 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

This paper presents the developments of the non-local sandwich beam theories that can be used to assess the micro- and macro-structural responses. The paper utilizes homogenization–localization, modified couple stress, and thick-faces beam theories. The homogenization derives prevailing differential equations from displacements through strains and stresses to external loading. This enables accurate localization process that recovers the microstructural effects from the homogenized solution and couples them to the global response. Some case studies are supported by computational experiments: the shortest beams have only four unit cell along their length. The present solution converges to the physically correct solution in the cases of infinite and zero shear stiffness. The limit of zero shear stiffness is important because there the traditional Timoshenko beam theory fails to predict the response correctly. Some difference in the solutions is observed on very short beams at intermediate rotation stiffness between the faces and the core, where the wavelengths of the global and local responses heavily interact. Present theory can be extended to different microstructures and to plates. The benefit of the present approach is that it is analytical which enables identification of the physical parameters.

All-metal sandwich panels with a low weight-to-stiffness ratio provide the means to develop cost and energy efficient large-scale thin-walled structures such as ships. However, the modeling and computational effort for a three-dimensional (3-D) sandwich structures may be significant especially in the limit state analysis of ship structures. This can become a significant drawback especially in the conceptual design stage. To this end, we determine the load–displacement curves until tensile instability for orthotropic web-core sandwich panels under multi-axial tension loading. Load–displacement curves are determined by dividing sandwich panel into discrete members and analyzing each of the constituents separately. Combined response is obtained by rule of mixtures. Tensile instability of panel is assessed using analytical necking criteria. Validity of the analytically obtained curves is established by comparing them with those obtained by a 3-D Finite Element Modeling (FEM) of the unit cell. Analyses with the unit cells show that while stiffness of the orthotropic sandwich panels is strongly direction dependent, tensile instability of panels is direction independent and governed by the faceplates. The analytical curves are then implemented to ABAQUS UGENS subroutine to describe the non-linear mechanical shell section behavior in the framework of Equivalent Single Layer (ESL) theory. The subroutine was used to simulate the response of idealized accommodation deck, with and without of a cut-out, of a passenger ship. Good agreement in response was found between 3-D FEM and ESL.

We develop a micropolar Timoshenko beam theory and use it to model web-core sandwich beams. The beam theory is derived by a vector approach and the general displacement solution to the governing sixth-order equations is given. A nodally-exact micropolar Timoshenko beam element is formulated using the solution. Bending and shear stiffnesses for a micropolar web-core sandwich beam are determined through unit cell analysis where the split of the shear forces into symmetric and antisymmetric parts plays a pivotal role. Static bending of web-core beams is studied using the micropolar model as well as modified couple-stress and classical Timoshenko beams. The 1-D micropolar results are in best agreement with 2-D web-core beam frame results. This is because the micropolar beam allows antisymmetric shear deformation to emerge at locations where the 2-D web-core deformations cannot be reduced to 1-D by considering only symmetric shear behavior., This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Action grant agreement No 745770 - SANDFECH - Micromechanics-based finite element modeling of sandwich structures. The preprint has been replaced by the accepted manuscript

The paper investigates the crack tip plasticity of laser stake-welded T-joints when the thickness is varied. For thin joints, mixed-mode condition is dominant, and the Minimum Plastic Zone Radius (MPZR) criterion is employed to estimate the crack initiation direction. Subsequently, the same first-order plastic zone size ry as defined by Irwin is kept equal for the selected load levels in the MPZR direction, and finite element (FE) analyses are carried out to investigate the stress gradient over ry comparing different thicknesses. The results allow us to explain why the fatigue slope m is different for thin and thick joints. Moreover, a new method is proposed that permits the number of cycles to failure of thin joints to be derived directly from the fatigue curve of thick joints, employing an effective J-integral. The results prove that the method is able to give an estimation of the number of cycles to failure for thin joints with acceptable error.

Modeling a periodic structure as a homogeneous continuum allows for an effective structural analysis. This approach represents a sandwich panel as a two-dimensional plate of equivalent stiffness. Known as the equivalent single-layer, the method is used here to analyze bifurcation buckling of three types of sandwich panels with unidirectional stiffeners in the core: truss-core, web-core and corrugated-core panels made of an isotropic material. The transverse shear stiffnesses of these panels can differ by several orders of magnitude, which cause incorrect buckling analysis when using the equivalent single-layer model with the first-order shear deformation theory. Analytical solution of the problem predicts critical buckling loads that feature infinite number of half-waves in the direction perpendicular to the stiffeners. Finite element model also predicts buckling modes that have non-physical, saw-tooth shape with infinite curvature at nodes. However, such unrealistic behavior is not observed when using detailed three-dimensional finite element models. The error in the prediction of the critical buckling load is up to 85% for the cases considered here. The correction of the equivalent single-layer model is proposed by modeling the thick-faces effect to ensure finite curvature. This is performed in the finite element setting by introducing an additional plate with tied deflections to the equivalent single-layer plate. The extra plate is represented with bending and transverse shear stiffness of the face plates. As a result, global buckling is predicted accurately. Guidelines are proposed to identify the sandwich panels where ordinary model is incorrect. Truss-core and web-core sandwich panels need the correction. Corrugated-core panels without a gap between plates in the core have smaller shear orthotropy and do not need the correction. Modeling the thick-faces effect ensures correct results for all cases considered in this study, and thus one should resort to this approach in case of uncertainty whether the ordinary equivalent single-layer model is valid.

This paper studies the main factors affecting the fatigue strength assessment of thin plates in large structures. The first part of study includes the influence of initial distortions, joints' flexibility and surrounding structure on structural stress analysis of welded joint. The second part covers the influence of joint and its geometrical properties on fatigue strength modelling. The third part includes also the material elastic-plastic behaviour and the influence of crack propagation. The results show that if the structural analysis considers secondary bending properly, the local elastic fatigue damage parameters such as J-integral range can be used to model fatigue strength at 2-5. million load cycles. However, to explain the slope variation of the fatigue resistance curve, the consideration of material elastic-plastic behaviour and short crack growth is needed. The strain-based crack growth simulations indicate that longer short crack growth period is the reason for the higher slope value. The importance of short crack growth is dependent on the weld notch geometry and plate thickness.

Beams with large thin-walled cross sections are not generally following the classical beam theories such as Euler-Bernoulli and Timoshenko theories. In free vibration, the cross section is deformed by the inertia induced body loads. These deformations may have significant effect on the beam modal frequencies, especially in applications involving non-structural masses. This paper presents a method to include the effect into vibration modal results obtained by the classical beam theories. Generalized mass and stiffness of the classical results are modified according to kinetic-, and strain energies of the cross section deformation. The method is validated in typical engineering case studies against fine mesh Finite Element Method and excellent agreement is found.

Due to the complexity of the ice-structure interaction, the knowledge about the ice-induced loads on a ship hull has been gained from full-scale measurements. As the instrumentation of the hull for the measurements is expensive, the extension is concerned at the planning phase. However, a narrow instrumentation can cause an error in the measurements, as the response of the adjacent frames with respect to the observed frame is not known. Thus, this paper studies the differences between loads measured from one or several frames on board S.A. Agulhas II. The difference results from the ability of the structure to transport loading internally. The results show that when the loads affect only an individual frame, the instrumentations give similar results. However, the difference increases with the load length and the loading that is determined can be 15% higher for one frame instrumentation for a structure similar to the aft shoulder of S.A. Agulhas II. Furthermore, the difference in the mean value and standard deviation of the measurement time histories can be up to 10%. The study presents a new efficient method to obtain a rough estimate of the possible difference in the measured results between extensive and narrow instrumentation.

Exact interior finite element for circular plates is developed using an elasticity solution.The solution includes Kirchhoff, Mindlin and Levinson plate theories as special cases.The exact finite element is developed both by force- and energy-based approaches.Interior stresses act as surface tractions on the plate edges.Analytical and numerical examples are presented. In this paper, a general elasticity solution for the axisymmetric bending of a linearly elastic annular plate is used to derive an exact finite element for such a plate. We start by formulating an interior plate problem by employing Saint Venants principle so that edge effects do not appear in the plate. Then the elasticity solution to the formulated interior problem is presented in terms of mid-surface variables so that it takes a form similar to conventional engineering plate theories. By using the mid-surface variables, the exact finite element is developed both by force- and energy-based approaches. The central, non-standard feature of the interior solution, and the finite element based on it, is that the interior stresses of the plate act as surface tractions on the plate edges and contribute to the total potential energy of the plate. Finally, analytical and numerical examples are presented using the elasticity solution and the derived finite element.

In this work, we present an exact 3D plate solution in the conventional form of 2D plate theories without invoking any of the assumptions inherent to 2D plate formulations. We start by formulating a rectangular plate problem by employing Saint Venant’s principle so that edge effects do not appear in the plate. Then the exact general 3D elasticity solution to the formulated interior problem is examined. By expressing the solution in terms of mid-surface variables, exact 2D equations are obtained for the rectangular interior plate. It is found that the 2D presentation includes the Kirchhoff, Mindlin and Levinson plate theories and their general solutions as special cases. The key feature of the formulated interior plate problem is that the interior stresses of the plate act as surface tractions on the lateral plate edges and contribute to the total potential energy of the plate. We carry out a variational interior formulation of the Levinson plate theory and take into account, as a novel contribution, the virtual work due to the interior stresses along the plate edges. Remarkably, this way the resulting equilibrium equations become the same as in the case of a vectorial formulation. A gap in the conventional energy-based derivations of 2D engineering plate theories founded on interior kinematics is that the edge work due to the interior stresses is not properly accounted for. This leads to artificial edge effects through higher-order stress resultants. Finally, a variety of numerical examples are presented using the 3D elasticity solution.

Finite element models of microstructure-dependent geometrically nonlinear theories for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two constituent material, the von Karman nonlinearity, and the strain gradient effects are developed for the classical and first-order plate theories. The strain gradient effects are included through the modified couple stress theory that contains a single material length scale parameter which can capture the size effect in a functionally graded material plate. The developed finite element models are used to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on the bending response of functionally graded circular plates with different boundary conditions. (C) 2015 Elsevier Masson SAS. All rights reserved. AcknowledgmentsThefirst author gratefully acknowledges the support of thiswork by the FidiPro Professorship from Aalto University (Finland)and Chair of Excellence from University Carlos III of Madrid (Spain).The second author acknowledges the“Finland Distinguished Pro-fessor (FiDiPro) -program: Non-linear response of large, complexthin-walled structures”supported by Tekes and industrial partnersNapa, SSAB, Deltamarin, Koneteknologiakeskus Turku and MeyerTurku to Aalto University, while the third author gratefully ac-knowledges the support of the Chair of Excellence granted to thefirst author at University Carlos III of Madrid.

The present work considers the evolution of the crack tip plasticity ahead of the crack-like notches with varying the loading condition and the thickness of laser stake-welded T-joints. A general procedure is described for the fatigue assessment of the joints under different loading conditions and made of different thicknesses. The new method is based on the analyses of the first order plastic radius (according to Irwin) for pure mode I, and it is generalized to mixed mode (I+II) by using the Minimum Plastic Zone Radius (MPZR) criterion. The method is validated, in the end, with a case study and explains why the fatigue slope m varies for joints loaded under bending/tension and when the thickness is reduced. The results prove that the procedure gives an excellent estimation of the fatigue life.

This paper studies the frequency of ice loads of varying lengths and the occurrence probability of their magnitudes in full-scale. In these measurements, the four frames were instrumented with shear strain gauges on the Polar Supply and Research Vessel S.A. Agulhas II. The experiments were carried out on first-year ice in the Baltic Sea. An influence coefficient matrix based on analytical and numerical analyses was used to determine the load length in the horizontal direction. Rayleigh separation was used to define the load amplitudes. The measurements show that the ice loading has to be long in order for the shear-load maximum on a single frame to occur. Furthermore, the statistical study showed that the Weibull distribution gives the best fit to the measured loads on a frame. The probability distribution of the ice loads on a frame is exponential-like for short loads and lognormal-like for long loads.

Previously, it has been shown that the grain size distribution plays an important role in the mechanical properties of welded steel. In the previous investigation, the volume-weighted average grain size has been shown to capture the influence of grain size distribution, resulting in a better fitting Hall–Petch relationship between grain size and hardness. However, the previous studies exclude the effects arising from local variation in grain size. In this paper, the grain size measurement methods are extended for the characterisation of the local grain size variation, which is significant for welded joints and can have an adverse effect on mechanical properties. The local gradient of grain size variation and its dependency on measurement direction are considered. In addition, examples of grain size and hardness variation are shown for S355 base metal and two weld metals, and characteristic differences are highlighted and discussed. The coarse-grained areas of a heterogeneous microstructure are found to have lower hardness than fine-grained areas. However, the surrounding microstructure, i.e. local grain size gradient, has an influence on the measured hardness values.

A geometricallynonlinear modelfor periodic sandwich structures based on the modified couple stress Timoshenko beam theory withvon Kármánkinematics is proposed. Constitutive relations for the couple stress beam are derived assuming an antiplane core and then extended for a generic periodic cell. A micromechanical approach based on the structural analysis of a unit cell is proposed and utilized to obtain thestiffness propertiesof selected periodic sandwich beams. Then, alocalizationscheme to predict the stress distribution over the faces of the selected beams is defined. The present model is shown to be equivalent to the thick-face sandwich theory for a linear elastic antiplane core cell. Numerical studies validate the present model against three-dimensionalfinite element modelsand the thick-face sandwich theory, and compare it with the conventional Timoshenko and couple stress Euler–Bernoulli beam theories. The present model is shown to predict deflections, stresses and buckling loads with good accuracy for different periodic cell setups. The model is able to describe elastic size effects in shear-flexible sandwich beams and the core stiffness influence onmembrane and bending stressresultants.

Constitutive equations are derived for a 1-D micropolar Timoshenko beam made of a web-core lattice material. First, a web-core unit cell is modeled by discrete classical constituents, i.e., the Euler-Bernoulli beam finite elements (FE). A discrete-to-continuum transformation is applied to the microscale unit cell and its strain energy density is expressed in terms of the macroscale 1-D beam kinematics. Then the constitutive equations for the micropolar web-core beam are derived assuming strain energy equivalence between the microscale unit cell and the macroscale beam. A micropolar beam FE model for static and dynamic problems is developed using a general solution of the beam equilibrium equations. A localization method for the calculation of periodic classical beam responses from micropolar results is given. The 1-D beam model is used in linear bending and vibration problems of 2-D web-core sandwich panels that have flexible joints. Localized 1-D results are shown to be in good agreement with experimental and 2-D FE beam frame results., This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Action grant agreement No 745770 - SANDFECH - Micromechanics-based finite element modeling of sandwich structures. The preprint has been replaced by the accepted manuscript

A 2-D micropolar equivalent single-layer (ESL), first-order shear deformation (FSDT) plate model for 3-D web-core sandwich panels is developed. First, a 3-D web-core unit cell is modeled by classical shell finite elements. A discrete-to-continuum transformation is applied to the microscale unit cell and its strain and kinetic energy densities are expressed in terms of the macroscale 2-D plate kinematics. The hyperelastic constitutive relations and the equations of motion (via Hamilton's principle) for the plate are derived by assuming energy equivalence between the 3-D unit cell and the 2-D plate. The Navier solution is developed for the 2-D micropolar ESL-FSDT plate model to study the bending, buckling, and free vibration of simply-supported web-core sandwich panels. In a line load bending problem, a 2-D classical ESL-FSDT plate model yields displacement errors of 34-175% for face sheet thicknesses of 2-10 mm compared to a 3-D FE solution, whereas the 2-D micropolar model gives only small errors of 2.7-3.4% as it can emulate the 3-D deformations better through non-classical antisymmetric shear behavior and local bending and twisting., This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Action grant agreement No 745770 - SANDFECH - Micromechanics-based finite element modeling of sandwich structures. The preprint has been replaced by the accepted manuscript