# Finite element analysis

# Finite element analysis

### Abstract

Finite element analysis (FEA) was performed using a two dimensional (2D) model to examine the characteristics of a seat belt latch plate and to come up with an optimal design. The finite element method allowed for the analysis of the stress distribution and analysis of the finite elements and their nodes. Peak stresses where observed where the loads where applied. It was found that peak stresses reduced and became fairly distributed across the model when the transition points near where the loads were applied increased in radii and the mesh density in the model increased. Also, the model obeyed Hook's law of elasticity, i.e. the material stayed within its elastic limit and so did not undergo plastic deformation.

### Introduction

A commercial software package called ABAQUS (Version 6.8) was used for this work. ABAQUS was used for the pre- and post-processing, and also for processing the Finite Element Analysis (FEA) of the component, assuming linear elastic behaviour (Budynas, 1999). A seat belt latch plate was modelled using two dimensional (2D) plane stress elements. A latch plate shown in Fig. 1 is the portion of the belt assembly which is inserted into the buckle.

The component was assumed to be 2D which allowed a plane stress condition to be assumed. A state of plane stress exists when one of the three principal stresses is zero (Lawrence, 2007). This occurs in the study carried out because one dimension is very small compared to the other two, i.e. the latch plate is thin.

In order to achieve shape optimisation a number of factors need to be taken under consideration. Firstly, the right element type needs to be chosen for the analysis to achieve reasonable results. This can be achieved by doing trial runs and refining the analytical approach by analysing the model to see if the results obtained are reasonable. Secondly, the geometry of the model must have the minimum dimensions to satisfy the safety requirements and the transition points reduced. These main points along with others are what will determine whether the component shape is optimised to an accepted standard.

Several different models were analysed with different elements which included three-node triangle and four-node quad elements. The latch plate was modelled using approximately 4334 elements.

### Analysis and Results

A finite element model of a latch plate was constructed (Fig. 2). The geometry of the component was taken as a typical automotive seat belt latch plate. The model was assigned a Young's modulus of Elasticity of 200 GPa and a Poisson's lateral to longitudinal strain ratio of 0.285. The von Mises yield criterion was used for the material. The actual material test certificate was obtained for the material using CES EduPack 2009, which enabled the results to be based on actual values rather than theoretical. The material used was high carbon steel and was modelled as a homogenous and elastic isotropic solid.

Fig. 2 illustrates the von Mises stress contours. Loading was via a 1500N uniformly distributed load on each of the two rectangular holes. A load of 1500N (~150 g's) was chosen as typical value a latch plate experiences. The 2D finite element model consisted of 4334 three-node triangular elements each having 2 degrees of freedom which were chosen using the plane stress option. A boundary condition was applied at the bottom of the latch plate whereby displacement in the X, Y direction was restrained.

At the beginning of the research several different models were analysed to assess the accuracy of the results obtained. Models with different solid element types, sizes were simulated before the final arrangement of the mesh and elements used was decided upon. The model was symmetrical about the vertical centre line in terms of geometry and loading. Because of symmetry the first portion of the component having a ‘tongue' was modelled separately and found to have a similar stress distribution and stress peak to the initial model.

The highest stress patterns were noted where the loads were applied. It can be seen from the contours that the maximum stress was ~1900N which is well within the elastic region of the material, 400MPa being the elastic limit. Also, the stress-strain plot showed linear behaviour. It can therefore be deduced that the material did not undergo plastic deformation and so obeyed Hook's law. It should be noted that material non-linearity occurs when the stress-strain curve relationship ceases to be linear and the material yields and becomes plastic (Chandrupatla, 2002). It should also be noted that the obtained results concerned the elastic behaviour of the component, thus it cannot be inferred that the ultimate strength increases as the peak stress is reduced.

Initially, the elements used were very coarse in order to see the results develop. It was then found that the stress distribution was naturally more distributed on the model and the maximum stress was lower by around 600 Pa than that compared to the denser mesh. The number of elements in the model were then increased until the stress values converged.

It was also found that when using quad elements the stress distribution was slightly greater than triangular elements, however the mesh irregularity in the quad mesh was greater which possibly affected the outcome of the results. Both types of element used had a good aspect ratio of 1, i.e. the sides of the elements were equilateral (Chandrupatla, 2002).

### Discussion

The purpose of the study was to model a seat belt latch plate using FEA. Different element types were explored including triangular and quadrilateral and both showed a fair stress distribution. The values obtained for the maximum stress were reasonable since the results obtained were similar to that of the applied load. The FE model supported the hypotheses that the behaviour of the material is elastic and linear and so obeyed Hook's law.

Possible solutions in optimising the design could have been to change the shape of the component and remove regions where no stress levels are recorded. Reducing sharp transitions near where the loads were applied reduced the peak stresses when using three-node triangular elements rather than four-node quad elements. Another possibility could have been to use fine discretisation in regions where high gradients of stress are expected and reduce the mesh density where stress concentrations are low, i.e. use a coarser mesh in these regions. Also, maintaining the aspect ratio of the elements as close as possible to 1, i.e. equilateral sides and by sustaining the corner angles near 90o will increase the accuracy of the solution by keeping the stress contours near continuous. The simplified forces and constraints could have slightly limited the conclusion that could be drawn from a stress analysis.

It should be noted that errors are inevitable when using FEA; examples include (Budynas, 1999):

- Computational errors: due to round-off errors from the computer.
- Discretisation errors: using a finite number of elements to model the component introduce errors.

To overcome these problem triangular elements were used which provided higher-order stain distribution and thus keeping the strain constant (Chandrupatla, 2002).

An alternative study could have been to develop a 3D model. The quantitative results produced could have been more accurate, since the component thickness despite its small size compared to its length and width would have been taking into account. The values used in the investigation for the Young's modulus and Poisson's ratio were minimum, another investigation could have been to use the upper limits of those constants.

The finite element analysis can be seen to provide advantages in terms of time and expense over full scale testing and can produce a more complete picture of stress, strain and displacement (Lawrence, 2007), i.e. the higher the number of nodes the greater the accuracy in describing stress, strain and displacement within the elements.

### Conclusion

The study carried out for the FEA of a seat belt latch presented reasonably accurate results for the standard 2D elements used. Results were obtained for the stress distribution using the software ABAQUS. It was found that the stress and strain values converged to definite values as the number of elements and hence number of nodes in the model increased. Overall, it can be concluded that in order for the design to be optimised it required that the mesh density be reasonable high and that the transition points near where the applied loads are present be reduced by increasing their radii, this would allow for the stress peaks to considerably fall and for the stresses to be more distributed across the model. Optimisation was easy in theory but took a few iterations to get results.