UDC 539.319.6-434.1

RESIDUAL STRESSES IN CYLINDRICAL ARTICLES

S. I. Karatushin,1 D. V. Spiridonov,1 and Yu. A. Pleshanova1

Translated from Metallovedenie i Termicheskaya Obrabotka Metallov, No. 6, pp. 53 – 55, June, 2013.

A method of calculating residual stresses that arise in solid or hollow cylinders following thermochemical

treatment is considered. Computer calculations are performed by means of the ANSYS program.

Key words: residual stresses, thermochemical treatment, solid cylinder, hollow cylinder, ANSYS

program.

INTRODUCTION

Residual stresses have long drawn the interest of re-

searchers as a consequence of the fact that they exert a major

influence on the service characteristics of the parts of en-

gines [1]. However, even for articles of simple shape, a com-

plete calculation of stresses was impossible before the ap-

pearance of the method of finite elements. In turn, the

method of finite elements yielded tangible results only with

the development of computational mathematics. All these

advances were ultimately implemented in engineering analy-

sis programs, one of which is ANSYS [2 – 5].

Through the use of the ANSYS Workbench software

module it is possible to calculate all the components of the

stress-strain state at each point of the volume of a body. Arti-

cles of cylindrical form are widely used in engineering appli-

cations and include cylindrical springs, shafts, axles, and

other parts of machines and mechanisms.

The most common of the various methods of hardening

is casehardening with carbon. Casehardening with carbon in

particular will be considered as an example in the present ar-

ticle. Similar methods of calculation may be applied for any

type of thermochemical treatment.

Volumetric variations in subsequent heat treatment are a

source of residual stresses. In carburizing of a surface an in-

crease in the volume of material occurs as a consequence of

an increase in the quantity of carbon in a surface layer, but

the basic increase in the volume of a casehardened layer oc-

curs in a � � � transformation, i.e., in heat treatment. The

increase in volume that occurs in carburizing may be entirely

ignored not only in view of its small magnitude, but also be-

cause of relaxation of stresses throughout an entire volume

due to the high temperature of the process of casehardening.

The objective of the present study is to calculate the re-

sidual stresses that arise in the course of casehardening with

carbon in solid and hollow cylinders made of steel 20Kh.

COMPUTATION TECHNIQUE

To solve the problems involved in calculation of the

stress-strain state, the following initial data will be deter-

mined experimentally or specified:

(1) distribution of carbon across the width of the

casehardened layer of a part;

(2) coefficient of linear expansion of casehardened layers

with different carbon contents caused by a � � � transfor-

mation.

These basic parameters are determined in accordance

with a technique set forth in a previous report [6]. Using the

ANSYS program it is possible to introduce any known cha-

racteristic of the physical properties of a material. Concrete

properties of steel 20Kh (yield strength, Poisson’s coeffi-

cient, concentration of carbon at different distances from the

surface, rate of increase of volume of casehardened layer in

the course of heat treatment) are input into the program when

calculating the residual stresses. A three-dimensional (solid)

or two-dimensional (axisymmetric) model is used to calcu-

late the stress-strain state. The use of volumetric models is

justified for parts of low symmetry. The model is created as a

compound model in one of the graphics programs and is

translated into ANSYS in the Parasolid format. The more

complex the model and the greater its number of parts, the

greater is the length of time of the solution. Computer sys-

tems with high performance must be used for complex

models.

Axisymmetric two-dimensional models are entirely ac-

ceptable for symmetric parts such as bodies of revolution. In

this case a model is created directly in the ANSYS program.

Metal Science and Heat Treatment, Vol. 55, Nos. 5 – 6, September, 2013 (Russian Original Nos. 5 – 6, May – June, 2013)

339

0026-0673/13/0506-0339 © 2013 Springer Science + Business Media New York

1D. F. Ustinov Baltic State Technical University “VOENMEKh,”

St. Peterburg, Russia (e-mail: karatsi@bk.ru).

For the parts that will be studied in the present article —

a solid cylinder 60 mm in diameter and a hollow cylinder

with outer diameter 72 mm and inner diameter 32 mm — a

variant of a solution in the form of an axisymmetric problem,

i.e., a two-dimensional model, is used. In this case, the solu-

tion is realized quite rapidly even with comparatively small

dimensions of the final elements. The volume of the file is

reduced roughly 10-fold.

RESULTS OF CALCULATIONS AND DISCUSSION

Results of a calculation for the articles being considered

here with an indication of, for example, certain magnitudes

of normal stresses are presented in Fig. 1.

The case-hardened layer is specified with step 0.25 mm.

As in [1], the calculations are presented for steel 20Kh. The

distribution of the stresses in a plane perpendicular to the

axis of the solid cylinder is shown in Fig. 2.

It is interesting to note that for these types of configura-

tions of articles, the tangential and axial normal stresses dif-

fer only very slightly in terms of magnitude and coincide in

terms of distribution law. The tangential stresses are not pre-

sented in the graph due to the lack of practical significance.

The thinner is the casehardened layer, the greater is the

level of the compression stresses and, correspondingly, the

level of the tensile stresses. The distribution pattern of the ra-

dial stresses changes. There is very little information in the

literature about tensile stresses, which balance the field of

compression stresses, though their role is no less important.

The distribution of stresses in a cylinder with an internal

opening is presented in Fig. 3a. In this case the radial stresses

are negative, as was assumed. As for plates, the position of

the maximum of the tensile stresses for cylindrical articles

corresponds to the full depth of casehardening. The effective

depth of casehardening in this case does not possess a dis-

tinct position in the graph. The program being used here

makes it possible to introduce a residual austenite parameter

and thereby estimate the influence of austenite on the distri-

bution of the residual stresses. However, there is a problem

here. There are practically no reliable data in the literature on

the distribution of the quantity of residual austenite by thick-

ness of the casehardened layer. The stresses in a case har-

dened layer that promote the existence of residual austenite

induce blurring of the diffraction lines and, correspondingly,

sharply reduce the precision with which the layer is deter-

mined. The distribution of the stresses in a hollow cylinder

based on the content of residual austenite is presented in

Fig. 3b. The distribution of residual austenite is conjectured

on the basis of generalized data from the literature and re-

lated studies.

CONCLUSIONS

Through the use of the ANSYS Workbench software

package it becomes possible to solve problems that arise in

the analysis of residual stresses in articles of practically any

shape where initial experimental data on the physical proper-

ties of the material of these articles are available. For thermo-

mechanical treatment these properties comprise the specific

volumes of structures and the transverse distribution of satu-

rating elements. For many cases that may be realized in prac-

340 S. I. Karatushin et al.

à b

Fig. 1. Results of a calculation in the form of an axisymmetric prob-

lem for a solid (a) and hollow (b ) cylinder.

60

50

40

30

20

10

– 100

– 200

– 300

– 400

– 500

– 600

– 700

– 800

– 900

�n , ÌPà

30 20 10 6 4 2 1.0 0.60.4 0.2 0.1

h, mm

Radial

Òàngential

and axial

Fig. 2. Distribution of normal stresses (�n

) in a solid case-hardened

cylinder (h — distance from surface).

tical applications, these data may be found in the literature or

determined experimentally. The volume of information that

may be obtained with the use of ANSYS is significantly

greater than that obtained by means of a solution performed

using analytic methods [7] even with the use of computer

programming. The precision of a calculation is assured with

the use of adaptive refinement of the net of final elements.

This yields an acceptable relative error expressed in terms of

the magnitude of the deformation energy, which should not

exceed 10%. Unfortunately, there are no reliable data in the

literature on the complete picture of the stress-strain state of

actual parts. There are no experimental studies in this area

while the computational studies deal only with a narrow

class of questions related to the stress-strain state. The results

that have been obtained in the present study do not contradict

the individual results in the cited literature.

The significance of the present method is that it makes it

possible to analyze in a comparatively simple fashion

changes in the stress-strain state that occur upon the applica-

tion of external forces.

REFERENCES

1. Ya. D. Vishnyakov and V. D. Piskarev, Management of Residual

Stresses in Metals and Alloys [in Russian], Metallurgiya, Mos-

cow (1989), 254 p.

2. K. A. Basov, ANSYS in Examples and Problems [in Russian],

Komp’yuter Press, Moscow (2002), 224 p.

3. O. M. Ogorodnikova, Structural Analysis in the ANSYS Environ-

ment [in Russian], UGTU, Ekaterinburg (2004).

4. A. B. Kaplun, E. M. Morozov, and M. A. Olfer’eva, ANSYS in the

Hands of the Engineer. Practical Manual [in Russian], URSS,

Moscow (2003), 272 p.

5. K. A. Basov, The Graphics Interface of the ANSYS Complex [in

Russian], DMK Press, Moscow (2006), 248 p.

6. S. I. Karatushin, D. V. Spiridinov, and Yu. A. Pleshanova, “Simu-

lation of residual stresses in the process of casehardening,” Izv.

Vysh. Ucheb. Zaved., Mashinostroenie, No. 3, 49 – 52 (2012).

7. I. A. Kazakovtsev and S. M. Shlyakhov, “Estimation of stress-

strain state of a hollow shaft casehardened along the contour with

lengthwise varying casehardening,” Izv. Vysh. Ucheb. Zaved.,

Mashinostroenie, No. 3, 33 – 40 (2008).

Residual Stresses in Cylindrical Articles 341

0.2 0.4 0.6 1.0 2 4 6 10 20 0.2 0.4 0.6 1.0 2 4 6 10 20

– 10

– 20

– 30

– 40

– 50

– 60

– 10

– 20

– 30

– 40

– 50

– 60

120

100

80

60

40

20

120

100

80

60

40

20

– 100

– 200

– 300

– 400

– 500

– 600

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– 100

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– 600

– 700

– 800

– 900

�n , ÌPà �n , ÌPà

�r , ÌPà �r , ÌPà

h, mm h, mm

Radial

Radial

Òàngential

and axial

Òàngential

and axial

à b

Fig. 3. Distribution of stresses in a hollow cylinder (external diameter 72 mm, internal diameter 32 mm) with casehardened in-

ternal surface (a) and with the presence of about 25% residual austenite (b ) (h — distance from internal surface).