Problem Statement: The retaining ring shown in the sketch must open by a distance of 0.125 + ID1/650 inch, as measured by the change in distance between the holes. The ring is thin (0.035 inches) and can be idealized as being in a state of plane stress. The number of times the ring must be installed and uninstalled is very low, so the main consideration is yielding of the material, which would loosen the fit of the ring. Use the Von Mises effective stress as a measure of how close to yielding the material is. Suggest a suitable material for the ring, and recommend changes to the design if necessary.

Modeling Approach: The sketch of the given part was made on ABAQUS, using global co-ordinate system. For the ease of analysis, the dimensions were changed from Inches to Millimeter. The thickness of the given part is significantly smaller than the other dimensions, hence it can be assumed to be in a state of plane stress. This assumption also led to the part to be sketched as a 2D shell element. Analysis: For the purpose of analysis, following factors were to be taken into account:

1. Suitable material for the part 2. Boundary condition 3. Mesh element type

Material Selection: The material selected for analyzing the part was 50 CrV4 (spring steel grade), it is one of the most commonly used material for the retaining rings. This material has Modulus of Elasticity, E=210GPa, Yield strength= 1200MPa, and poison ratio=0.3. Boundary Conditions: The type of boundary condition used for the given problem, is displacement boundary condition, because there is no force or pressure acting on the part. The top left end of the ring was fixed by making all the degree of freedoms equal to zero, and the top right end of the part was given a horizontal displacement of 5.3mm, all other degree of freedom on this node were kept equal to zero.

Mesh Element type: To perform the analysis, the choice of mesh element type was between Quad and triangular. Triangular element type was not used for the following reasons:

They have poor convergence rate They require extremely small mesh size to produce good results, this makes computation time very high.

Quad element type was chosen because they give the best quality result, and their accuracy is very high. The Q8 quad type was preferred over Q4 quad type because it gives more accuracy, this was achieved by using geometric order as Quadratic. To further reduce the computation time, the reduced integration option was not used. Upon analyzing the part using the above mentioned criteria, with seed size=0.7, the following result was obtained:

Figure 1: Failure when Spring Steel was used (seed size=0.7)

It is evident from the figure 1 that the maximum stress occurring is 1.612*10^3 MPa, and the yield strength ofthis material is 1200MPa. The failure criterion used for the analysis is Von Mises and it states that the maximum stress should be less the yield strength of the material. According to this criteria the part has failed, so in order to avoid this, other material having a slightly greater yield strength and a lower Modulus of Elasticity is chosen. The new material is Beryllium Copper (UNSC17200) having yield strength as 1205MPa and E=125GPa, poison ratio is has same value of 0.3. The analysis done using this material with the same seed size of 0.7, the following result was obtained:

Figure 2: Beryllium Copper (seed size=0.7) Figure 2 shows that the maximum stress occurring is 9.597*10^2MPa, which is well within the range of the material yield strength. The factor of safety was calculated to be 1.3. Design Change: No need for change in design was needed, as the maximum stress in given design is under the yield strength of the material. Mesh Convergence study: The following table was generated by using mesh element of different sizes:

Seed Size Maximum Stress(MPa)

1.5 966.6 1 961.2 0.9 960.7 0.8 960.3 0.7 959.7 0.6 959.6 0.55 959.4 0.5 959.1

Figure 3: Maximum stress (seed size=1.0)

Figure 4: Maximum stress (seed size=0.63)

Figure 5: Seed size vs. Maximum Stress plot

From figure 5, it can be seen that as the seed size reduces, the value of maximum stress converges to a single value. This means that going below a seed size of 0.6 is not recommended because the result will remain the same, but finer mesh sizes will cause the computation time to increase. So, to obtain the best result, mesh size of 0.6 is recommended. Conclusion: Upon analyzing the part, following conclusions can be drawn:

1. The material to be used is Beryllium Copper UNSC17200, this gives factor of safety equals to 1.3. 2. Mesh element type should be Quad, Q8 with reduced integration turned off 3. Optimum seed size to be used is 0.6

Material Beryllium Copper UNSC17200, E=125GPa, yield strength=1205MPa

Optimum Seed size 0.6

Mesh element type Quad, Q8

Summary Table

959960961962963964965966967

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Maxim

um Str

ess (M

Pa)

Seed size (mm)

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