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A method of predicting the fatigue life curve for misaligned welded joints
Int. J. Fatigue Vol. 18, No. 4, pp. 221-226, 1996 Copyright © 1996 Elsevier Science Limited Printed in Great Britain. All rights reserved 0142-1123/96/$15.00
ELSEVIER
PIhS0142-1123(96)00004-7
A method of predicting the fatigue life curve for misaligned welded joints Guan Deqing Department of Power Engineering, Changsha University of Electric Power, Changsha, 410077, People's Republic of China (Received 29 August 1995; revised 13 December 1995) The effects of material properties, manner of loading, geometric shape, value of misalignment, residual stress, stress ratio and short crack in notch root of misaligned welded joint on the fatigue life are considered respectively. By means of Peterson's equation and Topper's equation and the concept of the worst-case fatigue notch factor, two estimating equations of effective stress concentration factors have been adopted. A method for predicting the fatigue life curve of misaligned welded joints for middle range of cycle numbers is developed. As an example of application of the method, the fatigue life curves of several misaligned welded joints made of low-alloy structural steel are predicted. Good agreement between the predicting and experimental results is achieved. This method has significance for engineering applications. (Keywords: fatigue life curve; effective stress concentration factor; the worst-case fatigue notch factor)
INTRODUCTION
range of cycle numbers is presented. The model has successfully predicted several S-N curves of misaligned welded joints.
A weld is the most commonly used form of structure joints. The development of the engineering application of welding techniques in the industries of aerospace manufacturing, shipbuilding and offshore exploitation makes the research on the fatigue life of weld joints a problem with the utmost importance. Because many factors influence the fatigue life of a weld the examination of the fatigue behavior of the welded joint is very complex m. The misalignment which often exists in a butt weld will certainly influence the fatigue strength of the weld. Based on Peterson's equation 3 and Topper's equation 4, the worst-case notch concept 5,6 has been adopted, Guan Deqing and Wang Guanghai 7 have developed two equations for estimating the effective stress concentration factors for misaligned welded joints. It is an efficient method for predicting the fatigue strength of welded joint. In the assessment of the fatigue life one of the important relations is the fatigue life curve (S-N curve), which is generally obtained from experiment. However an S-N curve is usually time-consuming and costly from the fatigue test. Therefore to develop a predicting method for the S-N curve of welded joints will certainly be useful in engineering application and helpful in gaining economic efficiency. Many influencing factors such as material properties, manner of loading, geometric shape, value of misalignment, residual stress and short crack in notch root on the S-N curve of welded joints are considered, respectively. A method of predicting the fatigue life curve for the middle
THE WORST-CASE FATIGUE NOTCH FACTOR OF MISALIGNED WELDED JOINT The fatigue notch factor kf is an important parameter for the fatigue strength of a notch specimen. It may be determined by Peterson's equation (1) Kt-1 Kf= 1 + 1 + a/r
(1)
where Kt is elastic stress concentration factor, r is notch root radius, a is Peterson's material parameter which may be approximated by the expression 1.087× 10SSu-2(mm) for steel, Su is the ultimate strength of the material. The effects of material, manner of loading and the geometric shape of the notch specimen on fatigue strength are considered in the equation. Topper and EL Haddad 4 have analysed the influences of the material, manner of loading, short crack in the notch on the fatigue performance and established another equation of the fatigue notch factor by expression (2)
[
~=(1 lo
221
Kt-z
Kth
\Mo._l )2/7"r
(2)
Guan Deqing
222
in which K,h is the threshold stress intensity factor, and M is the geometry dependent constant, 0-_1 is the fatigue limit of the material when stress ratio R equal to -1. lo is a constant. The fatigue notch factor of the welded joint may be determined from expression (1), which is the simple and popular. In the investigation of paper 7, the crack in the toe root of welds may be regarded as a short crack. Thus, we may adopt Equation (2) to estimate Kf of welded joint. A general form ~'6 which can be used to describe Kt of welds is
Kt=A
+
B(~) ~
• ,• k t
rill
(3)
where A, B, A are constants whose values are determined by geometric shape of welds and the manner of loading, t and r are the thickness of the plate and notch root radius. The constants A and A are usually 1 and 1/2, respectively. Because of welding error and deformation, misalignment often exists in a butt weld. Figure I is a misaligned welded plate joint. The value of misalignment is 6. The fatigue strength will be changed when the misalignment takes place in a butt weld. The investigation of the fatigue performance of a misaligned welded joint has had significance for predicting fatigue life. The author 1° recently presented a relationship (4) about Kt of a misaligned welded plate joint based on three-dimensional finite element computation.
r
Figure 2 The relationship curve of Kf, K,, r related to Peterson's equation
related to Peterson's equation and Topper's equation are shown in Figures 2 and 3. Substituting Equation(3) into Peterson's equation (1), differentiating Kf with respect to r, and setting that expression equal to zero leads to the result that Kf is the maximum value Kfmp, as Equation (5). Similarly, the maximum value gfmw related to Topper's equation may be given by expression (6) Kfm p = 1 + ~
gfm T =
= I+B
%.
kfmp
(5)
M.
1 + ~-
(6)
Hence, we may calculate the worst-case fatigue notch factors of misaligned welded joints by expression (5) or (6).
t~
= 0.27 + 2.45(~ ) H3
(4) ESTIMATING METHOD OF THE S-N CURVE FOR MISALIGNED WELDED JOINTS
where r
0.0 ~< - ~< 0.3, 0.01 <~ - ~< 0.20. t
t
The fatigue notch factors of misaligned welded joints may be obtained from Equations (1) and (4) or Equations (2) and (4). Unfortunately, it is difficult to determine Kf for defects such as a weld toe, because the notch root radii of such notch are generally unknown or difficult to measure and what is worse, highly variable. Yung and Lawrence 5 and Lawrence 6 have developed the worst-case notch concept which considered that there is a maximum value of Kf. The use of the concept leads to a model which predicts that the fatigue strength of a weldment depends upon its size, as well as its shape, material properties and manner of loading. The relationship curves of Kf, Kt and r to a butt weld
i
A definition is introduced that the effective stress concentration factor/3 is the ratio between the fatigue limit tr_l of the smooth specimen and that of a misaligned welded joint o'_~w when the stress ratio R (the ratio between the minimum cycle stress and the maximum cycle stress) equal to -1. o'_ 1
(7)
/3 = - - .
O'_lw
• kt kfraT
.....................
•%
.:, ~
r
41o
r
Figure 3 The relationship curve of Kf, Kt, r related to Topper's equFigure 1 Misaligned welded joint
ation
Predicting a fatigue life curve
223
According to Guan Deqing and Wang Guanghai 7, two estimating relationships related to Equations (5) and, (6) between /3 and Kt- are [~= constant Ortf /3P -- O ' t f - Orr gfmp
(8) ~
dr ~f---- Orzf_ Orr gfmT
R=0.5 R=0 R = -0.5
(9)
in which /3p, /3v are effective stress concentration factors related to Peterson's equation and Topper's equation, respectively, O!f is the fatigue strength coefficient, Orr is the notch root residual stress. Based on a lot of experimental results, three assumptions may be adopted.
R = - 1.0
log N Figure 5 The S-N curves of different R values for a specified type of misaligned welded joint
Assumption one A fatigue life curve (S-N curve) in double logarithmic coordinates is decomposed into three parts of, low, middle and high range cycle numbers, as shown in Figure 4. We will aim at the middle cycle range of the S - N curve, its mathematical equation is written by logN = C - mlogora
A
= 1.o ~=1.5 13= 2.1)
(10)
where N is the fatigue life, Ora is stress amplitude, C is intercept of the curve, m may be regarded as the slope of the curve.
Assumption two For a specified type of misaligned welded joint, the S-N curves for different R values keep parallel to that for the case of R = - 1 , provided /3 takes a constant value. Assumption three When R = - 1 , all S-N curves for different types of joints and different /3 values intercross at the same point in the low cycle range. According to three assumptions, we have got two expressions
I~= 2.5
log N Figure 6
All S-N curves intercross at the same point
m = m(/3)
(11)
c = c(/3,R)
(12)
Equations (11) and (12) will be determined as long as we find two points on the S-N curve of the welded joints in Figure 4. The S-N curve of the smooth specimens can be estimated by the Basquin equation (13) o-a = o"f(2N) b
• * • •
Specimens Specimens Specimens Specimens
(13)
where b is the fatigue strength exponent. By means of Fuchs's suggestion, 8 the S - N curves of smooth specimens and notch specimens intercross at N = 500. The stress amplitude at point A (N= 500) OrA on the S-N curve of smooth specimens may be obtained from Equation (13)
of 8 = 0.15ram of 8 = 0.30ram of 8 = 0.60ram of 8 = 1.08 mm
O"A = OrPf(1000) b
~,3
(14)
Thus the stress amplitude at point A on the S-N curve of a welded joint may be calculated through Equation (14). It may be assumed that the stress amplitude OrB of smooth specimens at N = 2 x 106 is the fatigue limit. Hence
lm O
-- 2
OrB = O'_I = Or'r(4 X 106) b
(15)
The welded joint fatigue limit crBw at point B on the S-N curve of a welded joint may be obtained by Equations (7) and (15) I
I
I
I
I
I
1
2
3
4
5
6
log Figure
4
N
A fatigue life curve in double logarithmic coordinates
OrBw : Or-lw Or-I
/3
Guan Deqing
224 o'f
= -~- (4 × 106)b
(16)
• Specimens of b = 0.15 mm
According to the values of points A and B, the slope m of the S-N curve of a misaligned welded joint can be obtained as follows: 1
m-
(17)
- b + 0.27761og/3
We consider the effect of mean stress on long life fatigue strength. Through the Morrow equation 9 O"a
O"m
+-7- = 1
Or_lw
• Specimens of 6 = 0.30mm • Specimens of 6 = 0.60mm • Specimens o f b = 0.08mm
4 --
~'3~
e~
C)
(18)
O'f
in which mean stress orm is determined by Orm - -
I+R 1-R
I --
(19)
O"a
Substituting Equations (16) and (19) into Equation (18), we can get Or_ 1
(20)
era - / 3 + (1 + R)o'_l/[(1 - R)o~f]
Because the point (2 × 106, or_l/{/3 + (1 + R) o'_l/[(1-R) df]} is on the S-N curve of a welded joint, it should satisfy Equation (10). Hence C = 6.301 +
1
- b + 0.27761og/3
I
I
I
1
3
4
5
6
8 The experimental results of four groups of misaligned welded joints Figure
Through linear statistical analysis to experimental results in Figure 8, we can determine experimental constants C and m in Equations (10) as follows: n
n
~E
(21)
I 2
log N
o'_ i
log/3 + (1 + R)o-_l/[(1 - R)df]
I 1
Z
i=1
i=1
i=1
m=
so that we may estimate the effective stress concentration factors based on Equations (4), (5), (8) or (4), (6), (9), and predict the S-N curve of misaligned welded joints by means of Equations (17) and (21).
n
1ogO'aiE logN/-
, ~ (logo. i)2 _-1 n
i=l
1
~
m
C = - ~ logNi + n
EXPERIMENTAL RESULTS The fatigue experiments of four groups of misaligned welded joints made of low alloy structural steel have been accomplished. The material properties and other data are Su = 548 MPa, o-'f = 894 MPa, o_l = 245 MPa, b = -0.0854, Kth = 294 N mm -3/2, trr = 441 MPa. Four groups of the values of misalignment are ~ = 0 . 1 5 , 0.30, 0.60, 1.08 mm, respectively. Stress ratio R is 0.0 in the experiments. A specimen of a welded joint and load is shown in Figure 7. Experimental results are shown in Figure 8.
i=1
n
1ogo-ailOgNi
(22) logo'~i
"i--1 - -
"
" E logo-ai
(23)
i=l
where n is the number of specimens of every type of misaligned welded joint. In our experiments, n is five. O'ai is the stress amplitude of specimen i; Ni is fatigue life of specimen i. Meanwhile we consider the low limits of S-N curves which have 95% confidence limits. So the expressions of the S-N curves of misaligned welded joints may be written as logN = 13.84 - 3.961ogo'a for specimens of 6 = 0.15 mm
(24)
logN-- 13.70 - 4.041ogtra for specimens of 6 = 0.30 mm
4mm
(25)
logN = 13.31 - 4.0110go-a for specimens of 6 = 0.60 mm
I'Figure
7
122ram
-,--'~ 6 m m ~_I_ ~
122mm
A specimen of a misaligned welded joint
I
(26)
logN = 12.85 - 3.921ogO'a for specimens of 6 = 1.08 mm.
(27)
Here the unit of O'a is MPa. Using Equations (7), (18) and (19) we have obtained the effective stress concentration factor from the experiments as
=I
tr_ l --
o'a
( l + R ) o'_l (1 - R ) or#f
(28)
Predicting a fatigue life curve where o'a is stress amplitude when N is equal to 2 x 106 in Equations (24)-(27). COMPARISON OF THE PREDICTED S-N CURVES WITH THE EXPERIMENTAL RESULTS To investigate the accuracy of the predicting method for the S-N curve we compare the experimental results with the predicted results. Using the above constants in the experiments, and assuming the thickness of plate t = 4 mrn, the geometry dependent constant M is approximated as 1.12. First we may adopt equations (4), (5), (8) and estimate effective stress concentration factor /3p. Then we may substitute/3p into Equations (17) and (21), obtain slope mp and intercept Cp of the fatigue life curve for misaligned welded joints related to Peterson's equation. Similarly, by means of Equations (4), (6), (9) the effective stress concentration factor /3x can be obtained. Substituting /3r into Equations (17) and (21), we can get slope mx and intercept CT of the fatigue life curve for misaligned welded joints related to Topper's equation. The predictions are shown in Table 1. According to Equation (28), effective stress concentration factors from experiments can be given as in Table 1. Slope m and intercept C from experimental Equations (24)-(27) are shown in Table 1. The worst-case notch concept for welds is intended specifically for the crack initiation portion of the fatigue life. This, according to the concept, is more significant for longer lives. For a specified type of misaligned welded joint, with the same manner of loading and residual stress, the fatigue limit o_jw may be a certain value. Thus, although the experimental results are down to fatigue lives of 104 cycles in Figure 8, we consider that the difference between the fatigue limit, which may be determined by long lives fatigue experiments and nominal fatigue limit (which may be calculated by the S-N curve of the middle cycle range at N = 2 x 106), is small. From Table I we get that the predicted /3p, /3-r, mp, mT, Cp, CT of the misaligned welded joints are in agreement with the experimental results. It is observed that the greater the misalignment, the bigger the effective stress concentration factor. So that we should reduce the misalignment for rising fatigue strength.
225
1. A predicting method of the fatigue life curve for misaligned welded joints has been developed. The relationship between the slope rn of the S-N curve and effective stress concentration factor, and the relationship about the intercept C in that curve and effective stress concentration factor and stress ratio have been established, respectively. 2. As an important parameter for fatigue strength of weld, effective stress concentration factor may be estimated. It is determined by material properties, manner of loading, geometry, the value of misalignment, residual stress and short crack in notch root of the weld. 3. A few good examples of the predicting S-N curves for misaligned welded joints have been presented. 4. The greater the value of misalignment, the lower the fatigue strength of the weld. The misalignment should be reduced for rising fatigue strength. ACKNOWLEDGEMENTS The author would like to thank Professor Wang Guanghai of the China Ship Scientific Research Center for providing help for this study. The research project is supported by the National Natural Science Fund of China and the Provincial Natural Science Fund of Hunan, People's Republic of China. REFERENCES 1 2 3 4 5 6 7 8 9 10
Gurney, T.R. 'Fatigue of Welded Structures', Cambridge University Press, Cambridge, 1979 Rolfe and Barson 'Fracture and Fatigue Control in Structure Applications of Fracture Mechanics', Prentice-Hall, Englewood Cliffs, NJ, 1977 Peterson, R.E. 'Stress Concentration Factor', John Wiley and Sons, New York, 1974 Topper, T.H. and EL Haddad, M.H. 'Fatigue Strength Prediction of Notches Based on Fracture Mechanics' Proc. Int. Conf. on Fatigue Thresholds, Stockholm, 1981 Yung, J.Y. and Lawrence, F.V. Fatigue Fracture Engng. Mater. Struct. 1985, 18, 223 Lawrence, F.V. 'Estimating the fatigue crack initiation life of welds' ASTM STP648, 1978, 134 Guan Deqing and Wang Guanghai Engng. Mech. (Chinese) 1993, 11, 118 Fuchs, H.O. and Stephens, R.I. 'Metal Fatigue in Engineering', John Wiley & Sons, New York, 1980 Frost, N.E., Marsh, K.J. and Pook, L.P. 'Metal Fatigue', Oxford University Press, London, 1974 Guan Deqing and Yan Guoliang Comput. Mech. 1993, 2, 889
CONCLUSIONS The following conclusions can be drawn from this work.
NOMENCLATURE A a
Table 1 results
Comparison of the predictions with the experimental
8(mm)
0.15
0.30
0.60
1.08
,/3p /3r /3 (test) mp mr m(test) cp cT c(test)
3.06 2.73 2.77 4.54 4.84 3.96 14.76 15.55 13.84
3.30 2.94 3.34 4.35 4.64 4.04 14.30 15.04 13.70
3.81 3.39 4.08 4.05 4.29 4.01 13.50 14.14 13.31
4.71 4.18 4.96 3.67 3.87 3.92 12.52 13.04 12.85
B
b C
CT Ks Kfmp K freT
K.
Constant Peterson's material parameter Constant Fatigue strength exponent Intercept of fatigue life curve Intercept of fatigue life curve related to Peterson's equation Intercept of fatigue life curve related to Topper's equation Fatigue notch factor The worst-case fatigue notch factor related to Peterson's equation The worst-case fatigue notch factor related to Topper's equation Elastic stress concentration factor
226 gth
lo M m mp mT
N
N, n
R r
rm
& t
/3p
Guan Deqing Threshold stress intensity factor Constant Geometry dependent constant Slope of fatigue life curve Slope of fatigue life curve related to Peterson's equation Slope of fatigue life related to Topper's equation Fatigue life Fatigue life of specimen i Number of specimen Stress ratio Notch root radius The worst-case notch root radius Ultimate strength of material Thickness of plate Effective stress concentration factor Effective stress concentration factor related to Peterson's equation
3T ~5 Or_l O'_lw
O'a O'ai OrA O"B O'Bw O'tf O"m
Effective stress concentration factor realted to Topper's equation The value of misalignment Fatigue limit of the material Fatigue limit of weld Stress amplitude Stress amplitude of specimen i Stress amplitude at point A on fatigue life curve of smooth specimen Stress amplitude at point B on fatigue life curve of smooth specimen Stress amplitude at point B on fatigue life curve of weld Fatigue strength coefficient Mean stress Residual stress
ELSEVIER
PIhS0142-1123(96)00004-7
A method of predicting the fatigue life curve for misaligned welded joints Guan Deqing Department of Power Engineering, Changsha University of Electric Power, Changsha, 410077, People's Republic of China (Received 29 August 1995; revised 13 December 1995) The effects of material properties, manner of loading, geometric shape, value of misalignment, residual stress, stress ratio and short crack in notch root of misaligned welded joint on the fatigue life are considered respectively. By means of Peterson's equation and Topper's equation and the concept of the worst-case fatigue notch factor, two estimating equations of effective stress concentration factors have been adopted. A method for predicting the fatigue life curve of misaligned welded joints for middle range of cycle numbers is developed. As an example of application of the method, the fatigue life curves of several misaligned welded joints made of low-alloy structural steel are predicted. Good agreement between the predicting and experimental results is achieved. This method has significance for engineering applications. (Keywords: fatigue life curve; effective stress concentration factor; the worst-case fatigue notch factor)
INTRODUCTION
range of cycle numbers is presented. The model has successfully predicted several S-N curves of misaligned welded joints.
A weld is the most commonly used form of structure joints. The development of the engineering application of welding techniques in the industries of aerospace manufacturing, shipbuilding and offshore exploitation makes the research on the fatigue life of weld joints a problem with the utmost importance. Because many factors influence the fatigue life of a weld the examination of the fatigue behavior of the welded joint is very complex m. The misalignment which often exists in a butt weld will certainly influence the fatigue strength of the weld. Based on Peterson's equation 3 and Topper's equation 4, the worst-case notch concept 5,6 has been adopted, Guan Deqing and Wang Guanghai 7 have developed two equations for estimating the effective stress concentration factors for misaligned welded joints. It is an efficient method for predicting the fatigue strength of welded joint. In the assessment of the fatigue life one of the important relations is the fatigue life curve (S-N curve), which is generally obtained from experiment. However an S-N curve is usually time-consuming and costly from the fatigue test. Therefore to develop a predicting method for the S-N curve of welded joints will certainly be useful in engineering application and helpful in gaining economic efficiency. Many influencing factors such as material properties, manner of loading, geometric shape, value of misalignment, residual stress and short crack in notch root on the S-N curve of welded joints are considered, respectively. A method of predicting the fatigue life curve for the middle
THE WORST-CASE FATIGUE NOTCH FACTOR OF MISALIGNED WELDED JOINT The fatigue notch factor kf is an important parameter for the fatigue strength of a notch specimen. It may be determined by Peterson's equation (1) Kt-1 Kf= 1 + 1 + a/r
(1)
where Kt is elastic stress concentration factor, r is notch root radius, a is Peterson's material parameter which may be approximated by the expression 1.087× 10SSu-2(mm) for steel, Su is the ultimate strength of the material. The effects of material, manner of loading and the geometric shape of the notch specimen on fatigue strength are considered in the equation. Topper and EL Haddad 4 have analysed the influences of the material, manner of loading, short crack in the notch on the fatigue performance and established another equation of the fatigue notch factor by expression (2)
[
~=(1 lo
221
Kt-z
Kth
\Mo._l )2/7"r
(2)
Guan Deqing
222
in which K,h is the threshold stress intensity factor, and M is the geometry dependent constant, 0-_1 is the fatigue limit of the material when stress ratio R equal to -1. lo is a constant. The fatigue notch factor of the welded joint may be determined from expression (1), which is the simple and popular. In the investigation of paper 7, the crack in the toe root of welds may be regarded as a short crack. Thus, we may adopt Equation (2) to estimate Kf of welded joint. A general form ~'6 which can be used to describe Kt of welds is
Kt=A
+
B(~) ~
• ,• k t
rill
(3)
where A, B, A are constants whose values are determined by geometric shape of welds and the manner of loading, t and r are the thickness of the plate and notch root radius. The constants A and A are usually 1 and 1/2, respectively. Because of welding error and deformation, misalignment often exists in a butt weld. Figure I is a misaligned welded plate joint. The value of misalignment is 6. The fatigue strength will be changed when the misalignment takes place in a butt weld. The investigation of the fatigue performance of a misaligned welded joint has had significance for predicting fatigue life. The author 1° recently presented a relationship (4) about Kt of a misaligned welded plate joint based on three-dimensional finite element computation.
r
Figure 2 The relationship curve of Kf, K,, r related to Peterson's equation
related to Peterson's equation and Topper's equation are shown in Figures 2 and 3. Substituting Equation(3) into Peterson's equation (1), differentiating Kf with respect to r, and setting that expression equal to zero leads to the result that Kf is the maximum value Kfmp, as Equation (5). Similarly, the maximum value gfmw related to Topper's equation may be given by expression (6) Kfm p = 1 + ~
gfm T =
= I+B
%.
kfmp
(5)
M.
1 + ~-
(6)
Hence, we may calculate the worst-case fatigue notch factors of misaligned welded joints by expression (5) or (6).
t~
= 0.27 + 2.45(~ ) H3
(4) ESTIMATING METHOD OF THE S-N CURVE FOR MISALIGNED WELDED JOINTS
where r
0.0 ~< - ~< 0.3, 0.01 <~ - ~< 0.20. t
t
The fatigue notch factors of misaligned welded joints may be obtained from Equations (1) and (4) or Equations (2) and (4). Unfortunately, it is difficult to determine Kf for defects such as a weld toe, because the notch root radii of such notch are generally unknown or difficult to measure and what is worse, highly variable. Yung and Lawrence 5 and Lawrence 6 have developed the worst-case notch concept which considered that there is a maximum value of Kf. The use of the concept leads to a model which predicts that the fatigue strength of a weldment depends upon its size, as well as its shape, material properties and manner of loading. The relationship curves of Kf, Kt and r to a butt weld
i
A definition is introduced that the effective stress concentration factor/3 is the ratio between the fatigue limit tr_l of the smooth specimen and that of a misaligned welded joint o'_~w when the stress ratio R (the ratio between the minimum cycle stress and the maximum cycle stress) equal to -1. o'_ 1
(7)
/3 = - - .
O'_lw
• kt kfraT
.....................
•%
.:, ~
r
41o
r
Figure 3 The relationship curve of Kf, Kt, r related to Topper's equFigure 1 Misaligned welded joint
ation
Predicting a fatigue life curve
223
According to Guan Deqing and Wang Guanghai 7, two estimating relationships related to Equations (5) and, (6) between /3 and Kt- are [~= constant Ortf /3P -- O ' t f - Orr gfmp
(8) ~
dr ~f---- Orzf_ Orr gfmT
R=0.5 R=0 R = -0.5
(9)
in which /3p, /3v are effective stress concentration factors related to Peterson's equation and Topper's equation, respectively, O!f is the fatigue strength coefficient, Orr is the notch root residual stress. Based on a lot of experimental results, three assumptions may be adopted.
R = - 1.0
log N Figure 5 The S-N curves of different R values for a specified type of misaligned welded joint
Assumption one A fatigue life curve (S-N curve) in double logarithmic coordinates is decomposed into three parts of, low, middle and high range cycle numbers, as shown in Figure 4. We will aim at the middle cycle range of the S - N curve, its mathematical equation is written by logN = C - mlogora
A
= 1.o ~=1.5 13= 2.1)
(10)
where N is the fatigue life, Ora is stress amplitude, C is intercept of the curve, m may be regarded as the slope of the curve.
Assumption two For a specified type of misaligned welded joint, the S-N curves for different R values keep parallel to that for the case of R = - 1 , provided /3 takes a constant value. Assumption three When R = - 1 , all S-N curves for different types of joints and different /3 values intercross at the same point in the low cycle range. According to three assumptions, we have got two expressions
I~= 2.5
log N Figure 6
All S-N curves intercross at the same point
m = m(/3)
(11)
c = c(/3,R)
(12)
Equations (11) and (12) will be determined as long as we find two points on the S-N curve of the welded joints in Figure 4. The S-N curve of the smooth specimens can be estimated by the Basquin equation (13) o-a = o"f(2N) b
• * • •
Specimens Specimens Specimens Specimens
(13)
where b is the fatigue strength exponent. By means of Fuchs's suggestion, 8 the S - N curves of smooth specimens and notch specimens intercross at N = 500. The stress amplitude at point A (N= 500) OrA on the S-N curve of smooth specimens may be obtained from Equation (13)
of 8 = 0.15ram of 8 = 0.30ram of 8 = 0.60ram of 8 = 1.08 mm
O"A = OrPf(1000) b
~,3
(14)
Thus the stress amplitude at point A on the S-N curve of a welded joint may be calculated through Equation (14). It may be assumed that the stress amplitude OrB of smooth specimens at N = 2 x 106 is the fatigue limit. Hence
lm O
-- 2
OrB = O'_I = Or'r(4 X 106) b
(15)
The welded joint fatigue limit crBw at point B on the S-N curve of a welded joint may be obtained by Equations (7) and (15) I
I
I
I
I
I
1
2
3
4
5
6
log Figure
4
N
A fatigue life curve in double logarithmic coordinates
OrBw : Or-lw Or-I
/3
Guan Deqing
224 o'f
= -~- (4 × 106)b
(16)
• Specimens of b = 0.15 mm
According to the values of points A and B, the slope m of the S-N curve of a misaligned welded joint can be obtained as follows: 1
m-
(17)
- b + 0.27761og/3
We consider the effect of mean stress on long life fatigue strength. Through the Morrow equation 9 O"a
O"m
+-7- = 1
Or_lw
• Specimens of 6 = 0.30mm • Specimens of 6 = 0.60mm • Specimens o f b = 0.08mm
4 --
~'3~
e~
C)
(18)
O'f
in which mean stress orm is determined by Orm - -
I+R 1-R
I --
(19)
O"a
Substituting Equations (16) and (19) into Equation (18), we can get Or_ 1
(20)
era - / 3 + (1 + R)o'_l/[(1 - R)o~f]
Because the point (2 × 106, or_l/{/3 + (1 + R) o'_l/[(1-R) df]} is on the S-N curve of a welded joint, it should satisfy Equation (10). Hence C = 6.301 +
1
- b + 0.27761og/3
I
I
I
1
3
4
5
6
8 The experimental results of four groups of misaligned welded joints Figure
Through linear statistical analysis to experimental results in Figure 8, we can determine experimental constants C and m in Equations (10) as follows: n
n
~E
(21)
I 2
log N
o'_ i
log/3 + (1 + R)o-_l/[(1 - R)df]
I 1
Z
i=1
i=1
i=1
m=
so that we may estimate the effective stress concentration factors based on Equations (4), (5), (8) or (4), (6), (9), and predict the S-N curve of misaligned welded joints by means of Equations (17) and (21).
n
1ogO'aiE logN/-
, ~ (logo. i)2 _-1 n
i=l
1
~
m
C = - ~ logNi + n
EXPERIMENTAL RESULTS The fatigue experiments of four groups of misaligned welded joints made of low alloy structural steel have been accomplished. The material properties and other data are Su = 548 MPa, o-'f = 894 MPa, o_l = 245 MPa, b = -0.0854, Kth = 294 N mm -3/2, trr = 441 MPa. Four groups of the values of misalignment are ~ = 0 . 1 5 , 0.30, 0.60, 1.08 mm, respectively. Stress ratio R is 0.0 in the experiments. A specimen of a welded joint and load is shown in Figure 7. Experimental results are shown in Figure 8.
i=1
n
1ogo-ailOgNi
(22) logo'~i
"i--1 - -
"
" E logo-ai
(23)
i=l
where n is the number of specimens of every type of misaligned welded joint. In our experiments, n is five. O'ai is the stress amplitude of specimen i; Ni is fatigue life of specimen i. Meanwhile we consider the low limits of S-N curves which have 95% confidence limits. So the expressions of the S-N curves of misaligned welded joints may be written as logN = 13.84 - 3.961ogo'a for specimens of 6 = 0.15 mm
(24)
logN-- 13.70 - 4.041ogtra for specimens of 6 = 0.30 mm
4mm
(25)
logN = 13.31 - 4.0110go-a for specimens of 6 = 0.60 mm
I'Figure
7
122ram
-,--'~ 6 m m ~_I_ ~
122mm
A specimen of a misaligned welded joint
I
(26)
logN = 12.85 - 3.921ogO'a for specimens of 6 = 1.08 mm.
(27)
Here the unit of O'a is MPa. Using Equations (7), (18) and (19) we have obtained the effective stress concentration factor from the experiments as
=I
tr_ l --
o'a
( l + R ) o'_l (1 - R ) or#f
(28)
Predicting a fatigue life curve where o'a is stress amplitude when N is equal to 2 x 106 in Equations (24)-(27). COMPARISON OF THE PREDICTED S-N CURVES WITH THE EXPERIMENTAL RESULTS To investigate the accuracy of the predicting method for the S-N curve we compare the experimental results with the predicted results. Using the above constants in the experiments, and assuming the thickness of plate t = 4 mrn, the geometry dependent constant M is approximated as 1.12. First we may adopt equations (4), (5), (8) and estimate effective stress concentration factor /3p. Then we may substitute/3p into Equations (17) and (21), obtain slope mp and intercept Cp of the fatigue life curve for misaligned welded joints related to Peterson's equation. Similarly, by means of Equations (4), (6), (9) the effective stress concentration factor /3x can be obtained. Substituting /3r into Equations (17) and (21), we can get slope mx and intercept CT of the fatigue life curve for misaligned welded joints related to Topper's equation. The predictions are shown in Table 1. According to Equation (28), effective stress concentration factors from experiments can be given as in Table 1. Slope m and intercept C from experimental Equations (24)-(27) are shown in Table 1. The worst-case notch concept for welds is intended specifically for the crack initiation portion of the fatigue life. This, according to the concept, is more significant for longer lives. For a specified type of misaligned welded joint, with the same manner of loading and residual stress, the fatigue limit o_jw may be a certain value. Thus, although the experimental results are down to fatigue lives of 104 cycles in Figure 8, we consider that the difference between the fatigue limit, which may be determined by long lives fatigue experiments and nominal fatigue limit (which may be calculated by the S-N curve of the middle cycle range at N = 2 x 106), is small. From Table I we get that the predicted /3p, /3-r, mp, mT, Cp, CT of the misaligned welded joints are in agreement with the experimental results. It is observed that the greater the misalignment, the bigger the effective stress concentration factor. So that we should reduce the misalignment for rising fatigue strength.
225
1. A predicting method of the fatigue life curve for misaligned welded joints has been developed. The relationship between the slope rn of the S-N curve and effective stress concentration factor, and the relationship about the intercept C in that curve and effective stress concentration factor and stress ratio have been established, respectively. 2. As an important parameter for fatigue strength of weld, effective stress concentration factor may be estimated. It is determined by material properties, manner of loading, geometry, the value of misalignment, residual stress and short crack in notch root of the weld. 3. A few good examples of the predicting S-N curves for misaligned welded joints have been presented. 4. The greater the value of misalignment, the lower the fatigue strength of the weld. The misalignment should be reduced for rising fatigue strength. ACKNOWLEDGEMENTS The author would like to thank Professor Wang Guanghai of the China Ship Scientific Research Center for providing help for this study. The research project is supported by the National Natural Science Fund of China and the Provincial Natural Science Fund of Hunan, People's Republic of China. REFERENCES 1 2 3 4 5 6 7 8 9 10
Gurney, T.R. 'Fatigue of Welded Structures', Cambridge University Press, Cambridge, 1979 Rolfe and Barson 'Fracture and Fatigue Control in Structure Applications of Fracture Mechanics', Prentice-Hall, Englewood Cliffs, NJ, 1977 Peterson, R.E. 'Stress Concentration Factor', John Wiley and Sons, New York, 1974 Topper, T.H. and EL Haddad, M.H. 'Fatigue Strength Prediction of Notches Based on Fracture Mechanics' Proc. Int. Conf. on Fatigue Thresholds, Stockholm, 1981 Yung, J.Y. and Lawrence, F.V. Fatigue Fracture Engng. Mater. Struct. 1985, 18, 223 Lawrence, F.V. 'Estimating the fatigue crack initiation life of welds' ASTM STP648, 1978, 134 Guan Deqing and Wang Guanghai Engng. Mech. (Chinese) 1993, 11, 118 Fuchs, H.O. and Stephens, R.I. 'Metal Fatigue in Engineering', John Wiley & Sons, New York, 1980 Frost, N.E., Marsh, K.J. and Pook, L.P. 'Metal Fatigue', Oxford University Press, London, 1974 Guan Deqing and Yan Guoliang Comput. Mech. 1993, 2, 889
CONCLUSIONS The following conclusions can be drawn from this work.
NOMENCLATURE A a
Table 1 results
Comparison of the predictions with the experimental
8(mm)
0.15
0.30
0.60
1.08
,/3p /3r /3 (test) mp mr m(test) cp cT c(test)
3.06 2.73 2.77 4.54 4.84 3.96 14.76 15.55 13.84
3.30 2.94 3.34 4.35 4.64 4.04 14.30 15.04 13.70
3.81 3.39 4.08 4.05 4.29 4.01 13.50 14.14 13.31
4.71 4.18 4.96 3.67 3.87 3.92 12.52 13.04 12.85
B
b C
CT Ks Kfmp K freT
K.
Constant Peterson's material parameter Constant Fatigue strength exponent Intercept of fatigue life curve Intercept of fatigue life curve related to Peterson's equation Intercept of fatigue life curve related to Topper's equation Fatigue notch factor The worst-case fatigue notch factor related to Peterson's equation The worst-case fatigue notch factor related to Topper's equation Elastic stress concentration factor
226 gth
lo M m mp mT
N
N, n
R r
rm
& t
/3p
Guan Deqing Threshold stress intensity factor Constant Geometry dependent constant Slope of fatigue life curve Slope of fatigue life curve related to Peterson's equation Slope of fatigue life related to Topper's equation Fatigue life Fatigue life of specimen i Number of specimen Stress ratio Notch root radius The worst-case notch root radius Ultimate strength of material Thickness of plate Effective stress concentration factor Effective stress concentration factor related to Peterson's equation
3T ~5 Or_l O'_lw
O'a O'ai OrA O"B O'Bw O'tf O"m
Effective stress concentration factor realted to Topper's equation The value of misalignment Fatigue limit of the material Fatigue limit of weld Stress amplitude Stress amplitude of specimen i Stress amplitude at point A on fatigue life curve of smooth specimen Stress amplitude at point B on fatigue life curve of smooth specimen Stress amplitude at point B on fatigue life curve of weld Fatigue strength coefficient Mean stress Residual stress