Geometrical Considerations in the Modelling of Welding

temperature fields in the near weld zone region are most important and the far weld region can often be neglected except for providing heat sink. Size and shape of the weld pool are therefore of interest and hence microstructure and fluid flow calculations are crucial for near weld analyses.

On the other hand, it has in many cases been shown that the distribution of the heat input into the weld and thereby the prediction of shape and size of the weld pool has only minor effects when calculating residual stresses globally in real structures.

Therefore, it is also widespread acceptable to neglect fluid flow calculations and evolution of microstructure in the weld pool in these cases. This is also applicable when analysing distortions in a welded structure though the heat distribution and

4 See Enclosure B for case sheet on residual stress dependency of the distance to the transverse constraining of the plates. Applied process in the test case; Laser welding of AISI 304 steel sheets.

modelling of weld preparation and filler material in this case often play a significant role.

More important is the geometrical modelling and connectivity describing the constraints of the structure during welding. As mentioned in the introduction, most structures need to be modelled full three-dimensionally to adopt all deformations, i.e. angular distortions, bends, buckling, elongations and contractions etc. Taking into account the calculation time that normally can be expected with three-dimensional models, alternative geometrical modelling procedures are often sought for the actual structure.

In the following, the different two-dimensional assumptions used for welding are presented with the first being the one used in this thesis for the engine frame box application.

3.5.1 Plane Strain and Generalized Plane Strain

If the structure to be considered has one of the dimensions considerably longer compared with the other two dimensions, i.e. a prismatic body, and the load is not applied on the long dimension, a plane strain assumption can be used, Figure 3.6.

FIGURE 3.6 A prismatic body suitable for the plane strain assumption.

A plane strain hypothesis leads to an exaggeration of the stiffness in the orthogonal direction to the plane considered. This restraining gives an overestimate of the stresses in this out-of-plane direction increasing the tendency to plastic yielding.

With the generalized plane strain theory, also termed plane deformation, the model is assumed to lie between two bounding planes, see Figure 3.7. This hypothesis allows free translatory movements in the orthogonal direction of the transverse

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plane considered. In addition, rotations of the bounding planes about the in-plane axes are allowed. The extra degrees of freedom remedy the over-restraining of the ordinary plane strain model to some extent. The generalized plane strain formulation also allows an initial angle between the bounding planes to be defined permitting modelling of initial curvature in the out-of-plane or “axial” direction.

The deformation of the model is independent of position with respect to the orthogonal direction, thus causing direct (normal) strain of the “orthogonal” fibres of the model only.

FIGURE 3.7 Generalised plane strain model, as presented in ABAQUS [18].

The generalized plane strain assumption obviously is able to model the case of unequally distributed loading conditions across the plane that affects the long or out-of-plane direction. This is exactly the case in most welding applications where a two-dimensional assumption could be imagined applied.

The plane strain / generalised plane strain assumption is by far the most frequently used in numerical analysis of welding. It has for many years been computational manageable compared to 3D models. The different aspects of this kind of assumption are further discussed in [7]. Examples of uses of newer dates are [19,20,21,22].

3.5.2 Plane Stress

If a structure has one of the dimensions short compared to the other two dimensions, i.e. a thin slice, a plane stress assumption can be used. This is illustrated with Figure 3.8. The assumption implies that temperature, deformation etc. are assumed constant through the thickness and the stress normal to the plane is zero. With this assumption, all loads are in-plane and out-of-plane bending is ignored and obviously angular deformation cannot be predicted.

The plane stress approximation is less suitable for most analyses involving traditional arc-welding processes, but it could be used in connection with welding processes involving heat sources with roughly constant temperature distribution throughout the thickness as e.g. laser welding of thin plates. Additionally, plane stress does not imply boundary conditions on the plane surface such as thermal heat loss due to convection and radiation.

FIGURE 3.8 A thin plate suitable for plane stress assumption.

Despite the limitations, butt welded plates have been investigated using plane stress condition with success leaving out any detailed information of stresses in the weld metal and heat-affected zone. A very early example of this can be seen in Muraki et al. [23] from 1975.

3.5.3 Shell Element Formulation

The shell element formulation allows for boundary conditions on the surface (e.g.

convection on the surface unlike the plane stress formulation which only accounts for boundary conditions on the edge) and takes into account out-of-plane bending behaviour why this formulation is also more used than the plane stress assumption.

Indeed, it is also a 3D formulation and hence more descriptive but therefore also

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more computationally demanding. The shell theory has been used for many applications, e.g. in Michaleris and DeBiccari [21] for the prediction of welding distortion in large structures using structural elements (shells, beams and trusses), or the modelling of stresses in pipe girth welds, Dong et al. [24].

In recent years thorough work considering distortions in large and complex welded structures has appeared using a detailed 3D model of the local distortion and stress distribution around the weld and subsequently projecting this result as boundary condition on to a global model described by the less computational demanding shell elements, Mourgue [25] and Andersen [26].

An additional advantage of the shell element formulation is the ease with which the geometrical models can be made (enmeshed etc.) and altered compared to the time consuming preparation of a full 3D model. This is a crucial matter in terms of a design tool involving numerical modelling of welding.

3.5.4 Axisymmetric Deformation

If a structure is axisymmetric, i.e. a body of revolution generated by revolving a plane cross-section about an axis, and the load can be applied axial symmetrically, an axisymmetric deformation assumption can be used as shown in Figure 3.9. This gives three-dimensional deformations and stress fields with any given non-uniform load, though independent of the circumferential coordinate.

FIGURE 3.9 Plane defining axisymmetric model, rotated 360°.

Axisymmetric models supporting torsion loading are available and are often referred to as a generalized axisymmetric assumption. This can also be used for axisymmetric solids with twist. This modelling principle is shown in Figure 3.10 as presented in the documentation for ABAQUS. The shape for revolution is modelled in the x-y plane. A controlling node is used to define the twist by rotating the node around the axis of symmetry an angle of Φ^twist.

FIGURE 3.10 Representation of axisymmetric model, with twist in (b). After ABAQUS [18].

For welding, the axisymmetric approach, normally without twist and torsion loading, can be used as an approximation for girth welds if the weld speed is high compared to the dimension of the pipe and no near weld effects are of interest. It approximates the weld being cast in one and hence no starting and stopping condition are included. This is seldom of interest and therefore this assumption is rarely usable.

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