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Year: 2009 | Volume: 6 | Issue: 1 | Page No.: 140 - 146

M.M. Rahman, A.B. Rosli, M.M. Noor, M.S. M. Sani and J.M. Julie

**Abstract**

This study presents the effect of the spot weld and sheets thickness on the fatigue life of the of the spot-weld joints to predict the lifetime and location of the weakest spot-welds due to the variable amplitude loading conditions. A simple model was used to illustrate the technique of spot-weld fatigue analysis. Finite element model and analysis were carried out utilizing the finite element analysis commercial codes. Linear elastic finite element analysis was carried out to predict the stress state along the weld direction. It can be seen from the results that the predicted life greatly influence the sheet thickness, nugget diameter and loading conditions of the model. Acquired results were shown the predicted life for the nugget and the two sheets around the circumference of the spot-weld at which angle the worst damage occurs. The spot-welding fatigue analysis techniques are awfully essential for automotive structure design.

Fig. 1 and 2. Stress distribution at the edge of the spot weld nugget is assumed as shown in Fig. 1. In Fig. 1, t represents the thickness of the sheet steel, σ_{x} and τ are the normal and transverse shear stress under axial force P respectively. The corresponding structural stress distribution is shown in Fig. 2. The structural stress (σ) is expressed in Eq. 1:

σ
= σ _{m} + σ_{b} | (1) |

where, σ_{m} is the membrane stress component and σ_{b} is the bending stress component due to the axial force P in the x direction. The transverse shear stress can be calculated based on local structural shear stress distribution, however, the effect of transverse shear stress neglected since the spot weld does not experience significant transverse shear loads in general^{[15]}.

The structural stress is defined at a location of interest such as plane A-A
in Fig. 3 and the second reference plane can be defined along
plane B-B. Both local normal and shear stress along plane B-B can be obtained
from the finite element analysis. The distance in local x-direction between
plane A-A and B-B is defined as δ. The structural membrane stress and
bending stress must satisfy Eq. 2 and 3
for equilibrium conditions between plane A-A and B-B. Equation
2 shows the force balances in x-direction, evaluated along the plane B-B.
On the other hand, Eq. 3 shows moment balances with respect
to plane A-A at y = 0. When δ between planes A-A and B-B becomes smaller
then transverse stress τ in Eq. (3) is negligible. Therefore,
Eq. 2 and 3 can be evaluated at Plane A-A
in Fig. 3.

Fig. 1: | Local normal and shear stress in thickness direction at the edge of a spot weld |

Fig. 2: | Structural stress definition at the edge of spot weld nugget |

Fig. 3: | Structural stress calculation procedure for fatigue crack in thickness direction at the edge of the weld nugget |

(2) |

(3) |

**DEVELOPMENT OF FEM**

Traditionally, a very detailed finite element model of a spot welded joint
is required to calculate the stress states near the joint^{[17-19]}.
This model produces reasonable results but it requires a good amount of effort
for modeling and computational time.

Fig. 4: | FEM around spot weld nugget |

Therefore, the very detailed finite element modeling of spot welds is not
feasible for 3000- 5000 spot welds in a typical automotive body structure^{[12]}.
Instead of the detailed modeling of the spot welds, a simple beam element represents
a spot weld for fatigue calculation of the spot welds in a vehicle structure^{[12,14,20}].

For the mesh insensitive structural stress calculation, the specimen for a spot welded joint is modeled with shell/plate, beam and rigid elements. The circular weld mark in each plate is modeled by triangular shell elements and rigid beams forming a spoke pattern as shown in Fig. 4. The rigid beam elements are connected from the center node to the peripheral nodal points of the circular weld marks in the both plates. Then the center nodes of the circular weld marks in both plates are connected with a beam element. Fig. 4 shows a finite element mesh around a circular weld mark. The geometry of the circular weld mark is required in the finite element model since the structural stress is calculated along the periphery of the weld. The normal direction of the shell elements (weldline elements) along the outside of the weldline is important for the calculation of the structural stress. Here, the weldline is defined as the periphery of the weld mark as shown in Fig. 4. A beam element represents the weld nugget to connect the top and bottom sheet steels. The length of the beam element is determined to be equal to one half of the total thickness for two sheets.

**FINITE ELEMENT ANALYSIS**

The nodal forces and moments in a global coordinate system at each mesh corner
along the weld line (nugget periphery) with respect to the shaded elements in
Fig. 4 are directly obtained from a linear elastic finite
element analysis.

Fig. 5: | Local coordinate system at a grid point |

The forces and moments in the global coordinate system are then transferred into the local coordinate systems since the structural stresses are defined as those components normal to the weld line of the spot weld. Figure 5 shows a local coordinate system at a node used to convert the global forces and moments to local forces and moments on the weldline.

The nodal forces and moments in the local coordinate system are then converted
to the distributed forces in terms of line forces and moments using the assumption
that the work done by the nodal forces is equal to the work done by the distributed
forces. The transfer equations for the line forces and moments are derived along
the welding between to nodes on the weld periphery. The simultaneous equations
for converting local forces to line forces are shown in Eq. 4:

(4) |

where, f_{1}, f_{2}, f_{3}, … f_{n-1 }are the
line forces at nodal point 1,2, 3, …, n-1 and F_{1}, F_{2},
F_{3}, …, F_{n-1} are the nodal forces in local coordinate systems
at the nodal point 1, 2, 3, …, n-1. The line forces at nodal point n is the
same as the line force at nodal point 1 since the weldline along the nugget
periphery is closed.

Fig. 6: | Definition of the line forces at the nodal element |

The line forces and nodal forces are presented for a single element case in Fig. 6. The line moments at the nodal points can be obtained from the nodal moments in the local coordinate systems suing simultaneous equations similar to Eq. 4.

Linear static stress is calculated suing the line forces and moments at each
nodal point on the periphery of the nugget. The structural stress consists of
a membrane stress component (σ_{m}) and a bending stress component
(σ_{b}) at each nodal points as expressed in Eq.
5^{[15-16]}:

(5) |

where, t represents sheet thickness,
is the line force in the direction of
is the line moment about
axis in a local coordinate system as shown in Fig. 5. The
structural stress (σ) was shown to be constant even though the size of
the finite element mesh was changed^{[15-16, 21]}.

The specimen geometry and dimensions with the finite element meshes are shown
in Fig. 7. Eight nodal points are located along the weldline
of the spot weld in the finite element models for tensile shear and coach peel
specimens. The sheet thickness of the specimens was 0.2-1.2 mm and the diameter
of the spot weld was considered 2.5 mm to 8.5 mm in the finite element models.
One side of the specimen was constrained in all directions and the other side
of the specimen was constrained in all directions except the direction of the
loading that was applied at the center of the grip with RBE3 elements^{[22]}.
The RBE3 stands for rigid body element type 3. This element distributes the
loads on the reference node to a set of nodes connected to the RBE3 element
without adding extra stiffness in the model^{[23]}.

Fig. 7: | Dimensions and FEM for tensile shear and coach peel specimens |

Fig. 8: | Load-time histories |

The sheet-2 is loaded with 25 N loads in the X, Y and Z directions while the legs of the sheet-1 are clamped at the edges. The load-time histories are shown in Fig. 8.

**MATERIALS PROPERTIES**

The data on material properties required for the numerical calculations were
collected after extensive search through information of literatures and handbooks.
Table 1 shows the mechanical and fatigue properties of the
sheets and nugget in which the young’s modulus, poison’s ratio and density and
so on.

Table 1: | Mechanical and fatigue properties of the sheets and nugget |

Fig. 9: | Effect of spot diameter and sheet-1 thickness on the fatigue life |

**RESULTS AND DISCUSSION**

The aim of this study was to illuminate the effect of sheet thickness on the fatigue behavior of spot welds and in particular to investigate the use of fatigue life prediction approach. In this respect, the problem was a special one due to the geometry of the spot weld contains a stress singularity. The model clearly needs to be tested against more experimental data in a variety of situations, an exercise which is beyond the scope of this study.

Figures 9 and 10 show the effect of the sheet thickness and spot diameter on the fatigue life of the spot weld structure. Spot weld diameter of 2.5-8.5 mm and sheet thickness for 1 and 2 of 0.2 mm to 1.2 mm are considered in this study. It can be seen that from Fig. 9 and 10, the spot weld diameter and the thickness of the sheet metals are influences the fatigue life of the structure. It is observed that the fatigue life of the structure increases with the increases of the spot weld diameter and thickness of the sheet.

Figure 11 and 12 show the effects of
the loads and confidence of survival on the fatigue life on the spot weld structure.

Fig. 10: | Effect of spot diameter and sheet-2 thickness on the fatigue life |

Fig. 11: | Effects of the loads on the spot fatigue life |

Fig. 12: | Effect of the confidence of survival on the fatigue life |

From the obtained results, it can be seen from Fig. 11 that the fatigue life decreases linearly with the increases of loads, however, the increases of fatigue life with increases of spot weld diameter. The obtained results from Fig. 12, it is clearly seen that the fatigue life influences on the confidence of survival parameter which is based on the standard error of the S-N curves. The prediction of the fatigue life distribution with the range of probabilities of 50-97.5% is shown in Fig. 12.

**CONCLUSION**

A computational technique developed and has been applied to predict the fatigue life of spot welded structures in tensile loading. In this study, the effect of sheet and nugget diameter was investigated under variable amplitude loading. The behavior of diameter of spot weld and sheet thicknesses are very important parameters in stress distribution near spot welds. The acquired results seen that the spot diameter and thickness of the sheets are greatly influence the fatigue life of the spot welded structures. This application and related experiments will be the subject of further investigations. The effects of the loading and geometric configuration on the fatigue life of spot welded joints can be directly incorporated in the structural stress method.

**ACKNOWLEDGMENT**

The authors would like to express their thanks to the Universiti Malaysia Pahang for financial support under the project (No: RDU070347) and provides the laboratory facilities.

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