axisymmetric bar under bending was produced by Tipton 2° using Neuber's .... of a circuler hole' Phil Trans, Roy Sac London, Series A 229. (1930) pp 49-86. 17.
IntJ Fatigue9 No 3 (1987) pp 143-150
U n i v e r s a l f e a t u r e s of e l a s t i c notchtip s t r e s s fields G. Glinka and A. N e w p o r t
The results of linear elastic analyses of stress distributions near a wide variety of notches are presented. Notches under tension, tension and bending, and pure bending have been considered. It is demonstrated that notch-tip stress fields are similar to each other regardless of the notch geometry and the loading system. Universal functions describing the stress field in the notch-tip neighbourhood have been derived and tested against available analytical, numerical and experimental data. The universal expressions can be used to calculate stresses in the region x ~< 3p from the notch tip. These expressions can be particularly useful when using the weight function method to calculate stress intensity factors for cracks emanating from notches. Key words: fatigue; notch-tip stress distribution; stress concentration factor; notch geometry; loading conditions; prediction
Notation ~t
0 b d D K
Kt P r
S
st l
minor axis of an ellipse ligament width ahead of a notch width or diameter of a notched component stress intensity factor stress concentration factor force/load radial polar coordinate nominal stress nominal stress in the notch tip notch depth or notch half-length
It is well established that a notch in a component causes a localized stress concentration which may lead to early fatigue crack initiation and enhanced fatigue crack growth in the notch-tip region if the component is subjected to cyclic loading. One of the parameters used to characterize this local notch-tip stress field is the well-known stress concentration factor K,. The available handbooks 1~ make the determination of stress concentration factors quite simple for a variety of components and structures. However, it should be noted that, alone, the stress concentration factor can only be used to calculate the maximum stress in the notch-tip and does not provide any information about the other stress components, nor about the stress distribution in the notch-tip region. Consequently, although the stress concentration factor can be successfully used 3'4 for the prediction of fatigue crack initiation life, it is not sufficient to allow the stress intensity factor to be calculated for a crack emanating from a notch tip since the stress distribution ahead of the notch tip must also be known. Fatigue crack growth analysis requires the calculation
/¢
V
P O"
crx, %, or,
Cry,
distance from the notch tip angle of a V-noteh angular polar coordinate distance from the notch tip to the neutral axis in bending or bending and tension in relation to the nominal stress distribution S Poisson's ratio notch-tip radius stress stress components value of stress component o'y in the notch tip
of a stress intensity factor s which depends upon both the geometry of the cracked body and the stress distribution along the potential crack plane. It was shown by Bueckner 6 and Rice 7 that the stress intensity factor can be calculated by using the weight function method providing that the stress distribution in the potential crack plane is known.
P a r a m e t e r s c h a r a c t e r i z i n g n o t c h - t i p stress fields It is recognized l~z that the stress concentration factor in Equation (1), def'med as shown in Fig. 1, depends upon the global geometry of the notched body, the loading system and the local notch geometry:
(1) In contrast, the notch-tip stress distributions, when normalized by Kt, are very similar for a variety of notches and do not exhibit any strong dependency on the global geometry
0142-1123/87/03143-08 $3.00 © 1987 Butterworth & Co (Publishers) Ltd Int J Fatigue July 1987
148
¥
Y
st
Y
P
St
x
x -f-i
=
D
a
l-
b
Fig. 1 Stress concentration in notched elements: (a) under tensile loading and (b) under bending loading
width can be regarded as infinite, such that diP ~ ®. For notches in finite width plates the stress component o"x can be affected by the free edge ahead of the notch and Equation (3a) may overestimate the stress component cr~ at very short distances such as x ~< p. Conversely, Equation (3a) may underestimate the stress component cr~ at distances x > p for sharp notches in relatively wide plates. Thus the use of Equation (3a) has to be limitedto blunt notches with a wide ligament ahead of them, ie K¢ 3p. The stress component cry in the vicinity of a notch in a finite width plate quickly drops down below the nominal stress S ahead of the notch tip and therefore it was expected that Equation (3b) would overestimate the stress cry at larger distances, ie x > p. Thus Equation (3b) is limited to cases where the net and gross nominal stresses are almost equal, ied~D.
of the notched body. However, they do depend on the notchtip geometry, which can be characterized by the notch-tip radius p. This led several authors s-l° to conclude that the stress distribution in the notch-tip neighbourhood can be satisfactorily described by two parameters - - the stress concentration factor Kt and the notch-tip radius p. Schijve9 showed that the two-parameter description of the notch-tip stress field gave satisfactory results for several notch geometries. A two-parameter expression was also proposed by Santhanam and Bates) However, they tested the accuracy of their expressions against only a limited variety o f notches and in both investigations only notches under tensile loading were analysed. It should also be noted that most analyses 8-~°were devoted to only one stress component, cry. Nevertheless, further analysis has shown that the similarities between notch-tip stress fields indicated in References 9 and 10 are universal, and consequently it has been possible to derive universal expressions for the notch-tip stress field adequate for a wide variety of engineering applications.
Notch-tip stress fields near blunt notches The best known expressions for calculating stresses in the notch-tip neighbourhood are those derived for circular and elliptical notches in an infinite plate under uniform tensile loading. In the case of a circular notch in an infinite plate, the stress components in the plane 3' = 0 (Fig. 1) can be calculated from Equations (2a)-(2c) taken from Reference 11: crx =
srL 2jx\ p +
cry =
E 1(~ + l)-' S 1 + ~
1
- ~
+
')']
3(p )-4] + ~ + 1
cr~ = 0 for plane stress crz = v(crx + try) for plane strain
(2a) (2b) (2c)
Because of the similarities between notch-tip stress distributions, Equations (2a) and (2b) can be generalized by analogy to the expression proposed by Usami 12 for the stress component cry:
o
eI( +l)
cry= _ ~ [ 1 + ~ l ( p + 1)-2 + ~3(p + 1)-41
(3b)
However, due to the origin of Equations (3a) and (3b), they only provide relatively good estimates of notch-tip stresses around circular, semi-circular and blunt elliptical notches in wide plates, k in plates where theoretically the
144
N o t c h - t i p stress fields near sharp, deep notches The generality of the notch-tip stress field in the case of sharp, deep notches was clearly shown by Creager and Paris 13 who derived expressions for stresses in terms of the stress intensity factor K: -K
2v/
+
30
= c o s - J 1 2 L - sm " ~ sm " x/2nr K
eY
P
2r cos T
2~ + ~
(4a)
p 30 2r cos ~K
0 0 30-1 cos ~ [L + sin ~ sin ~ - J
crz = 0 crz = v(crx + %)
f o r plane stress for plane strain
(4b) (4c)
It was later shown by Glinka 1°'14 that the stress intensity factor K in Equations (4a) to (4c) can be replaced by the stress concentration factor Kt, and that the stress components cr, and cry occurring in the plane .7 = 0 can be written in the form:
gg~ crx = 2 x / ~
cry =
2x/~
+
+2
+
(5b)
Equations (5a) and (5b) give good estimations of notch-tip stress 1°'14 over the distance x ~< 3/9 for relatively sharp, deep notches in infinite and finite width plates with the ligament d >i 3p. For blunt notches such as those discussed earlier, Equation (5a) overpredicts the stress component cr~ at distances x > p while the stress component cry is usually underpredicted by Equation (5b) in such cases. Thus, Equations (5a) and (5b) can be recommended for relatively sharp notches in infinite and finite width plates where K, > 4.5 and d I> 3p. It is interesting to note that in the case of blunt notches the generalized Equation (3a) predicts the maximum value cr~ = ~K~S, at distance x -'- 0.414p from the notch tip. For sharp notches the maximum stress value cr, = Kfld3Vr3 is predicted by Equation (5a) as occurring at the distance x = p. However, both of these equations are reasonably accurate in the region of maximum cr, if the ligament ahead of the notch tip is wide enough, ie d >1 3p.
Int J Fatigue July 1987
Equations (3b) and (5b) overpredict and underpredict respectively the component of stress ~ry at distances x >/ p but both of them are reasonably good at distances x < p, It was also concluded that the distribution of the stress component O'y is less dependent on the ligament width d than is the distribution of the stress component ~,.
The analysis of existing data showed that the distribution of the stress component orx depends mainly on the ligament width d, being affected less by the stress concentration factor K,. Therefore Equation (3a) seems to be applicable for a variety of relatively blunt notches (K, ~< 4.5) provided that the ligament is satisfactorily wide, ie d >t 3p. The distribution of the stress component cry appears to be more strongly dependent on the stress concentration factor K, than on the ligament d, unlike a,. Here the mean values calculated from Equations (3b) and (5b) seem to be the best approximation for notches with stress concentration factor K, ~< 4.5. Blunt notches under tension, K, 4.5, better stress estimation was achieved by using Equations (9a) and (9b). The differences between accurate analytical solutions and the approximate solution were smaller than 5% in the case of peak Crx stresses. The errors in estimating the stress component Cry at a distance x = 3p were also below 5%. G o o d estimations of stresses based on Equations (8a) and (8b) were achieved for circular and elongated notches in finite width plates (see Figs 4 and 5). The results based on approximate equations (8a) and (8b) obtained for circular notches are shown in Fig. 4 together with the analytical solution obtained by Howland. 16 It is obvious that a better estimation of both stress components was achieved for the smaller notch where the width o f the ligament ahead of the notch was d = 4/). It is also apparent that the stress component Crx is affected more strongly by the ligament width d than is the stress component cTy. The greatest difference between the approximate expression (81)) and Howland's 16 analytical solution was below 15% for the case ofp/d -- 0.25. The flat plate with the elongated central notch (Fig. 5) was analysed by the finite element method, is It is apparent
146
~ O0
1
0.5
•
1.0
1.5
~
2.0
L_
25
J_
3.0
x/p Fig. 5 Stress distribution ahead of an elongated central notch in a finite width plate, p / d = 0.27, p / D = 0.09,/~ = 2.75
that the approximate equations (8a) and (8b) gave good stress estimation over the entire region 0 ~< x ~< 3p. The differences were below 5% for the stress component Cry,and 2% for the stress component Crx.
Edge n o t c h e s in s e m i - f i n i t e and f i n i t e w i d t h plates u n d e r tension First, the 'worst' case of the semi-circular notch in a semifinite plate under tension was analysed. The analytical solution for such a case was produced by Maunsell iv (see Fig. 6). As could be expected, Equation (Sa) describes the crx stress distribution well as it is almost identical to the distribution for a circular notch (Fig. 2). However, as in the central circular notch case, the stress component cry was underestimated by Equation (8b). The difference was below 20% at the distance x = 3p. Stress distributions around the symmetrical double Unotches in a finite width plate under tension are shown in Fig. 7. The approximate results obtained from Equations (8a) and (8b) were compared with finite element calculations. It is apparent that the Cry stress component was predicted adequately with an error below 10% over the entire region
Int J Fatigue July 1987
• Experimental data from Reference 19 mEquations (8a) and (8b)
• Analytical solution from Reference 17 Equations (8a) and (8b)
t
S 3.0
3.0 J b 2.0 2.0
I0
w
i.O o-x
•~
0 0
o.s
0
~.o
~.5
2.o
?-s
•
I 0.5
0
I 1.0
I 1.5
Fig. 6 Stress distribution ahead of a semi-circular notch in a semifinite plate under tension,/~ = 3
•
I 2.0
•
t 2.5
I 3.0
xlp
s.o
x/p
--o
Fig. 8 Stress distribution ahead of symmetrical V-notch with angle a = 60 °, p/d = 0.18, p/D = 0.04,/~ = 2.84
• Experimental data from Reference 19 mEquotions (So) and (Sb)
• Finite element calculations I~Equatlons (80) and (Sb) P
t 3.0 " 2.0'
b
o
2.0
P
x
a
'1 I
P
1.0
~
A
~-
e
0 °0
0.5
1.0
1.5
2.0
I ?-5
~
I 0.5
0
•
I 1.0
•
I 2.0
I 2,5
•
I-3.0
x/p
3.0
xlp
I 1.5
Fig. 9 Stress distribution ahead of a symmetrical V-notch with angle a = 120 °,
p/d =
0.18,
p/D
= 0.04,/~ = 2.33
Fig. 7 Stress distribution ahead of a symmetrical U-notch, /~ = 2.55,
p/d =
0.27,
p/D
= 0.09
0 ~< x ~< 3p. The component crx was underpredicted away from the notch tip. Similarly good results were achieved for the symmetrical V-notches shown in Figs 8 and 9. Equations (8a) and (8b) gave good stress estimations irrespective of the notch angles which were a = 60° and a = 120 °. The photoelastic experimental results shown in Figs 8 and 9 were taken from Reference 19. The discrepancy between the approximate and experimentally obtained results might also be caused by the accuracy of the experimental technique. Nevertheless, the differences were below 10% for the stress component cry over the entire region 0 ~< x ~< 3p.
Edge notches tension
under
combined
bending
and
For cases where the nominal stress 3" was not uniform, Equations (10a) and (10b) were used in the analysis. Two eccentrically loaded specimens containing single edge Unotches were analysed using the finite element method. The
Int J Fatigue
July
1987
geometries of the specimens were selected such that two different stress concentration factors were achieved. The first specimen, shown in Fig. 10, contained a notch with stress concentration factor K t = 1.77. The distance from the notch tip to the neutral axis was K" = 2.15/9. Again, good results were obtained from Equation (10b) in relation to the Oy stress component over the entire tensile section of the specimen. In the case of the crx stress component, the prediction based on Equation (10b) was low, although this may have been due to the coarse finite element mesh. The highest errors occurred near the neutral axis because Equations (10a) and (10b) do not account for the shifting of the neutral axis caused by stress concentration near a notch. However, it should be noted that a higher discrepancy than 10% occurred far from the notch tip where both stress components were relatively small. Similar results based on Equations (10a) and (10b) were obtained for the specimen with the higher stress concentration factor K, = 2.66 and ~¢ = 5.7p shown in Fig. 11. Again, a better description was achieved for the cry stress component with the error being largest approaching the neutral axis. In the case of the stress component cr, the highest error also occurred near the neutral axis.
147
• Finite element calculations --Equat ons (lOa) and (lOb)
e Analytical data from Reference 20 --Equations (lOa) and (lOb)
l
1.5~
1.5
P-
x =-
=[-•t ]_
E
1.0
b
'
\
d
o
Ii 0.5 o
0
Q5
ID
1.5
2.0
2.5
3.0
x/p Fig. 12 Stress distribution in a circumferentially notched bar under bending, 2p/d = 0.37, 2p/D = 0.27,/~ = 1.62, ~ = 2.7p ~0
0.5
ID
1.5
2.0
2.5
x/p Fig. 10 Stress distribution ahead of an edge U-notch under combined tension and bending, p/d = 0.27, p/D = 0.19, /~ = 1.77, 1¢ = 2.15p
• Finite element calculations ~ E q u a t i o n s (lOo) and (lOb)
Stress distributions in a threaded connection Both stress components o"x and O'y were analysed in the threaded connection shown in Fig. 14. It is worth noting that the local stress at the thread tooth bottom consists of two components - - the stress due to overall tensile loading and the stress due to the local bending of the teeth, z2 However, the gradient of the nominal stress S was very small in the analysed section. Therefore Equations (Sa) and (8b) derived for tension loading were used in the analysis. The finite element results and the approximate calculations based on Equations (8a) and (8b) were in good agreement over the entire region 0 ~< x ~< 3p. The differences between the accurate and the approximate estimation of stress component Crywere less than 5% at the distance x = 3p.
P
30
: p
r
1.0~ 0
x
-' o
;i
e I 0.5
0
I 1.0
I 1.5
I 2.0
~ 25
"
L 3.0
x/p Fig. 11 Stress distribution ahead of an edge U-notch under combined tension and bending, p/d = 0.1, p/D = 0.08, /~ = 2.66, ~ = 5.7p
Circumferential notches in cylindrical bars under bending r
The complete stress field for a circumferentially notched axisymmetric bar under bending was produced by Tipton 2° using Neuber's analytical solution. The analytical solution and the approximate equations (10a) and (10b) are shown in Fig. 12. Again it was found that both stress components can be adequately predicted over a major part of the tensile region of the net section. More apparent discrepancies occurred in the case of the stress component Cr, near the neutral axis. In order to find the overall effect of geometry on the stress distribution and the accuracy of approximate equation (10b), several other cases were analysed including different depths of circumferential notches. The results published by Lipson and Juvinall 2t were used for comparison. It is
148
apparent that using Equation (10b) it is possible to calculate the stress Cry with an accuracy of 5% or better over the region 0 ~< x ~< 3p for a wide variety of circumferential notches (see Fig. 13).
Discussion It is apparent that the near notch-tip elastic stress distributions are all very similar for a wide variety of geometric parameters and loading systems. Most of the available closed analytical solutions relate to notches in infinite or semi-finite plates or to very deep and sharp notches which can be analysed using fracture mechanics. The similarity between the notch-tip stress distributions makes it possible to derive approximate expressions of a universal character which can be used for notch-tip stress calculations. It was found that the universal equations derived above give a better estimation of the cry notch-tip stress component than in the case of the less important cr, component. This is due to the nature of the approximate expressions. The approximate equations for cry were derived by including solutions for both blunt and sharp notches while the approximate expression for or, resulted from modification of the analytical solution obtained for a circular hole in an infinite plate. It is also characteristic that o'~ exhibits a strong dependency on the width of the ligament d ahead of the notch tip and therefore approximate expressions for stress component cr~ should not be used for cases where dip < 3. Nevertheless, the highest errors
Int J Fatigue July 1987
• Analytical data from Reference21 for 2pId= 0.2, 2pID = 0.09, Kt = 200, ~ =10 o Analytical data from Reference21 for 2p/d = 0.4, 2p/D=O.17, At= 1.77, *¢=5 x Analytical data from Reference 21 for 2pId = 2, 2pID = 0.5, Kt = 1,23, *¢= I ~Equotlon (lOb) 20,
C" L5
0.5 0
kxkk,
0
I 0.5
~,xq 1.0
I 1.5
I 2.0
I 25
I 30
x/p Fig. 13 Stress distribution in circumferentially notched bars under bending
l e Finite element calculations Equations (8o) and (8b)
K, and the notch-tip radius p. This has made it possible to derive universal expressions for calculating stresses in the region ahead of a notch-tip. The universal expressions were derived on the basis of two analytical solutions derived for two extreme cases - - one solution derived for a blunt, circular notch in the infinite plate under tension and a second solution derived in terms of stress intensity factor for deep, sharp notches. Consequently, the approximate solutions show the worst accuracy for very blunt notches but good accuracy for the most common notches with stress concentration factors in the range 1.5 ~< Kt ~ 4.5. A better estimation of stress was usually achieved for the highest stress component, cry. The stress component crx exhibited a strong dependency on the ligament width d ahead of the notch and therefore it is suggested that the approximate expression for crx should not be used for notches with the ligament d < 3p. It was also shown that a correction for nominal stress gradient should be taken into account for notches under bending or tension and bending. References 1. Nueber, H. Kerbspannungslehre(Springer Verlag, BerlinHeildelberg, FRG, 1958) 2.
Paterson, R. E. Stress Concentration Design Factors (John Wiley, New York, USA, 1953)
3.
Wetzel, R. M. (lid) Fatigue Under Complex Loading: Analyses and Experiments (Sac Automotive Engrs, Warrandale, PA, USA, 1977)
4.
Glinka, G. and Stephens, R. I. 'Total fatigue life calculations in notched SAE 0030 cast steel under variable loading spectra' in J. C. Lewis and G. Sines (ads) Fracture Mechanics, ASTM STP 791 (American Society for Testing and Materials, 1983) pp 1-427-1-445
5.
Tada, H., Paris, P. and Irwin, G. The Stress Analysis of Cracks Handbook (Del Reseamh Corporation, St Louis, MO, USA, 1973)
6.
Bueckner, H. F. 'A novel principle for the computation of stress intensity factors' ZAMMSO No 9 (1970) pp 529-546
7.
Rice, J. R. "Some remarks on elastic crack-tip stress field' Int J Solids and Structures 8 (1972) pp 751-758
8.
Santhanam, A. T. and Bates, R. C. 'The influence of notch-tip geometry on the distribution of stresses and strains' Mater Sci and Engng 41 No 2 (1979) pp 107-114
9.
Schijve, J. "Stress gradients around notches" Fatigue of Engng Mater and Structures 5 No 2 (1982) pp 325-332
10.
Glinka, G. "Calculation of inelastic notch-tip strain-stress histories under cyclic loading' Engng Fracture Mech 22 No 5 (1985) pp 83,9-854
11.
Timoshenko, S. and Goodier, J. N. Theory of Elasticity (McGraw Hill, New York, USA, 1951)
12.
Usami, S. 'Short crack fatigue properties and component life estimation" in T. Tanka, M. Jono and K. Komai (ads) Current Research on Fatigue Cracks (The Sac Mater Sci, Japan, Kyoto, 1985)
13.
Creager, i . and Paris, P. C. 'Elastic field equations for blunt cracks with reference to stress corrosion cracking" /nt J Fracture3 No 2 (1967) pp 247-252
14.
Glinka, G., Ott, W. and Nowack, H. 'Elastoplastic plane strain analysis of stresses and strains at the notch root' J Engng Mater and Technol, ASME (to be published)
15.
Schijve, J. 'Stress gradients around notches" Report LR-297 (Delft University of Technology, The Netherlands, April 1980)
15.
Howland, R. C. J. 'On the stresses in the neighbourhood of a circuler hole' Phil Trans, Roy Sac London, SeriesA 229 (1930) pp 49-86
17.
Maunsell, F. G. 'Stesses in a notched plate under tension" Phil MagXXI No CXLII (1936) pp 765-773
b
1.0
O' 0
I 0.5
I 1.0
I 1.5 x/p
q 2.0
I, 2.5
~ • 3.0
Fig. 14 Distribution of stresses in a threaded connection, 2p/d = 0,060, 2p/D = 0.054, K; = 3.41
in estimating the crx component usually occur in the region x > p where the stress crz is relatively low. In all analysed cases the o'z and crystress distributions were predicted satisfactorily well and the approximate expressions can be used for engineering applications over the region 0 ~< x ~< 3p. However, it should also be pointed out that the whole analysis presented above has been addressed to symmetrical notches, /e for notches symmetrical with respect to the x axis. Therefore application to notches such as shafts with a shoulder fillet and weldments should be treated with caution. It is apparent that the approximate expressions for calculating the stress component crycan be applied with confidence throughout the notch-tip region 0 ~< x ~< 3p. Thus, these equations can be used for calculating stress intensity factors for cracks emanating from notches, and can be particularly useful when using the 'weight function' method. 23 Conclusions
Stress distributions near notches are similar and depend principally on two parameters: the stress concentration factor
Int J F a t i g u e J u l y 1 9 8 7
149
assessment of tethers' in W. D. Dover et al (eds) Fatigue and Crack Growth in Offshore Structures (The Institution of Mechanical Engineers, London, UK, 1986) pp 187-198
18.
Hensell, R. D. (ed) PAFEC 75 Manuals (PAFEC Limited, Nottingham, UK, 1984)
19.
Theocaris, P. S. and Marketos, E. "Elastic-plastic strain and stress distribution in notched plates under plane stress" J Mech and Phys of Solids11 (1963) pp 411-428
23.
Tipton, S. M. 'Fatigue behaviour under multiaxial loading in the presence of a notch: methodologies for the prediction of life to crack initiation and life spent in crack propagation' PhD thesis (Department of Mechanical Engineering, Standford University, Stanford, USA, 1985)
Authors
20.
21.
Lil~mn, C. and Juvinall, R. C. Handbook of Stress and Strength (The Macmillan Company, New York, USA, 1963)
22.
Glinka, G., Dover, W. D. and Topp, D. A. 'Fatigue
150
Shin, C. S. and Smith, R. A. 'Stress intensity factor for cracks emanating from sharp notches" Int J Fatigue 7 (April 1985) pp 87-93
The authors are with the London Centre for Marine Technology, Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK. Enquiries should be directed to Dr Glinka.
Int J Fatigue J u l y 1987
