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Performance of shell elements in modeling spot-welded joints
Finite Elements in Analysis and Design 35 (2000) 41}57
Performance of shell elements in modeling spot-welded joints W. Chen, X. Deng* Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA
Abstract The numerical performance of general shell elements in modeling spot-welded joints is investigated. Finite element meshes composed of purely three-dimensional (3D) elements and purely general shell elements are used to analyze the stress and deformation "elds in a symmetric coach-peel spot-welded specimen and their solutions are compared in detail. It is found that a properly re"ned "nite element mesh of general shell elements can produce stress and deformation solutions comparable to those generated by a similarly re"ned mesh of 3D solid elements. The "ndings of this paper have direct implications to the analysis of sheet metal structures with spot-welded joints (such as automotive structures). ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Finite element analysis; Shell elements; Spot-welded joints
1. Introduction Many types of sheet metal structures are built up using resistance spot welding (RSW) technologies. Automotive vehicles and many home appliances are examples of such structures. Although RSW is a rapid industrial process capable of mass production, many structures fabricated using RSW are not easy to analyze. For example, the body of a car usually contains several thousands of spot welds. To accurately account for the in#uence of each and every spot weld on the load transfer and structural rigidity of the body, the spot welds and the geometrical connectivity they provide must be modeled individually in structural analyses, which by itself is a time-consuming task. A common strategy in "nite element computation has been to model each sheet of a spot-welded joint using general shell elements and each spot weld using a single bar element, which connects the two sheets of a joint at two nodal points [1}3]. While this is a common and useful practice, there
* Corresponding author. Tel.: 001-803-777-7144; fax: 001-803-777-0106. E-mail address: [email protected] (X. Deng) 0168-874X/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 5 5 - 4
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are a number of associated issues that remain open. First, are shell elements appropriate for modeling possibly complex three-dimensional stress variations in a spot weld, especially near the weld nugget boundary where a crack-like geometric discontinuity exists? Second, if the answer to the "rst question is yes, then under what conditions can "nite element solutions obtained using shell elements be considered comparable to 3D solutions, assuming the weld nugget is modeled by connected shell elements instead of a single bar element? Third, if a single-bar element is used to model a weld nugget, what are the numerical strategies to ensure that the stress and deformation states in the neighborhood of the bar adequately represent those obtained using three-dimensional elements? The three questions above are in fact three staggered aspects of the same problem facing structural engineers who design, analysis, and evaluate spot-welded structures, namely the problem of how to perform the above tasks e!ectively, e$ciently and reliably using "nite element tools. The "rst question arises because of the dual-interpretation nature of spot-welded joints. From the point of view of fracture mechanics (e.g. [4}7]) the geometric discontinuity between two joined sheets just outside the weld nugget resembles that of a crack. Hence a stress singularity/concentration is naturally expected near the nugget boundary. As such, shell elements seem to be inappropriate in that region, and it has been suggested (e.g. [8]) that 3D elements be used within the weld nugget while plate elements (a special case of general shell elements) be used in the surrounding structure. On the other hand, since by de"nition sheet metals joined by spot welds have small thickness, the exposed surfaces of the joined sheets are traction-free and may not provide su$cient constraints in the thickness direction to maintain true crack-like stress and deformation states near the crack-like discontinuity. Consequently, spot-welded joints may behave more like a built-up shell/plate structure than a cracked three-dimensional solid. Therefore, it seems reasonable to model spotwelded joints with shell elements. Furthermore, it makes economical sense to use shell elements since it is computationally much more expensive (if not impossible) to use 3D solid elements for each and every spot weld. The second question is a natural extension of the "rst. In fact, these two questions are better answered simultaneously. As discussed in the preceding paragraph, shell elements appear to be reasonable and attractive for modeling spot-welded joints. It remains to be shown, however, that when both the nugget and the two sheets of a spot-welded joint are divided into shell elements, the resulting converged "nite element solutions of the stress and deformation "elds are comparable to those of converged solutions using 3D solid elements. The third question is a more practical question and has the most direct implications among the three in routine applications of the "nite element method to the design, analysis and evaluation spot-welded structures. If the "rst two questions are answered and they are positive, then the third question seeks the bottom line. It is certainly encouraging if shell elements are appropriate for modeling spot-welded joints and that solutions using shell elements can be comparable to those using 3D solid elements. However, from an economical and practical standpoint, it is absolutely essential in daily structural analysis that weld nuggets be represented with as few degrees of freedom as possible. A number of approaches have been discussed in the literature (e.g. [9]). The simplest approach, of course, is to model a nugget with a single-bar element, with each of its two ends being connected to a shell-element node on each of the two joined sheets. While it is expected that the stress and deformation "elds represented by the shell elements immediately surrounding the bar element will in general not be the true "elds, it is not clear whether the "elds at a certain
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distance away from the bar element can be made to approximate the true "elds. This is an important question * if such a good approximation can be made to exist, then it can be used as a boundary condition for characterizing the stress and deformation states within and around the spot weld. To our best knowledge, the questions raised above have not been properly addressed in the literature, although simpli"ed "nite element approaches, such as those using shell and bar elements, are being used daily in the industry. In light of the above discussions, the purpose of this paper is to provide some understanding for the "rst two questions discussed above (an investigation of the third question will follow the present study and will be reported separately). Detailed "nite element analyses will be carried out for a benchmark specimen with a single spot-welded joint. Finite element meshes consisting of all general shell elements and all 3D solid elements will be used to obtain converged solutions for the stress and deformation "elds in the specimen. These converged solutions will be compared carefully and will be used to answer the question of whether and how general shell elements can be applied to model adequately the stress and deformation "elds for spot-welded joints. The results reported below are obtained with the general-purpose "nite element code ABAQUS and under isotropic, homogeneous, and linearly elastic conditions. Linear elasticity is employed here because it is commonly used in large-scale automotive structural analyses, especially under fatigue loading conditions.
2. Benchmark problem description A symmetric coach-peel specimen (see Fig. 1) is used as the benchmark problem in this study. This specimen has two U-shaped sheets joined together by a single spot weld. Each of the left and right symmetric halves of the specimen resembles the conventional coach-peel specimen, but without the disadvantage of possible mechanical contact along the faying surface at the free end of the joint when the specimen is loaded. The stress state in the weld nugget of a symmetric coach-peel specimen is also expected to be similar to that in a cross-tension specimen, but the former can have an additional symmetry plane along the faying surface when the upper and lower plates are made to have identical dimensions. There are several advantages to using the symmetric coach-peel specimen as a benchmark problem. First, when the upper and lower plates are of the same geometry, this specimen has three mutually perpendicular symmetry planes, thus allowing us to simplify the problem domain and reduce the problem size. Second, because of the symmetry constraint along the faying surface (which coincides with the mid-plane of the weld nugget), the deformation of the nugget is minimized, making the rigid-nugget assumption a good approximation. From computational point of view, the use of a rigid nugget can reduce the size of the "nite element computation and provide a convenient connectivity between the joined plates. A rigid nugget also makes sense because it can properly model the enhanced rigidity of the joint when compared to the base metal plates. In general, the rigid-nugget assumption facilitates the use of all general shell elements in "nite element analyses of spot-welded joints. This is because, with this technique, the weld nugget does not need to be modeled explicitly * the rigid nugget and the symmetry requirement at the mid-plane provide a "xed-displacement boundary condition along the circular nugget boundary. Without this rigid-nugget assumption, the nugget must be modeled explicitly in the general shell
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analysis. That is, it must be divided into shell elements and elastic bar elements must be used to connect the element nodes in the nugget to the symmetry plane (the mid-plane) of the nugget, where the displacement in the thickness direction is zero. The di$culty of doing so is that the sti!ness of the bar elements is not easily known. To make sure that this simplifying assumption is adequate, its e!ect on the stress and deformation "elds around the spot-welded joint will be investigated in this study using 3D solid elements. Finally, because the symmetric coach-peel specimen bears so much similarity to the conventional coach-peel specimen and the cross-tension specimen, conclusions drawn from the former are expected to have a wider range of applicability. Fig. 2 outlines the in-plane geometry of the problem domain of a symmetric coach-peel specimen whose two joined plates have the same thickness. Because of the special symmetries possessed by the specimen, only one quarter of the upper plate in Fig. 1a is needed in the subsequent analyses. Thus only the top surface of the upper plate is shown in Fig. 2. The bottom surface of the upper plate is hidden and coincides with the faying surface between the two joined plates and with the mid-plane of the nugget. The circular boundary of the nugget has been designated as B4, and other boundaries as B1, B2, B3 and B5. Note that the vertical section of the upper plate along B1 has been omitted and replaced by distributed loads pointing out of the plane. This measure is introduced to further simplify the problem geometry and reduce the problem size, but it is not expected to modify the near-nugget "elds very much. The specimen dimensions, and the rectangular and polar coordinates are as shown in Fig. 2. The hidden z-axis is pointing outwards in the plate thickness direction, and the origin of the coordinate system is located at the center of the weld nugget (thus the mid-plane of the nugget is given by z"0.0 mm). The boundary conditions for the domain in Fig. 2 are as follows. By symmetry, the x-direction displacement component u must be zero along B3 and the y component v must be zero along B5. On the bottom surface of the nugget region, viz. z"0.0 mm and x2#y2)(4.0 mm)2, the z displacement component w must be zero. Along boundary B1, a uniformly distributed load is applied in the positive z direction. Since this is an elastic analysis, the actual magnitude of the total distributed load is not signi"cant. As such, we take the total load to be 4 N, so that the distribution along B1 is p "4/(12.5]1.0) N/mm2"0.32 MPa. 31
Fig. 1. Schematic of a symmetric coach-peel specimen: (a) side view, (b) top view.
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Fig. 2. In-plane geometry of the upper-right quarter of the top view shown in Fig. 1b for a symmetric coach-peel specimen.
When general shell elements are used to represent the domain, the nugget can be assumed to be rigid, as elaborated earlier. Then due to symmetry requirements at the bottom surface and along boundaries B3 and B5, the nugget must maintain zero deformation, resulting in a boundary condition of zero displacements and zero angles of rotation along the circular boundary B4. Consequently, the e!ect of the nugget can be replaced equivalent by the boundary conditions along B4 and the nugget region does not need to be discritized explicitly. On the other hand, when 3D solid elements are used to represent the domain, we consider a range of nugget behavior: (a) rigid nugget, (b) semi-rigid nugget, and (c) base-metal nugget. This range of study allows us to understand the e!ect of the rigid nugget assumption used in the general shell analysis. Speci"cally, the semi-rigid nugget is taken to have a Young's modulus of 173 GPa and a Poisson's ratio of 0.23. The base-metal nugget has the same elastic constants as the base metal, with a Young's modulus of 73 GPa and a Poisson's ratio of 0.3.
3. Finite element mesh design and convergence Fig. 3a shows a "nite element mesh composed of all general shell elements and Fig. 3b a mesh of all 3D solid elements. Note that the shell mesh does not need to contain the nugget region because of the rigid-nugget assumption. However, the 3D mesh does include the nugget region and will be used to study the e!ect of the rigid-nugget assumption. Except for the nugget region, the two meshes share the same in-plane mesh design, with smaller elements focused near the nugget
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Fig. 3. Finite element meshes for the upper plate of a simpli"ed symmetric coach-peel specimen: (a) a mesh of general shell elements, where the nugget is taken to be rigid and is not modeled explicitly; (b) a mesh of 3D solid elements, where the nugget is rigid, semi-rigid, or the same as the base metal.
boundary where stresses are expected to be higher than further away from the nugget. Within the nugget (Fig. 3b only) and immediately outside the nugget the domain is divided into annular rings of elements of similar sizes. For example, the two rings bordering the nugget boundary are divided into 20 elements. For the 3D mesh, the plate is discretized in the thickness direction by four equal layers of elements. In total, the shell mesh consists of 280 four-node general shell elements and 315 nodes, and the 3D mesh is composed of 1556 elements (mostly eight-node elements and some six-node elements) and 2075 nodes. To draw meaningful conclusions from the "nite element solutions, it must be demonstrated that the meshes used for obtaining the solutions are "ne enough to produce converged solutions. To this end, we have performed the following convergence calibrations for the meshes shown in Fig. 3. For
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Table 1 A comparison of de#ection solutions obtained using the original and re"ned shell meshes at various points (y coordinate values) along boundary B1 (unit: mm) y
0.00
2.50
5.00
7.50
10.0
12.5
Original mesh Re"ned mesh
1.5506E-02 1.5494E-02
1.5906E-02 1.5893E-02
1.6963E-02 1.6948E-02
1.8338E-02 1.8320E-02
1.9697E-02 1.9675E-02
2.0891E-02 2.0861E-02
the shell mesh in Fig. 3a, each element is subdivided into four to generate a re"ned mesh. Finite element solutions are then obtained for both the original and the re"ned meshes. Table 1 presents a comparison between the two solutions along the boundary B1 (see Fig. 2), where the shell de#ection in the z direction at various y coordinate values along B1 is shown. It is clear from the comparison that the maximum relative error is less than 1.0%, indicating that the original mesh indeed can produce converged solutions. To see if the 3D mesh in Fig. 3b can also lead to convergent "nite element solutions, a re"ned 3D mesh is created from the original mesh by reducing the size of each element by half in all dimensions. Thus an eight-node solid element is divided into eight elements of the same type, and a six-node element is divided into two eight-node elements plus four six-node elements. Consequently, the re"ned 3D mesh contains a total of 12 312 solid elements and 14 301 nodes. Finite element solutions have been obtained using the two meshes and stresses compared both near and away from the spot weld nugget. For example, Fig. 4a shows the through-thickness variation of the normal stress component p (S11 in the "gure) along the line x"4.14 mm and y"0.16 mm. This 11 line is just outside the nugget boundary and is near the boundary B5 (see Fig. 2). Similarly, Fig. 4b shows the through-thickness variation of S11 along the line x"4.14 mm and y"0.16 mm, which is further away from the nugget boundary and is near the intersection of boundaries B1 and B5 (see Fig. 2). It is clear from Fig. 4 that the stress variations according to the "nite element solutions of the two meshes are very close to each other, which demonstrate that these solutions are indeed converged.
4. Results and discussions In this section, the numerical performance of shell elements in modeling spot-welded joints will be evaluated through detailed comparisons of converged shell solutions with converged 3D solutions. Four types of comparisons will be performed: (a) overall stress distributions, (b) stress variations along radial and circular lines, (c) through-thickness stress variations, and (d) displacement variations along radial lines. Since the nugget is taken to be rigid in the shell analysis for reasons explained earlier, stress and displacement variations within the nugget are omitted from the "gures. The overall stress distributions over the bottom surface (given by z"0) of the upper plate are shown through the contour plots in Figs. 5}8. This particular surface is of interest because it is the faying surface of the two joined plates and is expected to have the highest stress levels in
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Fig. 4. A comparison of the through-thickness variation (in the z direction) of the normal stress component S11 based on two "nite element meshes: (a) along x"4.14 mm and y"0.16 mm; (b) along x"11.95 mm and y"0.47 mm.
the thickness direction. Since the surface is viewed from below, its geometry is di!erent (notice the location of the circular boundary of the nugget) from the top view of the specimen (see Fig. 2) * it is obtained by #ipping the view in Fig. 2 about the y-axis. The contours shown are for the von Mises e!ective stress (Fig. 5), the normal stresses p (denoted by S11 in Fig. 6) and p (denoted by S22 in 11 22 Fig. 7), and the shear stress p (denoted by S12 in Fig. 8). The following observations can be made 12 through comparisons of the contours from shell elements (Figs. 5a, 6a and 7a) with those from 3D elements (Figs. 5b, 6b and 7b). First, the contour shape given by shell elements is almost the same as that given by 3D elements. Second, the stress concentration sites suggested by the two types of solutions are the same. In particular, the Mises stress and S11 and S22 have concentration sites on the nugget boundary at about x"4.0 mm, y"0.0 mm and z"0.0 mm. On the other hand, S12 has its concentration site near the mid-point of the nugget boundary at an angular position of about h"303 to 403. Third, the overall stress level predicted by the shell solution is somewhat higher than that predicted by the 3D solution. Although the contour plots o!er an overall view of the stress distributions, many details are not easy to observe from these plots. Hence the stress variations along selected lines are presented in Fig. 9. Guided by the contour plots, a radial line denoted by h"2.253, which goes through the stress concentration zone, and a circular line given by R"4.14 mm, which lies next to the nugget
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Fig. 5. Stress contours for the von Mises e!ective stress on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 6. Stress contours for the normal stress component S11 on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 7. Stress contours for the normal stress component S22 on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 8. Stress contours for the shear stress component S12 on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 9. Comparisons of shell and 3D solutions for stress variations on the bottom surface of the upper plate: (a) along radial line h"2.253, (b) along the circular line R"4.14 mm.
boundary (at R"4.0 mm), are chosen. Both lines are on the bottom surface (i.e. z"0.0 mm) of the upper plate and are in the high stress regions. The stress variations along the radial line (the radial variations) are shown in Fig. 9a while those along the circular line (the angular variations) are shown in Fig. 9b. It is noted that the converged "nite element solutions from both the shell and the 3D meshes have employed the rigid-nugget simpli"cation. It is seen from these "gures that the shell solution consistently over-predicts the 3D solution. For the dominant stress component S11 (and hence the von Mises e!ective stress, see Fig. 11), the relative error at the maximum stress level is about near 9%. The angular stress variations (Fig. 9b) also con"rm the earlier observation that S11, S22 and S12 have their concentration zones, respectively, near h"03, between h"03 and h"403, and between h"303 and h"403. To shed light on the e!ect of the rigid-nugget assumption on predicted stress solutions, 3D "nite element solutions based on meshes with rigid, semi-rigid and base-metal nugget properties (recall de"nitions given earlier at the end of the problem description section) have been obtained. These solutions are compared in Figs. 10a and b, where Fig. 10a is for stress variations along the same radial line as in Fig. 9a, and Fig. 10b along the same circular line as in Fig. 9b. The comparison shows that the shape of the stress variation curves is very similar in all cases even though the magnitude of the stress quantities is somewhat a!ected by the assumed properties of the nugget. Among the three assumptions, the rigid-nugget assumption leads to the highest stress level, most likely due to an enhanced stress concentration at the nugget boundary. Because the same rigid-nugget assumption has been used for the shell analysis, the shell solution is expected to give
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Fig. 10. E!ect of the nugget assumptions on stress variations on the bottom surface: (a) along radial line h"2.253, (b) along the circular line R"4.14 mm.
Fig. 11. Comparisons of various solutions for the Mises stress variation on the bottom surface of the upper plate: (a) along radial line h"2.253, (b) along the circular line R"4.14 mm.
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the best comparison with the 3D solution with a rigid nugget, as shown in Fig. 9. For the von Mises e!ective stress, comparisons similar to those given in Figs. 9 and 10 are summed up in Fig. 11. The preceding comparisons are focused on the performance of shell solutions for computing stresses on the bottom surface of the upper plate. Although the bottom and top surfaces are in general of most interest to us (the maximum stress is expected to occur on these surfaces), it is also interesting to see how well the shell elements can represent stress variations in the plate thickness direction, especially in the stress concentration zones. To this end, Fig. 12 provides a comparison of the through-thickness S11 (the dominant stress component) variations predicted by shell and 3D "nite elements. The comparisons are performed at two points: one at x"4.14 mm and y"0.16 mm, which is in the concentration zone for S11, and the other at x"6.49 mm and y"0.25 mm, which is just outside the concentration zone. When shell elements are used, S11 (which is the normal stress due to bending) is predicted to vary linearly in the thickness direction. When compared with the 3D solution, this linear behavior is seen to represent well the actual 3D through-thickness variation. As expected, the accuracy of the shell solution is better away from the nugget boundary (see Fig. 12b) than near the nugget boundary (see Fig. 12a), where the shell stress over-shoots the 3D stress, con"rming previous observations. The reason is that away from the
Fig. 12. Comparisons of shell and 3D solutions for through-thickness stress variations: (a) along x"4.14 mm and y"0.16 mm, (b) along x"6.49 mm and y"0.25 mm.
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Fig. 13. Comparisons of shell and 3D solutions for the variation of the #exural displacement w on the bottom surface of the upper plate: (a) along h"03, (b) along h"453.
nugget the stress variation is controlled by the bending e!ect of the plate whereas near the nugget boundary the through-thickness behavior is in#uenced (although not substantially) by the presence of the nugget and the crack-like discontinuity between the joined plates. Finally, shell and 3D solutions for the #exural displacement w of the upper plate are compared in Fig. 13, where 3D solutions with both rigid nugget and base-metal nugget are presented. In particular, the displacement variation along the radial line h"03and z"0.0 mm is given in Fig. 13a while that along the radial line h"453and z"0.0 mm is given in Fig. 13b. It is interesting to observe that the shell solution is bounded by the 3D rigid-nugget solution from below and the 3D base-metal-nugget solution from above, such that the shell solution gives a very good approximation to the averages of the two 3D solutions.
5. Summary and conclusions The numerical performance of shell elements in "nite element modeling of spot-welded joints has been studied through detailed comparisons with 3D solid elements. The benchmark problem used in this study is a `symmetrica coach-peel spot-weld specimen, which facilitates the utilization of several simplifying symmetries and assumptions. The main conclusions of this study are as follows: First, converged shell element solutions provide an overall good approximation to converged 3D solid element solutions for spot-welded joints, even near the nugget boundary and in the stress concentration region.
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Second, shell element solutions generally predict a higher (about 9%) maximum stress than 3D element solutions with rigid nuggets. Third, the rigid-nugget assumption has minimal e!ects on the shape of stress distribution curves but it will lead to a higher stress level than a base-metal nugget assumption. Fourth, shell element solution for the #exural displacement is bounded by 3D element solutions with rigid and base-metal nuggets. The current study serves to illustrate the performance of shell elements in modeling spot-welded joints under converged conditions. In industrial practice, however, it is usually not possible to use re"ned meshes such as those used in this study. As such, there is a strong need to develop simpli"ed "nite element modeling techniques for spot-welded joints that will produce reasonable results with practical mesh designs. This is a challenge for further research in this area.
Acknowledgements This study was supported by an NSF/EPSCoR grant through Cooperative Agreement No. EPS-9630167.
References [1] News Trends, Optimizer locates welds to sti!en structures, Machines Design 66 (5) (1994) 18}19. [2] M. Chirehdast, T. Jiang, Optimal design of spot-weld and adhesive bond pattern, SAE Techn. Paper Ser. 960812 1996. [3] Y. Rui, R-J. Yang, C-J. Chen, Hari Agrawal, Fatigue optimization of spot welds, Body Des. Eng. (IBEC '96) (1992). 68}72. [4] L.P. Pook, Fracture mechanics analysis of the fatigue behavior of spot welds, Int. J. Fract. 11 (1975) 173}176. [5] S.I. Rokhlin, L. Adler, Ultrasonic method for shear strength prediction of spot welds, J. Appl. Phys. 56 (3) (1984) 726}731. [6] D. Radaj, Stress singularity, notch stress and structural stress at spot-welded joints, Eng. Fract. Mech. 34 (2) (1989) 495}506. [7] P.-C. Wang, K.W. Ewing, Fracture mechanics analysis of fatigue resistance of spot welded coach-peel joints, Fatigue Fract. Eng. Mater. Struct. 14 (9) (1991) 915}930. [8] D. Radaj, Z. Zheng, W. Mohrmann, Local stress parameters at the weld spot of various specimens, Eng. Fract. Mech. 37 (5) (1990) 933}951. [9] S.D. Sheppard, M. Strange, Fatigue life estimation in resistance spot welds: Initiation and early growth phase, Fatigue Fract. Eng. Mater. Struct. 15 (6) (1992) 531}549.
Performance of shell elements in modeling spot-welded joints W. Chen, X. Deng* Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA
Abstract The numerical performance of general shell elements in modeling spot-welded joints is investigated. Finite element meshes composed of purely three-dimensional (3D) elements and purely general shell elements are used to analyze the stress and deformation "elds in a symmetric coach-peel spot-welded specimen and their solutions are compared in detail. It is found that a properly re"ned "nite element mesh of general shell elements can produce stress and deformation solutions comparable to those generated by a similarly re"ned mesh of 3D solid elements. The "ndings of this paper have direct implications to the analysis of sheet metal structures with spot-welded joints (such as automotive structures). ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Finite element analysis; Shell elements; Spot-welded joints
1. Introduction Many types of sheet metal structures are built up using resistance spot welding (RSW) technologies. Automotive vehicles and many home appliances are examples of such structures. Although RSW is a rapid industrial process capable of mass production, many structures fabricated using RSW are not easy to analyze. For example, the body of a car usually contains several thousands of spot welds. To accurately account for the in#uence of each and every spot weld on the load transfer and structural rigidity of the body, the spot welds and the geometrical connectivity they provide must be modeled individually in structural analyses, which by itself is a time-consuming task. A common strategy in "nite element computation has been to model each sheet of a spot-welded joint using general shell elements and each spot weld using a single bar element, which connects the two sheets of a joint at two nodal points [1}3]. While this is a common and useful practice, there
* Corresponding author. Tel.: 001-803-777-7144; fax: 001-803-777-0106. E-mail address: [email protected] (X. Deng) 0168-874X/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 5 5 - 4
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are a number of associated issues that remain open. First, are shell elements appropriate for modeling possibly complex three-dimensional stress variations in a spot weld, especially near the weld nugget boundary where a crack-like geometric discontinuity exists? Second, if the answer to the "rst question is yes, then under what conditions can "nite element solutions obtained using shell elements be considered comparable to 3D solutions, assuming the weld nugget is modeled by connected shell elements instead of a single bar element? Third, if a single-bar element is used to model a weld nugget, what are the numerical strategies to ensure that the stress and deformation states in the neighborhood of the bar adequately represent those obtained using three-dimensional elements? The three questions above are in fact three staggered aspects of the same problem facing structural engineers who design, analysis, and evaluate spot-welded structures, namely the problem of how to perform the above tasks e!ectively, e$ciently and reliably using "nite element tools. The "rst question arises because of the dual-interpretation nature of spot-welded joints. From the point of view of fracture mechanics (e.g. [4}7]) the geometric discontinuity between two joined sheets just outside the weld nugget resembles that of a crack. Hence a stress singularity/concentration is naturally expected near the nugget boundary. As such, shell elements seem to be inappropriate in that region, and it has been suggested (e.g. [8]) that 3D elements be used within the weld nugget while plate elements (a special case of general shell elements) be used in the surrounding structure. On the other hand, since by de"nition sheet metals joined by spot welds have small thickness, the exposed surfaces of the joined sheets are traction-free and may not provide su$cient constraints in the thickness direction to maintain true crack-like stress and deformation states near the crack-like discontinuity. Consequently, spot-welded joints may behave more like a built-up shell/plate structure than a cracked three-dimensional solid. Therefore, it seems reasonable to model spotwelded joints with shell elements. Furthermore, it makes economical sense to use shell elements since it is computationally much more expensive (if not impossible) to use 3D solid elements for each and every spot weld. The second question is a natural extension of the "rst. In fact, these two questions are better answered simultaneously. As discussed in the preceding paragraph, shell elements appear to be reasonable and attractive for modeling spot-welded joints. It remains to be shown, however, that when both the nugget and the two sheets of a spot-welded joint are divided into shell elements, the resulting converged "nite element solutions of the stress and deformation "elds are comparable to those of converged solutions using 3D solid elements. The third question is a more practical question and has the most direct implications among the three in routine applications of the "nite element method to the design, analysis and evaluation spot-welded structures. If the "rst two questions are answered and they are positive, then the third question seeks the bottom line. It is certainly encouraging if shell elements are appropriate for modeling spot-welded joints and that solutions using shell elements can be comparable to those using 3D solid elements. However, from an economical and practical standpoint, it is absolutely essential in daily structural analysis that weld nuggets be represented with as few degrees of freedom as possible. A number of approaches have been discussed in the literature (e.g. [9]). The simplest approach, of course, is to model a nugget with a single-bar element, with each of its two ends being connected to a shell-element node on each of the two joined sheets. While it is expected that the stress and deformation "elds represented by the shell elements immediately surrounding the bar element will in general not be the true "elds, it is not clear whether the "elds at a certain
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distance away from the bar element can be made to approximate the true "elds. This is an important question * if such a good approximation can be made to exist, then it can be used as a boundary condition for characterizing the stress and deformation states within and around the spot weld. To our best knowledge, the questions raised above have not been properly addressed in the literature, although simpli"ed "nite element approaches, such as those using shell and bar elements, are being used daily in the industry. In light of the above discussions, the purpose of this paper is to provide some understanding for the "rst two questions discussed above (an investigation of the third question will follow the present study and will be reported separately). Detailed "nite element analyses will be carried out for a benchmark specimen with a single spot-welded joint. Finite element meshes consisting of all general shell elements and all 3D solid elements will be used to obtain converged solutions for the stress and deformation "elds in the specimen. These converged solutions will be compared carefully and will be used to answer the question of whether and how general shell elements can be applied to model adequately the stress and deformation "elds for spot-welded joints. The results reported below are obtained with the general-purpose "nite element code ABAQUS and under isotropic, homogeneous, and linearly elastic conditions. Linear elasticity is employed here because it is commonly used in large-scale automotive structural analyses, especially under fatigue loading conditions.
2. Benchmark problem description A symmetric coach-peel specimen (see Fig. 1) is used as the benchmark problem in this study. This specimen has two U-shaped sheets joined together by a single spot weld. Each of the left and right symmetric halves of the specimen resembles the conventional coach-peel specimen, but without the disadvantage of possible mechanical contact along the faying surface at the free end of the joint when the specimen is loaded. The stress state in the weld nugget of a symmetric coach-peel specimen is also expected to be similar to that in a cross-tension specimen, but the former can have an additional symmetry plane along the faying surface when the upper and lower plates are made to have identical dimensions. There are several advantages to using the symmetric coach-peel specimen as a benchmark problem. First, when the upper and lower plates are of the same geometry, this specimen has three mutually perpendicular symmetry planes, thus allowing us to simplify the problem domain and reduce the problem size. Second, because of the symmetry constraint along the faying surface (which coincides with the mid-plane of the weld nugget), the deformation of the nugget is minimized, making the rigid-nugget assumption a good approximation. From computational point of view, the use of a rigid nugget can reduce the size of the "nite element computation and provide a convenient connectivity between the joined plates. A rigid nugget also makes sense because it can properly model the enhanced rigidity of the joint when compared to the base metal plates. In general, the rigid-nugget assumption facilitates the use of all general shell elements in "nite element analyses of spot-welded joints. This is because, with this technique, the weld nugget does not need to be modeled explicitly * the rigid nugget and the symmetry requirement at the mid-plane provide a "xed-displacement boundary condition along the circular nugget boundary. Without this rigid-nugget assumption, the nugget must be modeled explicitly in the general shell
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analysis. That is, it must be divided into shell elements and elastic bar elements must be used to connect the element nodes in the nugget to the symmetry plane (the mid-plane) of the nugget, where the displacement in the thickness direction is zero. The di$culty of doing so is that the sti!ness of the bar elements is not easily known. To make sure that this simplifying assumption is adequate, its e!ect on the stress and deformation "elds around the spot-welded joint will be investigated in this study using 3D solid elements. Finally, because the symmetric coach-peel specimen bears so much similarity to the conventional coach-peel specimen and the cross-tension specimen, conclusions drawn from the former are expected to have a wider range of applicability. Fig. 2 outlines the in-plane geometry of the problem domain of a symmetric coach-peel specimen whose two joined plates have the same thickness. Because of the special symmetries possessed by the specimen, only one quarter of the upper plate in Fig. 1a is needed in the subsequent analyses. Thus only the top surface of the upper plate is shown in Fig. 2. The bottom surface of the upper plate is hidden and coincides with the faying surface between the two joined plates and with the mid-plane of the nugget. The circular boundary of the nugget has been designated as B4, and other boundaries as B1, B2, B3 and B5. Note that the vertical section of the upper plate along B1 has been omitted and replaced by distributed loads pointing out of the plane. This measure is introduced to further simplify the problem geometry and reduce the problem size, but it is not expected to modify the near-nugget "elds very much. The specimen dimensions, and the rectangular and polar coordinates are as shown in Fig. 2. The hidden z-axis is pointing outwards in the plate thickness direction, and the origin of the coordinate system is located at the center of the weld nugget (thus the mid-plane of the nugget is given by z"0.0 mm). The boundary conditions for the domain in Fig. 2 are as follows. By symmetry, the x-direction displacement component u must be zero along B3 and the y component v must be zero along B5. On the bottom surface of the nugget region, viz. z"0.0 mm and x2#y2)(4.0 mm)2, the z displacement component w must be zero. Along boundary B1, a uniformly distributed load is applied in the positive z direction. Since this is an elastic analysis, the actual magnitude of the total distributed load is not signi"cant. As such, we take the total load to be 4 N, so that the distribution along B1 is p "4/(12.5]1.0) N/mm2"0.32 MPa. 31
Fig. 1. Schematic of a symmetric coach-peel specimen: (a) side view, (b) top view.
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Fig. 2. In-plane geometry of the upper-right quarter of the top view shown in Fig. 1b for a symmetric coach-peel specimen.
When general shell elements are used to represent the domain, the nugget can be assumed to be rigid, as elaborated earlier. Then due to symmetry requirements at the bottom surface and along boundaries B3 and B5, the nugget must maintain zero deformation, resulting in a boundary condition of zero displacements and zero angles of rotation along the circular boundary B4. Consequently, the e!ect of the nugget can be replaced equivalent by the boundary conditions along B4 and the nugget region does not need to be discritized explicitly. On the other hand, when 3D solid elements are used to represent the domain, we consider a range of nugget behavior: (a) rigid nugget, (b) semi-rigid nugget, and (c) base-metal nugget. This range of study allows us to understand the e!ect of the rigid nugget assumption used in the general shell analysis. Speci"cally, the semi-rigid nugget is taken to have a Young's modulus of 173 GPa and a Poisson's ratio of 0.23. The base-metal nugget has the same elastic constants as the base metal, with a Young's modulus of 73 GPa and a Poisson's ratio of 0.3.
3. Finite element mesh design and convergence Fig. 3a shows a "nite element mesh composed of all general shell elements and Fig. 3b a mesh of all 3D solid elements. Note that the shell mesh does not need to contain the nugget region because of the rigid-nugget assumption. However, the 3D mesh does include the nugget region and will be used to study the e!ect of the rigid-nugget assumption. Except for the nugget region, the two meshes share the same in-plane mesh design, with smaller elements focused near the nugget
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Fig. 3. Finite element meshes for the upper plate of a simpli"ed symmetric coach-peel specimen: (a) a mesh of general shell elements, where the nugget is taken to be rigid and is not modeled explicitly; (b) a mesh of 3D solid elements, where the nugget is rigid, semi-rigid, or the same as the base metal.
boundary where stresses are expected to be higher than further away from the nugget. Within the nugget (Fig. 3b only) and immediately outside the nugget the domain is divided into annular rings of elements of similar sizes. For example, the two rings bordering the nugget boundary are divided into 20 elements. For the 3D mesh, the plate is discretized in the thickness direction by four equal layers of elements. In total, the shell mesh consists of 280 four-node general shell elements and 315 nodes, and the 3D mesh is composed of 1556 elements (mostly eight-node elements and some six-node elements) and 2075 nodes. To draw meaningful conclusions from the "nite element solutions, it must be demonstrated that the meshes used for obtaining the solutions are "ne enough to produce converged solutions. To this end, we have performed the following convergence calibrations for the meshes shown in Fig. 3. For
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Table 1 A comparison of de#ection solutions obtained using the original and re"ned shell meshes at various points (y coordinate values) along boundary B1 (unit: mm) y
0.00
2.50
5.00
7.50
10.0
12.5
Original mesh Re"ned mesh
1.5506E-02 1.5494E-02
1.5906E-02 1.5893E-02
1.6963E-02 1.6948E-02
1.8338E-02 1.8320E-02
1.9697E-02 1.9675E-02
2.0891E-02 2.0861E-02
the shell mesh in Fig. 3a, each element is subdivided into four to generate a re"ned mesh. Finite element solutions are then obtained for both the original and the re"ned meshes. Table 1 presents a comparison between the two solutions along the boundary B1 (see Fig. 2), where the shell de#ection in the z direction at various y coordinate values along B1 is shown. It is clear from the comparison that the maximum relative error is less than 1.0%, indicating that the original mesh indeed can produce converged solutions. To see if the 3D mesh in Fig. 3b can also lead to convergent "nite element solutions, a re"ned 3D mesh is created from the original mesh by reducing the size of each element by half in all dimensions. Thus an eight-node solid element is divided into eight elements of the same type, and a six-node element is divided into two eight-node elements plus four six-node elements. Consequently, the re"ned 3D mesh contains a total of 12 312 solid elements and 14 301 nodes. Finite element solutions have been obtained using the two meshes and stresses compared both near and away from the spot weld nugget. For example, Fig. 4a shows the through-thickness variation of the normal stress component p (S11 in the "gure) along the line x"4.14 mm and y"0.16 mm. This 11 line is just outside the nugget boundary and is near the boundary B5 (see Fig. 2). Similarly, Fig. 4b shows the through-thickness variation of S11 along the line x"4.14 mm and y"0.16 mm, which is further away from the nugget boundary and is near the intersection of boundaries B1 and B5 (see Fig. 2). It is clear from Fig. 4 that the stress variations according to the "nite element solutions of the two meshes are very close to each other, which demonstrate that these solutions are indeed converged.
4. Results and discussions In this section, the numerical performance of shell elements in modeling spot-welded joints will be evaluated through detailed comparisons of converged shell solutions with converged 3D solutions. Four types of comparisons will be performed: (a) overall stress distributions, (b) stress variations along radial and circular lines, (c) through-thickness stress variations, and (d) displacement variations along radial lines. Since the nugget is taken to be rigid in the shell analysis for reasons explained earlier, stress and displacement variations within the nugget are omitted from the "gures. The overall stress distributions over the bottom surface (given by z"0) of the upper plate are shown through the contour plots in Figs. 5}8. This particular surface is of interest because it is the faying surface of the two joined plates and is expected to have the highest stress levels in
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Fig. 4. A comparison of the through-thickness variation (in the z direction) of the normal stress component S11 based on two "nite element meshes: (a) along x"4.14 mm and y"0.16 mm; (b) along x"11.95 mm and y"0.47 mm.
the thickness direction. Since the surface is viewed from below, its geometry is di!erent (notice the location of the circular boundary of the nugget) from the top view of the specimen (see Fig. 2) * it is obtained by #ipping the view in Fig. 2 about the y-axis. The contours shown are for the von Mises e!ective stress (Fig. 5), the normal stresses p (denoted by S11 in Fig. 6) and p (denoted by S22 in 11 22 Fig. 7), and the shear stress p (denoted by S12 in Fig. 8). The following observations can be made 12 through comparisons of the contours from shell elements (Figs. 5a, 6a and 7a) with those from 3D elements (Figs. 5b, 6b and 7b). First, the contour shape given by shell elements is almost the same as that given by 3D elements. Second, the stress concentration sites suggested by the two types of solutions are the same. In particular, the Mises stress and S11 and S22 have concentration sites on the nugget boundary at about x"4.0 mm, y"0.0 mm and z"0.0 mm. On the other hand, S12 has its concentration site near the mid-point of the nugget boundary at an angular position of about h"303 to 403. Third, the overall stress level predicted by the shell solution is somewhat higher than that predicted by the 3D solution. Although the contour plots o!er an overall view of the stress distributions, many details are not easy to observe from these plots. Hence the stress variations along selected lines are presented in Fig. 9. Guided by the contour plots, a radial line denoted by h"2.253, which goes through the stress concentration zone, and a circular line given by R"4.14 mm, which lies next to the nugget
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Fig. 5. Stress contours for the von Mises e!ective stress on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 6. Stress contours for the normal stress component S11 on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 7. Stress contours for the normal stress component S22 on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 8. Stress contours for the shear stress component S12 on the bottom surface of the upper plate (viewed from below): (a) from shell element solutions; (b) from 3D element solution.
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Fig. 9. Comparisons of shell and 3D solutions for stress variations on the bottom surface of the upper plate: (a) along radial line h"2.253, (b) along the circular line R"4.14 mm.
boundary (at R"4.0 mm), are chosen. Both lines are on the bottom surface (i.e. z"0.0 mm) of the upper plate and are in the high stress regions. The stress variations along the radial line (the radial variations) are shown in Fig. 9a while those along the circular line (the angular variations) are shown in Fig. 9b. It is noted that the converged "nite element solutions from both the shell and the 3D meshes have employed the rigid-nugget simpli"cation. It is seen from these "gures that the shell solution consistently over-predicts the 3D solution. For the dominant stress component S11 (and hence the von Mises e!ective stress, see Fig. 11), the relative error at the maximum stress level is about near 9%. The angular stress variations (Fig. 9b) also con"rm the earlier observation that S11, S22 and S12 have their concentration zones, respectively, near h"03, between h"03 and h"403, and between h"303 and h"403. To shed light on the e!ect of the rigid-nugget assumption on predicted stress solutions, 3D "nite element solutions based on meshes with rigid, semi-rigid and base-metal nugget properties (recall de"nitions given earlier at the end of the problem description section) have been obtained. These solutions are compared in Figs. 10a and b, where Fig. 10a is for stress variations along the same radial line as in Fig. 9a, and Fig. 10b along the same circular line as in Fig. 9b. The comparison shows that the shape of the stress variation curves is very similar in all cases even though the magnitude of the stress quantities is somewhat a!ected by the assumed properties of the nugget. Among the three assumptions, the rigid-nugget assumption leads to the highest stress level, most likely due to an enhanced stress concentration at the nugget boundary. Because the same rigid-nugget assumption has been used for the shell analysis, the shell solution is expected to give
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Fig. 10. E!ect of the nugget assumptions on stress variations on the bottom surface: (a) along radial line h"2.253, (b) along the circular line R"4.14 mm.
Fig. 11. Comparisons of various solutions for the Mises stress variation on the bottom surface of the upper plate: (a) along radial line h"2.253, (b) along the circular line R"4.14 mm.
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the best comparison with the 3D solution with a rigid nugget, as shown in Fig. 9. For the von Mises e!ective stress, comparisons similar to those given in Figs. 9 and 10 are summed up in Fig. 11. The preceding comparisons are focused on the performance of shell solutions for computing stresses on the bottom surface of the upper plate. Although the bottom and top surfaces are in general of most interest to us (the maximum stress is expected to occur on these surfaces), it is also interesting to see how well the shell elements can represent stress variations in the plate thickness direction, especially in the stress concentration zones. To this end, Fig. 12 provides a comparison of the through-thickness S11 (the dominant stress component) variations predicted by shell and 3D "nite elements. The comparisons are performed at two points: one at x"4.14 mm and y"0.16 mm, which is in the concentration zone for S11, and the other at x"6.49 mm and y"0.25 mm, which is just outside the concentration zone. When shell elements are used, S11 (which is the normal stress due to bending) is predicted to vary linearly in the thickness direction. When compared with the 3D solution, this linear behavior is seen to represent well the actual 3D through-thickness variation. As expected, the accuracy of the shell solution is better away from the nugget boundary (see Fig. 12b) than near the nugget boundary (see Fig. 12a), where the shell stress over-shoots the 3D stress, con"rming previous observations. The reason is that away from the
Fig. 12. Comparisons of shell and 3D solutions for through-thickness stress variations: (a) along x"4.14 mm and y"0.16 mm, (b) along x"6.49 mm and y"0.25 mm.
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Fig. 13. Comparisons of shell and 3D solutions for the variation of the #exural displacement w on the bottom surface of the upper plate: (a) along h"03, (b) along h"453.
nugget the stress variation is controlled by the bending e!ect of the plate whereas near the nugget boundary the through-thickness behavior is in#uenced (although not substantially) by the presence of the nugget and the crack-like discontinuity between the joined plates. Finally, shell and 3D solutions for the #exural displacement w of the upper plate are compared in Fig. 13, where 3D solutions with both rigid nugget and base-metal nugget are presented. In particular, the displacement variation along the radial line h"03and z"0.0 mm is given in Fig. 13a while that along the radial line h"453and z"0.0 mm is given in Fig. 13b. It is interesting to observe that the shell solution is bounded by the 3D rigid-nugget solution from below and the 3D base-metal-nugget solution from above, such that the shell solution gives a very good approximation to the averages of the two 3D solutions.
5. Summary and conclusions The numerical performance of shell elements in "nite element modeling of spot-welded joints has been studied through detailed comparisons with 3D solid elements. The benchmark problem used in this study is a `symmetrica coach-peel spot-weld specimen, which facilitates the utilization of several simplifying symmetries and assumptions. The main conclusions of this study are as follows: First, converged shell element solutions provide an overall good approximation to converged 3D solid element solutions for spot-welded joints, even near the nugget boundary and in the stress concentration region.
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Second, shell element solutions generally predict a higher (about 9%) maximum stress than 3D element solutions with rigid nuggets. Third, the rigid-nugget assumption has minimal e!ects on the shape of stress distribution curves but it will lead to a higher stress level than a base-metal nugget assumption. Fourth, shell element solution for the #exural displacement is bounded by 3D element solutions with rigid and base-metal nuggets. The current study serves to illustrate the performance of shell elements in modeling spot-welded joints under converged conditions. In industrial practice, however, it is usually not possible to use re"ned meshes such as those used in this study. As such, there is a strong need to develop simpli"ed "nite element modeling techniques for spot-welded joints that will produce reasonable results with practical mesh designs. This is a challenge for further research in this area.
Acknowledgements This study was supported by an NSF/EPSCoR grant through Cooperative Agreement No. EPS-9630167.
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