How to Achieve Quick and Accurate FE Solution Small Radius Removal and Element Size Zhichao Wang - Sr. Lead Engineer Don Draper - Manager Jianxiong Chen Sr. Engineering Specialist Applied Mechanics Dept., Emerson Climate Technology Abstract It is common in FE modeling to remove small features such as small rounds or fillets to reduce the model size, save computer time, and avoid meshing difficulties. Removing a small feature at a sharp corner induces a strong singularity and stress concentration. FE solutions vary dramatically with the element size close to the stress concentration zone. This often creates confusion in the selection of element size. The influence of element size at the stress concentration with the small radii removed on the FE solution was studied in this paper. An Intermediate Element Size (IES) concept was proposed. The IES for a range of small radii was calculated to obtain a rational solution at the stress concentration. Examples are provided to illustrate the application of the IES concept. Introduction FEA has been used extensively to investigate stress concentrations and singularities induced by small features such as borehole, notches, small rounds, fillets and cracks since 1970 s. There is a wealth of Research papers on the topic of numerical solutions of stress singularities due to the complexity of stress and strain fields close to a stress concentration. However, not much attention has been given to the influence of ignoring or removing small features on the FE solution. It is a common practice in engineering FEA to remove or ignore small features such as small rounds and fillets to reduce the model size, to save computer time, and to avoid meshing problems. Removing small radii at concave sharp corner generally creates a stronger stress and strain gradient. The stress results vary dramatically at the sharp corner with element size when the small feature is removed. This often creates confusion in the selection of element size to obtain a rational solution at the location of interest. An accurate solution is critical for any material that is sensitive to the stress concentration. The issue remains, Can we obtain a reliable or accurate solution with the small feature removed? The influence of eliminating a small radius on the solutions was thus studied in this paper. An Intermediate Element Size (IES) concept was proposed. The IES for a range of small radii was calculated to obtain a reliable solution when a small radius was removed from the model. It is popular in Engineering to obtain quick and less accurate solutions in design and failure analysis. This often leads to an increase in the test costs and slows down the product design and development. This paper will introduce a concept of how to achieve quick and accurate FE solutions with the small radii removed in the FE Models. Application examples of the IES concept are provided. Intermediate Element Size Figure 1 showed the meshes of two 3-D L shape samples with B=1in, H=3B, L=2.5B, and the thickness t=b. Sample (a) has the radius r 0 =0.04 in 1 and (b) with the radius removed. A symmetric boundary condition was used on the x=0 and y=0 planes. A unit tensile stress was applied on the top surface. The bounded maximum stress of sample (a) is 3.846 Psi. The near field solution of the stress of Figure 1 (b) can be expressed as [2 ~3], 1 The units in this paper are lb., in., and Psi
Ki σ ij = f 1(r) f ( θ ) = f ( θ ) (i = I, II, III ) (1) 0.4555 (2πr) Where K I, K II, K III are generalized type I, II, and III stress intensity factors. R, θ are the location of the polar coordinates as shown in Figure 2. K I and K II correspond to in-plane loading while K III results from out of plane loading. It can be seen that the near field stress is a continuous function except at points on the line along the sharp corner. The maximum stress of the sample shown in Figure (b) is unbounded at the sharp corner. Therefore, the FE solution is a function of the element size at the sharp corner as shown in Figure 3. The influence of element size on the solution for a 2-D plane with a steep slope can be found in Paper [4]. It is convenient to call the elements that have a common node at the sharp corner as corner elements. It is obvious that the maximum stress at the sharp corner approaches infinitely when the corner element size approaches zero. The stress result decreases monotonically with the increase of the size of the corner elements. The solution is lower than 3.846 Psi (which is the result with the radius) when the corner elements becomes larger. Based on the Intermediate Value Theorem, there must be an Intermediate element size that will provide an accurate solution of the sample (a) with the small radius (3.846 Psi). Τhis Intermediate element size we herein define as IES. The existence of IES is certain because: (1) the actual stress and strain field is a continuous function except at the sharp corner; (2) the locations of the Gauss points move continuously with the increase or decease of the corner element size; and (3) the location of Gauss points are inside of the element mesh away from the singularity. As a result, an excellent approximation of the correct stress solution for a part can be achieved using this IES although the radius is removed. Figure 1. 3-D Sample Mesh (a) with Radius, (b) with Radius Removed
Figure 2. Sharp Corner With Vertex Angle In Polar Coordinate System [2] Figure 3. Influence of Element Size and IES IES and Fillet Radius Most engineering models are 3-Dimensional in which small feature removal is often required. A simple method to calculate the IES is to use a 2-D plane stress or plane strain models. However 3-D models as shown in Figure 1 or 5 will create a more accurate solution for the calculation of IES. This is due to the 3- D effect as discussed in paper [1]. A Small Feature is a relative concept that the small feature dimension is small relative to the global dimension of the volume attached to the small feature. For example, we say the radius r 0 in Figure 1 or 5 is small when r 0 is small relative to global dimension B and H. The radius r 0 would not be a small feature and can not be removed if r 0 is in the same order as B and H no matter how small it is. The IES for r 0 =0.01~0.1B was calculated using both 3-D L samples and 3-D shafts as shown in Figure 1 and 5. Figure 4 (a) illustrated the variation of maximum stress with the dimension of the small radius r 0 for both the 3-D L and the 3-D shaft samples. Note, the ratio of the small radius to the global dimensions is different between the two types of samples to make the discussion more generic. The IES results for the two types of samples are plotted in Figure 4 (b) and summarized in Table 1 and 2. Figure 6 illustrated the deviation of the IES between the two types of samples. It is expected that the IES is more
consistent with the decrease of the radius r 0. The accuracy in using IES is not necessarily getting lower for a larger radius r 0 although the difference of IES is larger. This is because the solution is less sensitive to a larger radius r 0 and IES as shown in Figure 4 (a) and (b). Figure 4. Maximum Stress with respect to r 0 (a) and IES (b) Figure 5. Shaft With and Without Radius for the Calculation of IES
Table 1. IES Obtained Using 3-D Sample Radius σ 1 IES σ 1 Difference 0.01 6.646 0.01535 6.642 0.06% 0.015 5.533 0.0235 5.532 0.02% 0.02 4.947 0.0308 4.94 0.14% 0.03 4.248 0.0445 4.247 0.02% 0.04 3.846 0.0572 3.84 0.16% 0.06 3.318 0.0832 3.315 0.09% 0.08 3.004 0.108 2.99 0.46% 0.1 2.795 0.131 2.792 0.11% Table 2. IES Obtained Using 3-D Shaft Models Radius σ 1 IES σ 1 Difference 0.01 7.4072 0.0155 7.4027 0.06% 0.015 6.1923 0.0238 6.1416 0.83% 0.02 5.4803 0.0315 5.4447 0.65% 0.03 4.6084 0.047 4.5953 0.29% 0.04 4.0779 0.062 4.0927 0.36% 0.06 3.4699 0.095 3.4334 1.06% 0.08 3.1004 0.124 3.0832 0.56% 0.1 2.8454 0.152 2.8435 0.07%
Figure 6. IES vs. r 0 Singularity Zone and Adjacent Element Size Effect Equation (1) showed the stress distribution in the vicinity of the sharp corner. Figure 7 illustrated the distribution of stress vs. the distance r to the sharp corner. The curve will shift up or down at higher or lower loads or loading type but the distribution f 1 (r) will not change. It is apparent that the stress gradient deceases very fast away from the sharp corner. The IES for r 0 from 0.01 to 0.1 in Table 2 changes from 0.0154 to 0.15. Compared with the IES, the stress singularity zone (about 0.005, in which the stress vs. r curve shows a steep slope change) is small. The zone is within the IES for r 0 from 0.01 to 0.1. This demonstrates that the IES of corner elements will have strong influence on the solution as discussed above. And the elements adjacent to and further away from the corner elements will have less influence on results. Take the 3-D L model as an example, for IES=0.05, the maximum difference in σ 1 is only about 1% for the adjacent element size ds = (1~2.5) IES as shown in Figure 8. Figure 7. Influence Zone of Stress Singularity Figure 8. Influence of adjacent Element Size
Examples In this section we apply IES obtained using the 3-D shaft sample (Figure 5, Table 2) and calculate the maximum stress for four type load cases. As given in table 2, the IES for r 0 =0.06 is 0.095. Note, to show the validity of IES in engineering analysis, we are using the IES obtained from 3-D tensile sample to solve torsion, bending as well as combined loads problems. In Figure 9, the shaft is subjected to a tensile load, P=10 Psi, (a) with the radius and (b) with the radius removed. In Figure 10, the shaft is subjected to torsion, M Z =10 lb-in. Similarly, Figure 11 showed the bending load case with M X =10 lb-in. The last example, in Figure 12 the shaft is subjected to combined loads with F X =10 lb, M X =10 lb-in, and M Z =10 lbin. Figure 13 plotted the maximum stress of bending (a) and combined loads (b) along the arc for o θ =0~180. Results are summarized in Table 3. It can be seen that the solutions obtained using IES are consistent with that without removing the radius. The maximum difference in the maximum stress for the worst case, torsion, is less than 8%. The reason we can use the IES from a tensile sample is the similarity of the stress and strain field close to the sharp corner and the localization of the singularity as discussed above. The CPU time for the sample with the radius is 3 times higher than that of the sample with the radius removed. This is a small model and it shows the great potential in time saving for 3-D large models. Using IES we can save both meshing hours and CPU time and yet not lose accuracy. A good quality mesh is required to achieve an accurate solution when the small radii are not removed. This can be seen in Figure 14 and 15. In Figure 14, the 3-D L sample with r 0 =0.08 is subjected of unit stress on the top similar to Figure 1 (a). Figure 15 showed the influence of the element size at the radius on the maximum stress results. The maximum error is about 5% for the first row element size ranges from 0.003 to 0.05. In reality, it is going to be hard to achieve this quality mesh, see Figure 14, due to the complexity of actual part geometry. Table 3. Comparison of FE Maximum Stress Using IES with Radius Removed * Radius σ1 (r0 = 0.06) σ 1 (IES = 0.095) Difference NER * /CPU NES * /CPU Times Tension 34.699 34.334 1.06% 6241/314.85 2317/112.02 2.8 Torsion 11.994 13.024 7.91% 6241/314.85 2317/113.06 2.78 Bending 35.984 35.417 1.6% 6241/315.04 2317/113.24 2.78 Combined 50.791 51.155 0.71% 6241/315.04 2317/113.06 2.78 * With radius Figure 9. Shaft Subjected of Tensile Load, 10Psi. (a) r 0 =0.06, (b) IES=0.095
Figure 10. Shaft Subjected of Torsion, M z =10lb-in (a) r 0 =0.06, (b) IES=0.095 Figure 11. Shaft Subjected of Bending, M x =10lb-in (a) r 0 =0.06, (b) IES=0.095 Figure 12. Combined Loads, F z =10lb, M x =10lb-in, M z =10lb-in (a) r 0 =0.06, (b) IES=0.095
Figure 13. Comparison of Maximum Stress at the radius, r 0 =0.065, with That Calculated Using IES=0.095, θ = 0 ~ 180, (a) Bending, (b) Combined Load Figure 14. Influence of element size on maximum Stress, 3-D L sample with r 0 =0.08 Figure 15. Maximum Stress vs. Element size dl=0.003~0.05 Recommendation As discussed above, the small feature is a relative concept. It is the ratio of the dimension of the small radius relative to the global dimension that defines the magnitude of the stress concentration. The stress concentration factor as well as the solution of stress and strain will change when the ratio of the small radius to the global dimension changes. Huge amount of work may be required to calculate the IES for a large range of r0 and the ratio of r0 to global dimension B, H, and L. It is recommended that a simple model be created based on the actual geometry and loads of the parts. Only a short time is required to create the simple models to calculate IES, especially using ANSYS WorkBench or APDL macros. The same models can be used again and again thus it becomes a convenient tool to calculate the IES. In the examples above, we computed the maximum stress using the same IES for different types of loads. It can be seen that the difference in the results for different load types are different. This may be due to the fact
that the function f( θ) in Equation (1) is different although the dominant function f1(r) is the same for all load types. A more accurate solution is expected when a closer loading condition is used in the calculation of IES. The same order of elements and the number of gauss points should be used for both calculating and using IES to solve problems. Element aspect ratio will affect the distribution of the Gauss points. Thus the same element aspect ratio should be used in the calculation and the application of IES. Summary and Conclusion An IES concept is proposed that has the potential to simplify meshing and save both meshing and computer time. The IES for radius r 0 =0.01~0.1 of a 3-D L-example with B=1, L= 2.5B H=3B and a 3-D shaft example with B=1, D= 4B H=4B was calculated. The adjacent element size has less influence on the stress results since the stress singularity zone is small. Four examples are provided using the IES calculated from tensile models. Solutions obtained using IES are consistent with that without having the radius removed. Results showed that reliable solutions can be achieved using the IES procedure. A tool may be added to ANSYS using the IES concept for users to select corner element size automatically that would save meshing and computer time and improve analysis quality. References 1. Zhenhuan Li, Wanlin Guo, Zhenbang Kuang, Three-dimensional elastic stress fields near notches in finite thickness plates, International Journal of Solids and Structures 37 (2000) 7617~7631. 2. Andrzej Seweryn, Krzysztof Molski, Elastic Stress Singularities And Corresponding Generalized Stress Intensity Factors For Angular Corners Under Various Boundary Conditions, Engineering Fracture Mechanics Vol. 55, No. 4, Pp. 529-556, 1996. 3. Martin L. Dunn, Wan Suwito, And Shawn Cunningham, Fracture Initiation At Sharp Notches: Correlation Using Critical Stress Intensities, Int. J. Solids Structures, Vol. 34, No. 29, Pp. 3873-3883, 1997. 4. Scott A. Ashford and Nicholas Sitar, Effect of Element Size on the Static Finite Element Analysis of Steep Slopes, Int. J. Numer. Anal. Meth Geomech, 2001, 25, 1361-1376.