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Fatigue strength assessment of scallops —an example for the application of nominal and local stress approaches
MarineStructures8 (1995)423-447
ELSEVIER
© 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0951-8339/95/$9.50 0951-8339(94)00029-8
Fatigue Strength Assessment of Scallops an Example for the Application of Nominal and Local Stress Approaches
Wolfgang Fricke* & Hans Paetzoldt *Germanischer Lloyd, P.O. Box 111606, D-20416 Hamburg, Germany tlnstitut ft~r Schiffbau, L/immersieth 90, D-22305 Hamburg, Germany (Received 7 December 1994)
ABSTRACT Scallops, i.e. small cut-outs in ship structural members, are frequently subjected to high cyclic stresses, which require fatigue strength to be considered during structural design. The paper summarizes the results of several fatigue tests for various types of scallops under axial loading. Further types and the effect of complex load combinations are evaluated using local approaches for the fatigue strength assessment, in particular the notch stress approach for plate edges and the structural or hot-spot stress approach for welded connections. After a review of the approaches, the analysis of local stresses by the finite element method and by stress concentration factors (SCFs) is described. Regression formulae for SCFs, derived from parametric stress analyses, are presented for elliptical and rectangular scallops with rounded corners. The approaches are illustrated by examples from which conclusions for the practical application a r e drawn. Key words." fatigue, scallop, stress concentration, notch stress, structural stress, h o t - s p o t stress, c u t - o u t , w e l d toe.
1 INTRODUCTION
Scallops and other small cut-outs at the connection between the web of a girder or stiffener and its flange or plating are arranged for several 423
424
W. Fricke, H. Paetzold
reasons. In many cases they are used to ensure that the stiffener butt can be properly welded. The risk of incomplete weld penetration or slag inclusion at the intersection between web and flange is avoided. Usually, half-round scallops are arranged, Fig. la. Longer scallops with oval or rectangular shape and rounded corners are used especially where additional margins have to be provided for block assembly and/or where certain hole sizes are required, e.g. for drain and air holes in tanks. Other reasons for the arrangement of scallops are the reduction of welding length and the related plate distortion. In this case series of scallops are normally arranged interrupting the fillet welding, as shown in Fig. lb. This is of particular advantage in the case of thin-plated structures. But, when applying the mechanized fillet welding process, intermittent fillet welds between web and plate are not appropriate. In this case, preference is given to cut-outs arranged in a way as shown in Fig. lc. Scallops may be subjected to high stresses. Under axial loading increased stresses may occur at the rounded corners as well as at the ends of the fillet weld between web and flange, as indicated in Fig. 2. Other load components such as shear forces in the web or pressure loads on the flange may also produce high local stresses. Under cyclic loading fatigue cracks may occur at the locations mentioned. The reported service performance of scallops indicates problems in some cases. 1'2 For this reason, a rational fatigue strength assessment of scallops including all load effects is required. This is particularly important in view of the increased optimization and utilization of structural
(a) ~x~,'N'N\\\\\\\\\\\\\\\\,
K\\\\\\\\\\\\\\\\\~\
(b) t\\~\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\~\\\~
Fig. 1. Types of scallops.
[ x. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~
425
Fatigue strength assessment o f scallops
"
+
÷
-
~
D
~\\\\\\\XN\\\\\\\\\\x,
i
Fig. 2. Typicalloadingof scallops.
members in ships, which have been made possible by improved analysis methods and mesh generation techniques. The approaches to the fatigue strength assessment have also been further developed during recent years. In addition to the traditional nominal stress approach, local approaches such as the hot-spot stress approach for welded joints and the notch stress approach for plate edges have reached the stage of practical application. As scallops include both types of notches, all approaches mentioned will--after a short introductory description--be applied to this type of detail to draw further conclusions for the applicability of scallops in various situations. 2 APPROACHES TO FATIGUE STRENGTH ASSESSMENT 2.1 Nominal stress approach
The fatigue strength assessment--especially for welded structures--is in most cases based on the nominal stress an, which can be calculated using sectional forces or moments in the structural component, e.g. F trn=~
or
M trn=-~
(1)
426
IV. Fricke, H. PaetzoM
where F, M = force and m o m e n t in cross section A, W = area and modulus of cross section. In connection with scallops and other cut-outs, the nominal stress is normally based on the total (gross) section, i.e. disregarding the cut-out. If fatigue tests are performed, the nominal stress refers to the section properties of the specimen. Several fatigue tests have been performed with models containing scallops, being subjected to axial loading. A re-analysis of such tests has been presented by Gurney & Maddox 3 showing a relatively broad scatter band containing 95% of all test results. The analysis of the test data has resulted in the classification of scallops into the detail categories used in relevant codes. For instance, detail category 71 has been selected for scallops in IIW-recommendations 4 and Eurocodes, 5 which means that the design S-N curve based on a probability of survival of 97-7% is characterized by a reference value AaR = 71 N / m m 2 at two million stress cycles. The inverse slope exponent is normally assumed for welded structures as
mo=3. Further tests have been performed during the past few years at the Institute of Naval Architecture of the University of Hamburg. In these tests the fatigue behavior of specific scallops has been investigated, including: - - half-round scallops at butt-welded ends of bulb-plate stiffeners 6 series of relatively long scallops of rectangular shape and rounded corners and of elliptical shape 7 relatively large scallops with well-rounded corners. 8 -
-
-
-
The results are plotted in Fig. 3 together with the scatter band evaluated by Gurney and Maddox for this detail. It turns out that the new test results are mainly within the lower range of the scatter band. The wide scattering of the results is to be attributed to the different geometrical shapes, i.e. the proportion of length and depth as well as the radius, and to fabrication methods. From the test results obtained for axial loading, the following conclusions can be drawn with respect to practical application: -
-
-
-
in most cases the crack initiation under axial loading occurs in the plate at the weld toe; no difference is found in the fatigue behavior between elliptical scallops and those with the normal rectangular shape. This applies to intermittent fillet welding. Preference should therefore be given to the normal shape in case of manual, intermittent welding;
Fatigue strength assessment of scallops 6 7 89105 I I I I I _4- ~ ~
2 3 4 5 6 7 89106 I I l~,,.I I I I I I H~ 160×7 tt:m:::::mm~'~
427 3 L
2 I
4 I
5 678 I I I I I
• ~ R 2 5
2 o
- 2
~ R 4 0
zx ~---------'l Scallop . t . . - a - ~ l . ~ 40x90 0 !""--'~
~
102 9 liP 160x7
Seall°~ 40x90
~'~
8
/
$catterband of Gurney and Maddox 3
Z
102 9
7
6 I L ILl 6 789105
I 2
I 3
I 4
I I lilt 5 6 789106
I 2
I 3
I 4
I I I II 5 678
Cycles
Fig. 3. Fatigue test result 6"7, 8 for scallops under axial loading.
--
in case of stiffeners with low depth, half-round scallops with a small radius (r = 25 mm) should be chosen. A larger radius (r = 40 mm) to facilitate welding is not recommendable; - - closing of the cut-outs, e.g. by overlapping parts, leads to an increased fatigue life; 8 --- the fatigue strength may fall below the reference value AaR = 71 N / m m 2 of the S - N curve, if a slope exponent mo --- 3 is assumed. The differences between the scallops shown in Fig. 3 can be verified by a refined fatigue strength assessment based on local approaches, which will be described in the following sections. The notch stress approach for plate edges will be applied to the rounded corners of the cut-out, while the hotspot stress approach is used for the assessment of the weld toe. Both approaches allow--contrary to the existing test data based on the nominal stress approach--the fatigue strength assessment for arbitrary shapes and for other load components such as shear and pressure loads. 2.2 ]Notch stress approach for plate edges Methods for the prediction of the crack initiation life of notched structural components are based on the hypothesis that the structural element at the crack location, which is subjected to a certain stress and strain history, behaves in the same manner as a small laboratory specimen, subjected to
428
W. Fricke, H. PaetzoM
the same stress and strain history. Cyclic material properties obtained from tests on unnotched specimens are used to evaluate the material behaviour in the notch root, taking into account the elastic as well as the elastic-plastic stress and strain. The approach, which is described in more detail in other references,9 consists generally of three evaluation steps: 1. evaluation of the cyclic material behaviour; 2. relationship between applied load and notch strain based on mechanical principles; 3. damage accumulation and crack initiation of the material. The local elastic-plastic material behaviour is determined by the cyclic material law, describing the relationship between the local stress amplitude aa and strain amplitude ea, which may differ considerably from the material behaviour under (first) monotonic loading. In most cases, the cyclic material law can be approximated by the sum of the elastic and plastic par t as follows: ~ = ~,et + e~,pt = -~- + ~,~7]
(2)
where E is the modulus of elasticity. For the details in question, the material constants K' and n' are required for the normal hull structural steel (NT 24) and the frequently used higher-tensile steel (HT 36) including the effect of a thermally cut plate edge. Extensive investigations 1° resulted in the mean values given in Table 1 for specimens with a plate edge produced by flame cutting or else by underwater-plasma cutting. The relationship between the applied load--represented by the nominal stress amplitude an,~--and the notch strain amplitude ea (step 2) can be established experimentally or numerically. Numerical solutions are of great importance, which can be obtained by the finite element technique, or alternatively, by approximation formulae such as proposed, inter alia, by Neuber: 11 Table 1 Material Constants K' and n' for Steels with Thermally Cut Plate Edge Material
N T 24
H T 36
min. yield stress [N/mm 1] K' [N/ram2] n'
235 981 0.1747
355 1103 0.1604
Fatigue strength assessment of scallops 1000
"\ ~ [ ] ,,~,, ~.~
4O0 300
=
\~. ~j~
500 ¢,1
429
~--~-~) 50~
var. cutting methods I°
¢,1
400300 = (~)-55'16 flame
cut 9 20(3
200 Normal strength steel (NT 24) / I I I 10~10 102 103 104 105
&~&mO
var. cuttinl~ methods lu Higher-tensile steel (HT 36)
I
106
I
10~101 102
107
Nc
I
103
I
104
I
105
1106
107
Nc
Fig. 4l. Damage parameter life curve according to Smith et al. t3 for specimenswith thermally cut plate edge. O'a "ga - - (0"n'a" g,)__,2
(3)
E where K t is the theoretical elastic stress concentration factor. If this is unknown due to problems in defining the nominal stress in complex structures, the product (trn, a " K t ) can be replaced by the amplitude of the theoretical elastic notch stress ak, a: tTk, a = tTn,a" g t .
(4)
For selected load amplitudes, the material law (2) together with Neuber's rule (3) yields the local elastic-plastic stress and strain amplitudes either by iteration or graphically. In the case of very high loads, where the total net section is stressed above the elastic limit, Neuber's rule has to be extended by an additional factor to account for general yielding. 12'9 The crack initiation life N c of the material under cyclic stress and strain is described by life curves using either the total strain or a special damage parameter which takes into account the influence of mean stress. Widely used is the damage parameter P s w r according to Smith e t al.: 13 P s w r = x/trmax . e~ . E
(5)
where the mean stress is considered in the maximum value of the local elastic-plastic stress amax. Figure 4 shows the damage parameter--life curves for both steels mentioned.
430
W. Fricke, H. Paetzold
2.3 Hot-spot stress approach for welded joints The hot-spot stress approach has firstly been introduced in the fatigue strength assessment of tubular joints in offshore structures. High local stresses may occur at the toe of the welded connection which are often magnified by bending stresses in the tubular shells. The stresses on the shell surface in the vicinity of the hot spots are extrapolated to the weld toe and assessed with a special S - N curve based on hot-spot stresses. Parametric formulae for stress concentration factors (SCFs) have been published in the literature. The hot-spot stress approach has also been applied to c o m m o n plated structures. Here, the hot-spot stress is considered in a similar way as in tubular joints as the highest value of structural or geometric stress, including axial and bending stresses in the plate or shell and omitting the local stress increase due to the weld toe. Applications to ship structural details have firstly been shown by Matoba et al. 14 Fricke & Petershagen 15 derived the reference value Aasn of the design S - N curve from Radaj's notch stress approach for welded joints 16 using fatigue test data of several basic joints and a relationship for the effect of weld toe angle. 17 For 90% survival probability, AasR becomes:
(_~) 1/2.1 AasR, 90 = 135-
[N/mm 2]
(6)
where 0 is the weld toe angle in degrees. A reduction by about 10% has to be taken into account if the reference value is based on 97.7% survival probability. In this case the reference value becomes AasR = 100 N / m m 2 for 45 ° fillet welds. It has been proposed 15 to apply a quadratic extrapolation of the plate surface stress in front of the weld toe because it seems to be impossible to define two stress points for linear extrapolation being generally valid for all possible situations. In several cases, e.g. with stiffener ends, the increase in structural stress may be highly nonlinear, depending on the actual geometric parameters. In scallops, the linear extrapolation could cause further problems, because two hot-spots exist so that the second stress point could be affected by the other hot spot or even be located outside the hole. In the case of a high stress gradient within the plate and quadratic extrapolation, a further increase in permissible stress should be allowed as already included in rules of Germanischer Lloyd 18 taking into account the prolonged crack propagation phase in this case.
Fatigue s t r e n g t h a s s e s s m e n t o f scallops
431
2
b
I
0 -
b
IV
u2
o Measured value
Fig. 5. Structural stress in the plate below a half-round scallop.
Figure 5 shows the stress distribution in the plate below a half-round scallop, calculated with a plate-element model 6 together with measured stresses which agree quite well with the calculated ones. The ratio between the hot-spot stress as at the weld toe and the nominal stress an, i.e. the hotspot stress concentration factor Ks =
--as
(7)
O"n
is about Ks = 1-8. The above m e n t i o n e d reference value AasR = 1 0 0 N / m m 2 of the design S - N curve for 45 ° fillet welds, based on hot-spot stresses, makes it possible to derive a corresponding value for nominal stress, which becomes AaR = 100/1.8 = 5 6 N / m m 2. This value is slightly below the lower bound of the scatter band at N=2.106 shown in Fig. 3. Better agreement is reached if the increase in permissible stress due to the non-linear extrapolation and high bending portion mentioned is taken into account. Furthermore, the stresses shown in Fig. 5 have been obtained from a test specimen with a relatively slender plate strip below the scallop. A further decrease of stress will occur in a continuous plating 6 so that the corrected results agree quite well with the detail classification given in the rules.
W. Fricke, H. Paetzold
432 (a) Plate elements
(b) 3D-elements
Fig. 6. Finite element models for the stress analysis of scallops.
3 STRESS ANALYSIS 3.1 Finite-element-method
Numerical methods such as the finite element (FE) method today allow a detailed stress analysis even of very complex structures. They can be well used for the analysis of local stresses in critical areas of scallops for the subsequent fatigue strength assessment. The following aspects should be considered when modelling the structure: - - a fairly fine mesh is necessary at the rounded plate edge of the cutout with regard to the high stress gradient; - - the elements of the flange or plating must be able to model the local bending occurring particularly under shear and local pressure loads. Models of plate elements or 3-dimensional elements as shown in Fig. 6 are suitable if the stress at the weld toe is assessed with the hot-spot stress approach. If plate elements are used (Fig. 6a), the modelling of the fillet weld may create some problems. On the one hand, the elements representing the weld should be sufficiently stiff to clamp the adjacent plate. On the other hand, the axial stiffness, governed by the cross section of the elements, should not be enlarged in order to avoid overstressing of the plate below the scallop under axial loads.
Fatigue strength assessment of scallops
433
The actual stiffness properties are better represented by models with 3dimensional elements as shown in Fig. 6b. Plate bending can be modelled well by isoparametric elements having mid-side nodes. The only problem requiring special attention is the evaluation of the structural or hot-spot stress; in the plate at the weld toe. If the above mentioned definition of the structural or hot-spot stress-being; considered as a plate or shell stress--is followed, the stress linearized in the plate thickness direction is required. This can be obtained from isoparametric 3-dimensional elements by arranging only one layer of elements with two integration points in the thickness direction. The stresses at the plate surface, obtained in this way, can then be extrapolated to the weld toe, yielding the required hot-spot stress os. 3.2 Stress concentration factors
The use of stress concentration factors can considerably facilitate the practical application. Scallops according to Fig. lc can approximately be considered as cut-outs in infinite plates as long as the remaining web near the flange is not too slender. A verification example will be shown in Section 4.1. For the assessment of the other, more complex scallops interrupting the welding, a great number of finite element analyses have been performed by systematically varying the geometric and load parameters. The results for two basic types of scallops shown in Fig. 7 will be described in this section in more detail: a) rectangular scallop with rounded corners, and b) elliptical scallop. The following geometric parameters have been varied:
(a) Rectangular scallop
(b) Elliptical scallop L
I~
b
=_
b
(t) 1
_I U
Side 1
Side 2
a = 0'4; b' = (8 + b/d)/I 8; h = a . %/2 Fig. 7. Types of scallops investigated.
7
IV. Fricke, H. Paetzold
434 Load case 1
......
._J
,,,
Load case 3
Load case 2
:_:_:_:
u._::
~._ .._.~..._..~.....i.~.-~ ~
-:I
.'-.~" I
_ ,
1
I
"
I
O"M
Load c a s e 5 Section A - A
Load c a s e 4
.~A
+
Section A - A
+ +', •
I-~ A "
++
I
b ! = 50 tg Fig.
+ +,+ +++, I~
8.
bg
I
,++, ,+,
=l
Basic load cases considered.
ratio between length b and depth d - - ratio between corner radius r and depth d (only for rectangular scallops) - ratio between web thickness t and flange thickness tg - ratio between length b and flange thickness tg. - -
The ranges of variation are given in the Appendix. Other data such as the throat thickness, a, of the fillet weld and the relative distance, b', between the toes of the elliptical scallop have been kept constant as defined in Fig. 7. Five load cases have been considered, see Fig. 8: 1. axial stress a s in the girder or stiffener where the scallop is arranged 2. average shear stress z in the web of the girder or stiffener 3. bending stress aM in the girder or stiffener, defined by half the difference in stress between the mid-plane of the flange and the upper edge of the scallop 4. pressure p on the flange 5. additional shear stress Zp in the flange due to a breadth deviating from the assumptions in load case 4. The axial, shear and bending stresses are nominal stresses calculated on the basis of sectional forces and moments at the location of the scallop, assuming gross section properties, a s and aM can be derived from stresses at in the flange and au at the upper edge of the scallop by the following equations:
Fatigue strength assessment of scallops al + flu
'~N- ---2-O-l
aM -
--
435
(8)
O- u
2
(9)
The effect of the pressure p on local stresses is considered assuming a continuous plating as flange having a ratio bg/tg = 50 between the spacing bg of the webs and the plate thickness tg see Fig. 8. In case of larger or smaller ratios bg/tg, the change in load transferred from the plating into the web can be taken into account by the additional shear stress Tv in load case :5, which may be calculated as follows: P (~-50) ,p=~.
(10)
The finite element calculations have been performed with models of girders having a large web height compared to the depth of the scallop, i.e. the effect of the web height is neglected. Isoparametric 3-dimensionalelements as shown in Fig. 6b have been used for the modelling. The loads except for the pressure have been applied far away from the scallop to avoid detrimental effects on the results. In the case of shear stress, the loads have been applied such that the associated bending moment has a zero crossing at the location of the scallop. Local stresses have been evaluated with respect to stress, concentration factors at three positions, as indicated in Fig. 7:. 1. at the rounded plate edge of the cut-out 2. at the upper toe of the fillet weld located at the cut-out (only for rectangular scallops) 3. at the lower toe of the fillet weld located on the flange. At the weld toes (positions 2 and 3), the hot-spot stress concentration factor Ks has been evaluated, while the theoretical stress concentration factor Kt (~0) has been evaluated at the rounded plate edge (position 1) as a function of the circumferential angle ~0. In elliptical scallops, q~ is defined by the corresponding point in the transformed circle. This definition has been chosen to improve the numerical evaluation of the results and the inteq~olation between different shapes. From the results, regression formulae for the stress concentration factors have been derived, which are summarized in the Appendix. The accuracy of the regression formulae, compared with the results of the parametric finite element computations, was found to be in the order of 3-8%. Only a few values showed deviations of more than 20% in one or two load cases. The local stress for any load combination can finally be calculated using the following equations:
436
w. Fricke, H. Paetzold
ak (¢p) = KtN (¢P) " aN 4- K,Q(qg) . "c + KtM(qg) " aM + gtp(q}) "p q- gtQp(q)) " rp
(11) as = KsN" aN + KsQ • ~ + KsM" aM + Ksp" p + KsQp " "Cp
(12)
These equations are valid for side 1 of the scallops as defined in Fig. 7, but can also be used for side 2, if the signs of Kto(q~) and KsQ are changed. 4 EXAMPLES OF APPLICATION
4.1 Scallops under axial loading Longitudinal structural members in the upper and lower flange of the hull girder m a y be stressed by very high cyclic stresses, particularly in naval ships, where stillwater stresses are very low and permissible stresses are to a large extent utilized by wave-induced hull girder bending stresses. Considered are three types of scallops with a depth of 50 m m in axially stressed longitudinals of bulb plate type (HP 200 × 9). The scallops are located at a small distance (20 mm) away from the plate so that continuous fillet welding is possible, see Fig. 9. The edge stress distribution shown in Fig. 9 was calculated by means of the F E - m e t h o d yielding the stress concentration factor Kt. It is interesting to note that the plate has a restraining effect on the adjacent local stresses so that the highest K/-value occurs on the opposite side in the case of the symmetrical cut-outs. This value can be alternatively determined using well-known results for standard cutouts in infinite plates. F o r instance, the stress concentration factor Kt = 1-99 evaluated for the elliptical scallop would be Kt = 2.0 in an infinite plate. In the following, the third scallop shown in Fig. 9 with the oval-elliptical shape will be further analysed because it shows a better performance as a drain hole with only slightly increased local stresses. The m a x i m u m nominal stress amplitude is assumed to be an, a = 150 N / m m 2 in the structure made of normal hull structural steel (NT 24). As local elastic-plastic straining has to be expected within the highest load cycle, its contribution to total damage will be evaluated on the basis of the notch stress approach described in Section 2.2. Stress concentration factor Nominal stress amplitude Elastic notch stress amplitude
Kt = 2 . 1 6
an, a = 150 N / m m 2 ak, a = a,,,a " Kt = 3 2 4 N / m m 2
O"n -~ c o n s
t = lOmm With oval cutout 100x50
Ellipse 100x50
With a ovalelliptical cutout 1 0 0 x 5
0
~ g~
Kt = 1.56
Kt max= 1.99
•
_~ ~ , ~ K t N 3
1.93
147
-- 1.54
~'/I////I////////////~ 4~
Fig. 9. Edge distribution for various scallops under axial loading.
,.,..I
W. Fricke, H. PaetzoM
438
Neuber's rule ~a = 0.5096" /
400
t
15a
{]r a
+ L6++mm .~
Ill'
ea = "-~ +
30G
Cyclic material law Material: steel NT 24 E = 2 . 0 6 . 105 N / m m 2 K' = 981 N/ram 2 n ' = 0.1747
200
r~
"F/ ! 0
,,T
1
I1
I
I
I
I
i
2
3
4
5
6
Strain [%ol
Fig. 10. L o c a l s t r e s s - s t r a i n v a l u e a c c o r d i n g to N e u b e r f o r Kt = 2.16 a n d crn,a = 1 5 0 N / m m 2.
Local notch stress amplitude Local notch strain amplitude
O"a = 268 N / m m 2 •a = 1.896.10 -3
The local notch stress and strain amplitudes have been determined graphically using Neuber's rule, 11 see Fig. 10. The damage parameter according to Smith et al.,12 if zero mean stress is assumed, becomes:
Pswr = X/am~x -e~. E = 324 N / m m 2 The n u m b e r Nc o f load cycles until crack initiation is according to Fig. 4: (PswT~
Nc = \ 2--~ J
- 4.812
= 25472 cycles
Thus, the damage caused by one cycle (to be used in the linear damage calculation becomes: DI -
1 - 3.926.10 -5 Nc
Fatigue strength assessment of scallops
439
q0=p.a
llllllllllllll11111111l @NO
i .~! a]2 ]~
I x
a=900mm
tg = 18 m m
1 = 5400 m m Fig. 11.
Example of tanker longitudinal
ed loading The second example deals with a longitudinal in the bottom structure of tanker under axial and lateral loads, see Fig. 11, where the effect of the location of a rectangular or elliptical scallop, as shown in Fig. 7, on local stresses will be investigated. The pressure p acting from inside (i.e. the difference between inner and outer pressure) causes a trapezoidal line load with a max. value qo = P" a ( a = spacing between longitudinals). Reduced loads occur near the transverse members which take over part of the: load. Assuming symmetry at the transverse members, the following shear force Q(x) and bending moment M(x) are acting in the longitudinal: a x<-:
-2
Q(x) = yqo( t - ga- 2 ~ ) M(x)=qo-~ a
(13)
~ / / ) 2 ( 2 - ~ / / ) + 6 " - lx ( 1 - ~ ) -a2 ( / )
1
32/la
(14)
l
g_
(15)
1-7
a,]
- (2l) -1
(16)
W. Fricke, H. Paetzold
440
The associated nominal stresses according to the load case definition in Fig. 8 are: 0"m - -
2
+
z --
a~-
(17)
-~- 0"N0
Q(x)
(18)
As
M(x) ( 1 2 VV,
1) #u
(19)
where: Wt = section modulus for point l in the mid-plane of the plating (= 13.106mm 3) Wu = section modulus for point u at the upper edge of the scallop (= 26.10 6 m m 3) A~ = sectional area of the web (= 5500 mm 2) trN0 = additional axial stress in the longitudinal due to hull girder bending. In this example, a scallop with a length b = 140 mm and a depth d = 70 mm is assumed together with the above given values of section properties. Figure
150 m ~N
~N0
e~
E
E z
I00
t~ t: b
50
0.
.
.
.
.
.
.
.
.
1000
. . . . . .
t
2000
x lmml Fig. 12. N o m i n a l stresses in the longitudinal.
------"-'~-~ (1/2)
441
Fatigue strength assessment o f scallops
51)I)
--~-'"O~ ' "'O'~~
400
300
z b 200
100
I
I
10o0
2000 x Imm]
0/2)
Fig. 13. Local stresses for scallops at different locations in the longitudinal.
12 shows the distribution of nominal stresses along the longitudinal for a pressure p = 0.15N/mm 2 and additional hull girder bending stress aN0 =: 130 N / m m 2. The max. local stresses shown in Fig. 13 have been calculated :for a rectangular scallop (with radius r = 35 mm) and for an elliptical scallop using the regression formulae for stress concentration factors given in the Appendix. At the ends of the longitudinal, a reduction of the pressure load due to the reduced effective plate breadth bg/tg < 50 has been taken into account by an additionally acting (negative) plate shear stress zp according to eqn. (10). The results for this load combination show that - - the local stresses are considerably increased near the supports of the longitudinal, which is mainly due to increased shear stresses
442
W. Fricke, H. Paetzold
- - the local stresses are generally lower in the - - the weld toe on the plate is always the elliptical scallop, while the rounded plate scallop becomes more critical near the itudinal. The permissible local stress range stress cycles is about
elliptical scallop critical position of the edge of the rectangular supports of the long-
Aap given in rules 18 for wave-induced
- - 420 N/mm 2 for structural stresses at weld toes and 440 N/mm 2 for plate edges of normal hull structural steel cut with machine quality, -
-
if a positive effect of mean stress is neglected. Under the assumption that the given nominal stresses correspond to the highest stress range in the spectrum, the elliptical scallop would be permitted over the full length of the longitudinal, while the rectangular scallop would not be allowed near the transverse webs.
5 CONCLUSIONS Scallops in the structural components of ships have to be assessed with respect to fatigue strength to avoid cracks during service life. Difficulties in the assessment arise due to the variety of geometric configurations and due to the loading pattern of the structural member characterized by various combinations of normal, shear, bending and local pressure loads. Local approaches such as the notch stress approach for plate edges and the structural or hot-spot stress approach for welded connections have reached a stage which allows their application to the problems mentioned. Stress concentration factors have shown to be very useful in practical application. From the theoretical and experimental investigation of several scallops, the following conclusions can be drawn: - - a fairly good performance can be expected from scallops which are arranged at a sniall distance away from the plate allowing continuous welding between web and plating as shown in Fig. lc. Optimal are elliptical shapes which can be analysed with sufficient accuracy using the stress concentration factors for the cut-out in an infinite plate; if scallops interrupting the fillet weld are arranged, for instance, at a stiffener butt (Fig. la) or in order to apply intermittent welding (Fig. l b), the type of loading determines the performance of elliptical and rectangular scallops with rounded corners. Under axial -
-
Fatigue strength assessment of scallops
-
-
443
loading, both types show similar local stresses and fatigue behaviour, while the elliptical scallop is superior under combined loading, particularly if shear stresses are acting in the web; half-round scallops at stiffener butts show a better fatigue behaviour under axial loading than elliptical and rectangular cut-outs. The size should be kept small especially in stiffeners with relatively low depth.
Generally it has to be emphasized that the fatigue problems, particularly in welded connections, may grow with increasing optimization and utilization of the structural members in ships and more frequent use of higher-tensile steel. The fatigue strength assessment by refined methods, as described in this paper, are expected to contribute to the design of fatigueresistant and durable ship structures.
ACKNOWLEDGEMENT The investigations described were financially supported in several research projects by the German Institutions DFG, FDS and BMFT, which is highly appreciated. The authors gratefully acknowledge the fruitful discussions with Professor H. Petershagen and the assistance of Mr S. Pohl in carrying out the systematic finite element calculations.
REFERENCES 1. Youshaw, R., Nondestructive inspection of longitudinal stiffener butt welds in commercial vessels. Ship Structure Committee, Report No. 295/ 1980. 2. Munse, W. H., Fatigue criteria for ship structure details. SNAME Extreme Loads Response Syrup., Arlington, 1981. 3. Gurney, T. R & Maddox, S. J., A re-analysis of fatigue data for welded joints in steel. Rep. E/44/72, The Welding Institute, Abington, 1972. 4. NN., Design recommendations for cyclic loaded welded steel structures. Welding in the World, 20 (1982), 153-165. 5. NN., Eurocodc 3, Design of steel structures. European Committee for Standardization, April 1992. 6. Fricke, W., Investigation of the fatigue strength of bulb plate stiffener joints. Schiff& Hafen, 38 (1986), 50-54 (in German). 7. Paetzold, H., Fatigue strength of ship structural details. FDS-Report No. 159/1985, Forschungszentrum des Deutschen Schiffbaus, Hamburg 1985 (in German). 8. Paetzold, H., Design and strength of sectional joints of HP sections under consideration of manufacturing influences. FDS-Report 222/1990,
444
W. Fricke, H. Paetzold
Forschungszentrum des Deutschen Schiffbaus, Hamburg 1990 (in German). 9. Fricke, W. & Paetzold, H., Application of the cyclic strain approach to the fatigue failure of ship structural details. J. Ship Research, 31 (1987), 177185. 10. Paetzold, H., Damage parameter--life curves for normal and high strength steels. Report No. 507, Institut ffir Schiffbau, Hamburg 1990 (in German). 11. Neuber, H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law. J. Applied Mechanics, Trans. ASME, 28 (1961), 544-550. 12. Seeger, T. & Heuler, P., Generalized application of Neuber's rule. J. Testing and Evaluation, 8 (1980), 199-204. 13. Smith, K. N., Watson, P. & Topper, T. H., A stress-strain function for the fatigue of metals. J. Materials, 5 (1970), 767-778. 14. Matoba, M. et al., Evaluation of fatigue strength of welded steel structures-- hull's members, hollow section joints, piping and vessel joints. IIW-Doc. XIII-1082-83, International Institute of Welding 1983. 15. Fricke, W. & Petershagen, H., Detail design of welded ship structures based on hot-spot stresses. In Practical Design of Ships and Mobile Units, ed. by J.B. Caldwell and G. Ward, Elsevier Appl. Science Publ., London and New York 1992. 16. Radaj, D., Notch stress proof for fatigue resistant welded structures. IIWDoc. XIII- 1157-85, International Institute of Welding. 17. Olivier, R. & Seeger, T., The effect of weld toe angle on fatigue strength of welded joints--unified evaluation of published test results. FG Werkstoffmechanik of TH Darmstadt, Rep. FF-2, 1989 (in German). 18. Germanischer Lloyd: Rules and Regulations, I - - S h i p Technology, Part 1 Seagoing Ships, Chapter 1--Hull Structures. Hamburg 1992.
A P P E N D I X : R E G R E S S I O N F O R M U L A E F O R STRESS CONCENTRATION FACTORS
a) Rectangular scallop with rounded corners Geometry parameters and validity ranges (Fig. 7a).
6 = b/d = b/tg
p = r/d 0 = tg/t
(2.0-4.0) (4.0-12.0)
Theoretical stress concentration factor Kt at position 1: 5 --
G
i=l
(0.25-0.5) (1.0-2.0)
445
Fatigue strength assessment of scallops Ci : X i I • (~ xi2 . fl xi4 . p xi6 . oxi8
f l (~P) = 1 + cos (2¢p)
j~(~p) = sin (2¢p)
j~(cp) = 1 - cos (2¢p)
fs(~P) = sin (4¢p)
f3(~P) = 1 - cos (4¢p) Coefficients LC
Xil, i
1
2 3 4 5 1
2 3 4 5
.e~ri2 ...
f o r l o a d cases (LC) 1-5:
Xil
Xi2
Xi4
Xi6
Xi8
0.9931 -0.5390 -0.0687 0-3026 0-1541
-0.099 0.0 0.0 -0-045 0.0
0.0 0.026 0.0 0-0 -0.070
-0.180 0.181 0.141 -0.584 0.229
-0-059 -0-030 -0.040 -0-176 0-043
-0.5408 -0.1676 -0.4421 -1.6739 0.0187
0.792 0.198 0-525 0.346 -0.163
-0.127 0.264 -0.731 0.074 0.187
-0.643 -1.114 -0.248 -0.146 0.163
-0.037 -0.123 -0-190 0.0 0.0
-0.7840 0-0930 0-4920 -0.1009 -0-2328
0.0 0.244 0.0 0.290 0.185
-0.067 -0.138 0.057 -0.273 -0-098
-0-159 -0.941 0.753 -0.849 0.203
-0.062 0-0 -0-044 0-0 -0-089
-0.8634 14.3126 -0.0914 6.1520 -2.3106
-0.783 0.597 0.909 0-808 -0.649
1.612 0-640 1.540 0.227 0.247
0.496 -0.063 -1-157 -1.258 -0.868
0.631 1.101 1.041 0.998 0.853
-0.7053 2.1792 -0.1726 0-4467 0.0394
-0.673 0.604 0.388 0.585 -0.772
0-358 0-160 0-465 0.060 1.204
0-153 -0.048 -0.249 -1-088 0.0
0.318 0-821 0.345 0.986 -1.014
H o t - s p o t stress c o n c e n t r a t i o n f a c t o r Ks at position 2 a n d 3:
Ks =: X1
• fix2 . (fl(x4 +xl0 .~i+Xll .fl+x,2 "p"}-XI3 .'0) +
Coefficients X1, X2 . . . for l o a d cases 1-5:
X5). ( p X 6 +
X 7 ) . 0x8 _~_X14.
W. Fricke, H. Paetzold
446
Pos. LC 1 2 3 4 5
Xi
X2
X4
-1.0435 0.327 0.046 2.3649 0.448 -0.728 -1.5138 -0.418 0.226 3.6046 -0.888 2.534 3.1806 0.908 0.476
1 2 3 4 5
1.6566 1.7948 0.4094 7.6088 0.3360
3(5
X6
X7
0.0 0.0 0.0 0.0 0.0
-0.168 -0.048 0.352 0.764 -0.057
0.731 1.918 0.601 2.299 0.0
-0.208 0.156 0.0 0.0 -0.327 -0.224 0-0 -0.214 -0.691 0.663 0-0 0.0 0.011 1.468 5.689 -0.502 -0.055 1.816 -0.771 0.0
0.0 0.0 0.0 0.0 0.0
Xa
Xto
Xll
Xl2
X13
-0-543 -0.048 0.0 -0.108 0.067 1.827 -0.076 0.0 0.0 0.284 0.111 -1.241 0.311 0.057 0.0 -0.366 -0.112 1-213 2.996 0.177 -0.041 0.322 -0.628 0.0 0.298 -0.070 0.0 0.340 0.102 0.0 -0.134 -0.569 0.054 0.774 0.614
0-0 0.0 0.0 0.0 0-0 0.102 0.0 0.259 0.0 0.637 0.0 0-0 0-0 0.0 0.0 0.0 -0.010 0.877 -0.149 0.0 0.0 0.0 -0.017 -0.125 -2.007
b) Elliptical scallop G e o m e t r y parameters and validity ranges (Fig. 7b)
6 = b/d fl = b'/tg
0 = tg/t
(1.0 ... 4.0) ( 2 . 6 7 . . . 8.0 2.85...8.57 3.20...9.60 (1.0 ... 2.0)
for 6 = 1 forfi=2 forfi=4)
Theoretical stress concentration factor Kt(q~) at position 1. 5
Kt(tp) = ~
C~ . f~ (q~)
i=1
Ci : X i 1 . ((~ (xi2+ x,8,6+ xi9 "fl-~- XilO "6) .j[_ X i 3 ) " (fl Xi4 + X i 5 ) "ox,6
3q (tp) = 1 + cos (2~p)
J~(q~) = sin (2¢p)
J~(tp) = 1 - cos (2tp)
fs(tp) = sin (4q~)
f3(tp) = 1 - cos(4tp) Coefficients
X/I , Xi2
...
Xt4
for load cases (LC) 1-5
Fatigue strength assessment of scallops LC 1
~l
i
~2
~3
~4
~5
~6
~8
~9
~10
0.0 0.0 0.0 0.0 0.0
0-0 0-092 0-0 0-0 -0-359
0-0 0.0 0.0 0.0 0.0
-0-046 0-0 0-0 -0-295 0-335
0-0 0.0 0.0 0.0 0.0
0-0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.853 0.0 -0-123 0.215 -0.775 0-696 0-048 -0.631 -0.046 0-196 0-830 0.0 1.011 -0.840 -0.176
0.0 0.0 0.0 0-0 0.0
0-0 0.0 0.0 0.0 -0-113 0.0 0.0 -0.037 0-038 0.0 -0.056 0.065 0-082 -0.172 0.0 -0.178 0-151 -0.070 0.0 -0.046
0.0 0.0 0.0 0-0 0-0
-0-050 -0-052 0-0 0-0 0-057
1-3153 -0.533 -0.5459 -0-622 O.1132 0.748 0-3129 0.439 0.1701 -1.861 -0.3693 -0.6740 -1-7232 -1-1852 2-0244
447
-0.9863 -0.405 0.1056 0.0 0.3656 -1.404 -0.0905 1.277 -0.3752 -0.312
0.0 0.0 0.0 0.0 0.0
-2.3085 0.751 1.770 7.8350 3.027 5-830 -4-2153 3.535 2 - 3 5 1 3-8262 -0.156 3-299 0.9992 0.813 -0-090
-0.055 -0.248 0.050 -0.216 0.080 0.816 0-329 0.382 0-415 1-566
0-0 0-0 0-0 0.0 0.0
-0.5414 0.452 0 - 0 6 1 0-359 0.239 1-2468 0.972 0.0 0-350 1 . 2 6 3 -0-9612 1.303 -0.523 0.283 0.554 0.8340 -0-101 0.335 -0.100 0.463 0.5290 0-602 -0.701 0.786 -0.223
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0-0 0-0 0-0
0.893 -0.338 0.0 0-201 0-987 -0.373 0.0 0-0 0~982 -0.402 0-0 0.0 1.206 0.287 -0-042 0.018 1.001 0.0 0-0 -0-078 0.678 -0.066 0.937 -0.073 0.965 -0.078 1.047 0-126 0.585 0-0
0-0 0-0 0-0 0.0 0.0
-0.080 -0.055 -0.068 0.0 -0.094
Hot-spot stress concentration factor Ks at position 3. Ks = X1 • (6(x2+x,o.O) + X 3 ) . (fix4 + X s ) . 0 x6 + Xll Coefficients X1, Xz ... for load cases 1-5: LC
Xt
X2
X3
X4
X5
)(6
Xlo
XII
1 2 3 4 5
1-7941 1-8846 1-8249 5-1097 0.1749
-0.403 -0.318 -0.250 0.315 0.102
0.0 0.0 0.0 4.410 0.412
0.058 0.300 0.188 1-064 1-749
0.0 0.0 0.0 4.349 3.697
-0-400 -0-701 -0.064 0.323 -0.117
0.0 0.0 0.0 -0.777 -0.642
0.769 0.573 -0.928 0.0 -2.953
ELSEVIER
© 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0951-8339/95/$9.50 0951-8339(94)00029-8
Fatigue Strength Assessment of Scallops an Example for the Application of Nominal and Local Stress Approaches
Wolfgang Fricke* & Hans Paetzoldt *Germanischer Lloyd, P.O. Box 111606, D-20416 Hamburg, Germany tlnstitut ft~r Schiffbau, L/immersieth 90, D-22305 Hamburg, Germany (Received 7 December 1994)
ABSTRACT Scallops, i.e. small cut-outs in ship structural members, are frequently subjected to high cyclic stresses, which require fatigue strength to be considered during structural design. The paper summarizes the results of several fatigue tests for various types of scallops under axial loading. Further types and the effect of complex load combinations are evaluated using local approaches for the fatigue strength assessment, in particular the notch stress approach for plate edges and the structural or hot-spot stress approach for welded connections. After a review of the approaches, the analysis of local stresses by the finite element method and by stress concentration factors (SCFs) is described. Regression formulae for SCFs, derived from parametric stress analyses, are presented for elliptical and rectangular scallops with rounded corners. The approaches are illustrated by examples from which conclusions for the practical application a r e drawn. Key words." fatigue, scallop, stress concentration, notch stress, structural stress, h o t - s p o t stress, c u t - o u t , w e l d toe.
1 INTRODUCTION
Scallops and other small cut-outs at the connection between the web of a girder or stiffener and its flange or plating are arranged for several 423
424
W. Fricke, H. Paetzold
reasons. In many cases they are used to ensure that the stiffener butt can be properly welded. The risk of incomplete weld penetration or slag inclusion at the intersection between web and flange is avoided. Usually, half-round scallops are arranged, Fig. la. Longer scallops with oval or rectangular shape and rounded corners are used especially where additional margins have to be provided for block assembly and/or where certain hole sizes are required, e.g. for drain and air holes in tanks. Other reasons for the arrangement of scallops are the reduction of welding length and the related plate distortion. In this case series of scallops are normally arranged interrupting the fillet welding, as shown in Fig. lb. This is of particular advantage in the case of thin-plated structures. But, when applying the mechanized fillet welding process, intermittent fillet welds between web and plate are not appropriate. In this case, preference is given to cut-outs arranged in a way as shown in Fig. lc. Scallops may be subjected to high stresses. Under axial loading increased stresses may occur at the rounded corners as well as at the ends of the fillet weld between web and flange, as indicated in Fig. 2. Other load components such as shear forces in the web or pressure loads on the flange may also produce high local stresses. Under cyclic loading fatigue cracks may occur at the locations mentioned. The reported service performance of scallops indicates problems in some cases. 1'2 For this reason, a rational fatigue strength assessment of scallops including all load effects is required. This is particularly important in view of the increased optimization and utilization of structural
(a) ~x~,'N'N\\\\\\\\\\\\\\\\,
K\\\\\\\\\\\\\\\\\~\
(b) t\\~\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\~\\\~
Fig. 1. Types of scallops.
[ x. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~
425
Fatigue strength assessment o f scallops
"
+
÷
-
~
D
~\\\\\\\XN\\\\\\\\\\x,
i
Fig. 2. Typicalloadingof scallops.
members in ships, which have been made possible by improved analysis methods and mesh generation techniques. The approaches to the fatigue strength assessment have also been further developed during recent years. In addition to the traditional nominal stress approach, local approaches such as the hot-spot stress approach for welded joints and the notch stress approach for plate edges have reached the stage of practical application. As scallops include both types of notches, all approaches mentioned will--after a short introductory description--be applied to this type of detail to draw further conclusions for the applicability of scallops in various situations. 2 APPROACHES TO FATIGUE STRENGTH ASSESSMENT 2.1 Nominal stress approach
The fatigue strength assessment--especially for welded structures--is in most cases based on the nominal stress an, which can be calculated using sectional forces or moments in the structural component, e.g. F trn=~
or
M trn=-~
(1)
426
IV. Fricke, H. PaetzoM
where F, M = force and m o m e n t in cross section A, W = area and modulus of cross section. In connection with scallops and other cut-outs, the nominal stress is normally based on the total (gross) section, i.e. disregarding the cut-out. If fatigue tests are performed, the nominal stress refers to the section properties of the specimen. Several fatigue tests have been performed with models containing scallops, being subjected to axial loading. A re-analysis of such tests has been presented by Gurney & Maddox 3 showing a relatively broad scatter band containing 95% of all test results. The analysis of the test data has resulted in the classification of scallops into the detail categories used in relevant codes. For instance, detail category 71 has been selected for scallops in IIW-recommendations 4 and Eurocodes, 5 which means that the design S-N curve based on a probability of survival of 97-7% is characterized by a reference value AaR = 71 N / m m 2 at two million stress cycles. The inverse slope exponent is normally assumed for welded structures as
mo=3. Further tests have been performed during the past few years at the Institute of Naval Architecture of the University of Hamburg. In these tests the fatigue behavior of specific scallops has been investigated, including: - - half-round scallops at butt-welded ends of bulb-plate stiffeners 6 series of relatively long scallops of rectangular shape and rounded corners and of elliptical shape 7 relatively large scallops with well-rounded corners. 8 -
-
-
-
The results are plotted in Fig. 3 together with the scatter band evaluated by Gurney and Maddox for this detail. It turns out that the new test results are mainly within the lower range of the scatter band. The wide scattering of the results is to be attributed to the different geometrical shapes, i.e. the proportion of length and depth as well as the radius, and to fabrication methods. From the test results obtained for axial loading, the following conclusions can be drawn with respect to practical application: -
-
-
-
in most cases the crack initiation under axial loading occurs in the plate at the weld toe; no difference is found in the fatigue behavior between elliptical scallops and those with the normal rectangular shape. This applies to intermittent fillet welding. Preference should therefore be given to the normal shape in case of manual, intermittent welding;
Fatigue strength assessment of scallops 6 7 89105 I I I I I _4- ~ ~
2 3 4 5 6 7 89106 I I l~,,.I I I I I I H~ 160×7 tt:m:::::mm~'~
427 3 L
2 I
4 I
5 678 I I I I I
• ~ R 2 5
2 o
- 2
~ R 4 0
zx ~---------'l Scallop . t . . - a - ~ l . ~ 40x90 0 !""--'~
~
102 9 liP 160x7
Seall°~ 40x90
~'~
8
/
$catterband of Gurney and Maddox 3
Z
102 9
7
6 I L ILl 6 789105
I 2
I 3
I 4
I I lilt 5 6 789106
I 2
I 3
I 4
I I I II 5 678
Cycles
Fig. 3. Fatigue test result 6"7, 8 for scallops under axial loading.
--
in case of stiffeners with low depth, half-round scallops with a small radius (r = 25 mm) should be chosen. A larger radius (r = 40 mm) to facilitate welding is not recommendable; - - closing of the cut-outs, e.g. by overlapping parts, leads to an increased fatigue life; 8 --- the fatigue strength may fall below the reference value AaR = 71 N / m m 2 of the S - N curve, if a slope exponent mo --- 3 is assumed. The differences between the scallops shown in Fig. 3 can be verified by a refined fatigue strength assessment based on local approaches, which will be described in the following sections. The notch stress approach for plate edges will be applied to the rounded corners of the cut-out, while the hotspot stress approach is used for the assessment of the weld toe. Both approaches allow--contrary to the existing test data based on the nominal stress approach--the fatigue strength assessment for arbitrary shapes and for other load components such as shear and pressure loads. 2.2 ]Notch stress approach for plate edges Methods for the prediction of the crack initiation life of notched structural components are based on the hypothesis that the structural element at the crack location, which is subjected to a certain stress and strain history, behaves in the same manner as a small laboratory specimen, subjected to
428
W. Fricke, H. PaetzoM
the same stress and strain history. Cyclic material properties obtained from tests on unnotched specimens are used to evaluate the material behaviour in the notch root, taking into account the elastic as well as the elastic-plastic stress and strain. The approach, which is described in more detail in other references,9 consists generally of three evaluation steps: 1. evaluation of the cyclic material behaviour; 2. relationship between applied load and notch strain based on mechanical principles; 3. damage accumulation and crack initiation of the material. The local elastic-plastic material behaviour is determined by the cyclic material law, describing the relationship between the local stress amplitude aa and strain amplitude ea, which may differ considerably from the material behaviour under (first) monotonic loading. In most cases, the cyclic material law can be approximated by the sum of the elastic and plastic par t as follows: ~ = ~,et + e~,pt = -~- + ~,~7]
(2)
where E is the modulus of elasticity. For the details in question, the material constants K' and n' are required for the normal hull structural steel (NT 24) and the frequently used higher-tensile steel (HT 36) including the effect of a thermally cut plate edge. Extensive investigations 1° resulted in the mean values given in Table 1 for specimens with a plate edge produced by flame cutting or else by underwater-plasma cutting. The relationship between the applied load--represented by the nominal stress amplitude an,~--and the notch strain amplitude ea (step 2) can be established experimentally or numerically. Numerical solutions are of great importance, which can be obtained by the finite element technique, or alternatively, by approximation formulae such as proposed, inter alia, by Neuber: 11 Table 1 Material Constants K' and n' for Steels with Thermally Cut Plate Edge Material
N T 24
H T 36
min. yield stress [N/mm 1] K' [N/ram2] n'
235 981 0.1747
355 1103 0.1604
Fatigue strength assessment of scallops 1000
"\ ~ [ ] ,,~,, ~.~
4O0 300
=
\~. ~j~
500 ¢,1
429
~--~-~) 50~
var. cutting methods I°
¢,1
400300 = (~)-55'16 flame
cut 9 20(3
200 Normal strength steel (NT 24) / I I I 10~10 102 103 104 105
&~&mO
var. cuttinl~ methods lu Higher-tensile steel (HT 36)
I
106
I
10~101 102
107
Nc
I
103
I
104
I
105
1106
107
Nc
Fig. 4l. Damage parameter life curve according to Smith et al. t3 for specimenswith thermally cut plate edge. O'a "ga - - (0"n'a" g,)__,2
(3)
E where K t is the theoretical elastic stress concentration factor. If this is unknown due to problems in defining the nominal stress in complex structures, the product (trn, a " K t ) can be replaced by the amplitude of the theoretical elastic notch stress ak, a: tTk, a = tTn,a" g t .
(4)
For selected load amplitudes, the material law (2) together with Neuber's rule (3) yields the local elastic-plastic stress and strain amplitudes either by iteration or graphically. In the case of very high loads, where the total net section is stressed above the elastic limit, Neuber's rule has to be extended by an additional factor to account for general yielding. 12'9 The crack initiation life N c of the material under cyclic stress and strain is described by life curves using either the total strain or a special damage parameter which takes into account the influence of mean stress. Widely used is the damage parameter P s w r according to Smith e t al.: 13 P s w r = x/trmax . e~ . E
(5)
where the mean stress is considered in the maximum value of the local elastic-plastic stress amax. Figure 4 shows the damage parameter--life curves for both steels mentioned.
430
W. Fricke, H. Paetzold
2.3 Hot-spot stress approach for welded joints The hot-spot stress approach has firstly been introduced in the fatigue strength assessment of tubular joints in offshore structures. High local stresses may occur at the toe of the welded connection which are often magnified by bending stresses in the tubular shells. The stresses on the shell surface in the vicinity of the hot spots are extrapolated to the weld toe and assessed with a special S - N curve based on hot-spot stresses. Parametric formulae for stress concentration factors (SCFs) have been published in the literature. The hot-spot stress approach has also been applied to c o m m o n plated structures. Here, the hot-spot stress is considered in a similar way as in tubular joints as the highest value of structural or geometric stress, including axial and bending stresses in the plate or shell and omitting the local stress increase due to the weld toe. Applications to ship structural details have firstly been shown by Matoba et al. 14 Fricke & Petershagen 15 derived the reference value Aasn of the design S - N curve from Radaj's notch stress approach for welded joints 16 using fatigue test data of several basic joints and a relationship for the effect of weld toe angle. 17 For 90% survival probability, AasR becomes:
(_~) 1/2.1 AasR, 90 = 135-
[N/mm 2]
(6)
where 0 is the weld toe angle in degrees. A reduction by about 10% has to be taken into account if the reference value is based on 97.7% survival probability. In this case the reference value becomes AasR = 100 N / m m 2 for 45 ° fillet welds. It has been proposed 15 to apply a quadratic extrapolation of the plate surface stress in front of the weld toe because it seems to be impossible to define two stress points for linear extrapolation being generally valid for all possible situations. In several cases, e.g. with stiffener ends, the increase in structural stress may be highly nonlinear, depending on the actual geometric parameters. In scallops, the linear extrapolation could cause further problems, because two hot-spots exist so that the second stress point could be affected by the other hot spot or even be located outside the hole. In the case of a high stress gradient within the plate and quadratic extrapolation, a further increase in permissible stress should be allowed as already included in rules of Germanischer Lloyd 18 taking into account the prolonged crack propagation phase in this case.
Fatigue s t r e n g t h a s s e s s m e n t o f scallops
431
2
b
I
0 -
b
IV
u2
o Measured value
Fig. 5. Structural stress in the plate below a half-round scallop.
Figure 5 shows the stress distribution in the plate below a half-round scallop, calculated with a plate-element model 6 together with measured stresses which agree quite well with the calculated ones. The ratio between the hot-spot stress as at the weld toe and the nominal stress an, i.e. the hotspot stress concentration factor Ks =
--as
(7)
O"n
is about Ks = 1-8. The above m e n t i o n e d reference value AasR = 1 0 0 N / m m 2 of the design S - N curve for 45 ° fillet welds, based on hot-spot stresses, makes it possible to derive a corresponding value for nominal stress, which becomes AaR = 100/1.8 = 5 6 N / m m 2. This value is slightly below the lower bound of the scatter band at N=2.106 shown in Fig. 3. Better agreement is reached if the increase in permissible stress due to the non-linear extrapolation and high bending portion mentioned is taken into account. Furthermore, the stresses shown in Fig. 5 have been obtained from a test specimen with a relatively slender plate strip below the scallop. A further decrease of stress will occur in a continuous plating 6 so that the corrected results agree quite well with the detail classification given in the rules.
W. Fricke, H. Paetzold
432 (a) Plate elements
(b) 3D-elements
Fig. 6. Finite element models for the stress analysis of scallops.
3 STRESS ANALYSIS 3.1 Finite-element-method
Numerical methods such as the finite element (FE) method today allow a detailed stress analysis even of very complex structures. They can be well used for the analysis of local stresses in critical areas of scallops for the subsequent fatigue strength assessment. The following aspects should be considered when modelling the structure: - - a fairly fine mesh is necessary at the rounded plate edge of the cutout with regard to the high stress gradient; - - the elements of the flange or plating must be able to model the local bending occurring particularly under shear and local pressure loads. Models of plate elements or 3-dimensional elements as shown in Fig. 6 are suitable if the stress at the weld toe is assessed with the hot-spot stress approach. If plate elements are used (Fig. 6a), the modelling of the fillet weld may create some problems. On the one hand, the elements representing the weld should be sufficiently stiff to clamp the adjacent plate. On the other hand, the axial stiffness, governed by the cross section of the elements, should not be enlarged in order to avoid overstressing of the plate below the scallop under axial loads.
Fatigue strength assessment of scallops
433
The actual stiffness properties are better represented by models with 3dimensional elements as shown in Fig. 6b. Plate bending can be modelled well by isoparametric elements having mid-side nodes. The only problem requiring special attention is the evaluation of the structural or hot-spot stress; in the plate at the weld toe. If the above mentioned definition of the structural or hot-spot stress-being; considered as a plate or shell stress--is followed, the stress linearized in the plate thickness direction is required. This can be obtained from isoparametric 3-dimensional elements by arranging only one layer of elements with two integration points in the thickness direction. The stresses at the plate surface, obtained in this way, can then be extrapolated to the weld toe, yielding the required hot-spot stress os. 3.2 Stress concentration factors
The use of stress concentration factors can considerably facilitate the practical application. Scallops according to Fig. lc can approximately be considered as cut-outs in infinite plates as long as the remaining web near the flange is not too slender. A verification example will be shown in Section 4.1. For the assessment of the other, more complex scallops interrupting the welding, a great number of finite element analyses have been performed by systematically varying the geometric and load parameters. The results for two basic types of scallops shown in Fig. 7 will be described in this section in more detail: a) rectangular scallop with rounded corners, and b) elliptical scallop. The following geometric parameters have been varied:
(a) Rectangular scallop
(b) Elliptical scallop L
I~
b
=_
b
(t) 1
_I U
Side 1
Side 2
a = 0'4; b' = (8 + b/d)/I 8; h = a . %/2 Fig. 7. Types of scallops investigated.
7
IV. Fricke, H. Paetzold
434 Load case 1
......
._J
,,,
Load case 3
Load case 2
:_:_:_:
u._::
~._ .._.~..._..~.....i.~.-~ ~
-:I
.'-.~" I
_ ,
1
I
"
I
O"M
Load c a s e 5 Section A - A
Load c a s e 4
.~A
+
Section A - A
+ +', •
I-~ A "
++
I
b ! = 50 tg Fig.
+ +,+ +++, I~
8.
bg
I
,++, ,+,
=l
Basic load cases considered.
ratio between length b and depth d - - ratio between corner radius r and depth d (only for rectangular scallops) - ratio between web thickness t and flange thickness tg - ratio between length b and flange thickness tg. - -
The ranges of variation are given in the Appendix. Other data such as the throat thickness, a, of the fillet weld and the relative distance, b', between the toes of the elliptical scallop have been kept constant as defined in Fig. 7. Five load cases have been considered, see Fig. 8: 1. axial stress a s in the girder or stiffener where the scallop is arranged 2. average shear stress z in the web of the girder or stiffener 3. bending stress aM in the girder or stiffener, defined by half the difference in stress between the mid-plane of the flange and the upper edge of the scallop 4. pressure p on the flange 5. additional shear stress Zp in the flange due to a breadth deviating from the assumptions in load case 4. The axial, shear and bending stresses are nominal stresses calculated on the basis of sectional forces and moments at the location of the scallop, assuming gross section properties, a s and aM can be derived from stresses at in the flange and au at the upper edge of the scallop by the following equations:
Fatigue strength assessment of scallops al + flu
'~N- ---2-O-l
aM -
--
435
(8)
O- u
2
(9)
The effect of the pressure p on local stresses is considered assuming a continuous plating as flange having a ratio bg/tg = 50 between the spacing bg of the webs and the plate thickness tg see Fig. 8. In case of larger or smaller ratios bg/tg, the change in load transferred from the plating into the web can be taken into account by the additional shear stress Tv in load case :5, which may be calculated as follows: P (~-50) ,p=~.
(10)
The finite element calculations have been performed with models of girders having a large web height compared to the depth of the scallop, i.e. the effect of the web height is neglected. Isoparametric 3-dimensionalelements as shown in Fig. 6b have been used for the modelling. The loads except for the pressure have been applied far away from the scallop to avoid detrimental effects on the results. In the case of shear stress, the loads have been applied such that the associated bending moment has a zero crossing at the location of the scallop. Local stresses have been evaluated with respect to stress, concentration factors at three positions, as indicated in Fig. 7:. 1. at the rounded plate edge of the cut-out 2. at the upper toe of the fillet weld located at the cut-out (only for rectangular scallops) 3. at the lower toe of the fillet weld located on the flange. At the weld toes (positions 2 and 3), the hot-spot stress concentration factor Ks has been evaluated, while the theoretical stress concentration factor Kt (~0) has been evaluated at the rounded plate edge (position 1) as a function of the circumferential angle ~0. In elliptical scallops, q~ is defined by the corresponding point in the transformed circle. This definition has been chosen to improve the numerical evaluation of the results and the inteq~olation between different shapes. From the results, regression formulae for the stress concentration factors have been derived, which are summarized in the Appendix. The accuracy of the regression formulae, compared with the results of the parametric finite element computations, was found to be in the order of 3-8%. Only a few values showed deviations of more than 20% in one or two load cases. The local stress for any load combination can finally be calculated using the following equations:
436
w. Fricke, H. Paetzold
ak (¢p) = KtN (¢P) " aN 4- K,Q(qg) . "c + KtM(qg) " aM + gtp(q}) "p q- gtQp(q)) " rp
(11) as = KsN" aN + KsQ • ~ + KsM" aM + Ksp" p + KsQp " "Cp
(12)
These equations are valid for side 1 of the scallops as defined in Fig. 7, but can also be used for side 2, if the signs of Kto(q~) and KsQ are changed. 4 EXAMPLES OF APPLICATION
4.1 Scallops under axial loading Longitudinal structural members in the upper and lower flange of the hull girder m a y be stressed by very high cyclic stresses, particularly in naval ships, where stillwater stresses are very low and permissible stresses are to a large extent utilized by wave-induced hull girder bending stresses. Considered are three types of scallops with a depth of 50 m m in axially stressed longitudinals of bulb plate type (HP 200 × 9). The scallops are located at a small distance (20 mm) away from the plate so that continuous fillet welding is possible, see Fig. 9. The edge stress distribution shown in Fig. 9 was calculated by means of the F E - m e t h o d yielding the stress concentration factor Kt. It is interesting to note that the plate has a restraining effect on the adjacent local stresses so that the highest K/-value occurs on the opposite side in the case of the symmetrical cut-outs. This value can be alternatively determined using well-known results for standard cutouts in infinite plates. F o r instance, the stress concentration factor Kt = 1-99 evaluated for the elliptical scallop would be Kt = 2.0 in an infinite plate. In the following, the third scallop shown in Fig. 9 with the oval-elliptical shape will be further analysed because it shows a better performance as a drain hole with only slightly increased local stresses. The m a x i m u m nominal stress amplitude is assumed to be an, a = 150 N / m m 2 in the structure made of normal hull structural steel (NT 24). As local elastic-plastic straining has to be expected within the highest load cycle, its contribution to total damage will be evaluated on the basis of the notch stress approach described in Section 2.2. Stress concentration factor Nominal stress amplitude Elastic notch stress amplitude
Kt = 2 . 1 6
an, a = 150 N / m m 2 ak, a = a,,,a " Kt = 3 2 4 N / m m 2
O"n -~ c o n s
t = lOmm With oval cutout 100x50
Ellipse 100x50
With a ovalelliptical cutout 1 0 0 x 5
0
~ g~
Kt = 1.56
Kt max= 1.99
•
_~ ~ , ~ K t N 3
1.93
147
-- 1.54
~'/I////I////////////~ 4~
Fig. 9. Edge distribution for various scallops under axial loading.
,.,..I
W. Fricke, H. PaetzoM
438
Neuber's rule ~a = 0.5096" /
400
t
15a
{]r a
+ L6++mm .~
Ill'
ea = "-~ +
30G
Cyclic material law Material: steel NT 24 E = 2 . 0 6 . 105 N / m m 2 K' = 981 N/ram 2 n ' = 0.1747
200
r~
"F/ ! 0
,,T
1
I1
I
I
I
I
i
2
3
4
5
6
Strain [%ol
Fig. 10. L o c a l s t r e s s - s t r a i n v a l u e a c c o r d i n g to N e u b e r f o r Kt = 2.16 a n d crn,a = 1 5 0 N / m m 2.
Local notch stress amplitude Local notch strain amplitude
O"a = 268 N / m m 2 •a = 1.896.10 -3
The local notch stress and strain amplitudes have been determined graphically using Neuber's rule, 11 see Fig. 10. The damage parameter according to Smith et al.,12 if zero mean stress is assumed, becomes:
Pswr = X/am~x -e~. E = 324 N / m m 2 The n u m b e r Nc o f load cycles until crack initiation is according to Fig. 4: (PswT~
Nc = \ 2--~ J
- 4.812
= 25472 cycles
Thus, the damage caused by one cycle (to be used in the linear damage calculation becomes: DI -
1 - 3.926.10 -5 Nc
Fatigue strength assessment of scallops
439
q0=p.a
llllllllllllll11111111l @NO
i .~! a]2 ]~
I x
a=900mm
tg = 18 m m
1 = 5400 m m Fig. 11.
Example of tanker longitudinal
ed loading The second example deals with a longitudinal in the bottom structure of tanker under axial and lateral loads, see Fig. 11, where the effect of the location of a rectangular or elliptical scallop, as shown in Fig. 7, on local stresses will be investigated. The pressure p acting from inside (i.e. the difference between inner and outer pressure) causes a trapezoidal line load with a max. value qo = P" a ( a = spacing between longitudinals). Reduced loads occur near the transverse members which take over part of the: load. Assuming symmetry at the transverse members, the following shear force Q(x) and bending moment M(x) are acting in the longitudinal: a x<-:
-2
Q(x) = yqo( t - ga- 2 ~ ) M(x)=qo-~ a
(13)
~ / / ) 2 ( 2 - ~ / / ) + 6 " - lx ( 1 - ~ ) -a2 ( / )
1
32/la
(14)
l
g_
(15)
1-7
a,]
- (2l) -1
(16)
W. Fricke, H. Paetzold
440
The associated nominal stresses according to the load case definition in Fig. 8 are: 0"m - -
2
+
z --
a~-
(17)
-~- 0"N0
Q(x)
(18)
As
M(x) ( 1 2 VV,
1) #u
(19)
where: Wt = section modulus for point l in the mid-plane of the plating (= 13.106mm 3) Wu = section modulus for point u at the upper edge of the scallop (= 26.10 6 m m 3) A~ = sectional area of the web (= 5500 mm 2) trN0 = additional axial stress in the longitudinal due to hull girder bending. In this example, a scallop with a length b = 140 mm and a depth d = 70 mm is assumed together with the above given values of section properties. Figure
150 m ~N
~N0
e~
E
E z
I00
t~ t: b
50
0.
.
.
.
.
.
.
.
.
1000
. . . . . .
t
2000
x lmml Fig. 12. N o m i n a l stresses in the longitudinal.
------"-'~-~ (1/2)
441
Fatigue strength assessment o f scallops
51)I)
--~-'"O~ ' "'O'~~
400
300
z b 200
100
I
I
10o0
2000 x Imm]
0/2)
Fig. 13. Local stresses for scallops at different locations in the longitudinal.
12 shows the distribution of nominal stresses along the longitudinal for a pressure p = 0.15N/mm 2 and additional hull girder bending stress aN0 =: 130 N / m m 2. The max. local stresses shown in Fig. 13 have been calculated :for a rectangular scallop (with radius r = 35 mm) and for an elliptical scallop using the regression formulae for stress concentration factors given in the Appendix. At the ends of the longitudinal, a reduction of the pressure load due to the reduced effective plate breadth bg/tg < 50 has been taken into account by an additionally acting (negative) plate shear stress zp according to eqn. (10). The results for this load combination show that - - the local stresses are considerably increased near the supports of the longitudinal, which is mainly due to increased shear stresses
442
W. Fricke, H. Paetzold
- - the local stresses are generally lower in the - - the weld toe on the plate is always the elliptical scallop, while the rounded plate scallop becomes more critical near the itudinal. The permissible local stress range stress cycles is about
elliptical scallop critical position of the edge of the rectangular supports of the long-
Aap given in rules 18 for wave-induced
- - 420 N/mm 2 for structural stresses at weld toes and 440 N/mm 2 for plate edges of normal hull structural steel cut with machine quality, -
-
if a positive effect of mean stress is neglected. Under the assumption that the given nominal stresses correspond to the highest stress range in the spectrum, the elliptical scallop would be permitted over the full length of the longitudinal, while the rectangular scallop would not be allowed near the transverse webs.
5 CONCLUSIONS Scallops in the structural components of ships have to be assessed with respect to fatigue strength to avoid cracks during service life. Difficulties in the assessment arise due to the variety of geometric configurations and due to the loading pattern of the structural member characterized by various combinations of normal, shear, bending and local pressure loads. Local approaches such as the notch stress approach for plate edges and the structural or hot-spot stress approach for welded connections have reached a stage which allows their application to the problems mentioned. Stress concentration factors have shown to be very useful in practical application. From the theoretical and experimental investigation of several scallops, the following conclusions can be drawn: - - a fairly good performance can be expected from scallops which are arranged at a sniall distance away from the plate allowing continuous welding between web and plating as shown in Fig. lc. Optimal are elliptical shapes which can be analysed with sufficient accuracy using the stress concentration factors for the cut-out in an infinite plate; if scallops interrupting the fillet weld are arranged, for instance, at a stiffener butt (Fig. la) or in order to apply intermittent welding (Fig. l b), the type of loading determines the performance of elliptical and rectangular scallops with rounded corners. Under axial -
-
Fatigue strength assessment of scallops
-
-
443
loading, both types show similar local stresses and fatigue behaviour, while the elliptical scallop is superior under combined loading, particularly if shear stresses are acting in the web; half-round scallops at stiffener butts show a better fatigue behaviour under axial loading than elliptical and rectangular cut-outs. The size should be kept small especially in stiffeners with relatively low depth.
Generally it has to be emphasized that the fatigue problems, particularly in welded connections, may grow with increasing optimization and utilization of the structural members in ships and more frequent use of higher-tensile steel. The fatigue strength assessment by refined methods, as described in this paper, are expected to contribute to the design of fatigueresistant and durable ship structures.
ACKNOWLEDGEMENT The investigations described were financially supported in several research projects by the German Institutions DFG, FDS and BMFT, which is highly appreciated. The authors gratefully acknowledge the fruitful discussions with Professor H. Petershagen and the assistance of Mr S. Pohl in carrying out the systematic finite element calculations.
REFERENCES 1. Youshaw, R., Nondestructive inspection of longitudinal stiffener butt welds in commercial vessels. Ship Structure Committee, Report No. 295/ 1980. 2. Munse, W. H., Fatigue criteria for ship structure details. SNAME Extreme Loads Response Syrup., Arlington, 1981. 3. Gurney, T. R & Maddox, S. J., A re-analysis of fatigue data for welded joints in steel. Rep. E/44/72, The Welding Institute, Abington, 1972. 4. NN., Design recommendations for cyclic loaded welded steel structures. Welding in the World, 20 (1982), 153-165. 5. NN., Eurocodc 3, Design of steel structures. European Committee for Standardization, April 1992. 6. Fricke, W., Investigation of the fatigue strength of bulb plate stiffener joints. Schiff& Hafen, 38 (1986), 50-54 (in German). 7. Paetzold, H., Fatigue strength of ship structural details. FDS-Report No. 159/1985, Forschungszentrum des Deutschen Schiffbaus, Hamburg 1985 (in German). 8. Paetzold, H., Design and strength of sectional joints of HP sections under consideration of manufacturing influences. FDS-Report 222/1990,
444
W. Fricke, H. Paetzold
Forschungszentrum des Deutschen Schiffbaus, Hamburg 1990 (in German). 9. Fricke, W. & Paetzold, H., Application of the cyclic strain approach to the fatigue failure of ship structural details. J. Ship Research, 31 (1987), 177185. 10. Paetzold, H., Damage parameter--life curves for normal and high strength steels. Report No. 507, Institut ffir Schiffbau, Hamburg 1990 (in German). 11. Neuber, H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law. J. Applied Mechanics, Trans. ASME, 28 (1961), 544-550. 12. Seeger, T. & Heuler, P., Generalized application of Neuber's rule. J. Testing and Evaluation, 8 (1980), 199-204. 13. Smith, K. N., Watson, P. & Topper, T. H., A stress-strain function for the fatigue of metals. J. Materials, 5 (1970), 767-778. 14. Matoba, M. et al., Evaluation of fatigue strength of welded steel structures-- hull's members, hollow section joints, piping and vessel joints. IIW-Doc. XIII-1082-83, International Institute of Welding 1983. 15. Fricke, W. & Petershagen, H., Detail design of welded ship structures based on hot-spot stresses. In Practical Design of Ships and Mobile Units, ed. by J.B. Caldwell and G. Ward, Elsevier Appl. Science Publ., London and New York 1992. 16. Radaj, D., Notch stress proof for fatigue resistant welded structures. IIWDoc. XIII- 1157-85, International Institute of Welding. 17. Olivier, R. & Seeger, T., The effect of weld toe angle on fatigue strength of welded joints--unified evaluation of published test results. FG Werkstoffmechanik of TH Darmstadt, Rep. FF-2, 1989 (in German). 18. Germanischer Lloyd: Rules and Regulations, I - - S h i p Technology, Part 1 Seagoing Ships, Chapter 1--Hull Structures. Hamburg 1992.
A P P E N D I X : R E G R E S S I O N F O R M U L A E F O R STRESS CONCENTRATION FACTORS
a) Rectangular scallop with rounded corners Geometry parameters and validity ranges (Fig. 7a).
6 = b/d = b/tg
p = r/d 0 = tg/t
(2.0-4.0) (4.0-12.0)
Theoretical stress concentration factor Kt at position 1: 5 --
G
i=l
(0.25-0.5) (1.0-2.0)
445
Fatigue strength assessment of scallops Ci : X i I • (~ xi2 . fl xi4 . p xi6 . oxi8
f l (~P) = 1 + cos (2¢p)
j~(~p) = sin (2¢p)
j~(cp) = 1 - cos (2¢p)
fs(~P) = sin (4¢p)
f3(~P) = 1 - cos (4¢p) Coefficients LC
Xil, i
1
2 3 4 5 1
2 3 4 5
.e~ri2 ...
f o r l o a d cases (LC) 1-5:
Xil
Xi2
Xi4
Xi6
Xi8
0.9931 -0.5390 -0.0687 0-3026 0-1541
-0.099 0.0 0.0 -0-045 0.0
0.0 0.026 0.0 0-0 -0.070
-0.180 0.181 0.141 -0.584 0.229
-0-059 -0-030 -0.040 -0-176 0-043
-0.5408 -0.1676 -0.4421 -1.6739 0.0187
0.792 0.198 0-525 0.346 -0.163
-0.127 0.264 -0.731 0.074 0.187
-0.643 -1.114 -0.248 -0.146 0.163
-0.037 -0.123 -0-190 0.0 0.0
-0.7840 0-0930 0-4920 -0.1009 -0-2328
0.0 0.244 0.0 0.290 0.185
-0.067 -0.138 0.057 -0.273 -0-098
-0-159 -0.941 0.753 -0.849 0.203
-0.062 0-0 -0-044 0-0 -0-089
-0.8634 14.3126 -0.0914 6.1520 -2.3106
-0.783 0.597 0.909 0-808 -0.649
1.612 0-640 1.540 0.227 0.247
0.496 -0.063 -1-157 -1.258 -0.868
0.631 1.101 1.041 0.998 0.853
-0.7053 2.1792 -0.1726 0-4467 0.0394
-0.673 0.604 0.388 0.585 -0.772
0-358 0-160 0-465 0.060 1.204
0-153 -0.048 -0.249 -1-088 0.0
0.318 0-821 0.345 0.986 -1.014
H o t - s p o t stress c o n c e n t r a t i o n f a c t o r Ks at position 2 a n d 3:
Ks =: X1
• fix2 . (fl(x4 +xl0 .~i+Xll .fl+x,2 "p"}-XI3 .'0) +
Coefficients X1, X2 . . . for l o a d cases 1-5:
X5). ( p X 6 +
X 7 ) . 0x8 _~_X14.
W. Fricke, H. Paetzold
446
Pos. LC 1 2 3 4 5
Xi
X2
X4
-1.0435 0.327 0.046 2.3649 0.448 -0.728 -1.5138 -0.418 0.226 3.6046 -0.888 2.534 3.1806 0.908 0.476
1 2 3 4 5
1.6566 1.7948 0.4094 7.6088 0.3360
3(5
X6
X7
0.0 0.0 0.0 0.0 0.0
-0.168 -0.048 0.352 0.764 -0.057
0.731 1.918 0.601 2.299 0.0
-0.208 0.156 0.0 0.0 -0.327 -0.224 0-0 -0.214 -0.691 0.663 0-0 0.0 0.011 1.468 5.689 -0.502 -0.055 1.816 -0.771 0.0
0.0 0.0 0.0 0.0 0.0
Xa
Xto
Xll
Xl2
X13
-0-543 -0.048 0.0 -0.108 0.067 1.827 -0.076 0.0 0.0 0.284 0.111 -1.241 0.311 0.057 0.0 -0.366 -0.112 1-213 2.996 0.177 -0.041 0.322 -0.628 0.0 0.298 -0.070 0.0 0.340 0.102 0.0 -0.134 -0.569 0.054 0.774 0.614
0-0 0.0 0.0 0.0 0-0 0.102 0.0 0.259 0.0 0.637 0.0 0-0 0-0 0.0 0.0 0.0 -0.010 0.877 -0.149 0.0 0.0 0.0 -0.017 -0.125 -2.007
b) Elliptical scallop G e o m e t r y parameters and validity ranges (Fig. 7b)
6 = b/d fl = b'/tg
0 = tg/t
(1.0 ... 4.0) ( 2 . 6 7 . . . 8.0 2.85...8.57 3.20...9.60 (1.0 ... 2.0)
for 6 = 1 forfi=2 forfi=4)
Theoretical stress concentration factor Kt(q~) at position 1. 5
Kt(tp) = ~
C~ . f~ (q~)
i=1
Ci : X i 1 . ((~ (xi2+ x,8,6+ xi9 "fl-~- XilO "6) .j[_ X i 3 ) " (fl Xi4 + X i 5 ) "ox,6
3q (tp) = 1 + cos (2~p)
J~(q~) = sin (2¢p)
J~(tp) = 1 - cos (2tp)
fs(tp) = sin (4q~)
f3(tp) = 1 - cos(4tp) Coefficients
X/I , Xi2
...
Xt4
for load cases (LC) 1-5
Fatigue strength assessment of scallops LC 1
~l
i
~2
~3
~4
~5
~6
~8
~9
~10
0.0 0.0 0.0 0.0 0.0
0-0 0-092 0-0 0-0 -0-359
0-0 0.0 0.0 0.0 0.0
-0-046 0-0 0-0 -0-295 0-335
0-0 0.0 0.0 0.0 0.0
0-0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.853 0.0 -0-123 0.215 -0.775 0-696 0-048 -0.631 -0.046 0-196 0-830 0.0 1.011 -0.840 -0.176
0.0 0.0 0.0 0-0 0.0
0-0 0.0 0.0 0.0 -0-113 0.0 0.0 -0.037 0-038 0.0 -0.056 0.065 0-082 -0.172 0.0 -0.178 0-151 -0.070 0.0 -0.046
0.0 0.0 0.0 0-0 0-0
-0-050 -0-052 0-0 0-0 0-057
1-3153 -0.533 -0.5459 -0-622 O.1132 0.748 0-3129 0.439 0.1701 -1.861 -0.3693 -0.6740 -1-7232 -1-1852 2-0244
447
-0.9863 -0.405 0.1056 0.0 0.3656 -1.404 -0.0905 1.277 -0.3752 -0.312
0.0 0.0 0.0 0.0 0.0
-2.3085 0.751 1.770 7.8350 3.027 5-830 -4-2153 3.535 2 - 3 5 1 3-8262 -0.156 3-299 0.9992 0.813 -0-090
-0.055 -0.248 0.050 -0.216 0.080 0.816 0-329 0.382 0-415 1-566
0-0 0-0 0-0 0.0 0.0
-0.5414 0.452 0 - 0 6 1 0-359 0.239 1-2468 0.972 0.0 0-350 1 . 2 6 3 -0-9612 1.303 -0.523 0.283 0.554 0.8340 -0-101 0.335 -0.100 0.463 0.5290 0-602 -0.701 0.786 -0.223
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0-0 0-0 0-0
0.893 -0.338 0.0 0-201 0-987 -0.373 0.0 0-0 0~982 -0.402 0-0 0.0 1.206 0.287 -0-042 0.018 1.001 0.0 0-0 -0-078 0.678 -0.066 0.937 -0.073 0.965 -0.078 1.047 0-126 0.585 0-0
0-0 0-0 0-0 0.0 0.0
-0.080 -0.055 -0.068 0.0 -0.094
Hot-spot stress concentration factor Ks at position 3. Ks = X1 • (6(x2+x,o.O) + X 3 ) . (fix4 + X s ) . 0 x6 + Xll Coefficients X1, Xz ... for load cases 1-5: LC
Xt
X2
X3
X4
X5
)(6
Xlo
XII
1 2 3 4 5
1-7941 1-8846 1-8249 5-1097 0.1749
-0.403 -0.318 -0.250 0.315 0.102
0.0 0.0 0.0 4.410 0.412
0.058 0.300 0.188 1-064 1-749
0.0 0.0 0.0 4.349 3.697
-0-400 -0-701 -0.064 0.323 -0.117
0.0 0.0 0.0 -0.777 -0.642
0.769 0.573 -0.928 0.0 -2.953