DYNAMIC MODELING OF SPOT WELDS USING THIN LAYER INTERFACE THEORY

Hamid Ahmadian: 1 DYNAMIC MODELING OF SPOT WELDS USING THIN LAYER INTERFACE THEORY H. Ahmadian 1, H. Jalali 1, JE Mottershead 2, MI Friswell 3 1 Iran University of Science and Technology, Faculty of Mechanical Engineering, Narmak, Tehran, 16844, IRAN 2 Liverpool University, Department of Engineering, Liverpool, L69 3BX, UK 3 University of Bristol, Department of Aerospace Engineering, Bristol, BS8 1TR, UK ahmadian@iust.ac.ir Abstract Spot welds are widely used for manufacturing of thin sheet components, and they are subject to complex dynamic loading conditions. Many procedures have been used to model the behaviour of spot welds. In the point-to-point connection they are modelled as beam elements, or a combination of them, connecting the shell elements that are used to model sheet metal. The main advantage of this approach is that because of the simplicity of the model of spot welds it produces a relatively simple finite element model of the structure. However the model neglects the stiffness of the face-to-face contact zone of sheet components in the vicinity spot welds. This paper addresses the modelling problem of a spot-welded structure under dynamic loading consideration. It employs a thin layer interface model with variable stiffness properties to represent the spot welds joint and face-to-face contact effects. The predictions of the resultant model are validated using experimental modal data of a test specimen. INTRODUCTION Resistance spot welds are widely used in the automotive industry in the fabrication of all manner of components and structures. A typical car body may have more than 2000 spot welds. Since spot-welds have a significant influence on the dynamical behaviour of automotive components and are subject to complex dynamic loading conditions, many procedures have been used to model their behaviour. Spot welds are usually modelled as beam elements connecting the shell elements that are used to

Hamid Ahmadian: 2 model sheet metals. Three arrangements of beam elements that are commonly used to model the spot welds are: point-to-point, point to patch, and patch-to-patch connections [1]. The patch-to-patch type connection is used to get a correct force transfer from the weld into the shell. The forces transmitted through the beam elements of this type of connections are used to calculate the structural stresses in the weld nugget and the sheet metal at intervals around the perimeter of the nugget. These stresses can then be used to make fatigue life predictions on the spot weld [2]. The main advantage of this approach is that the spot weld model is able to connect non-congruent meshes so that parts do not have to be re-meshed. The length of the spot weld and the sheet separation in the above modelling strategy is half the sum of the sheet thicknesses. In reality the sheet metals have a face-to-face contact in the vicinity of the weld in a large area compared to the area of weld nugget cross section while in the model they are connected only in the spot weld locations. The face-to-face contact provides a nonlinear stiffness in the sheet metal joint interface and its modelling is complicated. In practical FE models of automotive structures there is no scope for such detailed modelling. On the other hand, neglecting the stiffness introduced by the face-to-face contact between metal sheets reduces the accuracy of model dynamical response predictions. This paper proposes an equivalent linear joint interface model for the face-to-face contact area which models the shear stiffness produced by the friction between the two metal sheets and the normal stiffness due to face-to-face contact. The joint interface layer is assumed to have a thickness equivalent to half the sum of the sheet thickness and is modelled using solid elements. Two different material properties are assigned for the interface layer: one in the spot weld nugget locations to resemble the weld stiffness and the other for the rest of the layer to simulate the face-to-face contact stiffness. Recently this strategy has been used to predict the dynamical behaviour of bolted joints with successful results [3]. Due to similarities between the bolted joints and the spot welded joints such as the ratio of face-to-face contact area to the bolt or nugget cross section area and introduction of local stiffness by bolts or welds one expects similar results in the case of modelling spot welds with the joint interface layer. The paper is organised as follows. The thin joint interface layer theory is discussed and its modelling parameters are introduced in the next section. Matching the observed behaviour of a test specimen with the predictions of its corresponding FE model identifies these parameters. In section 3 the modal test results of a hatsectioned beam with two panels connected with spot welds from reference [4] are used to identify the structure interface layer parameters. The success of the identified model to match the experimental results indicates that the interface layer model captures all the dominant physical effects involved in the spot welds and the face-toface contacts within frequency range of interest.

Hamid Ahmadian: 3 THE JOINT INTERFACE LAYER THEORY In the finite element model of the hat-sectioned beam shown in figure (1), the joint produced by spot welds between sheet metals can be represented by interface elements. Interface elements have been developed to model the behaviour of joints with different loading conditions. Two groups of interface elements are commonly used: zero-thickness and thin-layer interface elements. In the zero-thickness interface elements [5] it is assumed that the interface has a zero thickness and a constitutive law, usually consisting of constant values for both the shear stiffness and the normal stiffness, is defined. The behaviour of the thin-layer interface [6] is assumed to be controlled by a narrow band or zone adjacent to the interface with different properties from those of the surrounding materials. The thin-layer element is treated as any other element of the finite element mesh and is assigned special constitutive relations. When an interface element is used to model a joint, a suitable constitutive relationship must be adopted. A number of interface constitutive models have been developed. Depending on the type of analysis performed, the interface physics may be represented by quasi-linear or nonlinear models. Quasi-linear models consider a constant value of stiffness over a range of interface displacements, until yield is reached at the capacity of the joint. In nonlinear models, the interface shear stressdisplacement relationship is represented by a mathematical function of higher order. The coupling between normal and shear deformations is often ignored but is included in most of the constitutive formulations found in the literature. Figure 1- The hat-sectioned beam structure (Dimensions in mm) In modelling the spot welded joints, we use thin layer interface elements. As shown in figure (2) the spot welds and the face-to-face contact areas are modelled using solid elements. Different properties can be assigned to the spot weld interface elements and the elements representing the face-to-face contact. However when the spot welds are close to each other and there are no local deformations between the spot welds in the frequency range of interest, one may assign similar parameters for the spot weld and the face-to-face contact models. All the elements used in modelling of the sheet metals have isotropic material properties, corresponding to steel, except for the interface elements. One should assign the interface elements to have a material property that allows appropriate stiffness in the joint. The general form of the constitutive equation for the interface layer has 21 parameters. Therefore by selecting

Hamid Ahmadian: 4 the entries of the constitutive matrix as updating parameters the joint physics can be identified. The selected parameters can be identified if they have a significant effect on the modal response of the model. Otherwise the process of identifying the parameters will be ill conditioned and physical realisable results would not be achieved. One may introduce relationships, or constraints, into the parameter Figure 2- The joint interface layer identification procedures to avoid ill conditioning [7-8]. In the face-to-face contact the behaviour of the joint is governed mainly by normal stiffness and shear stiffness, whilst coupling terms have mainly secondary effects. We also ignore the coupling between the shear deformations and the normal deformations in the spot-weld nugget zone. Therefore to avoid ill-conditioning problems in updating procedure we choose an isotropic constitutive law for the interface layer. By restricting the stress-strain relationship to be isotropic, we are still able to define stiffness in normal and tangential directions independently. The linear constitutive equations for isotropic CHEXA elements, which form the interface layer is: σ 2G xx λ+ σ yy σ zz = σ xy σ yz σ sym zx λ λ+ 2G λ λ λ+ 2G G G ε xx ε yy ε zz, xy ε ε yz G ε zx G( E 2G) λ= ( E 3G) (1) where E and G are the elastic and shear module respectively. Assigning these two as the updating parameters for each type of joint, i.e., spot weld nuggets and the face-toface joints, we allow appropriate shear and normal stiffness for the interface layer during updating.

Hamid Ahmadian: 5 PARAMATER IDENTIFICATION The first ten measured modes of the assembled structure with free boundary conditions are reported in reference [4]. The experimental natural frequencies are shown in table (1). Figures (3) to (12) show the first 10 mode shapes obtained from the model. The first five measured modes will be used to identify the unknown parameters of the joint interface model by tuning the parameters so that a good agreement between the predictions of the model and the measured data is achieved. The measured modes 6 to 10 are used to examine the predictability of the updated model and hence are not used in the updating. The identification procedure was performed using the Design Sensitivity Module available in MSC/NASTRAN 2001. The modulus of elasticity and shear modulus of the thin-interface elements were selected as the design variables. These parameters are assumed to be the same for the face-to-face contact and the spot-weld nuggets. The objective function for the optimisation procedure was defined as: 5 2 Wi( ω ai ω ei ) (2) i= 1 min / 1 where ω ei and ω ai are, respectively, experimentally measured and analytically determined natural frequencies, and W i is a real positive weighting factor. The design sensitivity procedure in MSC/NASTRAN is based on an iterative linearised eigenvalue sensitivity determined using the following expression: 2 ( K M) T φ 2 2 ai ω ai φ ai ai ei = T φ ai Mφ ai ω ω (3) where ai φ is the i th analytical mode shape, M is the mass matrix and M and K are the changes to the mass and stiffness matrices. As there was no preference between the modes, the weighing factors were set to unity, W i = 1, i=1,,5 in equation (2). The initial values for the properties of the interface layer were selected as 1% of the material properties of steel. A permissible range of variation for each parameter was also defined. The upper bound for the variation of the updating parameters was set to 1.01 of the initial value and the lower bound was set to 0.01 of the initial value. The selection of initial values and the bounds were based on the fact that the interface layer introduces a local softening effect and sharply reduces the normal stiffness represented by E and the shear stiffness represented by G at the joint compared to the other areas of the structure. The shear stiffness of the interface layer is decreased from initial value of G 0 = 8e8 Pa by one order of magnitude to G= 8.64e7 Pa while the change in the normal stiffness is marginal (E 0 =2.07e9 Pa to E=1.73e9 Pa). By

Hamid Ahmadian: 6 allowing both E and G to vary in updating, we observe excellent agreement between the model predictions and the test results as shown in Table (1). The accuracy of the updated model in re-producing the first five modes and predicting modes 6 to 10 which are not used in the updating with only two parameters ensures that the identified interface parameters have physical merit and can be used in the model for simulations of the dynamical behaviour of the structure in service. CONCLUSIONS A linear dynamic model for spot welded joints using the thin layer interface theory is developed. The parameters of the model are determined by an inverse approach. The model can be easily incorporated into existing commercial finite element codes with the ability to define the dominant physics of the joint. The method is demonstrated by identifying the interface parameters of a spot welded hat-section beam structure. The results show that at the interface layer the stiffness of the element is reduced significantly. This research provides insight on how the model should be constructed in the joint region. MODE NO. 1 2 3 4 5 6 7 8 9 10 MEASURED MODES 537.3 574.8 629.4 664.4 672.2 701.2 734.4 821.4 865.1 946.4 UPDATED MODES 531.2 582.0 616.4 668.3 669.6 677.9 734.6 813.6 865.0 908.7 ERROR % -1.1 1.2-2.0 0.5-0.3-3.3 0.02-0.9-0.01-3.9 Table 1- Measured modes and predictions of the updated model (Hz) REFERENCES [1]- J. Fang, C. Hoff, B. Holman, F. Mueller, and D. Wallerstein, Weld modeling with MSC/Nastran, 2nd MSC Worldwide Automotive User Conference, Dearborn, MI., 2000. [2]- Y.Zhang, D.Taylor, Optimisation of spot-welded structures, Finite Elements in Analysis and Design, 37, pp. 1013-1022, 2001. [3]- H. Ahmadian, M. Ebrahimi, J.E. Mottershead, and M.I. Friswell, Identification of Bolted Joint Interface Models'', ISMA2002: Noise and Vibration Engineering Conference, Katholieke University, Leuven, Belgium, Sep. 15-18,2002.

Hamid Ahmadian: 7 [4]- M. Palmonella, M.I. Friswell, C. Mares, and J.E. Mottershead, Improving spot weld models in structural dynamics, 19th Biennial ASME Conference on Mechanical Vibration and Noise, Chicago, Sept. 2002. [5] G. Beer, An isoparametric joint interface element for finite element analysis, International Journal for Numerical Methods in Engineering, 21, pp. 585-600, 1985. [6] T. D. Lau, B. Noruziaan and A. G. Razaqpour, Modelling of construction joints and shear sliding effects on earthquake response of arch dams, Earthquake Engineering and Structural Dynamics, 27, pp. 1013-1029, 1998. [7] H. Ahmadian, J. E. Mottershead and M. I. Friswell, Regularisation methods for finite element model updating, Mechanical Systems and Signal Processing, 12 (1), pp. 47-64, 1998. [8] M. I. Friswell, J. E. Mottershead and H. Ahmadian, Finite element modal updating using experimental test data: parameterisation and regularisation, Philosophical Transactions of the Royal Society of London, Series A, 359, pp. 169-186, 2001. Figure 3- First mode @ 531.2Hz Figure 4- Second mode @ 582.0Hz Figure 5- Third mode @ 616.4Hz Figure 6- Fourth mode @ 668.3Hz Figure 7- Fifth mode @ 669.6Hz Figure 8- Sixth mode @ 677.9Hz

Hamid Ahmadian: 8 Figure 9- Seventh mode @ 734.6Hz Figure 10- Eighth mode @ 813.6Hz Figure 11- Ninth mode @ 865.0Hz Figure 12- Tenth mode @ 908.7Hz