## 1. Introduction

Technological progress creates increasingly arduous conditions for rolling mechanisms. Advances in many fields including gas turbine design, aeronautics, space and atomic power, involve extreme operating speeds, load, temperatures, environments which increases power and load on machinery and demand high strength to weight ratio of the rolling element bearings. Also bearing stiffness is an important parameter in the designing. Bearing design calculations require a good understanding of the Hertzian contact stress due to which high stress concentration is produced which greatly influence the fatigue life and dominate the upper speed limits as in the case of solid rolling elements. Since being originally introduced, cylindrical rolling element bearings have been significantly improved, in terms of their performance and working life. A major objective has been to decrease the Hertz contact stresses at the roller–raceway interfaces, because these are the most heavily stressed areas in a bearing. It has been shown that bearing life is inversely proportional to the stress raised to the ninth power (even higher). Whereas making the rollers hollow which are flexible enough reduces stress concentration and finally increase the fatigue life of bearing.

Investigators have proposed that under large normal loads a hollow element with a sufficiently thin wall thickness will deflect appreciably more than a solid element of the same size. An improvement in load distribution and thus load capacity may be realized, as well as contact stress is also reduced considerably by using a bearing with hollow rollers. Since for hollow roller bearing no method is available for the calculation of hollowness, contact stresses and deformation. The contact stresses in hollow members are often calculated by using the same equations and procedures as for solid specimens. This approach seems to be incorrect.

Initially in the present work author has carried out sufficient literature review (Somasundar & Krishnamurthy, 1984; Harris & Aaronson, 1967; Bamberger, Parker & Dietrich, 1976; Bhateja & Hahn, 1980; Murthy & Rao, 1983; Hong & Jianjun, 1998; Zhao, 1998; Yangang, Raj & Qingyu, 2004; Darji & Vakharia, 2008) and market survey to understand the practical application of hollow cylindrical roller bearing and its advantages in comparison with solid roller bearing. It is concluded that bearing manufacturer are production these type of bearing as per the requirement, but no standard formula or catalogue is available through which user can directly select hollow cylindrical roller bearing. Thus no standard formula (method) is available to find the optimum hollowness for the given loading condition and dimensions of bearing. Hollowness of the roller bearing is mainly dependent of applied load, dimensions of roller and endurance limit of the material used. Calculation of exact contact pressure for the hollow roller requires a finite element approach, and this has not been carried out yet. Present work is aimed to identify optimum hollowness irrespective of the geometry of the bearing and applied load.

To meet the requirement, in the first part of the present work contact analysis has been carried out for contact between roller and flat. Dimensions of the rollers are calculated using equation of equivalent diameter corresponding to the five different cylindrical roller bearing i. e. 2206, 2210, 2215, 2220 and 2224 to get the large data range. Value of applied load is taken from minimum to maximum. Finite element analysis is carried out for the same roller-flat contact and results are compared with analytical solution given by Hertz. This step is required to check the feasibility of FE procedure. In the second part of the work FE analysis has been carried out for the same applied load and material for all five cases, only the change is rollers are taken hollow. Roller hollowness is ranging from 10% to the hollowness for which bending stress at the inner bore should not exceed endurance limit of the material is taken for the consideration. Flexural fatigue failures occurred in hollow roller when the maximum bending stress at the bore cross the limit of endurance limit of the material. The fatigue cracks always began in the bore of the hollow roller. Those that propagated to the roller surface resulted in surface cracks and spallig and finally it fails the bearing. Around seventy FE analysis are done to generate the large data range. Finally graphical solution has been proposed to identify optimum hollowness irrespective of geometry of bearing and material properties.

## 2. Analytical study of solid cylindrical roller bearings

In the present work five different cylindrical roller bearing of NU 22 series are selected. First of all load distribution (Fig. 1) is calculated by applying load equal to static load carrying capacity of the bearing and the values of contact pressure, deflection, contact width and von Mises stress induced in the roller-race interface are determined (SKF General Catalogue, 1989; Design Data, 1994; Harris, 2001; Shigley, 1983; Nortron, 2010; I. S. 9202, 2001; Harris & Kotzalas, 2007; Horng, Ju & Cha, 2000; Demirhan & Kanber, 2008; Kania, 2006). Taking modulus of elasticity E = 201330 N/mm^{2} and Poission ratio v = 0.277 for the bearing material AISI 52100 steel (Guo & Liu, 2002). These five cylindrical roller bearings are selected in such a way that the size of roller should be in step of 5 mm approximately. So we can get the wide data range of load distribution for further analysis. Considering this point in the present work 2206, 2210, 2215, 2220 and 2224 bearings are selected for analytical analysis. Contact behavior of all these bearing is studied using Hertz theory.

Before executing the FE analysis for cylindrical roller bearing to understand the contact behavior, it is possible to execute the same contact behavior for roller-flat interaction in place of roller-race as shown in Fig. 2. It is very easy to check the contact behavior of roller and flat. To check the contact behavior of roller and flat, if we will take the same diameter of roller which is used in the corresponding roller bearing then contact width will be changed. So comparison of contact behavior of roller and flat with roller-race as in case of bearing is not possible. Using the equation for equivalent diameter in the present work roller diameter is identified in such a way that for the same loading condition contact width of roller and flat interaction will remain same as the interaction of roller with race in bearing. These equivalent diameter of roller is designated by ‘Roller 1’ for NU 2206 bearing, ‘Roller 2’ for NU 2210 bearing, ‘Roller 3’ for NU 2215 bearing, ‘Roller 4’ for NU 2220 bearing and ‘Roller 5’ for NU 2224 bearing. Contact interaction between roller and flat plate is shown in Fig. 1, which is a part of the contact between inner-race and roller in the cylindrical roller bearing as described. This contact interaction is studied in detail by analytically using Hertz theory in the present work.

Table 1 shows the value of all these analytical results corresponding to roller-flat contact.

The induced von Mises stress in the cylinder/roller is less then the yield strength 1410.17 N/mm^{2} of roller material AISI 52100 steel (Guo and Liu, 2002).

## 3. FE analysis of solid cylinder and flat contact

### 3.1. Existing FE models

Since the first mathematical treatment of the contact problem of ideally smooth elastic solids, presented by Hertz in 1882, significant progress has been made in the field of contact mechanics. In particular, the deformation characteristics of semi-infinite elastic media subjected to concentrated and distributed surface traction have been elucidated, and analytical solutions for the contact pressure distributions and subsurface stress fields have been obtained for elastic bodies of different shapes and various interfacial friction conditions (Timoshenko & Godier, 1970). The results of these studies have been invaluable in the design of durable mechanical components, such as rolling element bearings (Komvopoulos & Choi, 1992). Existing FE Models like GW Model (Greenwood & Williamson, 1966), KE Model (Kogut & Etsion, 2002) and JG Model (Jackson & Green, 2005) are studied and finite element analysis for the present case is carried out.

### 3.2. Finite element analysis details

#### 3.2.1. Model description

In order to validate the relationship of load vs deflection, load vs contact width etc., an FE model of an un-profiled roller contacting a flat plate was set up. A sketch of the problem is presented in Fig. 3.

A commercial package ANSYS 9.0 was used to solve the non linear contact problem. Initially for Roller 1, first of all three dimensional axis symmetric model was developed to form the single asperity contact between half cylinder and flat plate as shown in Fig. 6. For bearing 2206 taking equivalent diameter of roller as 6.62 mm and length 12mm. Dimensions of flat plat are taken as 12 x 8 x 2 mm^{3}. The circular surface of cylinder and contact flat surface of plate was discretized by SOLID 185 elements. SOLID185 is used for the 3-D modeling of solid structures. It is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions see Fig. 4. The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain capabilities. It also has mixed formulation capability for simulating deformations of nearly incompressible elastoplastic materials, and fully incompressible hyperelastic materials.

#### 3.2.2. Mesh convergence

A converged solution is one that is nearly independent of meshing errors. An extremely coarse mesh would give a very approximate solution, which is far from reality. As the mesh is refined by reducing the size of the elements, the solution slowly approaches an exact solution. It should be noted that, in theory, the solution will not be exact until the mesh size is zero, which is obviously impossible. However, it is possible to fix a tolerance to the solution error and this can be achieved by solving the problem on several meshes. In order to ensure that the solution obtained is as close as possible to reality, solutions should be obtained from several meshes starting with a very coarse mesh and finishing with a very fine mesh. Once these solutions are available, many key quantities can be compared and plotted against mesh densities (or number of points) as shown in Fig. 5.

In order to investigate the convergence of the solutions, all models have been solved with increasing numbers of elements. The elements around the roller contact region are subdivided into number of elements as shown in Fig. 7. Although the stresses and displacements at different regions are investigated in this work the convergence check has been made for only point A as shown in Fig. 7 It is a common point of contact of roller and flat plate where induced von Mises stress should be investigated. The von Mises stress (σ_{VM}) converged with coarse to fine meshes as shown in Fig. 5. The number of elements and nodes of models will increase as size of model will increase.

It is very clear from the Fig. 5 that for the last three points value of von Mises stress is approximately same and its value are 242.09 N/mm^{2}, 259.94 N/mm^{2} and 263.52 N/mm^{2} for the corresponding values of 0.08, 0.05 and 0.03 element edge length.

The region of most interest is adjacent to the contact interface and has the greatest concentration of elements for lower interferences. Away from the contact region, the mesh becomes coarser to minimize the computational efforts. In the present all FE models in contact region element edge length is taken as 0.08 and in other area it is taken as 0.5, the details of which is shown in Fig 8. The total numbers of elements generated are 14120 and nodes generated are 16977 for this model.

#### 3.2.3. Contact model

In order to create contact models in ANSYS, a contact pair of elements must be created a contact element and a target element. ANSYS has general guidelines as to what line, surface, or volume these elements should be applied. Perhaps the most critical feature is the mesh size. For example, a large target element size and very fine contact element will not work. The sizes of the contact and target elements should be fairly close to one another. It is possible to get a solution to converge, but the results will most likely be incorrect. That is why there is a densely meshed region in both the bottom part of the half-cylinder and on the surface of the block shown in Fig. 8.

#### 3.2.4. Boundary condition and application of pressure

The boundary conditions are presented in Fig. 9. The nodes on the bottom surface of the flat plat all degree of freedoms are restricted and rigidly constrained from translating in the x, y and z direction. Where as on top surface of half cylinder uniform pressure of 68.16 N/mm^{2} is applied which is related with Qmax (5414.28 N) for bearing no. 2206

#### 3.2.5. Solutions

The solutions have been carried out by means of a PC. The hardware configuration consists or an Intel Pentium IV 2.53 GHz CPU with 1 GB of RAM. The models were solved in round 25 minutes to 7 hours. All models have been solved as 3D static with Newton Raphson option.

#### 3.2.6. Evaluation of the finite element model

To examine the appropriateness of the finite element mesh and modeling assumptions, such as the dimensions and fineness of the mesh and the imposed boundary conditions, finite element results for an elastic half-space indented by a rigid cylindrical asperity were compared with analytical results for line contacts. The FE model was first verified by comparing its output with the analytical results of the Hertz solution in the elastic regime. The verification included the contact pressure, contact stresses, deformation and contact width. For the evaluation of Finite Element Model initially von Mises stress criteria is taken for the consideration because it is the final output of analytical study as discussed in section 2. Also it is an important stress which should remain within limit with respect to yield stress of the material. Figure 10 shows the contour plot of von Mises stress for the applied load of 5414.28 N. As it is clear from the figure that at contact zone induced stress is higher which is marked by red colour. Figure 11 shows the detail view of contact zone with node numbers. Value of von Mises stress is to be identifying for the node no 5258 which is on contact surfaces and it is 1031.1 N/mm^{2}. Whereas analytical result gives 970.31 N/mm^{2} (Table 1). Thus the von Mises stress of FE model differs from the Hertz solution by 5.8% which is acceptable for the present analysis. The small differences between analytical and FEA solutions near the contact edge may be attributed to the fineness of the mesh. The favorable comparison of the results illustrates the suitability of the finite element model for the present analysis involving only global variables, such as von Mises stress, contact pressure, contact width and deformation.

Figure 12 shows the contour plot for shear stress distribution. It is clear from the FE analysis that value of induced shear stress is 457.88 N/mm^{2} and analytical result gives 485.16 N/mm^{2} with the percentage error of 5.6 % only.

Now using similar contact model and boundary conditions FE analysis has been carried out for all five rollers. Applied loads are taken as per the calculated load distribution among the rollers. Table 6 shows the validation of meshing scheme employed by comparing the analytical results with FEA.

Figure 13 to 16 shows the graphical comparison of Table 6. Four important parameters von Mises stress, Contact Pressure, Deformation and Contact Width are plotted for different applied load.

#### 3.2.7. Summary of FE analysis

The finite element method was used to analyze the contact mechanics aspects of nominally flat single-asperity surfaces and to identify the effect of important parameters like von Mises stress, contact pressure, contact width and deformation for the given load. On the basis of the presented results, the following major conclusions can be drawn.

The smaller error in the FE model is attributed to overall balance (static equilibrium) enforced by the FEM package. The smaller differences between analytical and FEA solutions near the contact edge may be attributed to the fineness of the mesh.

On the basis of the results discussed, it may be concluded that the finite element configuration shown in Fig. 10 and the invoked modeling approximations are acceptable for the purpose of the present analysis.

Figure 14 to 17 shows that the agreement between analytical and finite element results from different rollers and various load is appreciably good. The maximum disagreement between the FEA value and analytical values occurs at the lowest applied load. The accord between the FEA and analytical results gets progressively better as higher applied load. Thus smaller the interference the smaller number of contact elements are in effect, leading to a large error and visa versa.

## 4. Finite element analysis for hollow roller and flat contact

### 4.1. Finite element model

Different hollowness percentage ranging from 10% to 80% (in step of 10%) has been investigated for Roller 1 of diameter 6.62 mm which is equivalent of 8 mm diameter roller co-relate with bearing 2206. Figure 17 shows the finite element model for 40 % hollowness. Same surfaces as taken in the contact model of solid roller and flat i.e. outer surface of roller and top surface of flat plate are selected for contact element and target element respectively.

### 4.2. Meshing

Now in this case also for the contact surface element edge length is taken as 0.08. Also as discussed by Murthy & Rao (1983) that, in addition to the contact stresses at the outer contact zone, the hollow specimens are subjected to tangential stresses (bending stress) at inner surface. Thus for the inner surface element edge length should be high and is taken as 0.08 as shown in Fig. 18. For other remaining area it is taken as 0.5.

Same boundary condition and pressure is applied as discussed in section 3.2.4. For each hollowness 10% to 80%, FE model is developed as shown in Fig. 19, maximum applied load 5414.28 N is taken and results are observed.

## 5. Results and discussion

Due to thin section very less material is available to resist the force so von Mises stress is increase after 60% hollowness. Also at this stage plastic deformation will take place and failure will occur due to permanent deformation, which is not desirable and should be avoided.

An added criterion for evaluation in a bearing with hollow rollers is the roller bending stress. To evaluate the life integrals, the value of the fatigue limit stress must be known for the bearing component material. This can be determined by endurance testing of bearings or selected components. Performance analyses were conducted, using the von Mises stress as the fatigue failure-initiating criterion. Based on this subsequent study fatigue limit stress for bearing material AISI 52100 is 684 N/mm^{2} (Harris and Kotzalas, 2007). From Table 7 it is very clear that bending stress is continuously increase from 377.07 N/mm^{2} to 1721.4 N/mm^{2} as hollowness increase from 10% to 80% respectively. But the practical limit of this stress is 684 N/mm^{2}. So the hollowness should be restricted upto 52% which is clear from Fig. 20.

Thus for the present case of Roller 1, for the applied load of 5414.28 N the % hollowness of the hollow roller should not exceed 52%, otherwise induced stress at the bore of the roller will increase beyond the endurance limit and cause fatigue failure of roller.

Great care must be given to the smooth finishing of the inside surface of a hollow roller during manufacturing as the stress raisers that offer due to poorly finished inside surface will reduce the allowable roller hollowness ratios still further than indicated by Fig. 20.

Figure 21 shows the contour plot of maximum shear stress for 52% hollowness. The induced shear stress is 273.68 N/mm^{2} which is approximately half than the shear stress induced in solid roller for the same load of 5414.28 N. Thus reduction in shear stress gives improvement in fatigue life of bearing.

Figure 22 to 25 shows the effect of hollowness on different parameters.

The term “hollowness” as referred to here for the rollers in the ratio of the inner diameter to the outer diameter expressed as a percentage and is an important parameter in the design of the hollow roller bearing. It determines not only the overall bearing stiffness but also the amount of preload most desirable, the bearing’s load capacity and its life. In fact, hollowness is a control parameter used to optimize the bearing design. In the present case of Roller 1, load is applied in such a way that induced bending stress at inner bore should cross endurance limit of the material i.e. 684 N/mm^{2} for each hollowness. Result of FE analysis is shown in Fig. 26. The roller hollowness values from 10% to 80% have been analyze by Finite Element as discuss above for Roller 1 and the roller load, deflection and stress curves of Fig. 27 have been developed. The dotted line across these curves show the points of constant maximum roller bore stress for values of 684 N/mm^{2}.

Corresponding to each roller hollowness value, there is a specific optimum load for each bearing design, which is indicated in Table 8.

Figure 27 gives the best solution to find the optimum hollowness for verities of load. But this is not the final solution, because solution given in Fig. 27 is only applicable for Roller 1 i.e. equivalent roller of bearing 2206. As bearing/roller geometry will change again same procedure of FE analysis should be carried out and again optimum hollowness is to be identified. Thus similar analysis can also be carried out for remaining four rollers to find the optimum hollowness and results are given in Table 9. For each roller these analyses have been carried out upto the hollowness where induced bending stress should just cross the endurance limit.

From Fig. 28 it is clear that for

Roller 1 optimum hollowness should be 52% for the applied load of 5414.28 N,

Roller 2 optimum hollowness should be 39% for the applied load of 10397.88 N,

Roller 3 optimum hollowness should be 45% for the applied load of 21620.59 N,

Roller 4 optimum hollowness should be 29% for the applied load of 50656.34 N,

Roller 5 optimum hollowness should be 28% for the applied load of 73318.39 N.

It is very clear from the results and discussion of all five rollers that optimum value of hollowness is dependent on magnitude of applied load, bearing geometry i.e. diameter and length of roller and mechanical properties of material used. If the value of applied load will increase than hollowness should be reduced to maintain the bending stress within endurance limit of the material. Change in bearing geometry will change the applied pressure and resulted into change in hollowness. Thus the solution given in Fig. 26 and 27 is not a generalized solution and it can not be applicable to any bearing geometry for any load. It is applicable to specific type of bearing and for specific load only respectively. If applied load will change one can’t use the results shown in Fig. 28 and lengthy FE procedure should be again carried out to get the result in the form of hollowness.

### 5.1. Generalized graphical solution

To find the optimum hollowness for any material and for any applied load irrespective of bearing geometry, in the present work large data are generated by FE analysis. To get the generalized solution FE analysis for the hollowness percentage ranging from 1% to 95% (after 95% ANSYS solution was not supported) is carried out and following Table 10 has been developed. This table shows the values of bending stress corresponding to the applied pressure.

Table 10 is presented in graphical form in Fig. 29. This diagram shows the value of bending stress for different applied pressure with respect to hollowness ranging from 1% to 95%.

## 6. Conclusion

In case of solid roller bearing induced sub-surface stresses are the limiting criteria for the fatigue life of bearing whereas for hollow roller bearing bending stress is the limiting criteria. The bending stresses on the internal diameter of the roller in the plane of the loading forces are the most critical for destructions. In the present work graphical solution was developed to determine optimum hollowness of cylindrical roller bearing for which induced bending stress should be within the endurance limit of the material. Figure 29 shows the generalized diagram for bending stress vs applied pressure. Following are the major outcomes from this diagram.

For the same value of applied pressure, Fig 29 shows that there is very small variation in the value of bending stress by increase the hollowness from 10% to 30%.

If the hollowness increases from 1% to 95% the slop of line will also increase accordingly.

The durability of the bearings with hollow rollers operating on cycles not exceeding the maximum permitted level of bending stresses can be substantially greater than the durability of similar bearings with solid rollers.

For the applied load on equivalent size of roller initially applied pressure is to be calculated. As per the endurance limit of the material used and calculated applied pressure optimum hollowness can be identified from the diagram.

For the particular hollowness diagram gives the maximum limit of applied pressure and hence applied load. The developed graphical solution can be applicable for any material of bearing.