Mechanical Engineering Letters

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1 Bulletin of the JSME Mechanical Engineering Letters Vol., 016 Estimation of welding deformations and residual stresses based on the eigen-strain methodology Masaru OGAWA*, Tasuku YAMASAKI** and Haruo NAKAMURA*** *Faculty of Engineering, Yokohama National University 79-1 Tokiwadai, Hodogaya-ku, Yokohama-shi, Kanagawa , Japan **Research & Development Division, icad LIMITED Shibadaimon, Minato-ku, Tokyo , Japan ***Department of Mechanical Engineering, Tokyo Institute of Technology -1-1 Ookayama, Meguro-ku, Tokyo , Japan Received 16 April 016 Abstract The concept of assurance of structural integrity has been taken into accounted especially for energy-related structures where flaw evaluations are conducted based on the fracture mechanics and non-destructive inspections. In evaluations near welded locations, three-dimensional welding residual stresses must be given quantitatively. However, residual stresses only on surfaces can be measured by the X-ray technique. It is inadequate to apply the thermo-elasto-plastic simulation because it gives qualitative information owing to complexity of actual welding process. Moreover, it is required to predict welding deformations in the design process. Authors proposed a new method to evaluate both welding deformations and residual stresses based on the eigen-strain methodology, which is applicable to the finite element (FEM) analysis. In the proposed method, three-dimensional eigen-strains are estimated by the inverse analysis using the Truncated Singular Value Decomposition (TSVD) method and the penalty method. Numerical simulations are carried out for a butt-welded plate with larger angular deformation in the thickness direction. As a result, eigen-strains to express both welding deformations and residual stresses with higher accuracy could be estimated successfully. Key words : Inverse problem, Welding deformations, Residual stresses, Eigen-strain, Finite element method 1. Introduction Flaw evaluations based on the fracture mechanics and non-destructive inspections are conducted for large-scale structures including energy-related structures to assure structural integrity (Kobayashi, 1984). When its application is attempted to welded components, welding residual stresses must be evaluated quantitatively. Today, X-ray diffraction is widely used as non-destructive measurement technique on surface. However, evaluations of three-dimensional welding residual stress distributions are requested in actual structures. To make things worse, it is impossible to apply measured data by X-ray technique to finite element method (FEM) which is commonly used in the structural design of a mechanical product. Although welding simulation based on the thermo-elastic-plastic FEM analysis is effective to evaluated three-dimensional welding residual stresses (Yaghi, et al., 013), it gives only qualitative information due to complexity of welding process. Furthermore, predictions of welding deformations are requested in the design process. As non-destructive evaluation method of welding residual stresses, the bead flush method has been proposed (Nakamura, et al., 1995). As well, authors have been proposed a concept of evaluating welding deformations and residual stresses using image processing (Nakamura and Kondo, 006). Those methods are applicable to the FEM analysis because their concept is based on the eigen-strain methodology (Mura, 1987). Here, eigen-strains are defined to reproduce physical phenomena (welding displacements, stresses, strains, etc.), in this sense the response matrix [R] shown later in Eq. (1) can be regarded as a mapping function, and eigen-strains { * } in Eq. (1) should be regarded as unknown variables in mapping transformation. By using eigen-strain methodology, we can apply above methods to misfit strain problem (where Young s modulus of base and weld metals are different), fatigue or creep damage problem, Paper No

2 Ogawa, Yamasaki and Nakamura, Mechanical Engineering Letters, Vol. (016) as well as plastic, thermal, transformation and/or twin problems. It is noted they are not always equal to physical inherent strains as a sum of plastic, transformation, and thermal strains. A sophisticated destructive procedure to estimate residual stresses has been developed by Prof. Ueda s group (Ueda, et al., 1977) (Ueda, et al., 1979), however the method reproducing both welding deformations and residual stresses nondestructively has not been developed. In this study, a new method is proposed to evaluate welding residual stresses and welding deformations simultaneously from equivalent eigen-strains which are estimated from displacements and/or total strains measured by Digital Image Correlation (DIC) on plate surface. Here, equivalent eigen-strains are introduced to reproduce both residual stresses and deformations (Masuda and Nakamura, 010). Equivalent eigen-strain as a virtual parameter is hereinafter referred to as eigen-strain. In order to estimate eigen-strain distributions accurately, the Truncated Singular Value Decomposition (TSVD) method and the penalty method are applied in the inverse analysis (Kubo, 199). Estimation accuracy of the proposed method is evaluated via numerical simulations for a butt-welded plate.. Formulation of the proposed method.1 The eigen-strain method The surface 3-D deformations except for near welding bead and the surface (-D) total strains (both normal and shear total strains calculated from surface displacement s gradient based on the fundamental strain definition), {u } T can be calculated from 3-D eigen-strain vector { * } in whole location including weld metal and heat affected zone (see sec...1 regarding their full components), as follows: u * [ R]{ ε } ε (1) where [R] is a response matrix which can be determined using Young s modulus, Poisson s ratio and dimensions of a structure. In Eq. (1), {u } T consists of only non-weldment surface data, as displacements on weldment (where no metal existed in the groove before weld) cannot be defined. As { * } contains the data along weldment, Eq. (1) must be solved under the ill-condition. Also, preliminary simulation showed the total ( elastic) strain vector { } T is necessary and has an important role especially to evaluate residual stresses accurately. So, { } T is added in Eq.(1). Eigen-strains can be estimated from total strains and deformations, as follows: u { ε * } [ R] () ε where [R] + is the Moore and Penrose generalized inverse matrix (Kubo, 199) of [R].. Stabilization of the solution in this inverse analysis In this inverse analysis, three-dimensional eigen-strain distributions must be estimated from the surface information with measurement errors produced by DIC. To make things worse, it is not possible to measure them near the welded bead by DIC. Here, number of unknown parameters is reduced and stabilization methods are applied to stabilize the solutions...1 Reduction of unknown parameters In the previous study, only normal components of eigen-strains, { * x}, { * y} and { * z} were used to evaluate residual stresses for crack propagation prediction because unknown parameters must be reduced to stabilize solutions (Ogawa and Nakamura, 011). However, it is necessary to add shear components of eigen-strains to express in-plane shear deformations due to butt welding of plates (Masuda and Nakamura, 010). In this study, { * x}, { * y} and { * xy} are estimated to evaluate deformations and residual stresses. Note that eigen-strains of the y component { * y} are distributed in the z direction to express larger deformations in the thickness direction... Stabilization methods in this inverse analysis As stated above, Eq. (1) must be solved under the ill-condition. The TSVD method is known as a stabilization method to overcome the ill-posedness in the inverse analysis. However, it is difficult to determine the value of optimum

3 Ogawa, Yamasaki and Nakamura, Mechanical Engineering Letters, Vol. (016) rank to estimate both deformations and residual stresses accurately. In this study, eigen-strains are evaluated according to the following procedures. First, welding deformations and residual stresses are estimated separately with different values of optimized rank. Here, two optimized ranks corresponding to deformations and residual stresses are denoted as p and q, respectively. Second, estimated deformations with rank p and estimated elastic strains (or residual stresses) with rank q, for all nodes are substituted into {u }Tcalculated in Eq. (3) as prior knowledge. u u Π [ R]{ε } [ Rfull ]{ε* } ε measured ε calculated * (3) where, and are an evaluation function and a penalty factor, respectively. As well, [Rfull] is a response matrix between the eigen-strain vector and the vector constituted by deformations and elastic strains for all nodes. Third, eigen-strain distributions to minimize the evaluation function are estimated by the inverse analysis. Note that measured total strains in the first term in Eq. (3) are calculated from displacements observed by DIC. 3. Numerical simulation 3.1 Procedure to evaluate estimation accuracy Numerical simulations are carried out to prove the effectiveness of this method. First, equivalent eigen-strain distributions are determined by the inverse analysis for a welding simulation result based on thermal elastic-plastic FEM analysis. After that, exact welding deformations, exact strains, and exact residual stresses are calculated from exact eigen-strains. Second, measured values are made by adding random number as measurement errors into exact deformations and total strains on surface. Third, eigen-strains are estimated by the inverse analysis from the measured data. Finally, estimated and exact deformations and residual stresses are compared to evaluate estimation accuracy. Quick Welder (Research Center of Computational Mechanics, Inc.) was used in this welding simulation. As well, a commercial software, AMPS (Advanced Technologies Co., Ltd.) was used in FEM analyses. 3. FEM model An FEM model in this simulation is shown in Fig. 1. This is a half model of a butt-welded plate made of steel. The total number of nodes and elements are 1785 and 180, respectively. Four weld passes were conducted on a V-groove to make angular deformation larger in the welding simulation. Measured information was obtained from the top and bottom surfaces (z=10 mm and z=0 mm) excepting weld metal (y= mm at z=10 mm and y= mm at z=0 mm), where welding deformations and total strains could be obtained by DIC. Therefore, a total number of measured data becomes Note that a number of unknown parameter is Observation errors for displacements and total strains follow the normal distribution with a mean of 0 and a standard deviation of 3.3 m and 30, respectively. When there exist no observation errors, welding deformations and residual stresses could be estimated accurately. In this sense, the actual DIC method will become an excellent tool to evaluate welding displacement and residual stresses. Fig. 1 An FEM model. 3

4 Ogawa, Yamasaki and Nakamura, Mechanical Engineering Letters, Vol. (016) 3.3 Estimated results Estimated deformations in the y direction uy and estimated residual stresses in the x direction x on the top surface (z=10 mm) are shown in Figs. -4. Note that exact and estimated distributions are drawn in red and green, respectively, in those figures. The Exact values assumed in this analysis were determined by referring to multiple welding simulations. As shown in Fig., welding deformations could be expressed accurately without the TSVD method in this inverse analysis, but estimation accuracy of residual stresses was relatively poor. On the other hand, residual stresses were estimated accurately when a value of an optimized rank was set at 830 in the TSVD method as shown in Fig. 3. Unfortunately, it was difficult to estimate deformations accurately in this case (Fig. 3). In this method, however, accuracy of both welding deformations and residual stresses can be significantly improved by using Eq. (3) with a penalty factor of 10-3 (Fig. 4). Fig. uy and x on the top surface (z=10 mm) estimated without the TSVD method. Fig. 3 uy and x on the top surface (z=10 mm) estimated with an optimized rank of 830 in the TSVD method. Fig. 4 uy and x on the top surface (z=10 mm) estimated using Eq. (3) with a penalty factor of Furthermore, the Root Mean Square (RMS) values between exact and estimated results on the top surface were 4

5 Ogawa, Yamasaki and Nakamura, Mechanical Engineering Letters, Vol. (016) calculated to compare the estimation accuracy of this method. The RMS values of Figs. -4 calculated by using the following equation are listed in Table 1. m 1 RMS( s) ( sest, i sexact, i ) (4) m i 1 where, s est,i and s exact,i denote i-th estimated and exact deformations or residual stresses, respectively. The RMS values in Table 1 represent the effectiveness of the proposed method introducing the TSVD method as well as the penalty method. Although the values of optimized rank and penalty factor were determined by reference to exact deformations and residual stresses, the potential of this method could be successfully demonstrated. Table 1 RMS values of welding displacements and residual stresses. u y [mm] x [MPa] Without the TSVD method Only the TSVD method Both the TSVD method and the penalty method Conclusions An inverse analytical method to evaluate equivalent eigen-strains, which welding deformations and welding residual stresses can be expressed by, has been proposed and formulated. Then, numerical simulations were carried out to prove the effectiveness of the proposed method. Estimation accuracy of this method was improved for a butt-welded plate even with larger angular deformations by introducing the TSVD and penalty methods. References Kobayashi, H., Practical fracture mechanics (Assessment of integrity of structures), Transactions of the Japan Society of Mechanical Engineers, Vol.87, No.786 (1984), pp (in Japanese). Kubo, S., Inverse Problems (199), Baifukan (in Japanese). Masuda, K. and Nakamura, H., Improvement of the inverse analysis approaches for assessment of welding deformations and residual stresses by using thermo elasto-plastic welding simulation (nd report, Deformation analysis), Transactions of the Japan Society of Mechanical Engineers, Series A, Vol. 76, No. 769 (010), pp (in Japanese). Mura, T., Micromechanics of defects in solids (1987), pp. 1-15, Martinus Nijhoff Publishers. Nakamura, H., Naka, Y., Park, W. and Kobayashi, H., Computer aided non-destructive evaluation method of welding residual stresses by removing reinforcement of weld, Current Topics in Computational Mechanics, ASME Pressure Vessel and Piping, Vol.305 (1995), pp Ogawa, M. and Nakamura, H., proposal of an estimation method of welding residual stresses in welded pipes for risk-analysis-based assurance of structural integrity (nd report: Application of the L-curve and the artificial noise methods), Transactions of the Japan Society of Mechanical Engineers, Series A, Vol.77, No.774 (011), pp.8 9 (in Japanese). Ueda, Y., Fukuda, K., Nakacho, K. and Endo, S., Fundamental concept in measurement of residual stresses based on finite element method and reliability of estimated values, Theoretical and applied mechanics, University of Tokyo Press, No. 5 (1977), pp Ueda, Y., Fukuda, K. and Tanigawa, M., New measuring method of 3-dimensional residual stresses based on theory of inherent strain, The Society of Naval Architects of Japan, No. 145 (1979), pp (in Japanese). Yaghi, A. H., Hyde, T. H., Becker, A. A. and Sun, W., Finite element simulation of residual stresses induced by the dissimilar welding of a P9 steel pipe with weld metal IN65, International Journal of Pressure Vessel and Piping, Vols (013), pp