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Fatigue crack growth and failure of tubular members
Engineering
Fracture
Mechanics
Vol. 33,
No.
5, pp. 165-771,
1989
Printed in Great Britain.
0013-7944/89$3.00+ 0.00 0 1989Maxwell Pergamon Macmillan plc.
FATIGUE CRACK GROWTH AND FAILURE OF TUBULAR MEMBERS SOMSAK SWADDIWUDHIPONG,t
CHIH-YOUNG
LIAWS and KIAH-LIM
CHENG$
Department of Civil Engineering, National University of Singapore, Singapore 05 11, Republic of Singapore
Abstract-Fatigue crack growth and fatigue failure of tubular members subjected to both axial force and bending moments is investigated in the present paper. Initial semi-elliptical inside crack is assumed and the Paris equation is employed as the crack growth model. The stress intensity factor is determined through the influence function method. The net sectionfailure criterion is adopted for the prediction of the member life. Numerical studies are worked out and the normalized S-N curves for a range of parameters are presented.
INTRODUCTION FATIGUE failure initiating from surface cracks is one of the most commonly occurring failure modes of structures under fluctuating loads. The results of careful failure examination of several structures including plane and rocket chambers showed that crack propagation usually started from a surface crack defect. Considerable research work has been published on the analysis and mechanics of these cracked structural components. The technology in this area has reached the level where the residual life of these components with cracks of relatively simple geometry can be reliably and conservatively predicted (Cruse and Besuner[l]). A well compiled summary of the state of the art of this topic is documented in[2]. The prediction of fatigue life of cracked structural members is usually based on linear elastic fracture mechanics (LEFM) analysis. The crack growth rate, da/dN, is related to the cyclic changes in the crack tip stress intensity factor (AK). The relationship which depends on the materials, environment and loadings is usually determined empirically through extensive experimental investigation. The major task in this approach is the evaluation of the stress intensity factor, K. The conventional approach of determining K using finite element or boundary element methods is costly and rather impractical considering the fact that the crack geometry as well as the stress field undergo significant variation during the fatigue life time. Cruse and Besuner [l] and Besuner [3] based on some results by Rice [4] suggested an influence function method. The method provides an efficient mean for the determination of the root mean square stress intensity factor of the problem of current interest when the solutions of a cracked problem of similar kind are readily available. A failure criterion using stability analysis based on the J-integral and tearing modulus has been suggested by Tada et al.[5]. However, it is difficult to apply such a criterion to structures with only section properties given without knowing specifically the details, such as support conditions and length between the supports of the structural member. For the “load controlled” type of failure, it is more convenient and reasonable to employ the net section stress criterion suggested by Kanninen et a1.[6]. In the present study, the influence function method is applied to evaluate the crack growth of inside semi-elliptical circumferential surface cracks in tubular members. The loading conditions considered include both cyclic axial force and bending moments. The net section failure criterion is adopted as the basis indicating the end of the life of the members.
tSenior Lecturer. $Senior Lecturer. §Research Scholar.
766
SOMSAK SWADDIWUDHI~N~
CRACK
GROWTH
er al.
MODEL
For linear elastic fracture mechanics, it is generally accepted that the crack growth increment per cycle, da/dN, is a function of cyclic stress intensity factor range as proposed initially by Paris and Erdogan[7]: da dN = C(AK)“I
m
where a is the instantaneous crack length, N the number of load cycles, C and m the materials constants and AK the range of stress intensity factor fluctuation. Equation (1) implies that the relationship between da/dN and AK is linear if plotted on the log/log scale. In fact, the curve is usually sigmoidal as shown in Fig. 1. At the lower end, there exists a threshold value, A&, below which the crack growth will not proceed. At the upper end, the crack growth accelerates rapidly as the maximum value of stress intensity factor, Km,,, approaches &, the value for fracture. For most practical purposes, it is sufficient to state that eq. (I) is vahd for AK > AKt,, and da/dN = 0 for AK < AKth. As the majority of the life of the member is at the low value of lu,,,, the effect of the acceleration of the crack growth at the upper end on the fatigue life of the member is usually insignificant&]. A well compiled summary of the in-air fatigue crack growth data for a range of engineering materials was given by Pook[9]. Scott[lO] stated that the presence of water or other corrosive environment can accelerate the crack growth rate and increase the influence of the loading cycle and mean stress. For steels exposed to water environment, Bamford[l I] suggested an empirical relationship of the following form: da,
==O
AKiGAK,h
dai dN = C(ARj)”
Als, > AKtpc,,
where AK,
f2bi the subscript 3”’ implies the ‘7” direction of the check growth while ( -) represents the root mean square value. Equation (2) with a set of material constants, C = 9.14 x 10-12, m = 4 and
LogAK WiPafil Fig. 1. Relationship between da/dN and AK.
Fatigue and failure of tubular members
761
A& = 4.6 Ksi ,/% (5.06 MPa ,/-)m , as suggested by Harris et al. [12] for 304 stainless steel in water is adopted as the crack growth model in numerical studies of the present work. STRESS INTENSITY FACTORS
The influence function or Green’s function method suggested by Besuner[3] is adopted to evaluate the stress intensity factor of an inside circumferential surface crack in tubular member. A semi-elliptical crack possessing three degrees of freedom, one in the direction of the minor axis, X, and two in each direction of the major axis, y+ and y -, of the semi-ellipse as shown in Fig. 2 is assumed. Associated with each degree of freedom (DOF) is the root mean square (RMS) stress intensity factor defined as
R,=
s
hk(x,y)a(x,y)dA,
k =x,Y+,Y-
(3)
A
where A represents the crack area, h,(x, y) the influence function associated with each degree of freedom and a(x, y) the stress field of the untracked section. Cruse and Besuner[l] showed that the influence function which is independent of the loadings is obtained from h _ k-
--- 1 aa, [1 i
aA au 42 aw
H aakask
(4)
in which w is the crack opening displacement (COD) of half the crack for an arbitrary stress distribution, U the strain energy of the same stress distribution, ak the crack dimension, H = E/(1 - v2) for isotropic plane strain problem, E the Young’s modulus of elasticity and v the Poisson’s ratio. As the three dimensional solution for the embedded elliptical crack in an infinite body under uniform stress co is readily available, the stress intensity factor of the present problem can be conveniently determined through the method of influence function with correction factor. The strain energy and the crack opening displacement of the above base case are expressed, as U=
4na2a2b
(5)
3HE2
w=~[l-(gyg].
(al
Fig. 2. Section and crack geometry of tubular member.
168
SOMSAK SWADDIWUDHIPONG
et al.
where
Introducing
two functions, namely, g, = U/U’ and g, = w/w’, it can be shown that Ek =
j&a(x, sR
y) dA
4W
where
Ub) and hi is the influence function of the reference problem. In principle, eq. (7) can be used to evaluate the root mean square stress intensity factor, & of a three dimensional crack with arbitrary stress field provided that g, and g, are established. Numerical solutions of g, and g, were obtained for a number of different crack geometry by Harris et al.[l2]. Curve fitting technique was employed in the formulation of the two functions. Once the surface crack grows through the wall thickness, a through wall crack with two degrees of freedom (as shown in Fig. 2c), is assumed. The approximate influence function used in this case is the function for flat plates, i.e.
c!?_ty “,* tj;;i;b7.v 1
/z&=-1-
(84
-
[
with a geometry correction
factor f; = &,,,(A, b/t)
@b)
where k,, is the membrane stress intensity ratio of the cracked cylindrical shell and plate and j. a geometry parameter[ 131. FAILURE
CRITERION
If the dominating stress on a cracked section cannot be relaxed by the presence of cracks or deformation of the structural member, i.e. load controlled stress, a failure criterion based on the net section consideration is usually adopted[6]. Failure is assumed to occur when the applied force and moments on the cracked section exceed the resisting capacity of the net section under flow stress. In the present study, the average of the retained yield and ultimate stresses of the materials is taken as the flow stress. The limiting capacity of the cracked section is calculated based on the following assumptions. 1. The cracked area cannot sustain tensile stress but can resist compression. 2. To simplify the derivation of failure criterion, the cracked area is assumed to be of annular shape with depth and length equal to the length of the minor and major axes of the semi-elliptical crack, respectively. Figure 3 shows a circular section of internal radius "Ri"and outer radius “&” with an annular
ding
\compression
Fig. 3, Simplified crack geometry for establishing failure criterion.
Fatigue and failure of tubular members
769
crack of depth “a” and half length “b”. If the section is subjected to an axial force F and bending moment M acting at an angle /I with the minor axis of the crack, the equilibrium conditions require that F = arA, ec cosa =-cost e
(W (9’3
and M = nar[Q sin a - Q, sin/I]
(9c)
where A, = (Ri - RF) - Ri (7, - sin 7,) + Rf(y2 - sin yz) - A,, the net cross sectional area
(lOa) Q, = $ sin % [(Ri + ~2)~- Rf] 0 I
(W
Q =$[R,sin3($)-Rfsin3@)]
(104
y2=2cos-’
; 0 I where A, is the effective cracked area, e the eccentricity which is the shortest distance from the centre of the untracked section to the neutral axis and CIthe angle between the line perpendicular to the neutral axis and the direction of the applied binding moment as shown in Fig. 3. An iterative process is required in the evaluation of eq. (9). Once eq. (9) is evaluated, the interaction diagram for each cracked section can be generated. Some typical such diagrams for t/Ri = 0.02 are presented in Fig. 4 where the force and moment
Fig. 4. Typical interaction diagrams for tubular members with r/Ri = 0.02.
SOMSAK SWADDIWUDHIPONG
710
are normalized through the introduction
et al.
of the following two parameters,
namely,
Ff = na,(Ri - Rf)
(114
M,= $na,(Ri - Rf).
(I lb)
The application of this failure criterion is straight forward. At any stage of loading, if the values of the applied loads, F/Ff and M/M,, fall inside the interaction diagram, no failure occurs. otherwise the section deems to fail under that given load combination. NUMERICAL
STUDIES
Based on the approach described in previous sections, a computer program is written to predict the fatigue life of a tubular member with an initial circumferential crack for a given loading history. Some typical numerical results based on the properties of 304 stainless steel in water suggested by Harris et a/.[121 are presented in Fig. 5. They are in the form of normalized S-N or S’-N’ curves where S’ = FA/Ff or MA/Mf, N’ = (N)(C)(u,Ja(af)“’ and c1= (m/2 - 1). Figure 5(a) shows the variation of fatigue life of tubular members subjected to various level of applied forces. The minimum values of force and moment fluctuations, F, and M,, are set to zero. As expected, the higher the load fluctuation, the shorter the life of the member. The curve for MA/Mf = 0 is asymptotic to the horizontal axis as the member will not fail when no force is acting on the structure.
I
‘\
ke,O,
ERi
Id1
Fig. 5. Fatigue life of cracked tubular members.
4 F=O, I
$=O,
f
$=0.25,
f
fl=;,
T~0.25,
& = 0.05,
I
f/Ri = 0.2, except when specified otherwise.
Fatigue and failure of tubular members
771
In Fig. 5(b), the value of F, is set at FA - 0.2 F,, while M, is kept at zero value. The S’-N curves follow the same trend as observed earlier in Fig. 5(a) except moving to the left indicating a shorter fatigue life. This is expected as the section is relatively thinner and the initial crack length, b,,, is larger though the crack depth, a,,, is shallower. The effect of sizes of initial crack, a,, and b,, on fatigue life are illustrated in Fig. 5(c) and (d). The divergence of fatigue curves at the high end of FA/Ff implies that the fatigue life is more sensitive to the initial crack size when the member is subjected to high loads. Figure 5(e) shows the effect of crack locations on the fatigue life. When /3 = 0, the S’-N’ curve is very steep indicating that stresses induced by crack lying the bending axis is hardly affected by the magnitude of the bending moment. The three curves converge to one point on the horizontal axis as p is irrelevant when the applied bending moment vanishes. CONCLUSIONS A quantitative approach is used in this study to evaluate the fatigue life of cracked tubular structural members. Failure criterion of the circumferentially cracked members is based on the net section and flow stress considerations. Growth of the crack is evaluated from the root-mean-square stress intensity factor and the influence function approach. Typical fatigue life estimates for certain cases are presented in the form of normalized S-N curves. REFERENCES [I] T. A. Cruse and P. M. Besuner, Residual life prediction for surface cracks in complex structural details.
Aircraft 12, 369-375 (1975). [2] J. B. Chang (Ed.), Part-Through Crack Fatigue Life Prediction. ASTM, Philadelphia (1979). [3] P. M. Besuner, The influence function method for fracture mechanics and residual fatigue life analysis of cracked components under complex stress fields. Nucl. Engng Des. 43, 115-154 (1977). [4] J. R. Rice, Some remarks on elastic crack-tip stress fields. Int. J. Solids Sfruct. 8, 751-758 (1972). [5] H. Ta&, p. Paris and R. Gamble, A stability analysis of circumferenti~ cracks for reactor piping SyStemS. PrOC. Twelfrh Nat. Symp. on Fracture Mechanics, ASTM STP 700, 296-313 (1980). [6] M. F. Kanninen, D. Broek, G. T. Hahn, C. W. Marschall, E. F. Rybicki and G. M. Wilkowski, Towards an elastic-plastic fracture mechanics predictive capability for reactor piping. Nucl. JIngng Des. 48, 117-134 (1978). [7] P. Paris and F. Erdogan, Critical analysis of crack propagation laws. J. Basic Engng 85, 528-534 (1963). [8] T. C. Lindley and K. J. Nix, Metallurgical aspects of fatigue crack growth. Proc. Conf. on Fafigue Crack Growth, Cambridge, U.K., September 1984, pp. 53-74 (1986). [9] L. P. Pook, Fatigue-crack propagation, in Developments in Fracture Mechanics-l (Edited by G. G. Chell), pp. 183-220. Applied Science Publishers, London (1979). [lo] P. M. Scott, Effects of environment on crack propagation, in Developments in Fracture Mechanics-l (Edited by G. G. Chell, pp. 221-257. Applied Science Publishers, London (1979). [l l] W. H. Bamford, Fatigue crack growth of stainless steel piping in a pressurized water reactor environment. Press. Vess. Technol. 101, 73-79 (1979). [12] D. 0. Harris, E. Y. Lim and D. D. Dedhia, Probability of Pipe Fracture in the Primary Coolunr of a PWR Plant NlJREG/CR-2189 Vol. 5. U.S. Nuclear Regulatory Commission (1981). [13] F. Delale and F. Erdogan, Transverse shear effect in a circumferentially cracked cylindrical shell. Appl. Murh. 239-258 (1979).
(Receiued 18 July 1988)
Fracture
Mechanics
Vol. 33,
No.
5, pp. 165-771,
1989
Printed in Great Britain.
0013-7944/89$3.00+ 0.00 0 1989Maxwell Pergamon Macmillan plc.
FATIGUE CRACK GROWTH AND FAILURE OF TUBULAR MEMBERS SOMSAK SWADDIWUDHIPONG,t
CHIH-YOUNG
LIAWS and KIAH-LIM
CHENG$
Department of Civil Engineering, National University of Singapore, Singapore 05 11, Republic of Singapore
Abstract-Fatigue crack growth and fatigue failure of tubular members subjected to both axial force and bending moments is investigated in the present paper. Initial semi-elliptical inside crack is assumed and the Paris equation is employed as the crack growth model. The stress intensity factor is determined through the influence function method. The net sectionfailure criterion is adopted for the prediction of the member life. Numerical studies are worked out and the normalized S-N curves for a range of parameters are presented.
INTRODUCTION FATIGUE failure initiating from surface cracks is one of the most commonly occurring failure modes of structures under fluctuating loads. The results of careful failure examination of several structures including plane and rocket chambers showed that crack propagation usually started from a surface crack defect. Considerable research work has been published on the analysis and mechanics of these cracked structural components. The technology in this area has reached the level where the residual life of these components with cracks of relatively simple geometry can be reliably and conservatively predicted (Cruse and Besuner[l]). A well compiled summary of the state of the art of this topic is documented in[2]. The prediction of fatigue life of cracked structural members is usually based on linear elastic fracture mechanics (LEFM) analysis. The crack growth rate, da/dN, is related to the cyclic changes in the crack tip stress intensity factor (AK). The relationship which depends on the materials, environment and loadings is usually determined empirically through extensive experimental investigation. The major task in this approach is the evaluation of the stress intensity factor, K. The conventional approach of determining K using finite element or boundary element methods is costly and rather impractical considering the fact that the crack geometry as well as the stress field undergo significant variation during the fatigue life time. Cruse and Besuner [l] and Besuner [3] based on some results by Rice [4] suggested an influence function method. The method provides an efficient mean for the determination of the root mean square stress intensity factor of the problem of current interest when the solutions of a cracked problem of similar kind are readily available. A failure criterion using stability analysis based on the J-integral and tearing modulus has been suggested by Tada et al.[5]. However, it is difficult to apply such a criterion to structures with only section properties given without knowing specifically the details, such as support conditions and length between the supports of the structural member. For the “load controlled” type of failure, it is more convenient and reasonable to employ the net section stress criterion suggested by Kanninen et a1.[6]. In the present study, the influence function method is applied to evaluate the crack growth of inside semi-elliptical circumferential surface cracks in tubular members. The loading conditions considered include both cyclic axial force and bending moments. The net section failure criterion is adopted as the basis indicating the end of the life of the members.
tSenior Lecturer. $Senior Lecturer. §Research Scholar.
766
SOMSAK SWADDIWUDHI~N~
CRACK
GROWTH
er al.
MODEL
For linear elastic fracture mechanics, it is generally accepted that the crack growth increment per cycle, da/dN, is a function of cyclic stress intensity factor range as proposed initially by Paris and Erdogan[7]: da dN = C(AK)“I
m
where a is the instantaneous crack length, N the number of load cycles, C and m the materials constants and AK the range of stress intensity factor fluctuation. Equation (1) implies that the relationship between da/dN and AK is linear if plotted on the log/log scale. In fact, the curve is usually sigmoidal as shown in Fig. 1. At the lower end, there exists a threshold value, A&, below which the crack growth will not proceed. At the upper end, the crack growth accelerates rapidly as the maximum value of stress intensity factor, Km,,, approaches &, the value for fracture. For most practical purposes, it is sufficient to state that eq. (I) is vahd for AK > AKt,, and da/dN = 0 for AK < AKth. As the majority of the life of the member is at the low value of lu,,,, the effect of the acceleration of the crack growth at the upper end on the fatigue life of the member is usually insignificant&]. A well compiled summary of the in-air fatigue crack growth data for a range of engineering materials was given by Pook[9]. Scott[lO] stated that the presence of water or other corrosive environment can accelerate the crack growth rate and increase the influence of the loading cycle and mean stress. For steels exposed to water environment, Bamford[l I] suggested an empirical relationship of the following form: da,
==O
AKiGAK,h
dai dN = C(ARj)”
Als, > AKtpc,,
where AK,
f2bi the subscript 3”’ implies the ‘7” direction of the check growth while ( -) represents the root mean square value. Equation (2) with a set of material constants, C = 9.14 x 10-12, m = 4 and
LogAK WiPafil Fig. 1. Relationship between da/dN and AK.
Fatigue and failure of tubular members
761
A& = 4.6 Ksi ,/% (5.06 MPa ,/-)m , as suggested by Harris et al. [12] for 304 stainless steel in water is adopted as the crack growth model in numerical studies of the present work. STRESS INTENSITY FACTORS
The influence function or Green’s function method suggested by Besuner[3] is adopted to evaluate the stress intensity factor of an inside circumferential surface crack in tubular member. A semi-elliptical crack possessing three degrees of freedom, one in the direction of the minor axis, X, and two in each direction of the major axis, y+ and y -, of the semi-ellipse as shown in Fig. 2 is assumed. Associated with each degree of freedom (DOF) is the root mean square (RMS) stress intensity factor defined as
R,=
s
hk(x,y)a(x,y)dA,
k =x,Y+,Y-
(3)
A
where A represents the crack area, h,(x, y) the influence function associated with each degree of freedom and a(x, y) the stress field of the untracked section. Cruse and Besuner[l] showed that the influence function which is independent of the loadings is obtained from h _ k-
--- 1 aa, [1 i
aA au 42 aw
H aakask
(4)
in which w is the crack opening displacement (COD) of half the crack for an arbitrary stress distribution, U the strain energy of the same stress distribution, ak the crack dimension, H = E/(1 - v2) for isotropic plane strain problem, E the Young’s modulus of elasticity and v the Poisson’s ratio. As the three dimensional solution for the embedded elliptical crack in an infinite body under uniform stress co is readily available, the stress intensity factor of the present problem can be conveniently determined through the method of influence function with correction factor. The strain energy and the crack opening displacement of the above base case are expressed, as U=
4na2a2b
(5)
3HE2
w=~[l-(gyg].
(al
Fig. 2. Section and crack geometry of tubular member.
168
SOMSAK SWADDIWUDHIPONG
et al.
where
Introducing
two functions, namely, g, = U/U’ and g, = w/w’, it can be shown that Ek =
j&a(x, sR
y) dA
4W
where
Ub) and hi is the influence function of the reference problem. In principle, eq. (7) can be used to evaluate the root mean square stress intensity factor, & of a three dimensional crack with arbitrary stress field provided that g, and g, are established. Numerical solutions of g, and g, were obtained for a number of different crack geometry by Harris et al.[l2]. Curve fitting technique was employed in the formulation of the two functions. Once the surface crack grows through the wall thickness, a through wall crack with two degrees of freedom (as shown in Fig. 2c), is assumed. The approximate influence function used in this case is the function for flat plates, i.e.
c!?_ty “,* tj;;i;b7.v 1
/z&=-1-
(84
-
[
with a geometry correction
factor f; = &,,,(A, b/t)
@b)
where k,, is the membrane stress intensity ratio of the cracked cylindrical shell and plate and j. a geometry parameter[ 131. FAILURE
CRITERION
If the dominating stress on a cracked section cannot be relaxed by the presence of cracks or deformation of the structural member, i.e. load controlled stress, a failure criterion based on the net section consideration is usually adopted[6]. Failure is assumed to occur when the applied force and moments on the cracked section exceed the resisting capacity of the net section under flow stress. In the present study, the average of the retained yield and ultimate stresses of the materials is taken as the flow stress. The limiting capacity of the cracked section is calculated based on the following assumptions. 1. The cracked area cannot sustain tensile stress but can resist compression. 2. To simplify the derivation of failure criterion, the cracked area is assumed to be of annular shape with depth and length equal to the length of the minor and major axes of the semi-elliptical crack, respectively. Figure 3 shows a circular section of internal radius "Ri"and outer radius “&” with an annular
ding
\compression
Fig. 3, Simplified crack geometry for establishing failure criterion.
Fatigue and failure of tubular members
769
crack of depth “a” and half length “b”. If the section is subjected to an axial force F and bending moment M acting at an angle /I with the minor axis of the crack, the equilibrium conditions require that F = arA, ec cosa =-cost e
(W (9’3
and M = nar[Q sin a - Q, sin/I]
(9c)
where A, = (Ri - RF) - Ri (7, - sin 7,) + Rf(y2 - sin yz) - A,, the net cross sectional area
(lOa) Q, = $ sin % [(Ri + ~2)~- Rf] 0 I
(W
Q =$[R,sin3($)-Rfsin3@)]
(104
y2=2cos-’
; 0 I where A, is the effective cracked area, e the eccentricity which is the shortest distance from the centre of the untracked section to the neutral axis and CIthe angle between the line perpendicular to the neutral axis and the direction of the applied binding moment as shown in Fig. 3. An iterative process is required in the evaluation of eq. (9). Once eq. (9) is evaluated, the interaction diagram for each cracked section can be generated. Some typical such diagrams for t/Ri = 0.02 are presented in Fig. 4 where the force and moment
Fig. 4. Typical interaction diagrams for tubular members with r/Ri = 0.02.
SOMSAK SWADDIWUDHIPONG
710
are normalized through the introduction
et al.
of the following two parameters,
namely,
Ff = na,(Ri - Rf)
(114
M,= $na,(Ri - Rf).
(I lb)
The application of this failure criterion is straight forward. At any stage of loading, if the values of the applied loads, F/Ff and M/M,, fall inside the interaction diagram, no failure occurs. otherwise the section deems to fail under that given load combination. NUMERICAL
STUDIES
Based on the approach described in previous sections, a computer program is written to predict the fatigue life of a tubular member with an initial circumferential crack for a given loading history. Some typical numerical results based on the properties of 304 stainless steel in water suggested by Harris et a/.[121 are presented in Fig. 5. They are in the form of normalized S-N or S’-N’ curves where S’ = FA/Ff or MA/Mf, N’ = (N)(C)(u,Ja(af)“’ and c1= (m/2 - 1). Figure 5(a) shows the variation of fatigue life of tubular members subjected to various level of applied forces. The minimum values of force and moment fluctuations, F, and M,, are set to zero. As expected, the higher the load fluctuation, the shorter the life of the member. The curve for MA/Mf = 0 is asymptotic to the horizontal axis as the member will not fail when no force is acting on the structure.
I
‘\
ke,O,
ERi
Id1
Fig. 5. Fatigue life of cracked tubular members.
4 F=O, I
$=O,
f
$=0.25,
f
fl=;,
T~0.25,
& = 0.05,
I
f/Ri = 0.2, except when specified otherwise.
Fatigue and failure of tubular members
771
In Fig. 5(b), the value of F, is set at FA - 0.2 F,, while M, is kept at zero value. The S’-N curves follow the same trend as observed earlier in Fig. 5(a) except moving to the left indicating a shorter fatigue life. This is expected as the section is relatively thinner and the initial crack length, b,,, is larger though the crack depth, a,,, is shallower. The effect of sizes of initial crack, a,, and b,, on fatigue life are illustrated in Fig. 5(c) and (d). The divergence of fatigue curves at the high end of FA/Ff implies that the fatigue life is more sensitive to the initial crack size when the member is subjected to high loads. Figure 5(e) shows the effect of crack locations on the fatigue life. When /3 = 0, the S’-N’ curve is very steep indicating that stresses induced by crack lying the bending axis is hardly affected by the magnitude of the bending moment. The three curves converge to one point on the horizontal axis as p is irrelevant when the applied bending moment vanishes. CONCLUSIONS A quantitative approach is used in this study to evaluate the fatigue life of cracked tubular structural members. Failure criterion of the circumferentially cracked members is based on the net section and flow stress considerations. Growth of the crack is evaluated from the root-mean-square stress intensity factor and the influence function approach. Typical fatigue life estimates for certain cases are presented in the form of normalized S-N curves. REFERENCES [I] T. A. Cruse and P. M. Besuner, Residual life prediction for surface cracks in complex structural details.
Aircraft 12, 369-375 (1975). [2] J. B. Chang (Ed.), Part-Through Crack Fatigue Life Prediction. ASTM, Philadelphia (1979). [3] P. M. Besuner, The influence function method for fracture mechanics and residual fatigue life analysis of cracked components under complex stress fields. Nucl. Engng Des. 43, 115-154 (1977). [4] J. R. Rice, Some remarks on elastic crack-tip stress fields. Int. J. Solids Sfruct. 8, 751-758 (1972). [5] H. Ta&, p. Paris and R. Gamble, A stability analysis of circumferenti~ cracks for reactor piping SyStemS. PrOC. Twelfrh Nat. Symp. on Fracture Mechanics, ASTM STP 700, 296-313 (1980). [6] M. F. Kanninen, D. Broek, G. T. Hahn, C. W. Marschall, E. F. Rybicki and G. M. Wilkowski, Towards an elastic-plastic fracture mechanics predictive capability for reactor piping. Nucl. JIngng Des. 48, 117-134 (1978). [7] P. Paris and F. Erdogan, Critical analysis of crack propagation laws. J. Basic Engng 85, 528-534 (1963). [8] T. C. Lindley and K. J. Nix, Metallurgical aspects of fatigue crack growth. Proc. Conf. on Fafigue Crack Growth, Cambridge, U.K., September 1984, pp. 53-74 (1986). [9] L. P. Pook, Fatigue-crack propagation, in Developments in Fracture Mechanics-l (Edited by G. G. Chell), pp. 183-220. Applied Science Publishers, London (1979). [lo] P. M. Scott, Effects of environment on crack propagation, in Developments in Fracture Mechanics-l (Edited by G. G. Chell, pp. 221-257. Applied Science Publishers, London (1979). [l l] W. H. Bamford, Fatigue crack growth of stainless steel piping in a pressurized water reactor environment. Press. Vess. Technol. 101, 73-79 (1979). [12] D. 0. Harris, E. Y. Lim and D. D. Dedhia, Probability of Pipe Fracture in the Primary Coolunr of a PWR Plant NlJREG/CR-2189 Vol. 5. U.S. Nuclear Regulatory Commission (1981). [13] F. Delale and F. Erdogan, Transverse shear effect in a circumferentially cracked cylindrical shell. Appl. Murh. 239-258 (1979).
(Receiued 18 July 1988)