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Approximate crack growth estimate of railway wheel influenced by normal and shear action
Theoretical and Applied Fractu re Mechanics 15 ( 1991) 179-190 Elsevier
179
Approximate crack growth estimate of railway wheel influenced by normal and shear action A. M a r t i n Meizoso, J.M. M a r t i n e z E s n a o l a a n d M. F u e n t e s P6rez Centro de Estudios e Investigaciones Tdcnicas de Guipf~zcoa (CELT), San Sebastitln 20009, Spain and Escuela Superior de Ingenieros Industriales, Unioersidad de Navarra, San Sebastitln 20009, Spain
An elastic-plastic analysis of railway wheel life prediction is made for obtaining the residual stresses due to braking and alternating rolling contact stresses between the wheel and rail. These stresses are then used to obtain approximate crack border stress intensity factors Ki, K n and K m such that an equivalent combined load stress intensity factor can be derived by application of the strain energy density criterion. By including the effect of both the normal and shear action on the locomotive wheel, the results are indicative of those observed in service. While cracks may nucleate from thermal fatigue, it suffices to consider the cyclic load due to rolling contact for estimating the fatigue lives of locomotive wheels.
1. Introduction T h e c o n t a c t l o a d b e t w e e n the wheel a n d rail i n t r o d u c e s shear a c t i o n into the rim of the wheel even for the c o n d i t i o n o f p u r e rolling. Such a n effect b e c o m e s m o r e p r o n o u n c e d w h e n c o n t a c t friction is present. F o r a l o c o m o t i v e wheel, the t a n g e n t i a l forces are r e q u i r e d to b e as large as thirty-six (36) p e r c e n t of the weight. This corres p o n d s to the s i t u a t i o n o f a l o c o m o t i v e p u l l i n g a train on an incline where the c o m b i n e d n o r m a l a n d shear l o a d i n g effect is e x p e c t e d to b e a p p r e ciable. F o l l o w i n g the earlier w o r k [1] that dealt with p r e d i c t i n g r a i l w a y wheel life u n d e r n o r m a l load, this w o r k considers the effect of b o t h n o r m a l a n d shear a c t i o n o n c r a c k growth where a n equivalent stress intensity factor is d e f i n e d b y a p p l i c a t i o n of the strain energy d e n s i t y factor to include all three stress intensity factors K~, K H a n d g i l I.
3t wheel
section
2. Analytical modelling: stresses and intensity factor's T w o types of i n d u c e d stresses will b e c o n s i d ered; they are the residual stresses d u e to b r a k i n g a n d c o n t a c t stresses arising f r o m the wheel a n d 0167-8442/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
~.
Axial
Fig. 1. Finite element mesh of rail wheel section for determiningresidual stresses.
A. Martin Meizoso et al. / Crack growth of railway wheel
180
rail. The former will be computed from an elastic-plastic finite dement procedure [2,3]. Refer to Fig. 1 for the grid pattern of the rail wheel section used in the calculation. An elastic analysis [4,5] is used to obtain the Hertzian contact stresses such that the elliptical contact area and the corresponding pressure distribution are determined as functions of the wheel and rail radii in addition to the contact load magnitude. From a knowledge of the remote stresses, the crack edge stress intensity factors can be determined accordingly. A schematic of the crack configuration is shown in Fig. 3.
y -~----~
Fig. 3. Schematic of surface semi-elliptical crack in wheel.
2.1. Contact stresses
The contact pressure on the wheel is discretized into a finite number of point forces as illustrated in Fig. 2. Solutions to the concentrated normal and tangential forces acting on a semi-infinite elastic body are known, respectively, from the Boussinesq and Cerruti problem [5]. An accurate description of the state of affairs away from the local contact region is thus obtained. Near the contact, the point load is replaced by an equivalent uniform pressure distributed over a finite region [5]. It is the local stresses near the contact that would influence the onset of crack growth.
Referring to the coordinate axes x, y and z in Fig. 2, the stress components of interest are oxx, Oxy and oxz; they would exert mixed mode effects on crack growth. Contour plots of these stresses for a complete revolution of the wheel can be found in Figs. 4(a)-(c) inclusive. Table 1 summarizes the maximum positive and negative amplitudes of axe, Oxy and o~z. Negative o~ corresponds to a compressive action on the crack oriented in the yz-plane. Hence, the shear components O~y and oxz would lie in the crack plane and they are reduced because of compressions as friction between the crack surfaces. It is convenient to define an effective shear stress ~* such that
t; ~'*=
fOrOxx0 -#[Oxx[,
0,
foro~x<0; ~loxxl ~
(1)
in which • is the resultant shear given by r =
(2)
O~x2y+ Off~
In eq. (1), # is the coefficient of friction between the crack surfaces. The above model assumes that compressive normal stress or Oxx < 0 would correspond to crack closure. This is recognized as an approximation. More accurate assessment of crack closure would be beyond the objective of this y
5,57mm
8,09mm
N o r m a l load = I OOKN C i r c u m f e r e n t i a l load = 2 5 KN
Fig. 2. Diseretization of the rail-wheel load into points on contact surface.
Table 1 Numerical values of normal and shear stresses (MPa) Stresscomponent
Positive
Negative
o~x
99 497 841
- 1828 -497 - 841
oxy Oxz
A. Martin Meizoso et al. / Crack growth of railway wheel 2'5
30 m m
5
20
10
25
50
181
-1000-500-250
10
0
10
30
20
(a) stres, o'~ 25
~._
50
100
250
- 2 5 0 -100 -50
-25MPa
,t///l\\-~,
.°
10
0
10
_.~,o 20
Lo
(b) Strns O'~y
.0ram
-10
[ 30
mm
20
zo
1~
i'o
2'0
Fig. 4. Normal and shear stress (MPa) contours for a complete wheel revolution.
A. Martin Meizoso et al. / Crack growth o[railway wheel
182
Table 2 Coefficients in polynorrhals for a~*~,a *y and o* (MPa) and 2.5 mm < z < 34 mm
work. D i s p l a y e d in Figs. 5(a) to 5(c) inclusive are the stresses a * , a ~ a n d a * for/z = 1 as a f u n c t i o n o f the wheel r i m d e p t h in the z-direction at the prospective crack site. W h i l e a * = axx, o ~ a n d a ~ represent the p r o j e c t i o n o f ~-* a l o n g the y- a n d z-axis. Their variations c a n be r e p r e s e n t e d in p o l y n o m i a l s as follows:
+ dij l_-~, i, j = x ,
y, z
1000
Subscripts for oi~
o,°
a,j
h,)
i=x; j = x i=x; j=y i=x; j = z
0 -497.85 -1448.7 1369.2 0 -8.3684 408 -2116 3795.4 -2145.8 -26.42 1354.8 -6476.9 10656 -56947
C~j
(3)
(a)
500
~-
0
x
b
-500
b~ -1ooo
-1500
-2000
,~ 0
I 10
,
I 20
,
I 30
,.,. 40
DEPTH (ram) 200
(b)
1 O0
n ,>,,
--100
-200
.zk--.--..--___.
0
10
20
30
DEPTH ( m m ) Fig. 5. Normal and shear stress distribution with distance normal to wheel rim.
¢0
d~j
183
A. Martin Meizoso et al. / Crack growth of railway wheel
60
40 20 13_
0 b -20
-40 -60
I0
20
30
40
DEPTH (turn)
Fig. 5 (continued).
The coefficients o°, aij, bij . . . . . di# for i = x and j=x, y, z can be found in Table 2 for 2.5 mm~
o*=4z,
tion of the three-dimensional crack border stress field could also be made by using the finite element method [10], a consideration that would be left for the future.
o ~ = 120 - 42.2z, for z < 2.5 mm
(4)
When z exceeds 34 mm, the following expressions can be used: ox* = - 2 1 0 . 8 1 ' o~* = 326.4 1 ,
o ~ = 88.4 1 , for z > 34 mm
(5)
2.2. Stress intensity factors Approximate stress intensity factor expressions for K~ arising from o~ and K H and Kin from Oxy and o~ will be obtained, respectively, from the works in [6] and [7] for a complete elliptical crack in an infinite medium. The prevailing applied pressure is approximated by a second degree polynomial based on a least square fit. Since the actual crack configuration is interrupted by a free surface, additional correction factors [8] are introduced with the understanding that these results are subject to further refinements [9]. Direct determina-
3. Crack growth consideration Assessment of mixed mode cracking would be made for each point on the crack front by referring to the stress oxx normal to the crack plane and Oxy q-Oxz parallel to the crack surface. These stresses give rise to the stress intensity factors K~ and K H + Kin; they can be represented graphically as in Fig. 6 such that any 'stress state away from the vertical and horizontal axis would be subject to the combined influence of normal and shear stress.
3.1. Fracture criterion The combined influence of KI, KII and KIII could be visualized by an equivalent stress intensity factor Keq in Fig. 7 that traverses the surface of a distorted cone. Among many of the proposed fracture criteria, the strain energy density criterion or S-theory developed by Sill [11-14] is most expedient. When all three stress intensity factors
184
A. Martin Meizoso et al. / Crack growth of railway wheel
coefficients a,l (i, j = 1, 2, 31) are of the forms [13]
O-xx
/
Residual
|
KI/ I/"~"a --
,.
!
'\
/
\
/
\
/ / /
a33 - ~(1 + . ) with E and v being respectively, the Young's modulus and Poisson's ratio. The angle 0 is measured in a plane normal to the crack border. That is, 0 = 0 would correspond to the yz-plane in Fig. 2. Postulated in the S-theory is that crack would initiate in the direction determined by the relative minimum of the strain energy density factor S, say S,,io and that the onset of crack growth would occur when Smin reaches a critical value Sc~ which can be related to the Klc fracture toughness value:
\
/
\
/ /
\ \
/
\
/
(7)
32E
\ \
/
a22 = ~r(1 + v) [4(1 - v)(i - cos 0) + ( 3 cos O - 1)(1 + cos O)]
\
/
16E ~r(1 + v) sin 0(cos 8 - I + 2v)
\
\
/
a,2
8E
\
/
~r(l+v)(3-4v-cosO)(l+cosO)
xy + O-xz
(KIT +KIII) / V'~"0
\
/
[7
8E
au
\
Fig. 6. Schematic of cyclic load path.
(1 + v)(1 - 2v)K(~ are present, they could be combined into the strain energy density factor as
(6)
S = auK 2 + 2 a t 2 K , K , I + az2K 2 + a33 K 2 ,
In eq. (6), S is the amplitude of the strain energy density d W / d V that behaves as 1/r as r the distance from the crack border tends to zero. The
s. =
2~e
(8)
Since Scr is characteristic of the material, it can be used to predict the fracture behavior for situations where all three K-factors are present. In general, an equivalent K or Keq can be defined using the functional form of eq. (8), i.e.,
geq =
(1 + v)(1 - 2 v )
~
(9)
This quantity is calculated for each point on the crack border.
Keq
3.2. Load superposition The wheel is subjected to both residual and alternating stresses. These two effects can be reflected via the equivalent stress intensity factor by superposition: Keq = --eqK(r)-+ --eqK(f) ~ K I I
- KllI
Fig. 7. Equivalent stress intensity factor variation for a wheel revolution.
(10)
Note that the superscripts r and f refer, respectively, to residual and alternating stress. The sign for K~qr) can be positive or negative depending on
A. Martin Meizoso et al. / Crack growth of railway wheel
whether the alternating load exerts an opening or closing action to the crack surfaces. For each rolling cycle, K~q would vary along the crack border and only a portion of the crack would be subjected to the maximum intensity. This would imply that not all points on the crack border would initiate at the same time. Moreover, crack growth may take place in different planes at different locations. The foregoing two effects are not explicitly included in the schematic representation of Keq in Fig. 7 which assumes that all points on the crack border would reach the same maximum value at the same time, an assumption that is regarded to be conservative. Change in crack direction is weighed into crack growth in a manner to be discussed subsequently.
185
etc., on the crack and the condition SJrg = const. [12] would yield the new crack configuration which would no longer lie in the original crack plane. This process would require a three-dimensional finite element analysis of the crack problem. Instead, an approximate procedure would be adopted. Only the onset of out-of-plane crack growth is estimated. The new crack front is then projected [1] into the original yz-plane in Fig. 3. It is then fitted by an elliptically-shaped curve such that it suffices to use the semi-axes a and c for describing crack growth. Hence, the equivalent crack would always grow in the same plane. An estimate of fatigue crack growth is made by using the relation zaa
3. 3. Crack shape modelling
= Co(AKoq) ~
As stated earlier, the direction of crack growth, say 00 for a point on the crack border, would depend on the relative magnitudes of the three stress intensity factors K I, K n and Km. The condition aS/~O--0 for 02S/~02> 0 can be invoked on eq. (6) to find 00 for each point j = 1, 2,
where Co and m are empirical parameters determined experimentally for a given material and specimen type. The A K ~ would depend on AK~, AK u and AKIII. Negative values of K I are assumed to close the crack and hence would not yield crack growth. They are excluded from the
-
-
dlmtermal.for
fmttermol.for
~4
(11)
I Glro
Fig. 8. Computer program functional flow chart.
186
A. Martin Meizoso et al. / Crack growth of railway wheel
Table 3 Computational subroutines of revised program Subroutines
Functions
CONTAJU CRECE DATOS DXDY FATIGA GIRO INFORM INTEG INTEL JMSIH NEWANG NEWELP NK2K3 NKFKE NNEWMAN PINTEL PROPAG ROTURA
Contact stresses Update crack front Input data for main program Project crack growth of each border point Fatigue crack growth calculation Alter crack geometry: interchange x- and y-axis Print out of geometric configurations Evaluate definite integrals Evaluate complete elliptical integrals of the first and second kind Obtain equivalent stress intensity factor from K l, K H and K m Determine limiting angles for in-plane crack growth New crack front fitted with elliptical curvature Compute Kll and Kll I at crack border points for residual and rolling contact stresses. Compute K l Compute K x for semi-elliptical surface crack Search crack border nodes to fit tractions Unstable crack growth computation Check for global failure and switch in crack configuration from a semi-ellipse to a quarter-ellipse or quarter-circle Compute Kj for elliptical crack in infinite body in tension Compute equivalent stress intensity factor at points on crack border Second degree polynomial fit of nodal point stresses Select points to fit stresses Main program
SHAH SIH SUPER6 SUPERF TERMAL2
calculation. Shear action would change direction. Hence, the full range of _+K~I and + K m must be accounted for.
4. Computer subroutines Inclusion of the shear action on the wheel requires a minor modification of the computer program in [1]. Figure 8 is a flow chart that shows the various computational subroutines whose functions are described in Table 3. The main program is referred to as TERMAL2.
5. Discussion of results Referring to Fig. 2, a normal and tangential load of 100 kN and 25 k N prevail, respectively, on the contact area which is elliptically shaped with a semi-major axis of 8.09 m m and semi-minor axis of 5.57 mm. This corresponds to a wheel of 1,250
m m in diameter. The applied loads in terms of Oxy and oxz that give rise to the equivalent stress intensity factors K~q) and K ~ in eq. (10) are shown in Figs. 9(a), 9(b) and 9(c), respectively. Superscripts r and f are used on Oxx, Oxy and oxz to distinguish residual stress effects from those of alternating stress. Values of o~ (i = x, j = x, y, z) are taken from [1] for correct braking of 90 kW and 15 minutes duration. They are shown by the dotted lines. The peak values of oif (i = x, j = x, y, z) corresponding to those in Figs. 5(a) to 5(c) indicated by the open circles in Figs. 9(a) to 9(c) are used to check for the onset of unstable crack propagation. The solid lines represent contact loads from the wheel. Figure 10 shows the position of the crack which experiences an alternating load as the wheel rolls with a weight of 100 kN. Displayed in Fig. 11 are crack growth against load cycle N expressed in terms of the semi-elliptical crack dimensions a and c in Fig. 3. T w o sets of curves are presented; they correspond to CO= 6.979 x 10 -12 and m = 2.95 [15] in eq. (11) and to Oxx ,
A. Martin Meizoso et aL / Crack growth of railway wheel
187
20C BO
I
A~x
o'.: [ y , z ) n
7£
_,/[-
Stre== range u=ed to compute the =ubcdt|ca[ ~ack growth, ; y
x
b
(a) Stress O'xx
2~
~x f (Z) ~-~ Y
tL
~r 0
~xy
o'xr : 0 Y +'a"
-71"
0
b
(b) Strass O'xy
20(
_,~(z) z &O'~z
o
'S"
+71" 0 N
I
(c) Stres= O'~z
Fig. 9. Alternating load on wheel with surface crack.
estimates made by considering only K i and all three factors K I, KII and Kin. A threshold level of AKth = 4 MPaCm- has been assumed [15]. It is evident that the fatigue lives of the locomotive wheel are reduced appreciably when all three stress intensity factors are accounted for. The tangential load of 25 kN has a significant effect. This is also
reflected by the variations of Keq versus log10 N in Fig. 12 where the combination of K~, KI! and Knl resulted in much higher Kin. The presence of KiI and K m gives rise to out-of-plane crack growth which can be estimated from the strain energy density criterion. Figure 13 shows a family of semi-ellipses that correspond to
A. Martin Meizoso et aL / Crack growth of railway wheel
188
!250
80
i
;:°,7oo
60
KI, KI[ and KII I
~" 4 0
Kmdx
!
•
~s g
/
RAIL
Kmin
20
/ ~ ,~ ~
K I only
T 0
l 5
4
1O0 KN pure roiling
I 6
i
Log~oN
( cyg[e )
I 7
Fig. 12. Equivalent stress intensity factor variations with load cycle for wheel with and without tangential load.
Fig. 10. Reference angle for crack on wheel rim. 100 9O
....
K I ol~ly
- -
K I ,KII end Kll I
the crack growth profiles in the radial and axial direction. The out-of-plane crack border has been projected onto the original yz-plane that is normal to the wheel rim. Three different zones of crack orientation can be identified; they are referred to as A, B and C and explained in Figs. 14(a), 14(b) and 14(c), respectively. Initially, K l is more dominant and the crack grew in the original plane with O0 = 0 ° marked as zone A in Fig. 14(a). An intermediate zone B in Fig. 14(b) exhibits the influence of shear stresses (Oxy+ ox~) where an out-of-plane crack growth orientation angle of 00 = 82 o is predicted. This angle reduces to 00 = 72 ° in Fig. 14(c) labelled as zone C in Fig. 13 as the crack approaches unstable fracture. As
8O 70
~. 60
~ 5o 4 o
¢ / / i
4o
!1
30
t/ a
c
20
lO
/ 11 1I cl
/Y
o 4.5
5
5.5
6
6.5
7
7.5
Loglo N (cycle)
Fig. 11. Semi-elliptical crack growth as a function of load cycle in wheel with and without tangential load. 650
640
E
v
630
c
2 63
620
-6 ~5 o cr
610
600 _ 150
, 140
. 130
.
. 120
Axial
. 110
. 100
i
90
Distance (mm)
Fig. 13. Crack growth profiles projected onto plane normal to wheel rim.
80
m
189
A. Martin Meizoso et al. / Crack growth of railway wheel O'xx
O'xx
K, /,J2g - -
R~duol
,K, /V~-~
~
(p
0--
(K, + K, )/V'~T
,
F/
/
/ / ! / t / ! ! L
t /
I
X
J
82,
(K, + K, )/.,fi~-
0"=72"
im
O ~ + 0txz
(K, + K, )/~,/~
\ \ \ x x \ \ x x x 3
(a) Zone A
(b) Zone B
(e) Zone (2
Fig. 14. Path traversed by normal and shear stress intensity factors with reference to crack growth orientation.
mentioned earlier, the crack profiles in Fig. 13 correspond to those projected onto the plane normal to the wheel rim surface. Nevertheless, these features can be recognized from the actual fracture surfaces of a cracked UICR-7 wheel as shown in Figs. 15(a) and 15(b).
6. C o n c l u s i o n s
An approximate estimate of crack growth in a locomotive wheel has been made by assuming that the surface crack border stress intensity factors Kl, K u and Kll I could be obtained separately from a stress analysis of the wheel without the
crack. This is accomplished by introducing corrections to the K-factors of a completely embedded elliptical crack in an infinite body. Both normal and tangential loads transmitted from the wheel to the contact on the rail are considered and reflected by the factors K l, Kll and K l l I n e a r the crack border. Out-of-plane crack growth is considered, the assessment of which is made by projecting the new configuration onto the plane normal to the wheel rim in which the crack extends. Despite these approximations, the calculated crack surfaces resembled those on the fatigue fracture of an actual U I C R - 7 wheel. The validity of the analysis remains to be checked until more refinement results become available.
Fig. 15. Stereo pair of a cracked UICR-7 wheel.
190
A. Martin Meizoso et al. / Crack growth of railway wheel
The effect of thermal cycling due to braking applied on the wheel is approximately equivalent to that of rolling contact for short cracks when residual stress effects are appropriately included. As crack extends deeper into the wheel, thermal cycling effects become less significant because they are confined near the surface layer. Rolling contact loads would remain to be significant for the major portion of the wheel fatigue life. Consideration of thermal cycling alone [16-18] would not be sufficient for estimating the total life of railcar wheels.
Acknowledgements The authors wish to thank Construcciones y Auxiliar de Ferrocarriles (CAF, Spain) and the Association of American Railroads (AAR, USA), for supporting the present investigation. Valuable comments of Professor G.C. Sih are also acknowledged.
References [1] A. Martin Meizoso and J. Gil Sevillano, Life prediction of thermally cracked railway wheels: growth estimation of cracks with arbitrary shape, Theor. Appl. Fract. Mech. 9 (1988) 123-139. [2] J.M. Egafia and J.G. Gim~nez, Estudio elastoplhstico de s61idos de revoluci6n sometidos a cargas t&micas, I V Congreso Naeional sobre la Teorla de Meeanismos y M6quinas, 1981. [3] J.M. Egafia and J.G. Gim~nez, Analysis by the finite element method of thermoplastic problems in axisymmetrical solids, 2nd Int. Conf. in Numerical Methods for Nonlinear Problems, 1984.
[4] S. Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill: New York, 1951). [5] A.S. Saada, Elasticity: Theory and Applications (Pergamon: New York, 1974). [6] R.C. Shah and A.S. Kobayashi, Stress intensity factor for an elliptical crack under arbitrary normal loading, Eng. Fract. Mech. 3 (1971) 71-96. [7] F.W. Smith and D.R. Sorensen, Mixed mode stress intensity factors for semi-elliptical surface cracks, Report NASA CR-134684, Colorado State University, 1974. [8] J.C. Newman and I.S. Raju, An empirical stress intensity factor equation for the surface crack, Eng. Fract. Mech. 15 (1981) 185-192. [9] G.C. Sih and Y.D. Lee, Review of triaxial crack border stress and energy behavior, Theor. Appl. Fract. Mech. 12 (1989) 1-17. [10] G.C. Sih and C.T. Li, Initiation and growth characterization of comer cracks near circular hole, Theor. Appl. Fract. Mech. 13 (1990) 69-80. [11] G.C. Sih and B. Macdonald, Fracture mechanics applied to engineering problems--strain energy density fracture criterion, Eng. Fract. Mech. 6 (1974) 361-386. [12] G.C. Sih, Strain energy density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305-321. [13] G.C. Sih and B.C.K. Cha, A fracture criterion for threedimensional crack problems, Eng. Fract. Mech. 6 (1974) 699-723. [14] G.C. Sih and P.S. Theocaris, eds., Mixed Mode Crack Propagation (Sijthoff and Noordhoff: The Netherlands, 1981). [15] J.M.R. Ibabe, Propagaci6n de grietas por fatiga en estructuras ferritoperliticas (0.5% C), PhD Thesis, Escuela Superior de Ingenieros Industriales de San Sebastian, Universidad de Navarra, Spain, 1984. [16] J.L. Van Swaay, The mechanism of thermal cracking in railway wheels, Proc. 3rd Int. Wheelset Congress, Sheffield, 8/1-8, 8D/5 and 16D/2, 1969. [17] H.R. Wetenkamp, C.M. Sidebotton and H.J. Schrader, The effect of brake shoe action on thermal cracking and on failure of wrought steel railway car wheels, University of Ilhnois Bulletin 47, No. 77, 1950. [18] H.R. Wetenkamp and R.M. Kipp, Safe thermal loads for a 33 inch railroad wheel, Griffin Wheel Company, 1975.
179
Approximate crack growth estimate of railway wheel influenced by normal and shear action A. M a r t i n Meizoso, J.M. M a r t i n e z E s n a o l a a n d M. F u e n t e s P6rez Centro de Estudios e Investigaciones Tdcnicas de Guipf~zcoa (CELT), San Sebastitln 20009, Spain and Escuela Superior de Ingenieros Industriales, Unioersidad de Navarra, San Sebastitln 20009, Spain
An elastic-plastic analysis of railway wheel life prediction is made for obtaining the residual stresses due to braking and alternating rolling contact stresses between the wheel and rail. These stresses are then used to obtain approximate crack border stress intensity factors Ki, K n and K m such that an equivalent combined load stress intensity factor can be derived by application of the strain energy density criterion. By including the effect of both the normal and shear action on the locomotive wheel, the results are indicative of those observed in service. While cracks may nucleate from thermal fatigue, it suffices to consider the cyclic load due to rolling contact for estimating the fatigue lives of locomotive wheels.
1. Introduction T h e c o n t a c t l o a d b e t w e e n the wheel a n d rail i n t r o d u c e s shear a c t i o n into the rim of the wheel even for the c o n d i t i o n o f p u r e rolling. Such a n effect b e c o m e s m o r e p r o n o u n c e d w h e n c o n t a c t friction is present. F o r a l o c o m o t i v e wheel, the t a n g e n t i a l forces are r e q u i r e d to b e as large as thirty-six (36) p e r c e n t of the weight. This corres p o n d s to the s i t u a t i o n o f a l o c o m o t i v e p u l l i n g a train on an incline where the c o m b i n e d n o r m a l a n d shear l o a d i n g effect is e x p e c t e d to b e a p p r e ciable. F o l l o w i n g the earlier w o r k [1] that dealt with p r e d i c t i n g r a i l w a y wheel life u n d e r n o r m a l load, this w o r k considers the effect of b o t h n o r m a l a n d shear a c t i o n o n c r a c k growth where a n equivalent stress intensity factor is d e f i n e d b y a p p l i c a t i o n of the strain energy d e n s i t y factor to include all three stress intensity factors K~, K H a n d g i l I.
3t wheel
section
2. Analytical modelling: stresses and intensity factor's T w o types of i n d u c e d stresses will b e c o n s i d ered; they are the residual stresses d u e to b r a k i n g a n d c o n t a c t stresses arising f r o m the wheel a n d 0167-8442/91/$03.50 © 1991 - Elsevier Science Publishers B.V.
~.
Axial
Fig. 1. Finite element mesh of rail wheel section for determiningresidual stresses.
A. Martin Meizoso et al. / Crack growth of railway wheel
180
rail. The former will be computed from an elastic-plastic finite dement procedure [2,3]. Refer to Fig. 1 for the grid pattern of the rail wheel section used in the calculation. An elastic analysis [4,5] is used to obtain the Hertzian contact stresses such that the elliptical contact area and the corresponding pressure distribution are determined as functions of the wheel and rail radii in addition to the contact load magnitude. From a knowledge of the remote stresses, the crack edge stress intensity factors can be determined accordingly. A schematic of the crack configuration is shown in Fig. 3.
y -~----~
Fig. 3. Schematic of surface semi-elliptical crack in wheel.
2.1. Contact stresses
The contact pressure on the wheel is discretized into a finite number of point forces as illustrated in Fig. 2. Solutions to the concentrated normal and tangential forces acting on a semi-infinite elastic body are known, respectively, from the Boussinesq and Cerruti problem [5]. An accurate description of the state of affairs away from the local contact region is thus obtained. Near the contact, the point load is replaced by an equivalent uniform pressure distributed over a finite region [5]. It is the local stresses near the contact that would influence the onset of crack growth.
Referring to the coordinate axes x, y and z in Fig. 2, the stress components of interest are oxx, Oxy and oxz; they would exert mixed mode effects on crack growth. Contour plots of these stresses for a complete revolution of the wheel can be found in Figs. 4(a)-(c) inclusive. Table 1 summarizes the maximum positive and negative amplitudes of axe, Oxy and o~z. Negative o~ corresponds to a compressive action on the crack oriented in the yz-plane. Hence, the shear components O~y and oxz would lie in the crack plane and they are reduced because of compressions as friction between the crack surfaces. It is convenient to define an effective shear stress ~* such that
t; ~'*=
fOrOxx0 -#[Oxx[,
0,
foro~x<0; ~loxxl
(1)
in which • is the resultant shear given by r =
(2)
O~x2y+ Off~
In eq. (1), # is the coefficient of friction between the crack surfaces. The above model assumes that compressive normal stress or Oxx < 0 would correspond to crack closure. This is recognized as an approximation. More accurate assessment of crack closure would be beyond the objective of this y
5,57mm
8,09mm
N o r m a l load = I OOKN C i r c u m f e r e n t i a l load = 2 5 KN
Fig. 2. Diseretization of the rail-wheel load into points on contact surface.
Table 1 Numerical values of normal and shear stresses (MPa) Stresscomponent
Positive
Negative
o~x
99 497 841
- 1828 -497 - 841
oxy Oxz
A. Martin Meizoso et al. / Crack growth of railway wheel 2'5
30 m m
5
20
10
25
50
181
-1000-500-250
10
0
10
30
20
(a) stres, o'~ 25
~._
50
100
250
- 2 5 0 -100 -50
-25MPa
,t///l\\-~,
.°
10
0
10
_.~,o 20
Lo
(b) Strns O'~y
.0ram
-10
[ 30
mm
20
zo
1~
i'o
2'0
Fig. 4. Normal and shear stress (MPa) contours for a complete wheel revolution.
A. Martin Meizoso et al. / Crack growth o[railway wheel
182
Table 2 Coefficients in polynorrhals for a~*~,a *y and o* (MPa) and 2.5 mm < z < 34 mm
work. D i s p l a y e d in Figs. 5(a) to 5(c) inclusive are the stresses a * , a ~ a n d a * for/z = 1 as a f u n c t i o n o f the wheel r i m d e p t h in the z-direction at the prospective crack site. W h i l e a * = axx, o ~ a n d a ~ represent the p r o j e c t i o n o f ~-* a l o n g the y- a n d z-axis. Their variations c a n be r e p r e s e n t e d in p o l y n o m i a l s as follows:
+ dij l_-~, i, j = x ,
y, z
1000
Subscripts for oi~
o,°
a,j
h,)
i=x; j = x i=x; j=y i=x; j = z
0 -497.85 -1448.7 1369.2 0 -8.3684 408 -2116 3795.4 -2145.8 -26.42 1354.8 -6476.9 10656 -56947
C~j
(3)
(a)
500
~-
0
x
b
-500
b~ -1ooo
-1500
-2000
,~ 0
I 10
,
I 20
,
I 30
,.,. 40
DEPTH (ram) 200
(b)
1 O0
n ,>,,
--100
-200
.zk--.--..--___.
0
10
20
30
DEPTH ( m m ) Fig. 5. Normal and shear stress distribution with distance normal to wheel rim.
¢0
d~j
183
A. Martin Meizoso et al. / Crack growth of railway wheel
60
40 20 13_
0 b -20
-40 -60
I0
20
30
40
DEPTH (turn)
Fig. 5 (continued).
The coefficients o°, aij, bij . . . . . di# for i = x and j=x, y, z can be found in Table 2 for 2.5 mm~
o*=4z,
tion of the three-dimensional crack border stress field could also be made by using the finite element method [10], a consideration that would be left for the future.
o ~ = 120 - 42.2z, for z < 2.5 mm
(4)
When z exceeds 34 mm, the following expressions can be used: ox* = - 2 1 0 . 8 1 ' o~* = 326.4 1 ,
o ~ = 88.4 1 , for z > 34 mm
(5)
2.2. Stress intensity factors Approximate stress intensity factor expressions for K~ arising from o~ and K H and Kin from Oxy and o~ will be obtained, respectively, from the works in [6] and [7] for a complete elliptical crack in an infinite medium. The prevailing applied pressure is approximated by a second degree polynomial based on a least square fit. Since the actual crack configuration is interrupted by a free surface, additional correction factors [8] are introduced with the understanding that these results are subject to further refinements [9]. Direct determina-
3. Crack growth consideration Assessment of mixed mode cracking would be made for each point on the crack front by referring to the stress oxx normal to the crack plane and Oxy q-Oxz parallel to the crack surface. These stresses give rise to the stress intensity factors K~ and K H + Kin; they can be represented graphically as in Fig. 6 such that any 'stress state away from the vertical and horizontal axis would be subject to the combined influence of normal and shear stress.
3.1. Fracture criterion The combined influence of KI, KII and KIII could be visualized by an equivalent stress intensity factor Keq in Fig. 7 that traverses the surface of a distorted cone. Among many of the proposed fracture criteria, the strain energy density criterion or S-theory developed by Sill [11-14] is most expedient. When all three stress intensity factors
184
A. Martin Meizoso et al. / Crack growth of railway wheel
coefficients a,l (i, j = 1, 2, 31) are of the forms [13]
O-xx
/
Residual
|
KI/ I/"~"a --
,.
!
'\
/
\
/
\
/ / /
a33 - ~(1 + . ) with E and v being respectively, the Young's modulus and Poisson's ratio. The angle 0 is measured in a plane normal to the crack border. That is, 0 = 0 would correspond to the yz-plane in Fig. 2. Postulated in the S-theory is that crack would initiate in the direction determined by the relative minimum of the strain energy density factor S, say S,,io and that the onset of crack growth would occur when Smin reaches a critical value Sc~ which can be related to the Klc fracture toughness value:
\
/
\
/ /
\ \
/
\
/
(7)
32E
\ \
/
a22 = ~r(1 + v) [4(1 - v)(i - cos 0) + ( 3 cos O - 1)(1 + cos O)]
\
/
16E ~r(1 + v) sin 0(cos 8 - I + 2v)
\
\
/
a,2
8E
\
/
~r(l+v)(3-4v-cosO)(l+cosO)
xy + O-xz
(KIT +KIII) / V'~"0
\
/
[7
8E
au
\
Fig. 6. Schematic of cyclic load path.
(1 + v)(1 - 2v)K(~ are present, they could be combined into the strain energy density factor as
(6)
S = auK 2 + 2 a t 2 K , K , I + az2K 2 + a33 K 2 ,
In eq. (6), S is the amplitude of the strain energy density d W / d V that behaves as 1/r as r the distance from the crack border tends to zero. The
s. =
2~e
(8)
Since Scr is characteristic of the material, it can be used to predict the fracture behavior for situations where all three K-factors are present. In general, an equivalent K or Keq can be defined using the functional form of eq. (8), i.e.,
geq =
(1 + v)(1 - 2 v )
~
(9)
This quantity is calculated for each point on the crack border.
Keq
3.2. Load superposition The wheel is subjected to both residual and alternating stresses. These two effects can be reflected via the equivalent stress intensity factor by superposition: Keq = --eqK(r)-+ --eqK(f) ~ K I I
- KllI
Fig. 7. Equivalent stress intensity factor variation for a wheel revolution.
(10)
Note that the superscripts r and f refer, respectively, to residual and alternating stress. The sign for K~qr) can be positive or negative depending on
A. Martin Meizoso et al. / Crack growth of railway wheel
whether the alternating load exerts an opening or closing action to the crack surfaces. For each rolling cycle, K~q would vary along the crack border and only a portion of the crack would be subjected to the maximum intensity. This would imply that not all points on the crack border would initiate at the same time. Moreover, crack growth may take place in different planes at different locations. The foregoing two effects are not explicitly included in the schematic representation of Keq in Fig. 7 which assumes that all points on the crack border would reach the same maximum value at the same time, an assumption that is regarded to be conservative. Change in crack direction is weighed into crack growth in a manner to be discussed subsequently.
185
etc., on the crack and the condition SJrg = const. [12] would yield the new crack configuration which would no longer lie in the original crack plane. This process would require a three-dimensional finite element analysis of the crack problem. Instead, an approximate procedure would be adopted. Only the onset of out-of-plane crack growth is estimated. The new crack front is then projected [1] into the original yz-plane in Fig. 3. It is then fitted by an elliptically-shaped curve such that it suffices to use the semi-axes a and c for describing crack growth. Hence, the equivalent crack would always grow in the same plane. An estimate of fatigue crack growth is made by using the relation zaa
3. 3. Crack shape modelling
= Co(AKoq) ~
As stated earlier, the direction of crack growth, say 00 for a point on the crack border, would depend on the relative magnitudes of the three stress intensity factors K I, K n and Km. The condition aS/~O--0 for 02S/~02> 0 can be invoked on eq. (6) to find 00 for each point j = 1, 2,
where Co and m are empirical parameters determined experimentally for a given material and specimen type. The A K ~ would depend on AK~, AK u and AKIII. Negative values of K I are assumed to close the crack and hence would not yield crack growth. They are excluded from the
-
-
dlmtermal.for
fmttermol.for
~4
(11)
I Glro
Fig. 8. Computer program functional flow chart.
186
A. Martin Meizoso et al. / Crack growth of railway wheel
Table 3 Computational subroutines of revised program Subroutines
Functions
CONTAJU CRECE DATOS DXDY FATIGA GIRO INFORM INTEG INTEL JMSIH NEWANG NEWELP NK2K3 NKFKE NNEWMAN PINTEL PROPAG ROTURA
Contact stresses Update crack front Input data for main program Project crack growth of each border point Fatigue crack growth calculation Alter crack geometry: interchange x- and y-axis Print out of geometric configurations Evaluate definite integrals Evaluate complete elliptical integrals of the first and second kind Obtain equivalent stress intensity factor from K l, K H and K m Determine limiting angles for in-plane crack growth New crack front fitted with elliptical curvature Compute Kll and Kll I at crack border points for residual and rolling contact stresses. Compute K l Compute K x for semi-elliptical surface crack Search crack border nodes to fit tractions Unstable crack growth computation Check for global failure and switch in crack configuration from a semi-ellipse to a quarter-ellipse or quarter-circle Compute Kj for elliptical crack in infinite body in tension Compute equivalent stress intensity factor at points on crack border Second degree polynomial fit of nodal point stresses Select points to fit stresses Main program
SHAH SIH SUPER6 SUPERF TERMAL2
calculation. Shear action would change direction. Hence, the full range of _+K~I and + K m must be accounted for.
4. Computer subroutines Inclusion of the shear action on the wheel requires a minor modification of the computer program in [1]. Figure 8 is a flow chart that shows the various computational subroutines whose functions are described in Table 3. The main program is referred to as TERMAL2.
5. Discussion of results Referring to Fig. 2, a normal and tangential load of 100 kN and 25 k N prevail, respectively, on the contact area which is elliptically shaped with a semi-major axis of 8.09 m m and semi-minor axis of 5.57 mm. This corresponds to a wheel of 1,250
m m in diameter. The applied loads in terms of Oxy and oxz that give rise to the equivalent stress intensity factors K~q) and K ~ in eq. (10) are shown in Figs. 9(a), 9(b) and 9(c), respectively. Superscripts r and f are used on Oxx, Oxy and oxz to distinguish residual stress effects from those of alternating stress. Values of o~ (i = x, j = x, y, z) are taken from [1] for correct braking of 90 kW and 15 minutes duration. They are shown by the dotted lines. The peak values of oif (i = x, j = x, y, z) corresponding to those in Figs. 5(a) to 5(c) indicated by the open circles in Figs. 9(a) to 9(c) are used to check for the onset of unstable crack propagation. The solid lines represent contact loads from the wheel. Figure 10 shows the position of the crack which experiences an alternating load as the wheel rolls with a weight of 100 kN. Displayed in Fig. 11 are crack growth against load cycle N expressed in terms of the semi-elliptical crack dimensions a and c in Fig. 3. T w o sets of curves are presented; they correspond to CO= 6.979 x 10 -12 and m = 2.95 [15] in eq. (11) and to Oxx ,
A. Martin Meizoso et aL / Crack growth of railway wheel
187
20C BO
I
A~x
o'.: [ y , z ) n
7£
_,/[-
Stre== range u=ed to compute the =ubcdt|ca[ ~ack growth, ; y
x
b
(a) Stress O'xx
2~
~x f (Z) ~-~ Y
tL
~r 0
~xy
o'xr : 0 Y +'a"
-71"
0
b
(b) Strass O'xy
20(
_,~(z) z &O'~z
o
'S"
+71" 0 N
I
(c) Stres= O'~z
Fig. 9. Alternating load on wheel with surface crack.
estimates made by considering only K i and all three factors K I, KII and Kin. A threshold level of AKth = 4 MPaCm- has been assumed [15]. It is evident that the fatigue lives of the locomotive wheel are reduced appreciably when all three stress intensity factors are accounted for. The tangential load of 25 kN has a significant effect. This is also
reflected by the variations of Keq versus log10 N in Fig. 12 where the combination of K~, KI! and Knl resulted in much higher Kin. The presence of KiI and K m gives rise to out-of-plane crack growth which can be estimated from the strain energy density criterion. Figure 13 shows a family of semi-ellipses that correspond to
A. Martin Meizoso et aL / Crack growth of railway wheel
188
!250
80
i
;:°,7oo
60
KI, KI[ and KII I
~" 4 0
Kmdx
!
•
~s g
/
RAIL
Kmin
20
/ ~ ,~ ~
K I only
T 0
l 5
4
1O0 KN pure roiling
I 6
i
Log~oN
( cyg[e )
I 7
Fig. 12. Equivalent stress intensity factor variations with load cycle for wheel with and without tangential load.
Fig. 10. Reference angle for crack on wheel rim. 100 9O
....
K I ol~ly
- -
K I ,KII end Kll I
the crack growth profiles in the radial and axial direction. The out-of-plane crack border has been projected onto the original yz-plane that is normal to the wheel rim. Three different zones of crack orientation can be identified; they are referred to as A, B and C and explained in Figs. 14(a), 14(b) and 14(c), respectively. Initially, K l is more dominant and the crack grew in the original plane with O0 = 0 ° marked as zone A in Fig. 14(a). An intermediate zone B in Fig. 14(b) exhibits the influence of shear stresses (Oxy+ ox~) where an out-of-plane crack growth orientation angle of 00 = 82 o is predicted. This angle reduces to 00 = 72 ° in Fig. 14(c) labelled as zone C in Fig. 13 as the crack approaches unstable fracture. As
8O 70
~. 60
~ 5o 4 o
¢ / / i
4o
!1
30
t/ a
c
20
lO
/ 11 1I cl
/Y
o 4.5
5
5.5
6
6.5
7
7.5
Loglo N (cycle)
Fig. 11. Semi-elliptical crack growth as a function of load cycle in wheel with and without tangential load. 650
640
E
v
630
c
2 63
620
-6 ~5 o cr
610
600 _ 150
, 140
. 130
.
. 120
Axial
. 110
. 100
i
90
Distance (mm)
Fig. 13. Crack growth profiles projected onto plane normal to wheel rim.
80
m
189
A. Martin Meizoso et al. / Crack growth of railway wheel O'xx
O'xx
K, /,J2g - -
R~duol
,K, /V~-~
~
(p
0--
(K, + K, )/V'~T
,
F/
/
/ / ! / t / ! ! L
t /
I
X
J
82,
(K, + K, )/.,fi~-
0"=72"
im
O ~ + 0txz
(K, + K, )/~,/~
\ \ \ x x \ \ x x x 3
(a) Zone A
(b) Zone B
(e) Zone (2
Fig. 14. Path traversed by normal and shear stress intensity factors with reference to crack growth orientation.
mentioned earlier, the crack profiles in Fig. 13 correspond to those projected onto the plane normal to the wheel rim surface. Nevertheless, these features can be recognized from the actual fracture surfaces of a cracked UICR-7 wheel as shown in Figs. 15(a) and 15(b).
6. C o n c l u s i o n s
An approximate estimate of crack growth in a locomotive wheel has been made by assuming that the surface crack border stress intensity factors Kl, K u and Kll I could be obtained separately from a stress analysis of the wheel without the
crack. This is accomplished by introducing corrections to the K-factors of a completely embedded elliptical crack in an infinite body. Both normal and tangential loads transmitted from the wheel to the contact on the rail are considered and reflected by the factors K l, Kll and K l l I n e a r the crack border. Out-of-plane crack growth is considered, the assessment of which is made by projecting the new configuration onto the plane normal to the wheel rim in which the crack extends. Despite these approximations, the calculated crack surfaces resembled those on the fatigue fracture of an actual U I C R - 7 wheel. The validity of the analysis remains to be checked until more refinement results become available.
Fig. 15. Stereo pair of a cracked UICR-7 wheel.
190
A. Martin Meizoso et al. / Crack growth of railway wheel
The effect of thermal cycling due to braking applied on the wheel is approximately equivalent to that of rolling contact for short cracks when residual stress effects are appropriately included. As crack extends deeper into the wheel, thermal cycling effects become less significant because they are confined near the surface layer. Rolling contact loads would remain to be significant for the major portion of the wheel fatigue life. Consideration of thermal cycling alone [16-18] would not be sufficient for estimating the total life of railcar wheels.
Acknowledgements The authors wish to thank Construcciones y Auxiliar de Ferrocarriles (CAF, Spain) and the Association of American Railroads (AAR, USA), for supporting the present investigation. Valuable comments of Professor G.C. Sih are also acknowledged.
References [1] A. Martin Meizoso and J. Gil Sevillano, Life prediction of thermally cracked railway wheels: growth estimation of cracks with arbitrary shape, Theor. Appl. Fract. Mech. 9 (1988) 123-139. [2] J.M. Egafia and J.G. Gim~nez, Estudio elastoplhstico de s61idos de revoluci6n sometidos a cargas t&micas, I V Congreso Naeional sobre la Teorla de Meeanismos y M6quinas, 1981. [3] J.M. Egafia and J.G. Gim~nez, Analysis by the finite element method of thermoplastic problems in axisymmetrical solids, 2nd Int. Conf. in Numerical Methods for Nonlinear Problems, 1984.
[4] S. Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill: New York, 1951). [5] A.S. Saada, Elasticity: Theory and Applications (Pergamon: New York, 1974). [6] R.C. Shah and A.S. Kobayashi, Stress intensity factor for an elliptical crack under arbitrary normal loading, Eng. Fract. Mech. 3 (1971) 71-96. [7] F.W. Smith and D.R. Sorensen, Mixed mode stress intensity factors for semi-elliptical surface cracks, Report NASA CR-134684, Colorado State University, 1974. [8] J.C. Newman and I.S. Raju, An empirical stress intensity factor equation for the surface crack, Eng. Fract. Mech. 15 (1981) 185-192. [9] G.C. Sih and Y.D. Lee, Review of triaxial crack border stress and energy behavior, Theor. Appl. Fract. Mech. 12 (1989) 1-17. [10] G.C. Sih and C.T. Li, Initiation and growth characterization of comer cracks near circular hole, Theor. Appl. Fract. Mech. 13 (1990) 69-80. [11] G.C. Sih and B. Macdonald, Fracture mechanics applied to engineering problems--strain energy density fracture criterion, Eng. Fract. Mech. 6 (1974) 361-386. [12] G.C. Sih, Strain energy density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305-321. [13] G.C. Sih and B.C.K. Cha, A fracture criterion for threedimensional crack problems, Eng. Fract. Mech. 6 (1974) 699-723. [14] G.C. Sih and P.S. Theocaris, eds., Mixed Mode Crack Propagation (Sijthoff and Noordhoff: The Netherlands, 1981). [15] J.M.R. Ibabe, Propagaci6n de grietas por fatiga en estructuras ferritoperliticas (0.5% C), PhD Thesis, Escuela Superior de Ingenieros Industriales de San Sebastian, Universidad de Navarra, Spain, 1984. [16] J.L. Van Swaay, The mechanism of thermal cracking in railway wheels, Proc. 3rd Int. Wheelset Congress, Sheffield, 8/1-8, 8D/5 and 16D/2, 1969. [17] H.R. Wetenkamp, C.M. Sidebotton and H.J. Schrader, The effect of brake shoe action on thermal cracking and on failure of wrought steel railway car wheels, University of Ilhnois Bulletin 47, No. 77, 1950. [18] H.R. Wetenkamp and R.M. Kipp, Safe thermal loads for a 33 inch railroad wheel, Griffin Wheel Company, 1975.