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International Journal of Fatigue 23 (2001) 865–876 www.elsevier.com/locate/ijfatigue
A structural stress deﬁnition and numerical implementation for fatigue analysis of welded joints
P. Dong
*
Center for Welded Structures Research, Battelle, Columbus, OH 43016-2693, USA Received 10 December 2000; received in revised form 11 May 2001; accepted 12 June 2001
Abstract A mesh-size insensitive structural stress deﬁnition is presented in this paper. The structural stress deﬁnition is consistent with

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International Journal of Fatigue 23 (2001) 865–876www.elsevier.com/locate/ijfatigue
A structural stress deﬁnition and numerical implementation forfatigue analysis of welded joints
P. Dong
*
Center for Welded Structures Research, Battelle, Columbus, OH 43016-2693, USA
Received 10 December 2000; received in revised form 11 May 2001; accepted 12 June 2001
Abstract
A mesh-size insensitive structural stress deﬁnition is presented in this paper. The structural stress deﬁnition is consistent withelementary structural mechanics theory and provides an effective measure of a stress state that pertains to fatigue behavior of welded joints in the form of both membrane and bending components. Numerical procedures for both solid models and shell orplate element models are presented to demonstrate the mesh-size insensitivity in extracting the structural stress parameter. Conven-tional ﬁnite element models can be directly used with the structural stress calculation as a post-processing procedure. To furtherillustrate the effectiveness of the present structural stress procedures, a collection of existing weld S-N data for various joint typeswere processed using the current structural stress procedures. The results strongly suggests that weld classiﬁcation based S-N curvescan be signiﬁcantly reduced into possibly a single master S-N curve, in which the slope of the S-N curve is determined by therelative composition of the membrane and bending components of the structural stress parameter. The effects of membrane andbending on S-N behaviors can be addressed by introducing an equivalent stress intensity factor based parameter using the structuralstress components. Among other things, the two major implications are: (a) structural stresses pertaining to weld fatigue behaviorcan be consistently calculated in a mesh-insensitive manner regardless of types of ﬁnite element models; (b) transferability of weldS-N test data, regardless of welded joint types and loading modes, can be established using the structural stress based parameters.
2001 Elsevier Science Ltd. All rights reserved.
Keywords:
Structural stress; Finite element analysis; Welded joints; Fatigue; Notch stress; Stress concentration; Mesh-size sensitivity
1. Introduction
At present, fatigue design of welded structures isprimarily based on a nominal stress or hot spot stressapproach with a series of classiﬁed weld S-N curves [1–4], although a local stress or initiation-based fatigue lifeapproaches [5,6] provide an alternative method forfatigue life predictions of welded joints. Without goinginto a detailed discussion of the merits of the two differ-ent approaches, the premise of this paper is that the nom-inal stress or hot spot stress approach has been wellaccepted by major industries, and recommended bynumerous national and international codes and standards(e.g. [3,4]). A series of S-N curves corresponding to eachclass of joint types and loading mode were well docu-mented in some of the codes and standards. With such an
* Tel.:
+
1 614-424-4908; fax:
+
1 614-424-3457.
E-mail address:
dongp@battelle.org (P. Dong).
0142-1123/01/$ - see front matter
2001 Elsevier Science Ltd. All rights reserved.PII: S0142-1123(01)00055-X
approach, nominal stresses with appropriate geometric orstructural stress concentration factor (SCF) for a parti-cular class of joints must be determined against the cor-responding S-N curve to calculate fatigue damage. Twocritical issues remain unresolved in this context. First,both nominal stresses and geometric SCFs cannot bereadily calculated from ﬁnite element models due to theirstrong dependence on element size at weld disconti-nuities. Secondly, the selection of an appropriate S-Ncurve for damage calculation can be very subjective,since the weld classiﬁcations were based on not only joint geometry, but also dominant loading mode.There are numerous on-going international efforts toaddress the above two issues. A majority of the efforthas been on developing effective hot-spot stress extra-polation procedures and an ability to correlate variousavailable S-N curves (e.g., [7,8]). However, the extrapol-ation procedures available to date still lack consistencyfor general applications [9]. This is in part due to thefact that extrapolation procedures are based on the
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P. Dong / International Journal of Fatigue 23 (2001) 865–876
assumption that the surface stresses on a structural mem-ber provides an indication of the stress state at a weldfatigue prone location, such as a weld toe. This underly-ing assumption may become questionable if the struc-tural member is not a dominant load-transfer member ina joint. Under such circumstances, the surface stressesat some distance away from a weld toe may not be rel-evant to the stress state of concern. In addition, a refer-ence nominal stress in such a structural member may notbe readily identi
ﬁ
ed for conventional SCF calculations.Among the various extrapolation procedures proposedin the open literature (e.g., [7,8]), a typical one is basedon a linear extrapolation from stress values at both 0.4
t
and 1
t
from a weld toe [8,9], as shown in Fig. 1, where
t
represents the plate thickness of a structural member.The drawback in such an extrapolation scheme becomesimmediately clear in view of Fig. 1 in which some of the well-studied joints in the research community areillustrated. The stress concentration behaviors can becategorized into two types [10]: one is rather localizedstress concentration behavior (Type I) at weld toe, whilethe other is more global in length-scale (Type II). Inorder to correlate the fatigue behavior in various jointtypes, stress concentration behavior at the weld toe of various joint types must be captured. However, as shownin Fig. 1(b), any stress concentration effects in Type I
Fig. 1. Stress concentration behavior in welded joints.
joints cannot be captured in this extrapolation scheme,resulting in little stress concentration effects from thiscalculation. On the other hand, for Type II joints, Fig.1(b) shows that extrapolation from the two referencepositions (open circles) should provide some indicationof the concentrations at the weld toes. Then, one obviousquestion is if such calculation procedures provide areliable stress concentration measurement or hot spotstresses. As discussed in Neimi [9], the results are oftenquestionable due to the fact that these stresses can bestrongly dependent on mesh-size and loading modes.To improve the S-N curve approach (using eithernominal stresses or hot spot stresses) for welded struc-tures, a relevant stress parameter must satisfy the twobasic requirements: (a) mesh-size insensitivity in
ﬁ
niteelement solutions; (b) ability to differentiate stress con-centration effects in different joint types (e.g., butt jointsversus T-
ﬁ
llet cruciform joints) in welded structures. Inthe following, such a stress parameter is presented andthe corresponding
ﬁ
nite element procedures using bothsolid and shell element models are given. The validationof such a structural stress parameter is demonstrated byreprocessing a series of existing S-N data for joint typeslisted in Fig. 1(a).
2. Structural stress deﬁnition and formulation
As discussed in Dong [10] and Dong et al. [11], astructural stress de
ﬁ
nition that follows elementary struc-tural mechanics theory can be developed with follow-ing considerations:(a) It can be postulated that for a given local through-thickness stress distribution as shown in Fig. 2(a)obtained from a
ﬁ
nite element model, there exists acorresponding simple structural stress distribution asshown in Fig. 2(b), in the form of membrane andbending components that are equilibrium-equivalentto the local stress distributions in Fig. 2(a).(b) The structural stress distribution must satisfy equi-librium conditions within the context of elementarystructural mechanics theory at both the hypotheticalcrack plane [e.g., at weld toe in Fig. 2(a)] and anearby reference plane, on which local stress distri-butions are known a priori from typical
ﬁ
niteelement solutions. The uniqueness of such a struc-tural stress solution can be argued by consideringthe fact that the compatibility conditions of the cor-responding
ﬁ
nite element solutions are maintained atthis location in such a calculation.(c) While local stresses near a notch are mesh-sizesensitive due to the asymptotic singularity behavioras a notch position is approached, the imposition of the equilibrium conditions in the context of elemen-tary structural mechanics with respect to this regime
867
P. Dong / International Journal of Fatigue 23 (2001) 865
–
876
Fig. 2. Structural stresses de
ﬁ
nition for through-thickness fatiguecrack. (a) Local through-thickness normal and shear stress at weld toe,(b) Structural stress de
ﬁ
nition at weld toe.
should eliminate or minimize the mesh-size sensi-tivity in the structural stress calculations. This is dueto the fact that the local stress concentration closeto a notch is dominated by self-equilibrating stressdistribution, as discussed by Niemi [9].Along this line, the following typical situations are con-sidered:
2.1. Solid model with monotonic through-thick stressdistributions
As shown in Fig. 2(a), the stress distribution at the T-
ﬁ
llet weld toe is assumed to exhibit a monotonicthrough-thickness distribution with the peak stressoccurring at the weld toe. It should be noted that in typi-cal
ﬁ
nite element based stress analysis, the stress valueswithin some distance from the weld toe can change sig-ni
ﬁ
cantly as the
ﬁ
nite element mesh design changes(e.g., [9]), referred to as mesh-size sensitivity in thispaper. The corresponding statically equivalent structuralstress distribution is illustrated in Fig. 2, in the form of a membrane component (
s
m
) and bending component(
s
b
), consistent with elementary structural mechanicsde
ﬁ
nition:
s
s
s
m
s
b
. (1)The normal structural stress (
s
s
) is de
ﬁ
ned at a locationof interest such as Section A
–
A at the weld toe in Fig.2(b) with a plate thickness of
t
. In the above, the trans-verse shear (
t
m
) of the structural stress components [tobe calculated based on local transverse stress distributionfrom Fig. 2(a)] is not considered in the structural stressde
ﬁ
nition in the present discussions. In practice, thetransverse shear component can play an important role incontrolling crack propagation path if the remote loadingimposes a signi
ﬁ
cant transverse shear at the weld toe.A second reference plane can be de
ﬁ
ned along SectionB
–
B in Fig. 3(a), along which both local normal andshear stresses can be directly obtained from a
ﬁ
niteelements solution. The distance,
d
, represents the dis-tance between Sections A
–
A and B
–
B (in local
x
direction) at the weld toe. For convenience, a row of elements with same length of
d
can be used in the
ﬁ
niteelement model. By imposing equilibrium conditionsbetween Sections A
–
A and B
–
B, the structural stresscomponents
s
b
and
s
m
must satisfy the following con-ditions:
s
m
1
t
t
0
s
x
(
y
)
·
d
y
(2)
s
m
·
t
2
2
s
b
·
t
2
6
t
0
s
x
(
y
)
·
y
·
d
y
d
t
0
t
xy
(
y
)
·
d
y
. (3)Eq. (2) represents the force balances in
x
direction,evaluated along B
–
B and Eq. (3) represents moment bal-
Fig. 3. Structural stresses calculation procedure for through-thicknessfatigue crack.
868
P. Dong / International Journal of Fatigue 23 (2001) 865
–
876
ances with respect to Section A
–
A at
y
=
0. The integralterm on the right-hand side of Eq. (3) represents thetransverse shear force as an important component of thestructural stress de
ﬁ
nition. It is then follows that if element size (
d
) is small or transverse shear is negligible,the integral representations of
s
b
and
s
m
in Eqs. (2) and(3) can be directly evaluated at Section A
–
A in Fig. 3(a).
2.2. Solid model with ﬁnite fatigue crack depth
Often, a fatigue crack of a
ﬁ
nite depth is used as afatigue failure criterion (e.g. [9]), the correspondingstructural stress can be then de
ﬁ
ned in a similar mannerto that in Fig. 3. In Fig. 4, the depth of the fatigue crack
Fig. 4. Structural stresses de
ﬁ
nition for partial thickness (
t
1
) fatiguecrack weld toe. (a) Local normal stress distribution, (b) Structuralstress de
ﬁ
nition.
at failure is assumed to be
t
1
. Without losing generality,the structural stress procedures [10,11] can be effectivelydemonstrated using the example in Fig. 4. Note that forconvenience, the local
y
coordinate is de
ﬁ
ned as shownin Fig. 4(a), different from that in Fig. 3, At a horizontalcross section of depth
t
1
from the top surface, both nor-mal stress (
s
y
) and shear stress (
t
yx
) are present in gen-eral.By imposing equilibrium conditions between SectionsA
–
A and B
–
B, as well as the horizontal cross sectionin between, it can be shown that the structural stresscomponents (
s
b
and
s
m
) must satisfy the following equa-tions:
s
m
1
t
1
t
1
0
s
x
(
y
)
·
d
y
1
t
1
d
0
t
yx
(
x
)
·
d
x
(4)
s
m
·
t
21
2
s
b
·
t
21
6
t
1
0
s
x
(
y
)
·
y
·
d
y
d
t
1
0
t
xy
(
y
)
·
d
y
(5)
d
0
s
y
(
x
)
·
x
·
d
x
.Additionally, by considering the bottom element(spanning
t
–
t
1
) between Sections A
–
A and B
–
B, it canbe shown that
s
m
s
b
s
m
s
b
(6)
s
m
1
t
−
t
1
t
t
1
0
s
x
(
y
)
·
d
y
1
t
−
t
1
d
0
t
yx
(
x
)
·
d
x
. (7)As in Eqs. (2) and (3), the integrals in the above can beaccurately evaluated using
ﬁ
nite element solutions atcross sections away from the geometric discontinuity.The structural stress components
s
b
and
s
m
, including
s
b
and
s
m
can then be solved.
2.3. Solid model with non-monotonic through-thickness stress distributions
In thick section joints and some joint con
ﬁ
gurations,non-monotonic through-thickness or in-plane distri-butions may develop, as shown in Fig. 5(a). The corre-sponding structural stress de
ﬁ
nition can be consistentlyde
ﬁ
ned in a similar manner as that in Fig. 4. Note thatthe parameter
t
1
can be determined based on the positionat which the transverse shear stress changes direction, if there is no speci
ﬁ
ed crack depth as a failure criterion.Eqs. (4, 5) and (7) can be directly used for structuralstress calculations except a minor modi
ﬁ
cation of Eq.(6) as follows:
s
m
s
b
s
m
s
b
. (8)

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