Prediction of Fracture Loads for Gray Iron Bearing Housings

Patrick Tibbits, Ph.D. Life Fitness

Abstract

Gray cast iron exhibits very low notch sensitivity. A notched tensile specimen breaks at approximately the same load as a smooth specimen. Under the isotropic and homogeneous assumptions of continuum mechanics, the calculated stresses for notched specimens far exceed stresses calculated for un-notched specimens. Stress-based fracture criteria for brittle materials, including the maximum normal stress and modified Mohr criteria, therefore badly under-predict the breaking strength of notched cast iron specimens. Adjusting the breaking strength by a factor equal to the stress concentration factor matches the observed breaking load for the notched tensile specimen.

Calculation of the stress concentration factor requires taking the ratio of the maximum stress to the nominal stress. For the notched tensile specimen, nominal stress amounts to just the load divided by the minimum cross sectional area of the specimen. For the notched tensile specimen geometry, handbook values exist for stress concentrations for linear elastic materials. However, the nonlinear stress/strain curve for cast iron requires calculation of maximum stress by FEA.

For the more complex geometry of manufactured parts, such as a roller bearing housing, FEA suffices to calculate maximum stress, but no clear definition suggests itself for the nominal stress. This study describes a technique for estimation of nominal stress from a path plot of stress versus location across the critical cross section of a roller bearing housing. The estimated stress concentration factor then amounts to the ratio of maximum stress calculated in FEA to the estimated nominal stress. Estimated fracture loads then get calculated from FEA maximum stress compared to stress-based fracture criteria, and then multiplied by the stress concentration factor. The estimated fracture loads match experimentally determined fracture strengths for two sizes of housings with less than 10% error for two housing sizes, and match within 20% for a third housing size.

Introduction This study seeks to predict fracture load under monotonic loading, with no fatigue effects. A shaft, passing through a bearing which in turn gets pressed into a pillow block, applies the load upward, away from the base of the pillow block.

The pillow blocks consist of class 30 gray iron. Handbooks include stress-strain curves for this material in both compression and tension, as well as tensile and compressive strengths.

This study first reviews fracture criteria for brittle materials, then reviews the nonlinear stress-strain behavior of gray iron. Next a model of a gray iron notched tensile specimen demonstrates the notch insensitivity of gray iron. Discussion of notch insensitivity explains why typical fracture criteria for brittle materials underestimate fracture loads, and suggests a method to improve the prediction.

The study then reports the modeling and loading of several sizes of gray iron pillow blocks, as well as applying the method for fracture load prediction. The study gives special attention to estimation of stress concentration factor.

Tests provided fracture loads for three sizes of pillow block. This experimental data allows evaluation of the effectiveness of the method for fracture load prediction.

Fracture Criteria for Brittle Materials The maximum normal stress criterion equates the ultimate tensile strength of a material to the maximum, in absolute value, of the principal stresses (Reference 2).

|)||,||,max(|321 σσσσ =u ,

where subscript u denotes ultimate tensile strength, and subscripts 1, 2, and 3 denote principal stresses. The max normal stress criterion accurately predicts fracture in most brittle materials when the principal stress with largest magnitude is tensile.

Deviations from the max normal stress criterion arise when the tensile and compressive strengths of the material differ. For gray cast iron (Reference 1),

utuc σσ *3> ,

where subscript c denotes compression and t denotes tension. Figure 1 below plots, in a two-dimensional principal stress space, three fracture criteria for brittle materials (Reference 1). The figure shows as a rectangle the max normal stress criterion, adjusted for |||| utuc σσ > . Experimental points appear as circles.

Another kind of deviation from the max normal stress criterion arises when the normal stress with largest magnitude is compressive. The experimental points for this condition lie within the max normal stress rectangle. Gray cast iron can fracture at lower stress levels than predicted by the max normal stress criterion.

Figure 1 below shows two additional criteria for fracture of brittle materials. These criteria appear as lines cutting off two corners of the max normal stress criterion rectangle. The Modified Mohr fracture criterion better matches the experimental data.

Figure 1

The modified Mohr fracture criterion consists of the equation (Reference 3):

),,,,,max( 321321 σσσσ CCCut =

where

)](*|[|*)1(* 21211

σσσσσ

σ++−

−= m

mC

uc

ut

)](*|[|*)1(* 32322 σσσσ

σσ

++−−

= mm

Cuc

ut

)](*|[|*)1(* 13133 σσσσ

σσ

++−−

= mm

Cuc

ut

and

uc

utmσσ*21+= .

In other words, fracture occurs when any of the stresses 321321 ,,,,, σσσCCC equal the ultimate tensile

strength. Appendix I lists the APDL commands to calculate the modified Mohr fracture criterion from the stress results of an ANSYS model.

Gray Iron Stress-Strain Curve Figure 2 below shows the stress-strain curve for a class 30 gray cast iron. Class 30 refers to an iron having ultimate tensile strength or UTS = 30 ksi. The blue curve arises from ASM data (Reference 4). Magenta crosses show points on a spline fitted through the ASM data. The ANSYS manual contains APDL statements to define a class 30 gray iron stress-strain curve, and the red circles show those data points.

Appendix II lists the APDL which defines the stress-strain curve input to the ANSYS simulations for this study. The input stress-strain curve included the tensile points from the ASM data and the compression points from the ANSYS data.

The choice of ASM stress-strain data for the tensile curve, rather than the data from the ANSYS manual arose from the more conservative UTS of the ASM curve.

Figure 2

Salient features of the stress-strain curve include the inequality of tensile and compressive strength, and the nonlinear behavior except where strain remains < 0.001 or so.

Notch Insensitivity of Gray Iron

Figure 3 below shows schematically a notched and a smooth tensile specimen. The specimens have equal minimum area normal to the applied load, and would therefore experience equal nominal stress at equal applied loads, PnP = . For this study D = 1 inch, and r = 0.1 inch. The notch profile consists of a semi-circle.

Figure 3

Figure 4 below shows contours of the first principal stress for the notched tensile specimen. Areas in red, the stress concentrations, experience about 29 ksi, shading to blue areas with near zero stress. Values for

321 ,, CCC , calculated according to the modified Mohr fracture criterion, lie below the value for 1σ , so

the criterion predicts that the specimen will break when 1σ ~ 30 ksi.

Figure 4

At a given load the stress concentrations have higher stress than the nominal stress, where )/()( aMinimumAreLoadnom =σ . The un-notched or smooth tensile specimen experiences this

stress, )2*/()*4( DPnom πσ = . Figure 5 below shows, as a function of applied load, 1σ for the notched and smooth tensile specimens of Figure 3. Assuming UTS = 29 ksi, Figure 5 indicates that according to the modified Mohr criterion, the notched tensile specimen will break at a load nP = 16,000 lbf, and the smooth specimen will break at a load P = 23,500 lbf.

Figure 5

In experiments, notched and smooth tensile specimens of gray iron break at approximately the same load,

nP = P. The nonlinear stress analysis performed for the notched specimen, and interpretation of its results through the modified Mohr fracture criterion fail to predict the fracture load for the notched specimen. The low notch sensitivity of gray iron explains this discrepancy.

Notch sensitivity,

11

−−

=tkfkq ,

where nf PPk /= and nomtk σσ /= . The notch strength reduction factor, fk , just consists of the ratio of

smooth to notched specimen fracture loads, and the stress concentration factor tk consists of the ratio of

maximum stress in the notched specimen to the stress in the smooth specimen. When tfkk = , the notch

reduces the fracture load by the same factor by which it concentrates stress. When tfkk = , q = 1, and the

material has full notch sensitivity. When the notch has no effect on fracture load, 1=fk . When 1=

fk , q

= 0, and the material has no notch sensitivity. For gray iron, 0 < q < 0.2, so gray iron has very low notch sensitivity (Reference 5).

The graphite flakes in gray iron cause its low notch sensitivity. Figure 6 shows a gray iron microstructure at high magnification (Reference 7). In tension, the microscopic dark graphite flakes behave as cracks and concentrate stresses more effectively than a macroscopic notch.

Figure 6

Stress Concentration Factor for the Notched Tensile Specimen The decreasing slope, or modulus, of the tensile gray iron stress-strain curve means that as stress level increases, stiffness decreases. Load follows stiffness, so additional load tends to get carried by more lowly stressed areas of the cast iron specimen, lessening the stress concentration factor. Figure 7 below shows the stress concentration factor nomtk σσ /

1= , calculated as a function of load from the two curves of Figure 5.

Figure 7

Linear elastic calculation (Reference 6) of the stress concentration factor provides the estimate tlek = 2.2.

tlek lies very near the value for the first data point in Figure 7. At the very low load at the first data point, the cast iron has nearly constant modulus, so the linear elastic approximation holds.

Prediction of Fracture Load for the Notched Tensile Specimen

nP denotes the fracture load predicted by the modified Mohr criterion for the notched specimen. At an

applied load equal to nP , the stress concentration factor has value tk ~ 1.4. The similarity of tk to the ratio

of fracture loads nPP / = 23,500/16,000 = 1.47 suggests that, in the absence of notch sensitivity,

increasing the predicted fracture load by the factor tk can predict fracture loads, i.e. ntf PkP *= .

A General Method for Prediction of Fracture Load The experience with the notched tensile specimen suggests a method to estimate fracture loads for gray iron components.

In the procedure below, mMσ denotes ),,,,,max( 321321 σσσCCC , the stress calculated for the modified

Mohr fracture criterion. mMP denotes the fracture load predicted by the modified Mohr criterion.

Procedure for Estimating Fracture Load

1 - Define gray iron stress-strain curve.

2 - Construct and mesh the ANSYS model.

3 - Apply loads and constraints.

4 - Obtain stress results. Save the results for each sub-step.

5 - From the stresses for each sub-step, calculate mMσ .

6 - Estimate mMP , the load at which utmM σσ = .

7 - For an applied load of mMP , estimate tk .

8 - Estimate fracture load, mMtf PkP *= .

For the notched tensile specimen, estimating tk just required taking the ratio of the maximum 1σ for the

notched specimen to the nominal stress for the smooth specimen. Estimating tk presents a problem for geometries with no readily apparent calculation of nominal stress. A later section of this report presents a procedure for estimating tk from an ANSYS path plot.

Gray Iron Pillow Block Figure 8 shows a half-symmetry CAD model of a gray iron pillow block. Bolts on each side lock the pillow block to a flat surface bearing against the bottom of the pillow block

Figure 8

Loads on the Pillow Block Figure 9 shows a side view of the ANSYS model of the pillow block. Roller loads get distributed over nodes near each of the circumferential positions on the inside diameter of the pillow block at which rollers make contact. Bolt loads, which sum to the negative of the vertical components of the roller loads, get distributed over nodes near the bolt holes at each side of the pillow block.

Figure 9

The vertically upward direction of loading for this study lies opposite to the conventional downward loading applied to pillow blocks. The bearing industry denotes this loading as a “cap load”. Investigating maximum safe cap loads motivated the study.

Stress Results for the Pillow Block Figure 10 shows contours of the modified Mohr fracture criterion stress, mMσ , for the gray iron pillow block. The critical location has the maximum value of mMσ .

Figure 10

Estimation of PmM for the Pillow Block Figure 11 shows values for mMσ calculated for each substep. Interpolating between mMσ values for the

substeps whose values for mMσ bracket ut

σ provides an estimate for mMP .

Figure 11

Estimation of kt for the Pillow Block Figure 12 shows the CAD model of the pillow block once again. The figure shows the critical location, and also shows in red a path between the critical location and the inside diameter of the pillow block. The path follows the plane along which the pillow block breaks when cap-loaded in experiments.

Figure 12

Figure 13 shows the path plot for mMσ versus position along the path. The results plotted correspond to an

applied load of mMP , the fracture load estimated by the modified Mohr criterion. The ratio of the maximum value for mMσ to the average value, or membrane stress, estimates the stress concentration

factor tk .

Figure 13

Estimation of Fracture Load for the Pillow Block As outlined in the procedure above, and in keeping with the experience gained in modeling the notched tensile specimen, the product of the stress concentration factor and the modified Mohr criterion provides an estimate of the fracture load, mMtf PkP *= .

Repeating the procedure for several sizes of pillow block generated the data plotted in Figure 14. Figure 14 shows, for five sizes of pillow block, the fracture load predicted by the modified Mohr criterion, mMP , the

fracture load fP predicted by the product mMt Pk * , and the basic dynamic rating of the bearing, or BDR.

Showing the BDR puts the magnitude of the fracture loads in perspective, as good practice precludes applying loads above about 25% of the BDR.

Figure 14 also shows experimental fracture loads for three sizes of pillow block. Each point represents the average of six to ten tests.

Figure 14

Discussion For two of the pillow block sizes, predicted fracture loads match the experimental fracture loads within 10%. For a third size, the predicted fracture load matches fairly well the experimental fracture load, within 20%.

Conclusions At low loads, Roark’s linear elastic estimate of the stress concentration factor agrees with calculated maximum stress to nominal stress ratio for the notched tensile specimen, even in the case of gray iron’s nonlinear stress-strain behavior.

At higher loads, the stress concentration factor decreases from the linear elastic estimate.

For materials such as gray iron, that show little notch sensitivity, the modified Mohr and other stress-based brittle material fracture criteria under-predict fracture loads for components where stress concentrations occur.

Multiplying the modified Mohr predicted fracture load by the stress concentration factor predicts gray iron component fracture loads within 10 to 20%. The stress concentration factor gets calculated at the modified Mohr predicted fracture load.

An ANSYS path plot of stress versus location enables calculating the stress concentration factor for components whose geometry precludes estimation of nominal stress.

References

(Reference 1) Dowling, N.E., 1999, Mechanical Behavior of Materials, 2nd ed., p 261

(Reference 2) Dowling, N.E., 1999, Mechanical Behavior of Materials, 2nd ed., p 243

(Reference 3) Dowling, N.E., 1999, Mechanical Behavior of Materials, 2nd ed., p 270

(Reference 4) American Society for Metals, Atlas of Stress-Strain Curves

(Reference 5) Heywood, R.B., 1962, Designing Against Fatigue, p 130, Chapman and Hall

(Reference 6) Roarke, R.J., and Young, W.C., Formulas for Stress & Strain, 5th ed.

(Reference 7) Dowling, N.E., 1999, Mechanical Behavior of Materials, 2nd ed., p 58

Appendix I: APDL to Calculate Modified Mohr Fracture Criterion

! FRACTURE.INP

! calc the modified Mohr Fracture Criterion

! Dowling, p 263, 264

FINI

/POST1

ALLSEL

! ASM Handbook

! estimate UTS = 30000 ksi, UCS = -90000 ksi

UTS = 30000

UCS = -90000

M = 1 + 2*UTS/UCS

MOHRFAC = UTS/(UCS*(M-1)) ! always 1/2 with above def of M

MF2 = MOHRFAC*M

ETABLE, ERAS ! erase any old table

SABS, 0 ! take algebraic value in etable operations

ETABLE, S1, S, 1 ! store principal stresses

ETABLE, S2, S, 2

ETABLE, S3, S, 3

SABS, 0 ! take algebraic values in etable operations

! SADD, LabR, Lab1, Lab2, FACT1, FACT2, CONST

SADD, S1P2, S1, S2 ! (S1 + S2)

SADD, S1P3, S1, S3

SADD, S2P3, S2, S3

SADD, S1M2, S1, S2, 1, -1 ! (S1 - S2)

SADD, S1M3, S1, S3, 1, -1

SADD, S2M3, S2, S3, 1, -1

SABS, 1 ! take absolute values in etable operations

SADD, AS1M2, S1M2, S1M2, 0.5, 0.5 ! abs(S1 - S2)

SADD, AS1M3, S1M3, S1M3, 0.5, 0.5

SADD, AS2M3, S2M3, S2M3, 0.5, 0.5

SABS, 0 ! take algebraic values in etable operations

! calc C1, C2, C3, for modified Mohr criterion

SADD, C1, AS1M2, S1P2, MOHRFAC, MF2 ! C1 = MOHRFAC*abs(S1 - S2) + MF2*(S1 + S2)

SADD, C2, AS1M3, S1P3, MOHRFAC, MF2

SADD, C3, AS2M3, S2P3, MOHRFAC, MF2

SMAX, C12, C1, C2 ! get max of C1, C2, C3, S1, S2, S3

SMAX, C123, C12, C3

SMAX, S12, S1, S2

SMAX, S123, S12, S3

SMAX, SC123, S123, C123

ALLSEL

Appendix II: APDL for Class 30 Gray Iron Stress-Strain Curve

! Cast_Iron_TB.INP

TB, CAST, 1, , , ISOTROPIC

TBDATA, 1, 0.04 ! Plastic Poisson's ratio in tension

! manual TB, UNIAXIAL, 1, 1, 5, TENSION

! manual TBTEMP, 10 ! ANSYS manual

! manual TBPT, , 0.550E-03, 0.813E+04 ! Defines stress-strain, tension

! manual TBPT, , 0.100E-02, 0.131E+05

! manual TBPT, , 0.250E-02, 0.241E+05

! manual TBPT, , 0.350E-02, 0.288E+05

! manual TBPT, , 0.450E-02, 0.322E+05

TB, UNIAXIAL, 1, 1, 7, TENSION

TBTEMP, 10

TBPT, , 0.550E-03, 8.12807882E+3 ! modify 1st pt from ANSYS manual

! manual TBPT, , 0.550E-03, 0.813E+04 ! 1st pt from ANSYS manual curve

! atlas TBPT, , 0.0005237, 8529 ! replaces 1st pt from Atlas curve

TBPT, , 0.001573, 1.746E+04 ! ASM Atlas of Stress-Strain Curves data

TBPT, , 0.002622, 2.26E+04 ! replace remainder of ANSYS tension curve

TBPT, , 0.003672, 2.575E+04

TBPT, , 0.004721, 2.776E+04

TBPT, , 0.005771, 2.902E+04

TBPT, , 0.00682, 2.977E+04

TB, UNIAXIAL, 1, 1, 5, COMPRESSION

TBTEMP, 10 ! ANSYS manual

TBPT, , 0.203E-02, 0.300E+05 ! Defines stress-strain, compression.

TBPT, , 0.500E-02, 0.500E+05 ! matches ASM Atlas SS Curve in comp

TBPT, , 0.800E-02, 0.581E+05

TBPT, , 0.110E-01, 0.656E+05

TBPT, , 0.140E-01, 0.700E+05