Effect of Rim Thickness on Bending Stresses in Low ... · PDF fileEffect of Rim Thickness on Bending Stresses in ... gear rim thickness on tooth bending stresses in low addendum large

Effect of Rim Thickness on Bending Stresses in Low Addendum Large Spur Gears

Yesh P. Singh

Department of Mechanical Engineering The University of Texas at San Antonio

San Antonio, TX 78249-0670

Ravichandra Patchigolla Department of Mechanical Engineering The University of Texas at San Antonio

San Antonio, TX 78249-0670

Abstract A finite element based approach is used for determining the effect of gear rim thickness on tooth bending stresses in low addendum large spur gears. These gears are used in cement plants, sugar mills, ball mills, coal mills, kilns, grinding mills, copper converters, and anode furnaces, etc. A program is developed using ANSYS Parametric Design Language (APDL) to generate 1, 3, and 5 tooth segments of a large spur gear. A controlled meshing approach is developed using free and mapped meshing capabilities of ANSYS to generate two-dimensional and three-dimensional finite element models of the gear tooth segments. The controlled meshing approach employed here has the following advantages: it prevents high stresses at load location, avoids too many elements in the low stressed region, and generates a fine mesh in the fillet regions of the loaded gear tooth. The finite element models are analyzed in the position of Highest Point of Single Tooth Contact (HPSTC). The two and three-dimensional models of 1, 3 and 5 tooth segments are analyzed by varying β, which is the ratio of rim thickness to tooth height of the gear. The effect of web on stresses is studied in the 3-D models. Equivalent (von Mises) and bending stresses are obtained for different values of β in various models. Using a program written in APDL the gear tooth profile and rim surface bending stresses are arranged in the form of tables using Microsoft Excel and plots are generated using the capabilities of MATLAB program. Maximum bending and maximum von Mises stress plots are generated for varying values of β in two and three-dimensional models. Also, plots are generated to compare maximum stresses in 1, 3 and 5 teeth models. Introduction A number of researchers have worked on gear tooth failure and used experimental, analytical and numerical techniques to determine the stresses in the gear tooth. Most commonly used experimental techniques include photoelastic and strain gages, and finite element method was the mostly used numerical technique. Photoelastic technique was widely used for many years. Baud and Timoshenko1 introduced the photoelastic technique to examine the stress concentration effect at the gear tooth fillets. Sopwith and Heywood2 used photoelastic technique to develop a fillet stress formula that accounted for some pressure angle unbalance. Kelly and Pederson3 improved this formula by employing more realistic tooth shapes in their photoelastic models. Drago and Luthans4 conducted experiments using 2 and 3 dimensional photo elastic techniques to evaluate the combined effects of rim thickness and gear pitch diameter on tooth root and fillet stresses. They calculated stresses for the load applied at LPSTC, HPSTC and Pitch point along the tooth profile. The main drawback of this method is that the experimental investigation is time consuming and it is very difficult to construct and prepare the models for investigation. Many investigators have used different finite element approaches in evaluation of the gear tooth stresses for a long time. Wilcox and Coleman5 used analytical method of finite elements in analyzing the gear tooth stresses. Quadrilateral elements have generally been used in two dimensional models. They developed a new stress formula,

which is based on the stresses obtained from finite element analysis and takes tooth shape and loading condition into account in evaluating the tensile stress in the fillet region. Oda6 et al. analyzed the root stresses on the fillet of gear teeth as a two dimensional elastic problem by means of the FEM with typical triangular elements. They also measured these stresses experimentally with strain-gage method by carrying out a static bending test. These stresses were analyzed for spur gears of different rim thickness. The effects of rim thickness on root stresses and on the critical section were studied. Their results obtained by FEM confirm with results measured by strain gage. Chong7 et al. used finite element method to model the rack teeth as an example of thin-rimmed spur gear and to confirm the results of the approximate formula. They investigated the influences of radius of curvature of tooth fillet, pressure angle, and loading position on tooth flank on the tooth fillet and root stresses under a single and double tooth pair meshing. Chang8 et al. used SAP IV finite element technique to investigate the fillet and root section stresses for a variety of loading positions, mounting support, different fillet radii and rim thickness on a single tooth model. They also studied the surface stress distribution on the entire tooth profile for the tip and pitch point loading. Reddy9 et al. used 6-node isoparametric plane stress triangular element to build the finite element model of a thin rim spur gear. They calculated the effect of variation of rim thickness of a 5-tooth segment model on the location and magnitude of maximum bending stress value. Filiz and Eyercioglu10 evaluated the effects of module, contact ratio, fillet radius, pressure angle and teeth numbers of driving and driven gears on gear tooth stresses for three different loading conditions (i.e., point load, distribute load and simulated contact) using the finite element method. MSC/NASTRAN was used for finite element analysis. Based on their study, they modified Chabert and Tobe’s formula in order to include the effects of the above variables and presented a new formula that gave closest results to their study. Vijayarangan and Ganesan11, 12 employed the FE approach for stress analysis of a composite spur gear. They also used Lagrangian multiplier technique along with 2D FE method to evaluate tooth contact stresses. Triangular elements were used to discretize the spur gear tooth sector for the tooth contact stress analysis. Gordana13 determined the actual state of stress in the spur gear tooth root fillet by use of 3D finite element approach. His model used parabolic triangle, quad brick and parabolic tetrahedron solid elements and also used both h and p-convergence approaches. Shuting14 developed a 3D finite element model of a thin-rimmed gear using 11-node solid element to perform the deformation and bending stress analysis in a standard involute spur gear. Shuting15 also performed loaded tooth contact analysis of a 3 dimensional, thin-rimmed gear by combining a mathematical programming with 3D finite element method. The research work by most of the investigators implied that diametral pitch, shape of the tooth profile, highest location of full load on the tooth profile and fillet geometry of the gear tooth influenced the bending strength of the gear tooth. For the gears with thin rim, rim thickness is an other significant factor due to rim deflections. The gears considered by most of the authors are either small or have less number of teeth. In this study a large spur gear with 192 teeth and diametral pitch of 1 is considered. These gears are used in the kiln, grinding mill drives, mining industry’s etc. In such large and low addendum gears making a solid gear is not possible. Therefore rim thickness is optimized to achieve constant bending stress at the root. In this paper finite element modeling of the large spur gear is discussed. This paper also presents the results of the finite element analysis of 1, 3 and 5 teeth segments. Bending stresses on the tooth profile and rim bottom surface of the tooth subjected to loading are studied for 3 values ofβ . The effect of rim thickness on maximum value of bending stress and maximum von Mises stress is studied for 12 different values of β in 1, 3 and 5 teeth segments. The 3D models are also employed to study the effect of web. A web of 2 inch thickness has been considered and to include that affect, the nodes at the rim and web interface at 1 inch distance on either side from the middle of the face width were completely constrained. The bending stresses in these models were also compared with the 3D models without constraints at the rim-web interface. Comparisons for maximum von Mises stress and maximum bending stress was made for the following cases. 1. Comparison of tooth profile and rim bottom surface stresses in the middle tooth of 1, 3 and 5 teeth segments for

rim thickness equal to tooth height. 2. Comparison of maximum von Mises stress and maximum bending stress in 1, 3 and 5 teeth segments for 12

different values of β in 2D and 3D models. The comparison plots and the results of comparison are presented and an appropriate value of rim thickness is suggested. For simplicity results of the case with rim thickness equals tooth height are presented. The cases with

lowest and highest values of rim thickness and three dimensional analyses were also studied and the results of the study are listed in the reference16. Finite Element Modeling The gear used in this study is a large spur gear with 192 teeth and diametral pitch of 1. The addendum, dedendum, arc tooth and tooth space thickness are not standard. A program is developed using ANSYS Parametric Design Language (APDL) to generate 1, 3 or 5 teeth segment finite element models of the large spur gear.

Generation of Spur Gear Tooth Profile The gear tooth involute profile is modeled in ANSYS. In order to generate involute profile using ANSYS, 51 keypoints are created in the range of dedendum and addendum circle radii representing the involute profile. These points represent radii values in the range of dedendum and addendum circle. All these radii values are stored in an array ‘r’ of size 51x1 using “*VFILL” command. The coordinates of each point on the involute profile can be obtained by evaluating involute and pressure angle values at each radii value. Involute (inv) and pressure angle (θ ) values are evaluated using the following relations.

( ) θθθ −= taninv , (1)

Where ⎟⎠⎞

⎜⎝⎛ Φ×

= −

rrG cos

cos 1θ , (2)

θ and are in radians. Φ Knowing the involute and pressure angles at all the radii values from the above expressions, the arc tooth thickness at each radii value can be obtained from the following expression17.

( ) ⎥⎦

⎤⎢⎣

⎡−Φ+

××= θinvinv

rTrT

Gr 2

2 , (3)

Obtaining the tooth thickness at each radii value (Figure 1), the angle subtended by the half tooth thickness values at the center of the gear is obtained from the following relation.

rTr

×=

2α , where α is in radians. (4)

Figure 1 Geometry of Spur Gear Tooth

The coordinates of each point on the involute profile in the gear tooth coordinate system with y-axis coinciding with the gear tooth centerline are obtained from the following expressions.

)cos()sin(αα

×=×=

ryrx

(5)

All keypoints are plotted on GUI using the X and Y coordinate values and a B-spline is generated passing through all these keypoints. The shape of fillet has a direct effect on maximum bending stress developed at the root. Hence the root fillet geometry is vital. The generation process decides the shape of the root fillet. Form cutting operation is employed to manufacture large gears. The shape of the root fillet is a circular arc when form cutting is employed. This can be achieved by using line fillet option in ANSYS. The portion of the involute between the clearance and the root circle and circular arc in the tooth space are the entities selected for the line fillet option.

Generation of Finite Element Model The analysis is performed on two dimensional and three dimensional finite element models of 1, 3 and 5 teeth segments. This work used 4-node PLANE42 elements for the 2-D analysis. Four nodes having two degrees of freedom at each node define the element. Each node has translations in the nodal x and y directions. Plane stress with unit thickness option was used for the 2D analysis. While for the 3-D models, 8-node SOLID45 elements are used. Eight nodes having three degrees of freedom at each node define the element. Each node has translations in the nodal x, y, and z directions. The number of elements and nodes in two and three-dimensional models are tabulated in Table 1. Free and mapped meshing capabilities of ANSYS are employed to generate the finite element mesh. Refinement in the fillet region is performed using free mesh option and for the other regions of the gear tooth mapped meshing is employed.

Table 1 Number of elements in the Finite Element Model

MODELS

NUMBER OF ELEMENTS

NUMBER

OF NODES

NUMBER OF ELEMENTS (REFINED)

NUMBER OF NODES

(REFINED)

1-Tooth 2D 142 172 298 342

1-Tooth 3D 7384 9116 15496 18126

3-Teeth 2D 283 337 439 507

3-Teeth 3D 14716 17861 22828 26871

5-Teeth 2D 403 481 559 651

5-Teeth 3D 20956 25493 29224 34662

Figure 2 Area Model and Finite Element Model

The meshing (generation of finite elements) is initiated after the gear tooth sector is divided into regular four sided areas capable for mapped meshing. The area model and finite element model is shown in the Figure 2. The finite element mesh is generated in such a way to satisfy the following criteria so that it would reduce the overall element count in the model.

The size of the element should decrease when approaching the fillet (maximum stress region) from the addendum circle in the radial direction.

The size of the elements should increase when moving away from the fillet until middle of the rim portion and then decrease as approaching the rim surface in the radial direction.

Number of elements should be less in low stress regions. The middle of the tooth is a low stress region. More number of small size elements in the fillet region. Connectivity with the other teeth should be maintained.

Modifications in the Finite Element Model The finite element models are analyzed for the case of entire load acting at the HPSTC (Highest Point of Single Tooth Contact). To apply the load at a particular location on the involute profile there should be a node at that location. If the mesh does not have a node at this location the element that is intersected by the radius value of the HPSTC along the involute profile is determined. The edge length of the element along the profile is calculated and the arbitrary load is applied as a pressure on this edge (model has unit thickness). The following Figure 3 shows the pressure applied on the element edge, the arc of radius HPSTC intersecting the element and constraints in the rim. It can be seen that load is applied near the HPSTC but not exactly at that position. If the element size changes the position at which pressure is applied and moves further away from HPSTC, the load can be applied exactly at HPSTC by providing a node at that location. The area model in the Figure 2 is modified by providing a partition in the top portion of the tooth as shown in the Figure 4. The partition is provided at a radius of HPSTC by creating an additional key point on both the involute profiles. B spines are divided at this key point. Key points are also generated at the intersection of HPSTC arc with the lines inside the tooth profile. The resulting model is subjected to mapped meshing. The modified area and meshed area models are shown in the Figure 4 with a node at HPSTC.

Radius of HPSTC

Figure 3 Pressure Applied on the Element Edge, HPSTC Circle Arc, and Rim Constraints

FE model is analyzed by applying constraints and an arbitrary load (Figure 4) at the HPSTC. The results of the analysis showed high stress value at the point of application of load. High stress is produced at this location because load is applied along a single edge of the element. High stress at this location can be prevented by introducing a triangle element at the HPSTC, which enables distribution of the load along the slanted edges of the triangle. The apex of the triangle element coincides with the node at the HPSTC. The area model of the Figure 4 is modified in such a way to enable the generation of triangle element at this location. The modified area model is shown in the Figure 5. The analysis of this model produced a stress value, which was more than 50% lower than the stress produced by the model in Figure 4 at the node of load application. Based on the above discussion, meshed model of Figure 5 is chosen as the middle tooth for the actual analysis of 1, 3 and 5 teeth segment finite element models. The two dimensional finite element models are extruded to produce three dimensional finite element models.

HPSTC

Figure 4 Modified Area Model and Finite Element Model

Figure 5 Modified Area Model and Finite Element Model of Figure 3

Refinement in the Fillet Region ANSYS has the capability of adding more elements at any specified node or element location. The density of elements is given in terms of levels from 1 to 5. Value of one being minimal refinement at the specified location and 5 provides maximum refinement. The elements are generated in a haphazard manner and even violate shape limits. The Figure 6b shows the mesh using Ansys refinement. Shaded elements violated element shape limits. In order to avoid element shape warnings, lines in the fillet region represented by 1, 2 and 3 in Figure 6c are divided based on number of elements required in this region. Using the Free mesh option the elements are forced to form as a web (Figure 6d) and thus following a regular pattern. The division of the line segments is increased in the fillet region (Figure 6c) until the stress values in consecutive analysis is converged. The compressive side fillet region converges for 14 divisions and the tensile side converges for 6 divisions.

(a) (b)

(c) (d) (e) Figure 6 (a) Elements in the Fillet Area (b) ANSYS Refinement (c) Lines Attached to the Fillet (d) Refining by

Division of Line Segments on Compression Side (e) Refining by Division of Line Segments on Tension Side.

Results The terminology used to describe the position on profile and rim of the middle tooth is shown in the following Figure 7.

Figure 7 Terminology in the Gear Tooth

Comparison of Tooth Profile Stresses The Figure 8 shows the tooth profile stresses of the loaded tooth in one, three and five tooth segments 2-D models. T1 and T2 represent the corresponding points on the tooth profile shown in the Figure 7. In one tooth segment stress at T1 i.e., at the root, is tensile due to constraints. In three and five tooth segments stress at T1 is highly compressive due to bending of the rim and compression from the left tooth. At the nodes that are close to the addendum circle arc bending stress becomes minimum. When approaching the node at which load is applied i.e., HPSTC there is a sudden drop in the stress value. As one moves towards T2 on the profile, the stress at the fillet close to T2 is lower in magnitude when compared to the compressive stress at the fillet close to T1.

(a) (b)

(c)

Figure 8 Tooth Profile Stresses (a) One Tooth Segment (b) Three Teeth Segment (c) Five Teeth Segment

Figures 9, 10 and 11 shows the tooth profile stresses of the loaded tooth in one, three and five tooth segment 3-D models. TL and TR represent the corresponding edges on the tooth profile shown in the Figure 7. The 3D models are compared with the web nodes constrained model. One teeth segment does not show a significant change in the maximum stress value while in three and five teeth segment models the maximum stress reduces significantly due to the constraints in the rim and web interface. The stress in the nodes, located exactly above the constrained nodes in the root, reduces greatly. This effect can be seen only in the root circle and fillet because these nodes are located closer to the constrained nodes in the rim and web interface. The effect is more in the nodes located on the root than in the fillet.

(a) (b)

Figure 9 Surface Plot of Tooth Profile Bending Stress in One Tooth Model (β =1.0) (a) Without Web Constraints (b) With Web Nodes Constrained

(a) (b)

Figure 10 Surface Plot of Tooth Profile Bending Stress in Three teeth Model (β =1.0) (a) Without Web Constraints (b) With Web Nodes Constrained

(a) (b)

Figure 11 Surface Plot of Tooth Profile Bending Stress in Five teeth Model (β =1.0) (a) Without Web Constraints (b) With Web Nodes Constrained

Comparison of Rim Bottom Surface Stresses Bending stresses at the nodes along the circumference of the rim bottom edge of the tooth subjected to loading in one, three and five tooth segment are plotted in the Figure 12. In one tooth segment the stresses at R1 is compressive due to the constraints. While in three and five tooth segment models it is tensile due to the flexibility of the rim. The peak tensile stress on the rim bottom edge occurs at the nodes located radially below the compressive side fillet. For lower values of rim thickness the tensile stress in this region are as significant as the bending tensile stress at the fillet.

(a) (b)

(c)

Figure 12 Rim Bottom Surface Stresses (a) One Tooth Segment (b) Three Teeth Segment (c) Five Teeth Segment

(a) (b)

Figure 13 Surface Plot of Rim Bottom Surface Bending Stress in One Tooth Model (β =1.0) (a) Without Web Constraints (b) With Web Nodes Constrained

(a) (b)

Figure 14 Surface Plot of Rim Bottom Surface Bending Stress in Three teeth Model (β =1.0) (a) Without Web Constraints (b) With Web Nodes Constrained

.

(a) (b)

Figure 15 Surface Plot of Rim Bottom Surface Bending Stress in Five Teeth Model (β =1.0) (a) Without Web Constraints (b) With Web Nodes Constrained

Figures 13, 14 and 15 shows the rim bottom surface stresses of the loaded tooth in one, three and five teeth segment 3-D models. RL and RR represent the corresponding points on the rim bottom edge shown in the Figure 7. The 3D models are compared with the web nodes constrained model. The maximum value of the bending stress in the bottom surface of the rim with web constraints decreases when compared to the case of unconstrained rim. When moving along the face width the tensile stress on the RL side changes sign as it just reaches the constraints and follows a parabolic profile on the constrained nodes. Again as it just leaves the constraints it becomes tensile and remains tensile thereafter (symmetric). The same effect is seen in the RR side but vice versa.

Comparison of Maximum von Mises Stress The influence of rim thickness on maximum von Mises stress is summarized in the Figures 16, 17 and 18 in 2D and 3D models of one, three and five teeth segments. The von Mises stress was smaller in the case of 3D than the plane stress case. A stress difference of less than 10% was observed due to the stiffness in the axial direction, which could be accounted for lower stresses in the 3D model. These are tabulated in Table 2. The first column of the Table 2 represents the ratio of rim thickness to tooth height value.

Table 2 Comparison of 2-D and 3-D Models of 1, 3, and 5 Teeth Segments

Ratio (β)

1T -2D (σ’, psi)

1T -3D (σ’, psi)

3T-2D (σ’, psi)

3T-3D (σ’, psi)

5T-2D (σ’, psi)

5T-3D (σ’, psi)

0.5 11878 11270 19391 17574 26328 23759 0.6 11322 10696 15914 14517 20761 18882 0.7 10972 10328 13896 12736 17340 15832

0.75 10844 10192 13238 12139 16129 14752 0.875 10604 9936 12128 11119 13999 12819

1 10431 9756 11447 10479 12746 11673 1.125 10427 9748 11069 10139 12006 11014 1.25 10384 9704 10780 9870 11504 10540 1.35 10358 9678 10617 9708 11221 10274 1.4 10347 9668 10553 9640 11103 10163

1.45 10337 9658 10496 9580 10997 10063 1.5 10327 9649 10445 9526 10902 9973

0

5000

10000

15000

20000

25000

30000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim thickness to Tooth Height

Max

imum

von

Mis

es S

tres

s

Von-Mises Stress WithRefinement-2D-3TVon-Mises Stress withRefinement-2D-1TVon-Mises Stress WithRefinement-2D-5T

Figure 16 Comparison of 1, 3 and 5 Teeth Segment 2D Models

0

5000

10000

15000

20000

25000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim Thickness to Tooth height

Max

imum

Von

Mis

es S

tres

ses

Von-Mises Stress WithRefinement-3D-3TVon-Mises Stress withRefinement-3D-1TVon-Mises Stress WithRefinement-3D-5T

Figure 17 Comparison of 1, 3 and 5 Teeth Segment 3D Models Without Web Constraints

0

2000

4000

6000

8000

10000

12000

14000

16000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim thickness to tooth height

Max

imum

Von

Mis

es S

tres

s

Von-Mises Stress-1T-3D Web

Von-Mises Stress 5T 3D Web

Von-Mises Stress 3T 3D Web

Figure 18 Comparison of 1, 3 and 5 Teeth Segment 3D Models with Web Constraints

Comparison of Maximum Bending Stresses The influence of rim thickness on maximum bending tensile and compressive stress is summarized in the Figures 19, 20, 21, 22 and 23 in 2D models and 3D models with and without constraints in the web of one, three and five teeth segments.

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

10000

15000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim Thickness to the Tooth Height

Max

imum

Ben

ding

Str

esse

s

SX-2DC-5TSX-2DT-5TSX-2DC-3TSX-2DT-3TSX-2DC-1TSX-2DT-1T

Figure 19 Comparison of 1, 3, and 5 Teeth Segment 2D Models

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

10000

15000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim Thickness to the Tooth Height

Max

imum

Ben

ding

Str

esse

s

SX-3DC-5TSX-3DC-1TSX-3DC-3TSX-3DT-5TSX-3DT-1TSX-3DT-3T

Figure 20 Comparison of 1, 3, and 5 Teeth Segment 3D Models Without Web Constraints

Figure 21 Comparison of 1, 3, and 5 Teeth Segment 3D Models with Web Constraints

-20000

-15000

-10000

-5000

0

5000

10000

15000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim Thickness to the Tooth Height

Max

imum

Ben

ding

Str

esse

s

SX-3DC-web-3TSX-3DT-web-3TSX-3DC-3TSX-3DT-3T

Figure 22 Comparison of 3D Models With and Without Web Constraints in 3 Teeth Segment

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

10000

15000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Ratio of Rim Thickness to the Tooth Height

Max

imum

Ben

ding

Str

esse

s

SX-3DT-web-5TSX-3DC-5TSX-3DC-web-5TSX-3DT-5T

Figure 23 Comparison of 3D Models With and Without Web Constraints in Five Teeth Segment.

A comparison has been made to study the effect of rim thickness on maximum bending stresses in 1, 3 and 5 tooth segments for the models without web and with web. It can be seen from figures 22 and 23 that including the web for lower values of rim thickness in designing the gears could be more beneficial. Table 2 presents the transition β value of each case at which the gear maximum bending stresses does not vary significantly.

Table 3 Transition Value of β

Without Web Constraints With Web Constraints Number of Teeth in Segment

Tensile Compressive Tensile Compressive

One Tooth Segment 0.65 0.7 - -

Three Teeth Segment 0.75 1.4 0.7 1.1

Five Teeth Segment 0.8 1.5 0.65 1.15

Summary and Conclusion Finite element based approach was used to investigate the effect of the gear rim thickness on the tooth bending stresses in large spur gears. A program was developed using ANSYS Parametric Design Language (APDL), which can generate two dimensional or three dimensional finite element models of 1, 3 or 5 teeth segment with user defined rim thickness value. Models were constrained on the radial sides in the rim portion and also on the nodes located circumferentially along the bottom surface at the rim and web interface. The models are studied for the case of full load acting at Highest Point of Single Tooth Contact (HPSTC). A different meshing approach was developed in the generation of finite element grid. Bending stresses on the tooth profile and rim bottom surface are studied on the tooth subjected to loading for 3 values of β and the effect of rim thickness on maximum value of bending stress is studied for 12 different values of β in 1, 3 and 5 teeth segments. The tooth profile and rim bottom surface bending stresses in the 2D and 3D models for all values of rim thickness were almost equal. In 2D and 3D models the bending stresses at the nodes located radially below the compressive side tooth fillet in the rim bottom edge or surface are tensile. For lower values of rim thickness in five teeth segment models, these stresses are as significant as the maximum bending tensile stress at the fillet. The 3D models are also studied for the effect of web. A web of 2 inch thickness was considered and to include that effect the nodes at the rim and web interface at 1 inch distance on either side from the middle of the face width were completely constrained. The bending stresses in these models were also compared with the 3D models without constraints at the rim and web interface. One tooth segment did not show significant effect in the tooth bending stress values. In three and five teeth segments presence of web for lower values of rim thickness reduces the maximum bending stress significantly. Effect of rim thickness was studied on maximum von Mises stress and maximum bending stress. Results showed that in the one tooth segment there was no much variation in the location of maximum bending stress and also the stress value with the change in rim thickness. Five teeth segment showed higher bending stress than the three teeth segment due to more flexibility of the rim. In five teeth segment for lowest value of rim thickness the location of maximum bending tensile stress was at the root on left end constraints and as the rim thickness increased it was on the tensile side of the loaded tooth. In all the cases the location of maximum bending stress was closer to the root for lower values of rim thickness and as the rim thickness increased, this location moves up along the fillet. Through the investigations and results obtained, the main conclusions can be summarized as follows:

A difference of less than 10% was observed in the maximum von Mises stress values between two dimensional and three dimensional results, which can be accounted to the stiffness in the 3D models.

Analyses were performed to find the transition value of β at which the bending stress becomes stable in a large spur gear.

For high rim thickness, constraining the web nodes has a little effect on the bending stress. For low values of rim thickness, the presence of web reduces the maximum stress value. So including the web for lower values of rim thickness is prudent.

In this study the nodes located at the rim-web interface are completely constrained, which is conservative. As a future work the web may be included in the model. A rim-web interface of 2 inch thickness was considered in this study, effect of varying the web thickness may be studied.

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