CHAPTER 7 FINITE ELEMENT ANALYSIS OF SOLID ROLLER …shodhganga.inflibnet.ac.in/bitstream/10603/17483/12/12_chapter 7.pdf · CHAPTER 7 FINITE ELEMENT ANALYSIS OF SOLID ... analysis

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CHAPTER 7

FINITE ELEMENT ANALYSIS OF SOLID

ROLLER BEARING

7.1 MODELING AND MESHING

The two dimensional finite element model of the roller bearing of

specifications given in the Table 4.2 (Chapter 4) is developed for solid roller

bearing in ANSYS finite element tool. Because of the symmetrical shape of

the bearing, half of the model is taken for analysis. The roller is modeled with

four symmetrical quarter circles of radius 7.5 mm and glued together to get

symmetrical meshing of the complete model. The inner race and the shaft are

combined in the model and are modeled with two symmetrical quarter circles

of radius 41.75 mm and are glued together as shown in Figure 7.1. The outer

race is modeled with two partial quarter annulus of inner of 56.75 mm and

outer radius of 62.5 mm and glued together. The models of rollers, inner race

and outer race are meshed with PLANE42 elements with the roller effective

thickness of 20.4 mm as shown in Figure 7.2. All the bearing elements are

meshed with the PLANE42 elements of global element edge length of 0.4 mm

as shown in Figure 7.3 to get accurate results. Since the analysis is extended

for contact analysis, the elements at the contact interface between inner race

and roller and the interface between outer race and roller are refined with the

element edge length of 0.04mm as shown in Figure 7.4. The element can be

used either as a plane element (plane stress or plane strain) or as an

axisymmetric element (Kogut and Etsion 2003). The element is defined by

four nodes having two degrees of freedom at each node: translations in the

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nodal X and Y directions. The element has plasticity, creep, swelling, stress

stiffening, large deflection, and large strain capabilities. The option available

to suppress the extra displacement shapes is used in the meshed model.

Figure 7.1 2D model of the solid cylindrical roller bearing

Figure 7.2 Meshed model of the solid cylindrical roller bearing

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Figure 7.3 Meshed model of solid cylindrical roller

Figure 7.4 Refined mesh at contact interface

138

The geometry, node locations, and the coordinate system for

PLANE42 element are shown in Figure 7.5. The element input data includes

four nodes, a thickness and the orthotropic material properties. Orthotropic

material directions correspond to the element coordinate directions. Surface

loads are designated by a label and a key. The label indicates the type of

surface load and the key indicates where on the element the load acts. For

PLANE42 element type, the surface load list of "Pressure: face 1 (J-I), face 2

(K-J), face 3 (L-K), face 4 (I-L)" indicates that pressure loads are available on

4 faces: the line from node J to node I defines the element's face 1 (identified

on surface load commands with key = 1), and K-J (key = 2), L-K (key = 3),

and IL (key = 4). The nodal forces, if any, should be input per unit of depth

for a plane analysis (except for KEYOPT(3) = 3) and on a full 360° basis for

an axisymmetric analysis. KEYOPT(2) is used to include or suppress the

extra displacement shapes.

Figure 7.5 Geometry of the element PLANE42

The material properties of the PLANE42 elements like Young’s

modulus and Poisson’s ratio are defined for the meshed models as 2.3E5

N/mm2 and 0.3 respectively. Kogut and Etsion (2002) recommended mesh

refinement, and the mesh in the current analysis is orders of magnitude more

refine, as necessitated by mesh convergence.

X

Y

Element coordinate system(shown for KEYOPT(1)=1)

1

2

4

3

I

J

KL

X

Y

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7.2 PARAMETERS FOR CONTACT ANALYSIS

The multi-body contact phenomena are used to perform the contact

analysis of the roller bearing so as to ensure local iteration process converged

and increase the computational efficiency (Werner Schiehlen et al 2006). Two

contact pairs are developed in the FE model for inner race contact and outer

race contact by using contact manager of the tool. Contact elements are

modeled at the contact lines of inner race, roller and outer race. For the inner

race contact, the curved lines of inner race are taken as target surface and

modeled by TARG169 elements and the curved lines at top of the rollers are

taken as contact surface and modeled by CONTA172 elements. Similarly for

the outer race contact, the curved lines of outer race are taken as target surface

and modeled by TARGE169 elements and the curved lines at bottom of the

rollers are taken as contact surface and modeled by CONTA172 elements as

shown in Figure 7.6. The mash is constructed using 1287 contact elements at

the inner race contact and 1451 contact elements at the outer race contact.

CONTA172 is used to represent contact and sliding between 2D

"target" surfaces and a deformable surface, defined by this element. The

elements are located on the surfaces of 2D solid elements with mid side

nodes. It has the same geometric characteristics as the solid element face with

which it is connected. Contact occurs when the element surface penetrates one

of the target segment elements on a specified target surface. Coulomb friction

coefficient is given as 0.3 for steel roller contact with steel inner and outer

races (Prasanta Sahoo et al 2010). The geometry and node location of the

CONTA172 element is shown in Figure 7.7.

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Figure 7.6 Contact elements at inner and outer race interfaces

Figure 7.7 Geometry of the element CONTA172

The element is defined by three nodes (the underlying solid element

has mid side nodes). The element X-axis is along the I-J line of the element.

The 2D contact surface elements are associated with the 2D target segment

elements TARGE169 via a shared real constant set. For either rigid-flexible or

X

Y

Surface of Solid element

Contact element

Associated Target surface

Contact normal

J I

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flexible-flexible contact, one of the deformable surfaces must be represented

by a contact surface. This element supports various 2D stress states, including

plane stress, plane strain, and axisymmetric states. The stress state is

automatically detected according to the stress state of the underlying element.

TARGE169 is used to represent various 2D "target" surfaces for the

associated contact elements CONTA172. The contact elements themselves

overlay the solid elements describing the boundary of a deformable body and

are potentially in contact with the target surface, defined by TARGE169. This

target surface is discretized by a set of target segment elements and is paired

with its associated contact surface via a shared real constant set. It can be

imposed any translational or rotational displacement on the target segment

element. Also it can be imposed forces and moments on target elements. For

flexible targets, these elements will overlay the solid elements describing the

boundary of the deformable target body (Stolarski et al 2006).

Figure 7.8 Geometry of the element TARGE169

The target surface is modeled through a set of target segments;

typically, several target segments comprise one target surface. The target and

associated contact surfaces are identified by a shared real constant set. This

real constant set includes all real constants for both the target and contact

Surface-to-SurfaceContact Element

Node-to-Surface ContactElement

M LN

M

K

I JK

n

n

I Jn

Target Segment

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elements. Each target surface can be associated with only one contact surface,

and vice-versa. However, several contact elements could make up the contact

surface and thus come in contact with the same target surface. Likewise,

several target elements could make up the target surface and thus come in

contact with the same contact surface. The nodes are ordered so that, for a

2D surface, the associated contact elements CONTA172 must lie to the right

of the target surface when moving from target node I to target node J. The

mesh has extensively been verified for model convergence (Jackson et al

2005 and Quicksall et al 2004).

7.3 LOAD AND BOUNDARY CONDITIONS

The boundary conditions are applied such that, the displacement of

outside surface of the outer race is arrested in all directions and the

displacement of top surface of the inner race is arrested only in X direction.

The displacement of inner race in Y direction is kept free to get bearing

deformation (Biqiang Xu and Yanyao Jiang 2002). The rollers are positioned

between the inner and outer races and the degrees of freedom are kept free.

The radial load is applied on the top surface of the inner race as distributed

vertical downward load as shown in Figure 7.9. This boundary condition may

be valid for the modeling of asperity contacts for two reasons. One of the

reason is the asperities are actually connected to a much larger bulk material

at the base and will be significantly restrained there, and another reason is the

high stress region occurs near the contact, the boundary condition at the base

of the hemisphere will not greatly affect the solution because of Saint

Venant’s Principle. Since the problem is nonlinear, small load steps are used

to increment toward a solution in loading. It is important to assign a large

value of stiffness for these contact elements so that negligible penetration

occurs between the surfaces. However, using too high of a stiffness can result

in convergence problems. This work uses a stiffness that is approximately the

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elastic modulus multiplied by a characteristic length (approximately the

contact radius of the problem).

Figure 7.9 Load and boundary conditions of the FE model

In addition, if the penetration is greater than a defined value

(tolerance), the Lagrangian multiplier method is used. This ensures that the

penetration of the converged solution is less than the assigned tolerance. The

tolerance of the current work is set to 1% of the element width. The contact

elements thus apply forces to the nodes of the elements that are in contact

(Bourdon et al 1999b). There are two ways to simulate the contact problem.

The first applies a force to the rigid body and then computes the resulting

displacement. The second applies a displacement and then computes the

resulting contact force. In both methods, the displacement, stress, and strain in

the elastic body can be determined, as well as the contact pressure. In this

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work the latter approach is used, where the base nodes of the hemisphere are

displaced a distance approaching the target surface. This method is used

because the resulting solution converges more rapidly than the former. An

iterative scheme is used to solve for the solution, and many load steps are

used to enhance solution convergence (Cui et al 2008). The transient mode of

analysis is selected for various load steps and the results of roller deformation,

bending stress, and contact parameters are obtained for solid roller bearing.

7.4 FEA RESULTS OF SOLID ROLLER BEARING

Finite Element Analysis (FEA) is carried out on the solid roller

bearing model. By considering the experimental results, the critical load is

maximum for 17 numbers of rollers and the cage slip is maximum at 135 N

load and at 2000 rpm shaft speed. Hence the 17 number of roller bearing is

taken for finite element analysis. Initially the rollers are positioned so that one

of the rollers is placed exactly at the load line of the bearing and the other

rollers are placed with equal angular positions of 21.176o at both the sides of

the loaded roller as shown in Figure 7.10. The load and boundary conditions

of the bearing are applied and the results of deformation, contact status and

contact pressure are obtained. Then the rollers are positioned so that the

loaded roller is placed at the angular position with the increment of 1o from

the previous position and the other rollers are placed accordingly. The same

procedure is used until the angular position of roller reaches 10.588o, so that

two rollers are loaded and are equally spaced from the load line. The bearing

deformation for various angular positions of the rollers is obtained by finite

element analysis and the sample results are given in Table 7.1. The contact

status which includes far open, near contact, sliding, and sticking region of

roller positioned at equal angle of 10.588o from load line is shown in

Figure 7.11.

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The bearing deformation against the applied radial load for various

angular positions of rollers is plotted as shown in Figure 7.12. It shows that

the increased radial load increases the deformation of bearing for any given

angular positions of the roller. It is observed that the deformation of bearing

for the roller position at 0, 1, 2, 3, 4 and 5o are same for any given radial load.

The deformation of bearing for the roller position of 6o is reduced

significantly from the previous value. Hence it is concluded that, for the given

radial clearance of 0.05 mm, only one roller is loaded for the angular position

ranging from 0o to 5o from the load line. And maximum deformation is

obtained at these positions. When the angular position of the roller reached 6o,

two rollers located near the load line carries the bearing load and the

deformation is decreased because the applied load is shared by two rollers.

The bearing deformation is further reduced for the same applied loads with

increasing angular position of the roller from the load line because of the load

distributed to two adjacent rollers. The roller located near to the load line

carries more load than the other roller. It is also observed that the bearing

attains minimum deformation for the given radial loads at 10.588o because the

rollers are equally spaced and equally shares the bearing load. Hence the

specific roller load which is defined as the load acting on individual roller is

decreased and the deformation is correspondingly decreased with increasing

the angular position of the roller from load line. The maximum and minimum

deformation of bearing for the applied load of 1000 N is observed as 0.0162

and 0.0085 mm respectively. By neglecting the inertia forces, the bearing run

out for 1000 N load is expected as 0.0077 mm which is the difference

between maximum and minimum deformations (Harsha 2006). It is also

observed that the run out of the bearing is decreased with decreasing the

radial load.

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Figure 7.10 Position of rollers in the FE model

Table 7.1 Deformation at various angular position of roller

Radialload, N

Deformation, mm0 to 5deg 6 deg 7 deg 8 deg 9 deg 10 deg 10.588

deg0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

104.88 0.0053 0.0039 0.0037 0.0034 0.0032 0.0029 0.0028213.25 0.0076 0.0056 0.0052 0.0049 0.0046 0.0042 0.0039324.88 0.0094 0.0069 0.0065 0.0060 0.0057 0.0052 0.0048423.97 0.0107 0.0078 0.0074 0.0069 0.0065 0.0059 0.0055536.23 0.0121 0.0088 0.0083 0.0078 0.0073 0.0066 0.0062618.39 0.0130 0.0095 0.0089 0.0083 0.0078 0.0071 0.0067706.40 0.0139 0.0101 0.0095 0.0089 0.0083 0.0076 0.0071800.26 0.0147 0.0108 0.0101 0.0095 0.0089 0.0081 0.0076952.03 0.0161 0.0117 0.0110 0.0103 0.0097 0.0088 0.00831060.5 0.0170 0.0124 0.0116 0.0109 0.0102 0.0093 0.0087

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Figure 7.11 Contact status of bearing model

0

0.0018

0.0036

0.0054

0.0072

0.009

0.0108

0.0126

0.0144

0.0162

0.018

0 100 200 300 400 500 600 700 800 900 1000Radial load, N

10.588 deg

10 deg

8 deg9 deg

7 deg6 deg

0 to 5 deg

Figure 7.12 Deformation of bearing at various angular position of roller

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Figure 7.13 Contact pressure of the loaded roller (Radial load – 135 N)

Table 7.2 Contact pressure of loaded roller at various angular position

Radialload, N

Contact pressure, MPa0 to 5deg 6 deg 7 deg 8 deg 9 deg 10 deg 10.588

deg0 0.00 0.00 0.00 0.00 0.00 0.00 0.00

104.88 167.54 142.62 137.72 133.29 128.50 122.85 117.49213.25 238.90 203.37 196.38 190.06 183.23 175.17 167.53324.88 294.87 251.01 242.39 234.59 226.16 216.21 206.78423.97 336.85 286.75 276.89 267.99 258.36 246.99 236.22536.23 378.83 322.49 311.40 301.38 290.55 277.78 265.66618.39 406.82 346.31 334.41 323.65 312.02 298.30 285.28706.40 434.81 370.13 357.41 345.91 333.48 318.82 304.91800.26 462.80 393.96 380.42 368.18 354.95 339.34 324.54952.03 504.78 429.69 414.92 401.58 387.14 370.12 353.971060.5 532.76 453.52 437.93 423.84 408.61 390.64 373.60

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The contact pressure distribution of the loaded rollers when the

rollers are equally spaced from the load line and for the minimum radial load

of 135 N at where the maximum slip occurs is shown in Figure 7.13. It shows

that the maximum contact pressure of the roller is observed as 137.827 MPa

and the bearing deformation is 0.0033 mm for the radial load of 135 N. The

finite element results of contact pressure for various angular positions of the

rollers are obtained and are given in Table 7.2. The maximum contact

pressure of loaded roller is plotted against the applied radial load as shown in

Figure 7.14. It shows that the contact pressure is increased with increasing

radial load for any given angular position. The contact pressure for the roller

positions at 0o, 1o, 2o, 3o, 4o and 5o is same and maximum for the given radial

load. And the contact pressure is decreased with increasing angular position

of the rollers at any given radial load. It is observed that, only one roller is

loaded at the angular position between 0o and 5o and the specific load on

roller is maximum at these positions hence the contact pressure is

correspondingly increased and attains maximum at these positions. For the

angular positions ranging from 6o to 10.588o, the contact pressure is gradually

decreased for any given radial load because of two rollers sharing the total

bearing load results to decrease in specific load on roller. The minimum

contact pressure is obtained at the angular position of roller is 10.588o because

two rollers are equally spaced and equally shares the total bearing load at this

position. Hence the specific radial load on roller is minimum at the angular

position of 10.588o for any given radial load resulting in decreased contact

pressure. It is noted that the contact pressure for the critical load of 765 N and

the minimum radial load of 135 N is 310 MPa and 137 Mpa respectively.

Hence it is concluded that the cage slip is initiated at where the contact

pressure of the roller at inner race contact is equal to 310 MPa. Decreasing

contact pressure increases the cage slip and the cage slip attains maximum at

the contact pressure of 317 MPa.

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Table 7.3 Contact pressure for specific loads

Angularposition ofroller, deg

Contact pressure, MPa

135 N 300 N 500 N 765 N

0 190.08 283.36 365.81 452.481 190.10 283.39 365.85 452.532 190.14 283.44 365.92 452.623 190.21 283.56 366.07 452.804 190.31 283.70 366.25 453.035 190.44 283.90 366.51 453.356 161.81 241.21 311.40 385.187 156.25 232.92 300.70 371.948 151.22 225.43 291.02 359.989 145.79 217.32 280.57 347.04

10 139.37 207.77 268.23 331.7810.588 133.30 198.70 256.53 317.3111.176 139.37 207.77 268.23 331.7812.176 145.79 217.32 280.57 347.0413.176 151.22 225.43 291.02 359.9814.176 156.25 232.92 300.70 371.9415.176 161.81 241.21 311.40 385.1816.176 190.44 283.90 366.51 453.3517.176 190.31 283.70 366.25 453.0318.176 190.21 283.56 366.07 452.8019.176 190.14 283.44 365.92 452.6220.176 190.10 283.39 365.85 452.5321.176 190.08 283.36 365.81 452.48

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Table 7.4 Semi contact width of loaded roller at various angular position

Radialload,

N

Semi contact width (b), mm0 to 5deg 6 deg 7 deg 8 deg 9 deg 10 deg 10.588

deg0.01 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

104.88 0.0195 0.0166 0.0161 0.0155 0.0150 0.0143 0.0137213.25 0.0279 0.0237 0.0229 0.0222 0.0214 0.0204 0.0195324.88 0.0344 0.0293 0.0283 0.0274 0.0264 0.0252 0.0241423.97 0.0393 0.0335 0.0323 0.0313 0.0301 0.0288 0.0276536.23 0.0442 0.0376 0.0363 0.0352 0.0339 0.0324 0.0310618.39 0.0475 0.0404 0.0390 0.0378 0.0364 0.0348 0.0333706.40 0.0507 0.0432 0.0417 0.0404 0.0389 0.0372 0.0356800.26 0.0540 0.0460 0.0444 0.0430 0.0414 0.0396 0.0379952.03 0.0589 0.0501 0.0484 0.0468 0.0452 0.0432 0.04131060.5 0.0622 0.0529 0.0511 0.0494 0.0477 0.0456 0.0436

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700 800 900 1000Radial load, N

10.588 deg10 deg

8 deg9 deg

7 deg6 deg

0 to 5 deg

Figure 7.14 Maximum contact pressure of loaded roller at variousangular position

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135 N

300 N

500 N

765 N

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16 18 20 22Angular position, deg

Figure 7.15 Contact pressure of roller at various angular position

The contact pressure for specific loads like 135 N (shaft self

weight), 300 N, 500 N and 765 N (critical load for 17 number of rollers) is

obtained for the roller located at the angular position ranging from 0o to

21.176o. The finite element analysis results of contact pressure of loaded

roller at various angular positions and specific radial loads are given in

Table 7.3. When two rollers are located at different angular positions from the

load line and are loaded, the roller located near to the load line carries more

load than the load carried by other roller located at the other side of the load

line. The contact pressure in the roller nearer to load line is also

correspondingly higher than other roller. Both the rollers carry equal load and

contact pressure only at the angular position of 10.588o. The maximum

contact pressure occurring in the heavily loaded roller is taken for discussion.

The maximum contact pressure of the heavily loaded roller for the specified

loads are plotted against the angular position as shown in Figure 7.15. It is

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observed that the contact pressure is almost constant at the angular position

ranging from 0o to 5o. It shows that only one roller is loaded and the specific

load acting on the roller is also same at this range of angular position. The

contact pressure is decreased at the angular position between 5o and 6o for any

given radial load. The contact pressure is further decreased at the angular

position ranging from 6o to 10o. It is noted that the contact pressure is

minimum for any given radial load at 10.588o where the two successive

rollers are located at equally spaced angular position and carries equal but

minimum radial load. The pattern of the curve is symmetrical about the

angular position of 10.588o for any given radial load. It shows that further

increasing the angle increases the contact pressure because the successive

roller moves near to the load line and carries more specific load than previous

roller.

0

0.0065

0.013

0.0195

0.026

0.0325

0.039

0.0455

0.052

0.0585

0.065

0 100 200 300 400 500 600 700 800 900 1000Radial load, N

10.588 deg10 deg

8 deg9 deg

7 deg6 deg

0 to 5 deg

Figure 7.16 Semi contact width of loaded roller

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The contact pressure attains maximum value at the angular position

ranging from 16.176o to 21.176o because again only one roller carries the

entire radial load of the bearing. At this angular position, the maximum

contact pressure occurs in the successive roller. The contact pressure for

minimum radial load of 135 N and the critical load of 765 N is noted as 130

MPa and 317 MPa respectively. Hence it is concluded that the contact

pressure is minimum at the angular position where two successive rollers are

equally spaced with the load line.

max

2 'rQb P (7.1)

The semi contact width at inner race contact of the loaded roller is

theoretically calculated by substituting the applied radial load and maximum

contact pressure in the equation (7.1). In general the contact width of roller at

inner race contact is always less than the contact width of roller at outer race

contact because; the effective radius at inner race contact is greater than the

outer race contact (Jackson et al 2005). Because of focusing inner raceway

contact, the semi contact width of roller at inner race contact is taken for

discussion. The calculated values of semi contact width (Table 7.4) for

various angular positions of the rollers are plotted against the applied radial

load as shown in Figure 7.16. The curves demonstrate that the semi contact

width increases with increasing the radial load. It is observed that the semi

contact width of the loaded roller at the angular positions between 0o and 5o

are same and attains maximum value. This is true for any given radial load.

The semi contact width is gradually decreased with increasing the angular

position from the load line and reaches minimum value at the angular position

of 10.588o. Again it is proved that, only one roller is loaded at the angular

positions ranging from 0o to 5o and two rollers are loaded at the angular

positions ranging from 6o to 10.588o. The semi contact width is reduced and

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attains minimum value for any given radial load at the angular position of the

roller 10.588o because of the decreased specific load of roller. It is also noted

that the semi contact width for the critical load of 765 N and the minimum

radial load of 135 N is 0.0364 mm and 0.0156 mm respectively. From these

results, it can be concluded that, the cage slip is initiated when the semi

contact width is equal to 0.0364 mm and decreasing the semi contact width

increases the cage slip. The conclusion based on contact width is more

consistent than the conclusion based on contact pressure for any geometrical

conditions of the cylindrical roller bearing. In some specific geometrical

conditions like hollow roller in contact, increasing radial load increases the

contact width. But increasing radial load decreases the contact pressure

because of the increased contact width.