Wear, 45 (1977) 403 - 407 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

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Short Communication

Contact stress solutions using computer techniques

ROGER K. REEVES and DAVID W. HOEPPNER

College of Engineering, University of Missouri - Columbia, Columbia, MO. 65201 (U.S.A.)

(Received April 17, 1977)

The use of a computer program to solve example problems involving contact stresses is presented as an initial step in the development of com- puterized stress analyses involving fretting fatigue loadings, The u~~ation of the computer greatly simplifies the calculation of contact stresses.

In toduc tion Contact stresses play an important role in many design situations, such

as the fretting fatigue process, implying that unusual stress distributions may occur. In the example of fretting fatigue, the axial load is constantly changing, thus causing an inherent change in the normal load which gives rise to an additional stress distribution.

Little work has been done in the area of contact stresses since the work of Hertz [l] . The work of Hertz has been simplified by many sources such as Seely and Smith [2] , whose work in the area of contact is an extension of the earlier work of Thomas and Hoersch 131. Seely and Smith provide procedures by which stresses can be calculated for various contact loading schemes (point contact, crossed cylinder contact and line contact). These procedures may become useful in determining a method by which the con- stantly changing stress distribution could be calculated.

This paper is an initial step in developing procedures to be used in fretting fatigue distribution analyses and provides the basis for future work in the form of a computer program by which contact stresses can be cal- culated.

A program entitled CONTACT was developed. The general ease of con- tact is considered with options available for such special cases as point con- tact, crossed cylinder contact, and two aspects of line contact (i.e. a cylinder on a cylinder or on a flat plate). The equations used in the program are taken from the work of Seely and Smith [ZJ and used with the graphs shown in Fig. 1.

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,.03b Y

!3’, 3 o.ool

IO 100 200 500 IO00 5x103 10x103

(a) VALUES OF B/A

(b) VALUES OF B/A

Fig. 1. Values of contact coefficients for determining stresses and deflections between two bodies in contact. (Courtesy of John Wiley and Sons, Inc.)

In order to illustrate the use of the program, various example problems are given [2] .

Problem 1 Two semicircular disks of steel are brought into contact under a load

P of 1000 lb. The radii of curvature of the surfaces at the point of contact are as follows: R1 = 2 in, R; = 5 in, R, = 3 in and R; = 8 in. The angle a between the planes of minimum curvature is 60”. Compute the maximum principal stress, the maximum shearing stress, the maximum octahedral shearing stress and the location of the maximum shearing stress below the surface of contact in body 2 due to the applied load. E, = E, = 30 000 000 lb in-? and p1 = p2 = 0.25.

The solution of this problem with an explanation of the program is as follows:

ENTER 1 IF EXPLANATION OF THE PROGRAM IS DESIRED, 0 IF NOT

1

THIS PROGRAM USES A METHOD OF SOLVING FOR CONTACT STRESSES SET FORTH IN CHAPTER 11 OF “ADVANCED MECHANICS OF MATERIALS” BY SEELY AND SMITH.

405

TO USE THIS PROGRAM YOU MUST HAVE ACCESS TO THE GRAPHS ON P. 356 - 357 OF SEELY & SMITH OR FIGURE l(A) OF THIS PAPER. P = TOTAL FORCE EXERTED BY BODY 1 ON BODY 2 (POUNDS). Q = LOAD PER UNIT LENGTH FROM BODY 1 TO 2 (LBSJIN.). El, E2 = MODULI OF ELASTICITY (PSI). POl, PO2 = POISSON’S RATIO. Rl, R2 = MINIMUM RADII OF BODIES (INCHES). RlP, R2P = MAXIMUM RADII OF BODIES (INCHES). ALPHA = ANGLE BETWEEN PLANES CONTAINING Rl & R2 (DEGREES).

ENTER 1 FOR POINT CONTACT, 2 FOR CROSSED CYLINDERS, ALPHA = 90, 3 FOR LINE CONTACT.

1

SUPPLY Rl, RlP, R2, R2P, POl, P02, El, E2, P, ALPHA

2 538.25.25 3000000030000000100060

B/A = 1.60

SUPPLY CB, CSN, CSS, CSSG, CZS FROM GRAPHS

.78 .73 .24 .22 .56

MAXIMUM PRINCIPAL STRESS IS -251207.88 PSI MAXIMUM SHEARING STRESS IS 82588.88 PSI MAXIMUM OCTAHEDRAL STRESS IS 75706.44 PSI LOCATION OF SHEARING STRESS IS 0.021 INCHES

Problem 2 A steel railway car wheel 33 in in diameter rolls on a steel rail the top

surface of which has a cross radius of 12 in. The wheel pressure on the rail head is 25 000 lb. Calculate the maximum principal stress, the maximum shearing stress, the maximum octahedral shearing stress and the location below the surface of the rail at which the maximum shearing stress occurs. El = E2 = 30 000 000 lb inm2 and p1 = p2 = 0.25.

The solution of this problem is as follows:

ENTER 1 FOR POINT CONTACT, 2 FOR CROSSED CYLINDERS, ALPHA = 90, 3 FOR LINE CONTACT.

2

SUPPLY Rl, R2, POl, P02, El, E2, P

12 16.5.25.25 30000000 30000000 25000

B/A = 1.38

SUPPLY CB, CSN, CSS, CSSG, CZS, FROM GRAPHS

.81 .70 .24 .22 .53

MAXIMUM PRINCIPAL STRESS IS -182141.69 PSI MAXIMUM SHEARING STRESS IS 62448.59 PSI MAXIMUM OCTAHEDRAL STRESS IS 57244.54 PSI LOCATION OF SHEARING STRESS IS 0.120 INCHES

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Problem 3 Two steel cylinders of radii RI = 2 in and Rz = 6 in have their longi-

tudinal axes parallel and are pressed against each other by a load q of 1500 lb in-‘. Compute the maximum principal stress (where the z axis is directed into cylinder 2 parallel to the contact load and where the x and y axes are in the plane of contact, x being parallel to the longitudinal axis of the cylinders and y being perpendicular to the longitudinal axis), the maximum shearing stress, the maximum octahedral shearing stress and the location below the surface of cylinder 2 at which the maximum shearing stress occurs. E, = Ez = 30 000 000 lb inm2 and PI = p2 = 0.25.

The solution of this problem is given as follows:

ENTER 1 FOR POINT CONTACT, 2 FOR CROSSED CYLINDERS, ALPHA = 90,3 FOR LINE CONTACT.

3

ENTER 1 IF CYLINDER ON PLANE

SUPPLY Rl, R2, POl, P02, El, E2, Q

2 6 .25.25 30000000 30000000 1500

MAXIMUM PRINCIPAL STRESSES ARE X DIRECTION -35682.47 PSI Y DIRECTION -71364.94 PSI Z DIRECTION -71364.94 PSI

MAXIMUM SHEARING STRESS IS 21409.48 PSI MAXIMUM OCTAHEDRAL STRESS IS 19268.53 PSI LOCATION OF SHEARING STRESS IS 0.011 INCHES

Problem 4 Assuming the top surface of the rail in problem 2 to be flat and hori-

zontal with a width of 2 in, calculate the maximum principal stress (with the x, y and z axes oriented as in problem 3), the maximum shearing stress, the maximum octahedral shearing stress and the depth below the surface’of the rail at which the maximum shearing stress occurs. P = 25 000 lb, E, = E2 = 30 000 000 lb ine2 and p1 = p2 = 0.25.

The solution of this problem is given as follows:

ENTER 1 FOR POINT CONTACT, 2 FOR CROSSED CYLINDERS, ALPHA = 90, 3 FOR LINE CONTACT.

3

ENTER 1 IF CYLINDER ON PLANE

SUPPLY Rl, POl, P02, El, E2, Q

i6.5 .25.25 30000000 30000000 12500

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MAXIMUM PRINCIPAL STRESSES ARE X DIRECTION -31057.61 PSI Y DIRECTION -62115.22 PSI 2 DIRECTION -62115.22 PSI

MAXIMUM SHEARING STRESS IS 18634.56 PSI MAXIMUM OCTAHEDRAL STRESS IS 16771.11 PSI LOCATION OF SHEARING STRESS IS 0.101 INCHES

Conclusions and recommendations It is evident that the program CONTACT is a simplified version for

calculating contact stresses. However, further work in this area may be directed toward the solution of more complex contact situations which illus- trate the actual contact loadings of interest. Further computerized stress analyses using the contact load plus an additional load, such as fatigue in the case of fretting fatigue, may be investigated.

This program provides a starting point from which further computer techniques may be developed to analyze contact stresses. This type of analysis of complex contact problems is the method by which unusual stress distributions must be determined.

Acknowledgments The continued support of the Office of Naval Research on the fretting

fatigue program under the contract number ONR NO0014 75 C 0670 at the University of Missouri - Columbia is acknowledged. The assistance of Gregg Bogucki in the development of the contact stress program and the acquisi- tion of the reference material is greatly appreciated. The authors are grateful to Dr. P. Clarkin, the technical monitor of the ONR program for his con- tinuing encouragement and support.

References

1 H. Hertz, Miscellaneous Papers, translated by D. E. Jones, MacMillan, London, 1896, Chaps. V, VI and XV.

2 F. B. Seely and J. 0. Smith, Advanced Mechanics of Materials, Wiley, New York, 1952, Chap. 11.

3 H. R. Thomas and V. A. Hoersch, Stresses due to the pressure of one elastic solid upon another, University of Iilinois Experiment Station, Bulletin no. 212, July 1930.