Study on ultimate strength of ship plates with calculated weld-induced residual stress

Maritime Technology and Engineering Guedes Soares & Santos (Eds) 2015 Taylor & Francis Group, London, ISBN 978-1-138-02727-5 Study on ultimate strength of ship plates with calculated weld-induced residual stress B.Q. Chen & C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal ABSTRACT: The objective of this work is to assess the importance of an accurate description of the distribution of residual stress for the assessment of the ultimate strength of rectangular plate and stiffened panels. The ultimate strength is assessed for the plates with calculated welding residual stress by nonlinear finite element analyses and it is compared with the results of the cases taking into account the idealized residual stresses as well as the residual stresses calculated by Common Structural Rules. It is concluded that the shape of the idealized residual stress distribution is very similar to the calculated one, although its size can only be determined from the calculated results. For the specific case studied here the idealized distribution results in 2% higher ultimate strength, while the IACS CSR underestimates the post collapse behavior of the stiffened plate. The studies on the effects of initial geometrical distortion, boundary condition, element size and type, are also included. 1 Introduction Welding is one of the most important fabrication processes to join structural elements for a wide range of applications. In the shipbuilding industry, steel-plated structures are welded as assemblies of individual plate elements. Extremely high temperatures occurring in the complex welding process produce geometric distortions and significant residual stresses. Fracture, buckling, corrosion and other type of failures occur due to these extreme phenomena and the strengths of structures are reduced. It has been noted that the shape of geometric deformations can significantly affect the behaviour of plate elements at the ultimate strength and also after the ultimate strength is reached. In this regard, considerable research efforts have been devoted to investigate the effect of initial geometric distortion shape on collapse behavior of the plate. The shape of welding-induced geometric distortion in steel plate elements is normally very complex. Kmiecik et al. (1995) performed a detailed statistical and regression analyses based on up to 1998 measurements of plates of different ships. A double Fourier series was proposed to describe the initial imperfections of the plates. The effect of the initial deflection on the ultimate strength of the plate was clarified by Ueda & Yao (1985). They concluded that one component of the initial deflection which became stable above the buckling load played an important role in the collapse in the case of thin plates, while the maximum curvature of the initial deflection influenced the plastification process below the buckling load in thick plate. Guedes Soares et al. (2005) presented a study aiming at quantifying the influences of localized imperfections on the ultimate compressive strength of unstiffened plate. The effect of the position of the localized imperfection was important for the longitudinal direction but negligible in the transverse direction. The initial imperfections shape seemed to change considerably the value of the collapse strength and, more importantly, changed the effects of the localized imperfection. It has long been recognized that the ultimate compressive strength of plates and stiffened panels decreases due to the presence of welding-induced residual stress (Guedes Soares, 1988). Gordo & Guedes Soares (1993) developed an approximate procedure for determining load shortening curves for stiffened plates under uniaxial compression and showed the effect of residual stresses. Guedes Soares & Gordo (1996) derived the equations to assess the strength of plates subjected to biaxial compressive and lateral pressure loads, including both effects of initial distortions and residual stresses. Due to the difficulty in measuring residual stress, idealization of the residual stress distribution has been used in most of the researches (i.e. Faulkner, 1975; Guedes Soares, 1988; Khan & Zhang, 2011; Paik & Sohn, 2012). To calculate the residual stress induced in ship plates by the welding process, Chen et al. (2011, 2012) recently developed models and techniques for predicting the corresponding structural response. 513

Parametric studies based on numerical results were performed for different weld parameters including welding speed, plate thickness, heat input, and heat source type. The effect of convection coefficient and finite element meshes were studied as well by Chen et al. (2014a). Based on the previous study on the residual stress calculation, the objective of this work is to access the importance of an accurate description of the residual stresses for predicting the ultimate strength of steel-plated ship structures considering the weld-induced initial geometrical distortions and residual stresses, and to investigate the influence of residual stress on the compressive ultimate strength in longitudinal direction of ship structures. The residual stresses are calculated by a nonlinear Finite Element Method (FEM) considering a range of plate thicknesses. The obtained ultimate strength is compared with the results of the cases taking into account the idealized residual stresses as well as the residual stresses calculated by International Association of Classification Societies (IACS) Common Structural Rules (CSR). The studies on the effects of initial geometrical distortion, boundary condition, element size and element type, are also included. Figure 1. The shape of geometric distortions of the plate (magnified by 100 times). Figure 2. The boundary condition for the plate (Khan & Zhang, 2011). 2 Ultimate strength calculation considering idealized residual stress 2.1 Ultimate strength of a plate The dimensions of the studied plate are 4300*815*9.1 mm 3 that is identical to the work of Khan & Zhang (2011). The yield stress of the material is 315 MPa, the Young s Modulus is 205.8 GPa, and the Poisson ratio 0.3. The initial imperfection is modeled by a simple double-sinusoidal formula, ω Z = b 5π π a x b y sin sin (1) 200 where x and y are the coordinates in the longitudinal and transversal axis, respectively, a and b are the length and width of the plate. The number of half-waves in the longitudinal direction is considered as the nearest integer (5) to the aspect ratio (a/b = 4300/815). The maximum amplitude of the deflection is considered as 1/200 of the plate width. As shown in Figure 1, five half waves are observed in the longitudinal direction, and only one half wave appears in the transversal direction. The symmetric boundary conditions are used in four edges of the plate, as shown in Figure 2. All nodes in line AB is coupled in longitudinal direction. Figure 3. Idealized residual stress pattern in the plate (Khan & Zhang, 2011). A constant force equal to the yield stress times the area of the transversal section is applied in line AB. 2.1.1 Effect of initial residual stress Figure 3 displays a typical idealized pattern of the initial residual stress in a plate. For practical design purposes, the welding induced residual stress distributions of a plate element in the plate can be idealized as the combination of both tensile and compressive stress blocks. The equilibrium requirement for the tensile and compressive stresses provides a relationship between the magnitude of compressive residual stress in the plating and the widths η t of the tension zones each side of the weld. To study the effect of the initial residual stress, two plates with the same dimensions (4300*815*9.1 mm 3 ) have been modeled. ANSYS SHELL281 element that has eight nodes with six degrees of freedom at 514

each node is used in the analysis. Using 20 mm mesh size, the plate is divided to 8815 square elements. The symmetric boundary condition (see Fig. 2) is applied in both plates. Idealized residual stress (see Fig. 3) has only been considered in one of the models. Figure 4 displays the Finite Element (FE) model of the plate with initial residual stresses. The value of η is approximately equal to 25%. The load shortening curves of the two models are plotted in Figure 5. The ultimate strength of the imperfect plates without residual stress is higher compared to the plate considering the presence of welding-induced residual stresses. The corresponding strain at the ultimate strength also increases with an increase of residual stress. It could be explained that the presence of tensile residual stresses induces certain degrees of hardenings, hence the ultimate strengths are achieved at higher strain values. 2.1.2 Effect of initial geometrical distortion To study the effect of the pattern of initial geometrical deflection, another model (Case 2) with a different type of initial imperfection is considered as, π π ω b 5 Z = a x b y cos sin (2) 200 Figure 4. Initial residual stresses in plate. Figure 7. Load shortening curve of 9.1 mm thick plate (Case 3). Figure 5. Load shortening curve of 9.1 mm thick plate (Case 1). Figure 6. Load shortening curve of 9.1 mm thick plate (Case 2). Figure 8. The shape of geometric distortion in the stiffened panel model. (a) local mode of distortion in the plate and web for the panel section at x = 0 along the transverse direction, (b) shape of the stiffened panel plate along the longitudinal direction, (c) shape of the global mode of distortion along the longitudinal direction and (d) shape of the global mode of distortion in the plate, web and flange (Khan & Zhang, 2011). 515

Table 1. Scantling and material property of the stiffened panel with four stiffeners. A b t p h w t w b f t f E λ σ y 4300 mm 815 mm 17.8 mm 463 mm 8 mm 172 mm 17 mm 205.8 MPa 0.3 315 MPa where a = frame spacing along longitudinal direction, b = spacing between two stiffeners, t p = plate thickness, h w = web height, t w = web thickness, b f = breadth of flange and t f = flange thickness. The load shortening curves in the new case are shown in Figure 6. Similar to the first case, the plate with residual stress has slightly lower ultimate strength and larger corresponding strain at the ultimate strength. The calculated results are compared with the ones in ABAQUS by Khan & Zhang (2011). The behaviors of the load shortening curves are identical, while the results in ABAQUS are higher. The ultimate strength of the plate is slightly reduced in Case 2. However, the trend of the curve is observed different which means the initial geometrical distortion has an important influence on behaviour of the plate on the postcollapse stage. 2.1.3 Effect of boundary condition In Case 3, the boundary conditions have been modified as fixing UR1 DOFs in the longitudinal edges instead of UR2, and fixing UR2 DOFs in the shorter edges instead of UR1. Figure 7 presents the load shortening curves. The trends of the curves are similar to the first two cases but the magnitudes of the ultimate strength are much higher. A second peak value is observed in the plate considering residual stress after the ultimate strength in a larger corresponding strain. 2.2 Ultimate strength of a welded stiffened panel A stiffened panel has also been studied, with initial geometric distortions shown in Figure 8. The scantling and material property is shown in Table 1. The boundary conditions are presented in Figure 9. Figure 10 illustrates the idealized residual stress distribution in plate and stiffeners, where the residual stresses have been applied in the welded region of the web and plate. In the present study, 25% residual stress is considered. Using ANSYS SHELL181 element, The FE model of the plate considering initial geometrical distortions and residual stress is displayed in Figure 11. The load shortening curves of the stiffened panels with and without residual stress are plotted in Figure 12. Higher ultimate strength is observed due to the effect of the stiffeners. The slope of the postcollapse load shortening curve for the stiffened Figure 9. The boundary conditions in the stiffened panel model. Figure 10. Idealized residual stress pattern in the stiffened panel (Khan & Zhang, 2011). Figure 11. FE model of the stiffened plate, considering initial geometrical distortions and residual stress. 516

Figure 12. Load shortening curve of stiffened panel. panels without residual stress declines smoothly, whereas the strength keeps high values when the normalized strain increase from 1.0 to 1.5. The results in ABAQUS by Khan & Zhang (2011) have similar trends but have higher values. 3 Ultimate strength calculation considering calculated residual stress 3.1 Numerical approach based on welding simulation The finite element analysis of the welding process can be defined as a three dimensional coupled thermo-mechanical analysis. In the present work, the three-dimensional FE model was solved using ANSYS code. In some cases, the effect of the structural result on the thermal analysis is very small and can be even neglected. Thus, the indirect method, in which the thermal analysis and structural analysis are performed separately, can be utilized for these single directional coupled issues. Figure 13 illustrates the procedures of the numerical approach. The temperature distribution of all nodes can be obtained from the thermal analysis solved in the first step, and be used as body loads to applied in the some geometric model to do the mechanical analysis. To complete a thermal analysis using ANSYS, the eight-noded, 3D brick thermal element Solid 70 is used, whereas the element will be converted to the corresponding structural element for the mechanical analysis. By means of inputting commands to change thermal element type to structural type, the element Solid 70 can be automatically replaced by the equivalent structural element Solid 185 which is also an eight-noded, three-dimensional element but has plasticity, hyper-elasticity, stress stiffening, creep, large deflection, and large strain capabilities. Figure 13. Flow diagram of transient thermo-mechanical analysis model. Table 2. Welding conditions. Current Voltage Speed Efficiency 270 A 29 V 0.4 m/min 0.8 Table 3. Double ellipsoidal heat source model descriptors. a b c f c r f f f r 4 mm 4 mm 4 mm 16 mm 0.4 1.6 In this conversion, the element mesh that plays an essential role in the calculation remains the same as the thermal analysis geometric model. Instead of idealized initial imperfections, the distortion and residual stress obtained in the second step can be imported in the same geometric model to calculate its ultimate strength. 3.2 Parametric studies The stiffened plate analyzed has dimension 500*500*12 (plating) + 300*500*9 (web) mm 3 that is identical to the specimen tested by Deng et al. (2007). The temperature-variant thermal and mechanical properties of the material are adopted from Chen et al. (2014b). The welding starts from one side of the web before the other, with the conditions listed in Table 2. Table 3 illustrates the parameters of the double ellipsoidal heating source applied in the FE model. More than 2200 ºC temperature is observed during the welding process, as shown in Figure 14. As predicted by the double ellipsoidal model, the input heat energy affects more areas in the rear part of the Heat Affected Zone (HAZ). The temperature decreases rapidly outside the welding zone. 517

Figure 14. Temperature distribution in thermal analysis of welding process (plate thickness is 9 mm). The calculated residual stress in the mid-section of the plating is plotted in Figure 16. Three levels of residual stress (5%, 15% and 25%) are considered in the idealized models and are also shown in the figure. The 15% case has the best agreement with the calculated result in terms of the compressive stress and the width of the tensile region. The idealized curve is able to predict the basic characteristics of the residual stress distribution, but it neglects the transformations along the plate width. Furthermore, the choice of the level of residual stresses in the idealized model can only be made from a comparison with the shape of the calculated values, as there are no methods to accurately determine the level of residual stresses a priori. To calculate the ultimate strength of the stiffened plate, symmetric boundary condition is included in one end of the transversal section, as well as two edges parallel to the welding direction, whereas all the nodes in the other end sections are coupled in the translational degrees of freedom in the longitudinal direction. Considering the calculated residual stresses, the load shortening curves of the stiffened plate are plotted in Figure 17. The result calculated by Figure 15. Residual stress distribution in structural analysis of welding process (plate thickness is 9 mm). In the mechanical finite element analysis, which represents the second step, the stiffened plate is constrained along the welding direction at two nodes located in the two corners of the start crosssection, in order to prevent rigid body motion allowing deformation. The deformed patterns are observed symmetric, and linear from the weld to the edge of the plate width. There is almost no deformation in the side edges due to the constraints, while maximum vertical deformations occur around the weld. Residual stresses are stresses that remain after the heat gradient has been removed. In the welded plate, the most important residual stresses are the longitudinal residual stress. Figure 15 plots the longitudinal residual stress distribution in the stiffened plate. Significant tensile stress (344 MPa) is observed in the weld, whereas in the region away from the welding line, the longitudinal residual stresses become compressive. Figure 16. Residual stress distributions in the midsection of the plating. Figure 17. Load shortening curve of stiffened plate. 518

proposed numerical model is observed similar to that of the model with idealized residual stress, whereas the ultimate strength in the idealized model is 2.1% higher. It is concluded that a satisfactory strength behavior of the stiffened plate can be achieved using the idealized model, and the IACS CSR underestimates the post collapse behavior of the stiffened plate. 3.2.1 Effect of mesh size Finite element models of the stiffened plate with different element sizes were established in order to analyze the effect of the mesh size on the ultimate strength of the structure. Figure 18 plots the resultant ultimate strength with respect to mesh size. The models with 2.5, 5 and 10 mm element size are satisfactory while the 15 mm element is unacceptable in the present study, considering the plate dimensions 500*12*500 + 300*9*500 mm 3. 3.2.2 Effect of element type and slenderness Figure 19 displays the load shortening curves of both Solid (eight-noded element with three degrees of freedom) and Shell (four-noded element with six degrees of freedom) models. Comparing with the IACS prediction, better performance is observed with the solid model in terms of the initial slope and the ultimate strength. The shell model has similar post collapse behaviour as the IACS prediction but results in lower ultimate strength. The compressive strength of plate and stiffened panel is mainly governed by the slenderness that is defined as, σ β = b t E (3) where b is the plate width, t is the plate thickness, σ is the plate yield stress and E is the Young s modulus. Table 4 lists the studied plates considering various slenderness. Figure 20 demonstrates the ultimate strength of the stiffened plate considering different element types and slenderness. It is observed that the ultimate strength decreases when the slenderness increases. The solid model fits the IACS result very well in plates with small slenderness 1.59 (thinnest plate) but results in much bigger strength in larger slenderness cases. The shell model underestimates the ultimate strength of the structures. 3.2.3 Effect of residual stress Figure 21 plots the load shortening curves of the same stiffened plate (9 mm thickness) with Table 4. Study cases. b (mm) t (mm) σ 0 (MPa) E (MPa) β (-) Figure 18. Effect of mesh size on ultimate strength of stiffened plate. 500 12 300 2.06E+05 1.590 500 9 300 2.06E+05 2.120 500 6 300 2.06E+05 3.180 500 4 300 2.06E+05 4.770 Figure 19. Effect of element type on load shortening curve of stiffened plate. Figure 20. Effect of element type on ultimate strength of stiffened plate. 519

Figure 21. Effect of residual stress on load shortening curve of stiffened plate. Comparing with the imperfect plate, the plate with residual stress has slightly lower ultimate strength and a larger corresponding strain at the ultimate strength. The idealized residual stress distribution results in 2% higher ultimate strength but can still be used to assess a satisfactory strength behavior of the stiffened plate when taken into consideration the computational time. The IACS CSR underestimates the post collapse behavior of the stiffened plate. In the 4 cases studied, it is observed that the ultimate strength decreases when the slenderness increases. The 15% level residual stress reduces 1.7% ultimate strength of the stiffened plate and about 3 4% ultimate strength in the stiffened plates with larger slenderness. Acknowledgement The first author has been funded by a PhD scholarship from ABS, the American Bureau of Shipping. ReferenceS Figure 22. Effect of residual stress on ultimate strength of stiffened plate. and without considering the residual stress. It is observed that in the model with 15% residual stress, the nominal stress is lower. The value of ultimate strength with respect to the slenderness is shown in Figure 22. Due to the effect of residual stress, the ultimate strength reduces 1.69% in the stiffened plate, which has a slenderness of 1.59. The reductions are about 3 4% in larger slenderness cases. 4 Conclusions The results of the study, confirmed that the initial geometrical distortion has an effect on the post-collapse behaviour of the plate. The boundary conditions have a significant effect on the ultimate strength. The models with 2.5, 5 and 10 mm element size are satisfactory in the present study, considering the plate dimensions 500*12*500 + 300*9*500 mm 3. Solid element model results in better performance in terms of the ultimate strength than the Shell element model. ANSYS. Online Manuals, Release 112009. Chen, B.Q., Adak, M., Guedes Soares, C. 2011. Thermomechanical analysis of the effects of weld parameters in ship plates during welding process. Proceedings of the 2nd International Conference on Ship & Offshore Technology (ICSOT2011): 23 30. Chen, B.Q., Adak, M., Guedes Soares, C. 2012. Effect of weld parameters on the temperature-time history in steel plates. In: Guedes Soares, C., Garbatov, Y., Sutulo, S., Santos, T.A. (eds.) Maritime Technology and Engineering: 285 292. London, UK: Taylor & Francis Group. Chen, B.Q., Hashemzadeh, M., Guedes Soares, C. 2014a. Numerical and experimental studies on temperature and distortion patterns in butt-welded plates. International Journal of Advanced Manufacturing Technology 72: 1121 1131. Chen, B.Q., Hashemzadeh, M., Garbatov, Y., Guedes Soares, C. 2014b. Numerical and parametric modeling and analysis of weld-induced residual stresses. International Journal of Mechanics and Materials in Design. Epub ahead of print 27 June 2014, doi: 10.1007/s10999-014-9269-7. Deng, D., Liang, W., Murakawa, H. 2007. Determination of welding deformation in fillet-welded joint by means of numerical simulation and comparison with experimental results. J. Mater. Process. Technol. 183: 219 25. Faulkner, D. 1975. A review of effective plating for use in the analysis of stiffened plating in bending and compression, J. Ship Research, 19: 1 17. 520

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