Case Study 7, Page 1

ReSHAPE Case Study 7 Wheel Brace Stress Reduction

(Revised: 10 December, 2004, Author: David Bucca, Edited by Ryan Adams)

Version E04

ADVEA Engineering Pty. Ltd. 7 Calderwood Avenue, Wheelers Hill 3150, Victoria, Australia.

Tel: +61 3 9681 9094 Fax: +61 3 9681 9093

E-mail: info@advea.com http://www.advea.com

Case Study 7, Page 2

Case Study 7, Page 3

1. REDUCTION OF STRESSES IN A WHEEL BRACE

1.1. Problem Specification

A standard wheel brace used for the loosening of wheel nuts when changing a tyre is depicted in Figure 1.1-1. It was found that upon the application of an extensive amount of torque, there was a considerable stress concentration at the connection point adjoining the two arms. Particularly close to the yielding stress of the material, it is most desirable to change the shape of the brace in aim of reducing this stress build-up to an acceptable working level. Given that the wheel brace is made of steel and considerably heavy to begin with, it is also required that this task be achieved without a further increase in weight.

Figure 1.1-1 Standard Wheel Brace

1.2. Finite Element Model

The wheel brace was modeled in a pre-processor by the customer. Given that the area of interest only concerns the centre of the arms, it was most efficient and accurate to only model close to this region using solid tetrahedral elements from which rigid links connected to beam elements complete the geometry. The forces and constraint were applied to the ends of the beam elements Figure 1.2-1.

Case Study 7, Page 4

Figure 1.2-1 Detail of FE Model

1.3. Finite Element Analysis

For the initial analysis, the model was run using both NASTRAN and ReSHAPE, and the results were also verified via hand calculations. It was found that the maximum von Mises stress was 243 MPa and occurred at element 5025, in the red region as seen below in Figure 1.3-2. This region, as depicted, occurs on the under side of the centre-point where the arms of the Wheel Brace cross.

Figure 1.3-1 Topside, and underside of the brace

Case Study 7, Page 5

Figure 1.3-2 von Mises contour plot (close up, MAX = 243MPa)

It was found that the analysis results for both NASTRAN and ReSHAPE were identical, giving the same maximum Von Mises stress.

1.4. Reduction of Maximum Stress

Having analysed the original model of the wheel brace, ReSHAPE is employed to change the geometry of the model for a reduction of the maximum stress. Given that the model is composed of solid tetrahedral elements, PROCESS(GEOMETRY) is not suitable. Therefore Basis Vectors and Influence Functions will be used to determine the most suitable shape-change.

1.5. ReSHAPE Sensitivity Analysis

Before any iterative shape change is implemented, it is important to establish the areas that will be most sensitive to change to grasp a deeper understanding into the behaviour of the model. The command file used for the sensitivity analysis is:

RESHAPE(SENSITIVITY) RESPONSES STRESS(VAR=VMMAX,AVGW=97%,TARGET=-20%,TOL=3%)=ELEM(ALL) PROCESS() DOMAIN=ELEM(ALL) LOCKED=NODE(4652) END

Case Study 7, Page 6

The objective for the ReSHAPE analysis is to minimise the von Mises stress, calculated as a weighted average of stresses in all elements in which the stress is above 97% of the maximum stress in the whole component. Using a weighted average is important for models with high local stresses.

The LOCKED node seen above corresponds to the master node at the end of the beam on which the constraint (the wheel nut) is applied. The results found are depicted in Figure 1.5-1.

Figure 1.5-1 Sensitivity analysis contour plot

It can be seen that the most sensitive area is coincident with the location of the stress concentration, therefore suggesting that this should be the most suitable area for a shape change. It will be seen hereafter that this information became very useful for application of both the Basis Vector and Influence Function techniques, whereby only the sensitive regions were selected in the domain for shape changes to occur.

1.6. ReSHAPE using Basis Vectors

The virtual-loading conditions applied for this analysis consisted of three separate subcases, namely a force in the negative y-direction, a force in the negative z-direction and a pressure loading acting normal to the surfaces. These loading cases were analysed individually to generate separate influence vectors. Subsequent improvement of the structure showed that these three basis vectors produced a suitable shape change.

The command file used for the generation of the basis vectors is seen below. For each generation of a specific basis vector, the relevant sub-case was activated and the others commented out.

Case Study 7, Page 7

RESHAPE(ANALYSIS) CONTROL ANALYSIS(STATIC) SUBCASE=2 ! SUBCASE=3 ! SUBCASE=4 PROCESS(VECTOR) VCREATE(FILE=VECPIPE1.VEC) ! VCREATE(FILE=VECPIPE2.VEC) ! VCREATE(FILE=VECPIPE3.VEC) DOMAIN=ELEM(ALL) LOCKED(XYZ)=NODE(4652) END

Once the three displacement vectors were generated, the ‘improvement’ analysis was run using the following command file where SUBCASE=1 is the original load case of the model.

RESHAPE(IMPROVEMENT) CONTROL STEPS(N=50,S=2) SUBCASE=1 RESPONSES STRESS(VAR=VMMAX,AVGW=97%,TARGET=-20%,TOLERANCE=3%)=ELEM(ALL) PROCESS(VECTOR) VINPUT(FILE=VECPIPE1.VEC) VINPUT (FILE=VECPIPE2.VEC) VINPUT (FILE=VECPIPE3.VEC) DOMAIN=ELEM(ALL) LOCKED(XYZ)=NODE(4652) END

The results for the optimised shape can be seen below in Figure 1.6-1. As expected, the arms of the wheel brace simply expanded in this region thereby increasing the mass, resulting in a stress reduction. These results proved very successful at meeting the objective target of a 21% reduction of stress, with the maximum von Mises stress now at a value of 193 MPa at element 5025.

Case Study 7, Page 8

Figure 1.6-1 Changed shape for 21% reduction of maximum stress (scaled up 5x)

It can be seen that the stress concentration has not changed position, however now with high stresses also occurring at the constrained end where the diameter has remained the same. This 21% reduction in maximum stress has meant that quite a significant increase in weight has occurred. Ideally, a weight increase is to be avoided, hence a volume constraint is added to the command file. RESPONSES VOLUME(BOUND=CEILING,TOLERANCE=5%)=ELEM(ALL)

The CEILING parameter ensures that the volume cannot increase. The tolerance to a volume increase is 5%.

Using this constraint and running the same analysis procedure as before, the results can be seen in Figure 1.6-2. In this instance, the maximum value of Von Mises stress has reduced to

Case Study 7, Page 9

220 MPa occurring in element 5025. It is noted that the 20% reduction objective was not met given the bounds of the volume constraint. However this 9.5% reduction is a substantial decrease given the magnitude of stresses in question.

Figure 1.6-2 Changed shape for 9.5% reduction of maximum stress and unchanged volume (scaled up 5x) and stress contours

It can be seen from the stress contours of the new shape that the stress concentration is in a similar location but more spread as depicted by the pink band, however what is of more interest is what is happening at the constrained end. It is seen there that there is another localised stress concentration occurring on the underside of the connecting arm. Upon analysis of the stress values at this location, they were found to be in the order of 200 – 212 MPa. This suggests that the smoothing process could have ceased at 9% because the elements at this region cannot change. Given that, only the central region is of concern for stress

Case Study 7, Page 10

reduction. This places a limitation on the model. For better analysis a new model should be constructed with the connecting-arm mesh lengthened and therefore avoiding this second stress concentration area that jeopardises the objective and results. If a longer volume of the arm were meshed, the stress objective would most likely be achieved with the volume constraint, as greater shaping along the length would occur as predicted by these results.

Table 1.6-1 Reduction of stress for Basis Vector analysis

Model Maximum Von Mises stress (MPa)

Element number Percentage reduction

Original 243 5025 -

All elements

– no constraint

193 5025 21%

All elements

– volume constraint

220 5025 9.5%

1.7. ReSHAPE using Influence Functions

The sensitive region around the join was selected as the domain for PROCESS(FINCTION). The selected sensitive region is shown in Figure 1.7-1.

Figure 1.7-1 Selection of the active region

Case Study 7, Page 11

Using the volume constraint again , the command file for the iterative design improvement reads:

RESHAPE(IMPROVEMENT) ESET(ACTIVE)=18 CONTROL STEPS(N=50,S=.5) SUPER RESPONSES STRESS(VAR=VMMAX,AVGW=95%,TARGET=-20%,TOLERANCE=3%)=ELEM(ALL) VOLUME(BOUND=CEILING,TOLERANCE=5%)=ELEM(ALL) PROCESS(FUNCTION) DOMAIN(XYZ)=SET(ACTIVE) FCONTROL(ROT=YES)=NODES(4652THRU4655) FPARAM(BOUNDARY)=0.01 END

The command SUPER in the control block of the command file ensures that the part of the model not contained in the domain (i.e., not in the set EACTIVE) will be condensed during the first iteration. In subsequent steps of this run, and also in all further calculations, neither the stiffness nor sensitivities need be recalculated in the non-active elements. As long as the domain is not changed, the condensed model will be used. This allows, for example, investigating the effect of different control points a much more efficient process. The saving of computer time in subsequent runs can be substantial (80% for this model).

The commands FCONTROL and FPARAM are specific to process function.

FCONTROL specifies the nodes to be used as control points; in this case the four master nodes, connecting beams to solids, were selected. These nodes are not active (not being in the active region), therefore they cannot move. For this reason the parameter ROTATION=YES must be used, specifying that influence functions for the rotations will be created.

The FPARAM command with the BOUNDARY parameter controls the effect of the internal influence functions generated in the nodes at the boundary of the active region. The default value of 0.1 is too large for the relatively coarse mesh, resulting in a rough shape close to the boundary.

Figure 1.7-2 shows the changed model, Figure 1.7-3 shows the same in more detail, both in a true scale (1x).

Case Study 7, Page 12

Figure 1.7-2 Shape change for reduced stress and same volume

Figure 1.7-3 Detail of Figure 1.7-3

The maximum von Mises stress was reduced to 205 MPa. Without any volume constraint, the reduction was to 202.5 MPa. Figure 1.7-4 shows the stress contours. It is evident that the stress has spread from a concentrated location over a wider area.

Case Study 7, Page 13

Figure 1.7-4 Stress contours of the improved design

Table 1.7-1 Reduction of stress for Influence Function Analyses

Model Maximum Von Mises stress (MPa)

Element number Percentage reduction

Original 243 5025 -

no constraints 203 5025 16%

volume constraint 205 5025 15%

The results obtained from the influence functions method are comparable to the results from the basis vector process.

1.8. CONCLUSION

Using ReSHAPE, it was possible to successfully alter the geometry of the wheel brace model to reduce the maximum stress experienced upon the application of torque, without increasing its weight. Both the basis vectors and the influence functions methods provided suitable shape changes in terms of practicability and manufacturing given they were generally regular and symmetrical.