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A structural stress definition and numerical implementation for fatigue analysis of welded joints
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 elementary 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 or plate element models are presented to demonstrate the mesh-size insensitivity in extracting the structural stress parameter. Conventional finite element models can be directly used with the structural stress calculation as a post-processing procedure. To further illustrate the effectiveness of the present structural stress procedures, a collection of existing weld S-N data for various joint types were processed using the current structural stress procedures. The results strongly suggests that weld classification based S-N curves can be significantly reduced into possibly a single master S-N curve, in which the slope of the S-N curve is determined by the relative composition of the membrane and bending components of the structural stress parameter. The effects of membrane and bending on S-N behaviors can be addressed by introducing an equivalent stress intensity factor based parameter using the structural stress components. Among other things, the two major implications are: (a) structural stresses pertaining to weld fatigue behavior can be consistently calculated in a mesh-insensitive manner regardless of types of finite element models; (b) transferability of weld S-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 is primarily based on a nominal stress or hot spot stress approach with a series of classified weld S-N curves [1– 4], although a local stress or initiation-based fatigue life approaches [5,6] provide an alternative method for fatigue life predictions of welded joints. Without going into a detailed discussion of the merits of the two different approaches, the premise of this paper is that the nominal stress or hot spot stress approach has been well accepted by major industries, and recommended by numerous national and international codes and standards (e.g. [3,4]). A series of S-N curves corresponding to each class of joint types and loading mode were well documented in some of the codes and standards. With such an
* Tel.: +1 614-424-4908; fax: +1 614-424-3457. E-mail address: [email protected] (P. Dong).
approach, nominal stresses with appropriate geometric or structural stress concentration factor (SCF) for a particular class of joints must be determined against the corresponding S-N curve to calculate fatigue damage. Two critical issues remain unresolved in this context. First, both nominal stresses and geometric SCFs cannot be readily calculated from finite element models due to their strong dependence on element size at weld discontinuities. Secondly, the selection of an appropriate S-N curve 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 to address the above two issues. A majority of the effort has been on developing effective hot-spot stress extrapolation procedures and an ability to correlate various available S-N curves (e.g., [7,8]). However, the extrapolation procedures available to date still lack consistency for general applications [9]. This is in part due to the fact that extrapolation procedures are based on the
0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 0 5 5 - X
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P. Dong / International Journal of Fatigue 23 (2001) 865–876
assumption that the surface stresses on a structural member provides an indication of the stress state at a weld fatigue prone location, such as a weld toe. This underlying assumption may become questionable if the structural member is not a dominant load-transfer member in a joint. Under such circumstances, the surface stresses at some distance away from a weld toe may not be relevant to the stress state of concern. In addition, a reference nominal stress in such a structural member may not be readily identified for conventional SCF calculations. Among the various extrapolation procedures proposed in the open literature (e.g., [7,8]), a typical one is based on a linear extrapolation from stress values at both 0.4t and 1t 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 becomes immediately clear in view of Fig. 1 in which some of the well-studied joints in the research community are illustrated. The stress concentration behaviors can be categorized into two types [10]: one is rather localized stress concentration behavior (Type I) at weld toe, while the other is more global in length-scale (Type II). In order to correlate the fatigue behavior in various joint types, stress concentration behavior at the weld toe of various joint types must be captured. However, as shown in Fig. 1(b), any stress concentration effects in Type I
joints cannot be captured in this extrapolation scheme, resulting in little stress concentration effects from this calculation. On the other hand, for Type II joints, Fig. 1(b) shows that extrapolation from the two reference positions (open circles) should provide some indication of the concentrations at the weld toes. Then, one obvious question is if such calculation procedures provide a reliable stress concentration measurement or hot spot stresses. As discussed in Neimi [9], the results are often questionable due to the fact that these stresses can be strongly dependent on mesh-size and loading modes. To improve the S-N curve approach (using either nominal stresses or hot spot stresses) for welded structures, a relevant stress parameter must satisfy the two basic requirements: (a) mesh-size insensitivity in finite element solutions; (b) ability to differentiate stress concentration effects in different joint types (e.g., butt joints versus T-fillet cruciform joints) in welded structures. In the following, such a stress parameter is presented and the corresponding finite element procedures using both solid and shell element models are given. The validation of such a structural stress parameter is demonstrated by reprocessing a series of existing S-N data for joint types listed in Fig. 1(a).
2. Structural stress definition and formulation As discussed in Dong [10] and Dong et al. [11], a structural stress definition that follows elementary structural mechanics theory can be developed with following considerations:
Fig. 1.
Stress concentration behavior in welded joints.
(a) It can be postulated that for a given local throughthickness stress distribution as shown in Fig. 2(a) obtained from a finite element model, there exists a corresponding simple structural stress distribution as shown in Fig. 2(b), in the form of membrane and bending components that are equilibrium-equivalent to the local stress distributions in Fig. 2(a). (b) The structural stress distribution must satisfy equilibrium conditions within the context of elementary structural mechanics theory at both the hypothetical crack plane [e.g., at weld toe in Fig. 2(a)] and a nearby reference plane, on which local stress distributions are known a priori from typical finite element solutions. The uniqueness of such a structural stress solution can be argued by considering the fact that the compatibility conditions of the corresponding finite element solutions are maintained at this location in such a calculation. (c) While local stresses near a notch are mesh-size sensitive due to the asymptotic singularity behavior as a notch position is approached, the imposition of the equilibrium conditions in the context of elementary structural mechanics with respect to this regime
P. Dong / International Journal of Fatigue 23 (2001) 865–876
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a membrane component (sm) and bending component (sb), consistent with elementary structural mechanics definition: ss⫽sm⫹sb.
(1)
The normal structural stress (ss) is defined at a location of interest such as Section A–A at the weld toe in Fig. 2(b) with a plate thickness of t. In the above, the transverse shear (tm) of the structural stress components [to be calculated based on local transverse stress distribution from Fig. 2(a)] is not considered in the structural stress definition in the present discussions. In practice, the transverse shear component can play an important role in controlling crack propagation path if the remote loading imposes a significant transverse shear at the weld toe. A second reference plane can be defined along Section B–B in Fig. 3(a), along which both local normal and shear stresses can be directly obtained from a finite elements solution. The distance, d, represents the distance 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 finite element model. By imposing equilibrium conditions between Sections A–A and B–B, the structural stress components sb and sm must satisfy the following conditions:
冕 t
1 sm⫽ sx(y)·dy t
(2)
0
Fig. 2. Structural stresses definition for through-thickness fatigue crack. (a) Local through-thickness normal and shear stress at weld toe, (b) Structural stress definition at weld toe.
冕 t
0
should eliminate or minimize the mesh-size sensitivity in the structural stress calculations. This is due to the fact that the local stress concentration close to a notch is dominated by self-equilibrating stress distribution, as discussed by Niemi [9].
冕 t
t2 t2 sm· ⫹sb· ⫽ sx(y)·y·dy⫹d txy(y)·dy. 2 6
(3)
0
Eq. (2) represents the force balances in x direction, evaluated along B–B and Eq. (3) represents moment bal-
Along this line, the following typical situations are considered: 2.1. Solid model with monotonic through-thick stress distributions As shown in Fig. 2(a), the stress distribution at the Tfillet weld toe is assumed to exhibit a monotonic through-thickness distribution with the peak stress occurring at the weld toe. It should be noted that in typical finite element based stress analysis, the stress values within some distance from the weld toe can change significantly as the finite element mesh design changes (e.g., [9]), referred to as mesh-size sensitivity in this paper. The corresponding statically equivalent structural stress distribution is illustrated in Fig. 2, in the form of
Fig. 3. Structural stresses calculation procedure for through-thickness fatigue crack.
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ances with respect to Section A–A at y =0. The integral term on the right-hand side of Eq. (3) represents the transverse shear force as an important component of the structural stress definition. It is then follows that if element size (d) is small or transverse shear is negligible, the integral representations of sb and sm 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 finite depth is used as a fatigue failure criterion (e.g. [9]), the corresponding structural stress can be then defined in a similar manner to that in Fig. 3. In Fig. 4, the depth of the fatigue crack
at failure is assumed to be t1. Without losing generality, the structural stress procedures [10,11] can be effectively demonstrated using the example in Fig. 4. Note that for convenience, the local y coordinate is defined as shown in Fig. 4(a), different from that in Fig. 3, At a horizontal cross section of depth t1 from the top surface, both normal stress (sy) and shear stress (tyx) are present in general. By imposing equilibrium conditions between Sections A–A and B–B, as well as the horizontal cross section in between, it can be shown that the structural stress components (sb and sm) must satisfy the following equations:
冕
冕
t1
d
1 1 sm⫽ sx(y)·dy⫹ tyx(x)·dx t1 t1
(4)
0
0
冕
冕
t1
t1
t21 t21 sm· ⫹sb· ⫽ sx(y)·y·dy⫹d txy(y)·dy 2 6 0
(5)
0
冕 d
⫹ sy(x)·x·dx. 0
Additionally, by considering the bottom element (spanning t–t1) between Sections A–A and B–B, it can be shown that sm⫺sb⫽sm⬘⫹sb⬘
冕
t⫺t1
1 sm⬘⫽ t−t1
(6)
冕 d
1 sx(y)·dy⫺ t (x)·dx. t−t1 yx
0
(7)
0
As in Eqs. (2) and (3), the integrals in the above can be accurately evaluated using finite element solutions at cross sections away from the geometric discontinuity. The structural stress components sb and sm, including sb⬘ and sm⬘ can then be solved. 2.3. Solid model with non-monotonic throughthickness stress distributions
Fig. 4. Structural stresses definition for partial thickness (t1) fatigue crack weld toe. (a) Local normal stress distribution, (b) Structural stress definition.
In thick section joints and some joint configurations, non-monotonic through-thickness or in-plane distributions may develop, as shown in Fig. 5(a). The corresponding structural stress definition can be consistently defined in a similar manner as that in Fig. 4. Note that the parameter t1 can be determined based on the position at which the transverse shear stress changes direction, if there is no specified crack depth as a failure criterion. Eqs. (4, 5) and (7) can be directly used for structural stress calculations except a minor modification of Eq. (6) as follows: sm⫺sb⫽sm⬘⫺sb⬘.
(8)
P. Dong / International Journal of Fatigue 23 (2001) 865–876
Fig. 5. Structural stresses definition for non-monotonic throughthickness stress distribution.
A special case in this category is that the joint configuration and loading are symmetric with respect the neutral axis of the horizontal member in Fig. 5 such as Joints B and C in Fig. 1(a). In this case, the shear stress on the cross section along the symmetry line is zero as shown in Fig. 6. Thus, Eqs. (4) and (5) can be directly used to calculate the structural stress components sb and sm, by substituting t1=t/2 and tyx=0. 2.4. Shell/plate element models Shell or plate element models of complex structures are widely used for performing stress analyses for fatigue evaluation. The underlying principles of structural stress calculations are the same between the
869
Fig. 6. Structural stresses procedure for non-monotonic throughthickness stress distribution: (a) symmetry with respect to plate midthickness (t1=t/2); (b) structural stress definition.
shell/plate and solid element models, while special consideration will be given to shell/plate structural theory and its finite element implementation. It should be noted that shell/plate element solutions at geometric discontinuities (e.g., weld toe) converge only to the solutions described by the corresponding shell/plate theory. Consequently, the local stresses at a weld toe in actual structures are forced to obey the shell/plate theory used in the finite element model and are not the structural stresses sought in Eq. (1), even though a linear throughthickness stress distribution is maintained. Two general methods for structural stress calculations are presented below:
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P. Dong / International Journal of Fatigue 23 (2001) 865–876
2.4.1. Using stresses and stress resultants As shown in Fig. 7, stresses and nodal quantities from shell or plate element models are often defined in a global coordinate system (x, y, z), depending on the finite element codes used. Given the definition of the structural stress components in Eq. (1), it is the local coordinate system (x⬘, y⬘, z⬘) that is convenient for calculating the structural stresses with respect to a weld, with local x⬘ and y⬘ being perpendicular and parallel to the weld direction, respectively. Consistent with the solid element model approach (e.g., see Fig. 3), three components of the stress resultants (sectional forces and moments), i.e., fx⬘, fz⬘, and my⬘, at Section B–B in Fig. 7 can be used to calculate the structural stress components at Section A–A: fx⬘ 6(my⬘+d·fz⬘) . ss⫽sm⫹sb⫽ ⫹ t t2
(9)
In the above, a finite element formulation with six degrees of freedom at each node is assumed, i.e., six components of generalized forces at each node (three translational and three rotational). If stresses in the global coordinate system (x, y, z) are used, they must be transformed to the local coordinate system (x⬘, y⬘, z⬘) before Eqs. (2) and (3) can be used for the structural stress calculations. 2.4.2. Using element nodal forces In some applications, the reference section B–B in Fig. 7 may not be available. This situation arises if welds are rather close to each other or load transfer at a weld of interest is very localized. If the element sectional forces and moments (with respect to the reference element in Fig. 7) at Section A–A are available from a finite element solution, the equilibrium requirements
described by Eq. (9) are automatically satisfied within the accuracy of the finite element solutions. In view of typical finite element procedures in commercial codes, a general structural stress calculation procedure is presented below: With respect to the global coordinate system (x, y, z), the element stiffness matrix {Ke} can be obtained either directly from a finite element solution or formulated separately afterwards. The nodal displacements at a node within the reference element are typically described in the form of: {u}Ti ⫽{uxi, uii, uzi, qxi, qyi, qzi} where uxi, uyi, uzi represent the three translational displacements in x, y, and z directions at node i and qxi, qyi, qzi three rotational displacements, respectively. The subscript i takes 1, 2, …, n, with n being the number of the nodes in the element. The element nodal force vector, {Fe}Ti ⫽{Fxi, Fyi, Fzi, Mxi, Myi, Mzi, …}, i⫽1,2,…,n can be obtained by: {Fe}⫽{Ke}{u}.
(10)
The element nodal forces in the local coordinate system (x⬘, y⬘, z⬘) can then be computed as {Fe⬘}⫽{T}{Fe}
(11)
where the matrix {T} is the coordinate transformation matrix built up of directional cosines of angles formed between the two sets of axes in Fig. 7. Once the element nodal forces are obtained for the nodal positions along Section A–A in Fig. 7, the corresponding sectional forces and moments (fx⬘, fz⬘, and my⬘) can be calculated using appropriate shape functions. Then, Eq. (9) simply becomes at Section A–A: fx⬘ 6my⬘ ss⫽sm⫹sb⫽ ⫹ 2 . t t
(12)
Note that transverse shear effects in Eq. (9) are already taken into account in the finite element solution in this instance.
3. Numerical examples
Fig. 7. Structural stress procedures for a shell/plate element adjacent to a weld.
The structural stress procedures presented above [e.g., Eqs. (2–5, 9) and (12)] can be implemented as post-processing procedures to the finite element results for a structure using commercially available finite element codes. In what follows, a series of numerical examples will be presented to demonstrate the effectiveness of the structural stress procedures. Some typical joint details discussed in the open literature are considered here.
P. Dong / International Journal of Fatigue 23 (2001) 865–876
3.1. Plate lap fillet weld A typical lap fillet joint is shown in Fig. 8(a). A plane strain eight-node solid element model is used as shown in Fig. 8(b) and (c), illustrating two representative meshes with drastically different element sizes at the weld toes, among those used for mesh-size sensitivity investigation. Since large element sizes are to be investigated, parabolic elements with reduced integration are used for this joint, due to shear lock considerations if a strong bending action is present [11]. As the element size (a/b) varies from a/b=0.16t/0.1t to a/b=2t/1t, the structural stresses calculated at the weld toe according Eqs. (2) and (3) remains essentially identical. Fig. 8(d) summarizes the structural stress based SCF values calculated with different element mesh sizes. The SCF values were calculated using the structural stresses calculated using the present methods normalized
871
with respect to remote nominal stress, i.e., F/A, where F is remote loading and A the area of loaded member. It is important to note that once the mesh-size insensitivity is ensured, the structural stress should serve as an intrinsic stress parameter for a given geometry and boundary conditions, regardless of numerical procedures used. Then, it is natural to expect a similar structural stress value from a shell/plate element model for a geometrically similar joint. Indeed, this is the case, as evidenced in Fig. 9. The structural stress values obtained using the two shell element procedures, i.e., Eqs. (9) and (12), give the same results. The slightly higher structural stress values than those obtained by solid element models reflect, to a large extent, the simple representation of the fillet weld geometry in the shell model, as shown in Fig. 9(a). A proper definition of the shell element thickness in the weld area should further improve the shell element results. Attempts were not made here to optimize some of the detailed modeling issues. 3.2. Double plate lap fillet weld The double plate lap fillet weld shown in Fig. 10 serves as a special case of the proposed structural stress definition for non-monotonic through-thickness stress distributions, as shown in Fig. 5. The calculation procedures are illustrated in Fig. 6 by imposing the symmetry conditions at the base plate mid-thickness, as
Fig. 8. Structural stress and mesh insensivity–single plate lap joint: (a) model definition; (b) a representative FE model with fine mesh (0.16t/0.1t) at weld toe; (c) a representative FE model with coarse mesh (0.8t/t) at weld toe; and (d) structural stress SCF calculated from six FE models.
Fig. 9. Comparison of structural stresses calculated from both solid and shell models as a function of element size–single plate lap joint: (a) model definition; (b) structural stress results.
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P. Dong / International Journal of Fatigue 23 (2001) 865–876
Fig. 10. Structural stress and mesh insensivity–double plate lap fillet weld: (a) model definition; (b) structural stresses calculated from four FE models with various mesh-sizes at weld toe.
shown in Fig. 10(a). For this case, shell or plate element models cannot be not directly used. Instead, plane-strain solid element models (same as those in Fig. 8) are used for demonstrating the mesh-size insensitivity in the structural stress calculations. The structural stress results based on Eqs. (4) and (5) are shown in Fig. 10(b). The mesh-size insensitivity can be demonstrated from very small element size up to a/b=0.4t/0.5t. Unlike the single plate lap weld, the mesh-insensitivity of the structural stress results will be limited to less than 1/2t since a larger element size will not be able to capture the localized stress distribution in this joint type. 3.3. Rectangular hollow section joint Fig. 11(a) shows a typical rectangular hollow section (RHS) joint under the given loading of 1000 N. Fournode linear shell elements were used with a full integration scheme. The procedure discussed earlier [Eq. (9)] is used for calculating the structural stresses at the weld toe on the vertical member. The results are summarized in Fig. 11(b). The local stress values directly obtained from the shell element model are also shown for comparison purposes. Again, the mesh-size insensitivity of the current structural stress procedures is evident. It should be noted that the curved geometry of the
Fig. 11. Structural stress at weld toe for RHS joint (shell element model). (a) Stress distribution - Shell Element model, (b) comparison of local stress and structural stresses.
RHS limits the increase in element sizes along the weld direction without losing a proper representation of the actual geometry, particularly at corner positions. The element size represents the variations in the direction perpendicular to the weld.
4. Implications and discussions 4.1. Properties of the structural stresses As discussed in the previous sections, the structural stress parameter presented in this paper demonstrates that it can be robustly computed from typical finite element models used in practice. The mesh-size insensitivity in the structural stress calculations can be generally maintained using rather coarse mesh (up to multiple thicknesses) for monotonic through thickness stress distributions (Fig. 3) and up to 1/2t for non-monotonic through-thickness distributions such as those shown in Figs. 5 and 6. In former, other considerations in final finite element mesh design should be given in practice since interactions of welds and a proper representation of structural geometry should be considered. In the latter, the element size limitation can be removed if the structural stress based SCF can be used in conjunction with the structural stresses calculated from shell/plate element
P. Dong / International Journal of Fatigue 23 (2001) 865–876
873
models (nominal structural stresses in this context) in actual applications. By its very definition, the structural stress provides a global stress measure at a location of interest such as weld toe, which satisfies equilibrium conditions in the context of elementary structural mechanics. Consequently, the structural stress parameter is solely determined by joint geometric parameters, such as joint type, plate thickness (or reference fatigue crack depth), and weld shape under a given load transfer mode. Once it is determined in a mesh-insensitive manner, the structural stress parameter can be viewed as an equivalent structural stress state with respect to a hypothetical crack orientation such as a weld toe crack in a simple geometry. Both the stress concentration and load mode effects in the actual structure are fully contained in the structural stress parameter. It should also be noted that the average transverse shear stress [e.g., Eq. (3)] should also be viewed as a component of the structural stress parameter. The effects of the transverse shear component of the structural stress parameter are currently under investigation [14] for joints loaded with significant transverse shear loading. 4.2. Correlation between structural stresses and S-N data Structural stress calculations were performed for a series of selected weld details upon which well-documented S-N fatigue data can be found from a database at Battelle. These joint configurations are shown in Fig. 1(a) [Joints (A), (B), (C), (E), and (F)]. For comparison purposes, nominal stress and hot-spot stress extrapolation procedure based on 0.4t and 1.0t positions [1–4] were also used with the same finite element models, in addition to the structural stress procedures discussed in the above. A relatively small element size (about t/6) at the weld toe were used in all these calculations due to the hot spot stress extrapolation requirements, even though the present structural stress procedures are capable of generating the same results with much coarser models. Fig. 12 summarizes the results processed using the three stress definitions. To facilitate an one-to-one comparison, all plots in Fig. 12 are plotted with same length scale and ranges both for the abscissa and ordinate. As shown in Fig. 12(a), the nominal stress range versus life plot shows that each of the joint types essentially retains its own S-N data trend, as expected. The hot spot stress based plot in Fig. 12(b) did not provide any noticeable improvement. Note that some of the data points such as those for Joint E and Joint F have a lower nominal stress value (F/A) than 100 MPa and are outside of the plot in Fig. 12(a), but having higher hot spot stress and structural stress values than 100 MPa in Fig. 12(b) and (c). As discussed earlier, due to the highly localized stress
Fig. 12. Comparison of S-N data representations using various stress parameters published in the literature: (a) nominal stresses; (b) IIW hot spot stresses (0.4t/1.0t); (c) structural stresses.
concentration behavior in most of the joints (see Fig. 1), the stress extrapolations from positions at 0.4t and 1t from weld toe yield essentially the nominal stresses for almost of all the joint types, except Joint (F). However, the structural stress based plot in Fig. 12(c) for the same S-N data shows that the structural stress parameter provides more effective consolidation of the S-N data from the different joint types studied. The structural stress based SCFs (sm and sb values corresponding to unit remote tension) are listed in the parenthesis following each legend in Fig. 12(c). The results suggest that the structural stress parameter can be used to establish the transferability of the S-N data among the joint types. Further results will be reported separately [14] for more
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P. Dong / International Journal of Fatigue 23 (2001) 865–876
detailed considerations of various other joint types and corresponding S-N data. 4.3. Effects of bending and membrane components With a S-N curve approach, fatigue lives (in cycles) of welded joints are described as a function of either nominal or hot spot stress ranges. In the present approach as shown in Fig. 12(c), structural stress ranges are used as a stress parameter to relate to fatigue lives of the welded joints under remote tension. For all the joints analyzed in Fig. 12(c) under remote tension, the bending content (sb) of the structural stresses at weld toe varies from 0.214 to in Joint A to 0.522 to Joint F, normalized by remote tension stress. It is conceivable that the peak surface stress range parameter (sm+sb), although it can be calculated in a mesh-insensitive manner, may not be adequate to consolidate S-N data from drastically different remote loading modes, such as pure remote tension versus pure remote bending. In such situations, the corresponding structural stress components sm and sb will be drastically different as well. The effects of the membrane and bending effects on S-N behavior can be inferred from fracture mechanics considerations since stress intensity factor provides an effective one-parameter characterization of cracking driving force under linear elastic conditions. A large amount of published work demonstrates that crack growth rate can be related to the stress intensity factor (⌬K) range in a form of the Paris law. Once a structural stress is available for a joint geometry in terms of sm and sb in Eq. (1), the calculated structural stress transforms the actual geometry and loading effects for a given weld detail on to a simple geometry as shown in Fig. 13, where a represents a fatigue crack length and tr stands for a reference length corresponding to the final crack depth at failure. For such a simple crack geometry in Fig. 13, stress intensity factor solutions are readily available in the open literature. For the hypothetical crack plane in Fig. 13, the two structural stress components serve as an equivalent remote loading with respect to the hypothetical crack face spanning from a=0 to a=tr. Then, the corresponding stress intensity factor solution a for welded joint with structural stress components of sm and sb can be directly constructed
Fig. 13. Fracture mechanics interpretation of the structural stresses as fatigue propagation parameter.
using existing solutions such as those given by Tada et al., [15]. It can be shown that the Mode I stress intensity factor range for the crack geometry in Fig. 13 can be expressed as a function of the ranges of the structural stress components using superposition principle:
冋 冉冊
⌬K⫽⌬Km⫹⌬Kb⫽冑tr ⌬smfm
冉 冊册
a a ⫹⌬sbfb tr tr
(13)
The fm(a/tr) and fb(a/tr) are dimensionless functions of a/tr for the membrane and bending components, expressed as follows [15]:
冉冊 冋
冉冊 冉
冉冊 冋
冉
fm
fb
冪2tan2t
冊册
a a pa ⫽ 0.752⫹2.02 ⫹0.37 1⫺sin tr tr 2tr
pa r
pa cos 2tr
冪2tan2t .
冊册
pa a ⫽ 0.923⫹0.199 1⫺sin tr 2tr
4
3
(14)
pa r
pa cos 2tr
(15)
For a given combination of ⌬sm and ⌬sb, fm(a/tr) and fb(a/tr) are functions of a/tr. It should be pointed out that the K solutions using Eqs. (13)–(15) are strictly valid for joint types without symmetry with respect to the midplane of the horizontal member in Fig. 1(a), such as Joints D, G, and G⬘. For joints with symmetry, such as Joints A–C, E, and F, the K solutions using Eqs. (13)– (15) should be viewed as an approximation. The adequacy of the approximation for the current applications can be justified from the fact that the structural stress calculations have already taken into account of the stress gradient effects in the symmetrical joints (A–C, E, and F). Similar assumptions have been used in various early publications, e.g., [1,2]. A more refined approach by incorporating more appropriate functions fm(a/tr) and fb(a/tr) in place of Eqs. (14) and (15) is given by Dong et al. [16] for joints with horizontal symmetry. In typical fatigue testing of welded joints for S-N data generation, crack size information such as a/tr is often not available. For S-N data interpretation purposes, as a reasonable approximation, it may be assumed that a crack size at the beginning of a test is infinitesimally small (a/tr苲0) and reaches about a/tr苲1 (complete separation) at the final failure. (It is recognized that in most of fatigue tests, a final failure often occurs well before a/tr=1. More detailed considerations are discussed in [16].) Then, an effective ⌬K can be defined by considering an averaged stress intensity factor integrating over a=0苲tr. The average stress intensity factor range ⌬K can be obtained by integrating the expressions in Eqs. (14) and (15) over a/tr=0 to 1 as:
P. Dong / International Journal of Fatigue 23 (2001) 865–876 a/tr⫽1
⌬K⫽冑tr
冕冋 冉冊
a/tr⫽0
⌬smfm
875
冉 冊册 冉 冊
a a a ⫹⌬sbfb d . tr tr tr
The following simple expression can then be obtained: ⌬K⫽冑tr(33.17⌬sm⫹11.87⌬sb).
(16)
For a given combination of ⌬sm and ⌬sb, Eq. (16) provides the stress intensity factor based measurement of their relative contributions to the total ⌬K. Note that Eq. (16) is rather general since the structural stress components already contain the geometry and three dimensional loading effects from an actual welded joint, as discussed earlier. Naturally, if ⌬K serves as an effective parameter for crack propagation behavior, a single ⌬K⫺N log–log plot should be expected regardless of the relative magnitude of the membrane and bending components. However, in a log–log based S-N plot, if ⌬sm and ⌬sb are significantly different from those in Fig. 12(c), a slope change is anticipated since Eq. (1) provides only a peak structural stress measure at the surface with an equal weight on both components. To substantiate this augment, two sets of S-N data for Joints G and G⬘ by Huther et al. [12] and Maddox [13] were processed using both Eqs. (1) and (16). Joint G⬘ is the same as Joint G in geometry, but under horizontal tension instead of pure bending. The structural stress based SCFs for the bend component (sb) for Joints G and G⬘ are 1.206 and 0.383, respectively, as indicated in Fig. 14(a), drastically different from each other. The structural stress based S-N plot is shown in Fig. 14(a). It can be seen that the two sets of SN data exhibit two distinct bands separating membranedominant structural stress from the bending-dominant. However, if Eq. (16) is used, Fig. 14(b) shows that the fatigue data from the two drastically different membrane and bending combinations fall into essentially the same band. Fig. 14(c) summarizes the stress intensity factor ranges versus cycles at failure for all joints [Joints A though G in Fig. 1(a) and G⬘]. The results in Fig. 14(c) clearly suggest that the S-N data for Joints A through G⬘ can be effectively consolidated into a single band, even though both the joint types and loading are significantly different from one another. In calculating the structural intensity factor ranges for each of the joint types, Eq. (16) was used throughout the process once the meshinsensitive structural stresses are obtained for all joint types and loading mode. This can also be viewed as a validation of the structural stress based transformation process shown in Fig. 13 and its potential usefulness for general applications. Further discussions on the use of the relation in Eq. (13) or Eq. (16) and more refined definition of ⌬K parameters for consolidating existing S-N data be reported separately [16], in which structural stresses at notches
Fig. 14. Effects of bending and membarane components on S-N curve slope: (a) structural stress range; (b) stress intensity factor range; (c) stress intensity factor range for joints A–G⬘.
and detailed K solutions were also used [16]. It should be noted that the structural stress based ⌬K⫺N interpretation for fatigue behavior has been successfully used for consolidating S-N data of spot welds under drastically different loading conditions (e.g., [17,18]), where ⌬K, however, can be expressed in a simple form without explicit crack size dependency, as initially proposed by Zhang [17]. A recent independent evaluation on its effectiveness can be found in Gao et al. [19].
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5. Summary 1. A structural stress definition has been presented as an effective measure of the stress state at welded joints. The structural stress parameter in terms of both membrane and bending components is consistent with elementary structural mechanics theory and can be readily calculated in a mesh-insensitive manner from conventional finite element models. Its applications in characterizing weld toe cracking for a series of joint types were demonstrated. 2. The structural stress based interpretation of existing S-N data considered here suggests that transferability of S-N data among different joint types and loading conditions can be established. 3. The relative bending to membrane ratio of the structural stress components calculated at weld toe can provide additional insight on the S-N curve behavior, particularly for those joints tested under drastically different remote loading conditions. A simplified stress intensity factor range parameter (an integrated average) proposed in this study has demonstrated its potential in correlating such S-N curve behaviors in terms of the membrane and bending effects. 4. The structural stress procedure can be viewed as a transformation process that maps a complex stress state at a joint situated in an actual structure subjected to complex loading mode onto a simple structural stress state represented by membrane and bending components. Consequently, stress intensity factors can be readily calculated using existing solutions for a simple crack problem for various welded joints once the structural stress components become available.
[3] [4] [5]
[6]
[7]
[8]
[9] [10]
[11] [12]
[13]
[14] [15] [16]
[17] [18]
References [1] Hobbacher A. Fatigue design of welded joints and components: Recommendations of IIW Joint Working Group XIII–XV. Abington, Cambridge: Abington Publishing, 1996. [2] Hobbacher A. Basic philosophy of the new IIW recommendations
[19]
on fatigue design of welded joints and components. Welding in the World 1997;39(5):272–8. Code of practice for fatigue design and assessment of steel structures. BS7608, British Standards Institution, 1993. Design of steel structures—Part 1-1. ENV 1993-1-1. Eurcode 3, European Committee for Standardization, Brussels, 1992. Radaj D. Review of fatigue strength assessment of non-welded and welded structures based on local parameters. International Journal of Fatigue 1996;18(3):153–70. Lawrence FV, Mattos RJ, Higashida Y, Burk JD. Estimating the fatigue crack initiation life of welds. ASTM STP 1978;648:134–58. Stress determination for fatigue analysis of welded components. IIS/IIW 1221-93. Abington, Cambridge: Abington Publishing, 1993. Niemi E. Recommendations concerning stress determination for fatigue analysis of welded components. IIW-1458-92/XV-79792, 1992. Niemi E, Tanskanen P. Hot spot stress determination for welded edge gussets. IIW XIII-1781-99. 1999. Dong P. A robust structural stress parameter for characterizing fatigue behavior of welded joints. SAE Technical Paper Series: No. 2001-01-0086, Fatigue and Failure of Spot Welds and Weld Joints, March, 2001. Dong P, Zhang J, Hong JK. Structural stress calculation scheme. Battelle’s Patent Application (pending), 2000. Huther I, Gorski S, Lieurade HP. Longitudinal non-loaded welded joints geometric stress approach. Welding in the World 1999;43(3):20–6. Maddox SJ. Influence of tensile residual stresses on the fatigue behavior of welded joints in steel. Residual stress effects in fatigue. ASTM STP 1982;776:63–96. Dong P, Hong JK. A master fatigue damage curve approach for welded joints. Under preparation, 2001. Tada H, Paris P, Irwin G. The stress analysis of cracks handbook. 2nd ed. St Louis, MO: Paris Productions, 1985. Dong P, Hong JK, Cao Z. A mesh insensitive structural stress procedure for fatigue evaluation of welded structures. International Institute of Welding (IIW), IIW Doc. XIII-1902-01/XV1089-01, July, 2001. Zhang S. Stress intensities at spot welds. International Journal of Fracture 1997;88:167–85. Zhang J, Dong P, Gao Y. Evaluation of stress intensity factor based predictive technique for fatigue life of resistance spot welds. SAE Technical Paper Series: No. 2001-01-0830, Fatigue and Failure of Spot Welds and Weld Joints, March, 2001. Gao Y, Chucas D, Lewis C, McGregor IJ. Review of CAE fatigue analysis techniques for spot-welded high strength steel automotive structures. SAE Technical Paper Seires: No. 2001-01-0835, Fatigue and Failure of Spot Welds and Weld Joints, March, 2001.
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 elementary 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 or plate element models are presented to demonstrate the mesh-size insensitivity in extracting the structural stress parameter. Conventional finite element models can be directly used with the structural stress calculation as a post-processing procedure. To further illustrate the effectiveness of the present structural stress procedures, a collection of existing weld S-N data for various joint types were processed using the current structural stress procedures. The results strongly suggests that weld classification based S-N curves can be significantly reduced into possibly a single master S-N curve, in which the slope of the S-N curve is determined by the relative composition of the membrane and bending components of the structural stress parameter. The effects of membrane and bending on S-N behaviors can be addressed by introducing an equivalent stress intensity factor based parameter using the structural stress components. Among other things, the two major implications are: (a) structural stresses pertaining to weld fatigue behavior can be consistently calculated in a mesh-insensitive manner regardless of types of finite element models; (b) transferability of weld S-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 is primarily based on a nominal stress or hot spot stress approach with a series of classified weld S-N curves [1– 4], although a local stress or initiation-based fatigue life approaches [5,6] provide an alternative method for fatigue life predictions of welded joints. Without going into a detailed discussion of the merits of the two different approaches, the premise of this paper is that the nominal stress or hot spot stress approach has been well accepted by major industries, and recommended by numerous national and international codes and standards (e.g. [3,4]). A series of S-N curves corresponding to each class of joint types and loading mode were well documented in some of the codes and standards. With such an
* Tel.: +1 614-424-4908; fax: +1 614-424-3457. E-mail address: [email protected] (P. Dong).
approach, nominal stresses with appropriate geometric or structural stress concentration factor (SCF) for a particular class of joints must be determined against the corresponding S-N curve to calculate fatigue damage. Two critical issues remain unresolved in this context. First, both nominal stresses and geometric SCFs cannot be readily calculated from finite element models due to their strong dependence on element size at weld discontinuities. Secondly, the selection of an appropriate S-N curve 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 to address the above two issues. A majority of the effort has been on developing effective hot-spot stress extrapolation procedures and an ability to correlate various available S-N curves (e.g., [7,8]). However, the extrapolation procedures available to date still lack consistency for general applications [9]. This is in part due to the fact that extrapolation procedures are based on the
0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 0 5 5 - X
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assumption that the surface stresses on a structural member provides an indication of the stress state at a weld fatigue prone location, such as a weld toe. This underlying assumption may become questionable if the structural member is not a dominant load-transfer member in a joint. Under such circumstances, the surface stresses at some distance away from a weld toe may not be relevant to the stress state of concern. In addition, a reference nominal stress in such a structural member may not be readily identified for conventional SCF calculations. Among the various extrapolation procedures proposed in the open literature (e.g., [7,8]), a typical one is based on a linear extrapolation from stress values at both 0.4t and 1t 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 becomes immediately clear in view of Fig. 1 in which some of the well-studied joints in the research community are illustrated. The stress concentration behaviors can be categorized into two types [10]: one is rather localized stress concentration behavior (Type I) at weld toe, while the other is more global in length-scale (Type II). In order to correlate the fatigue behavior in various joint types, stress concentration behavior at the weld toe of various joint types must be captured. However, as shown in Fig. 1(b), any stress concentration effects in Type I
joints cannot be captured in this extrapolation scheme, resulting in little stress concentration effects from this calculation. On the other hand, for Type II joints, Fig. 1(b) shows that extrapolation from the two reference positions (open circles) should provide some indication of the concentrations at the weld toes. Then, one obvious question is if such calculation procedures provide a reliable stress concentration measurement or hot spot stresses. As discussed in Neimi [9], the results are often questionable due to the fact that these stresses can be strongly dependent on mesh-size and loading modes. To improve the S-N curve approach (using either nominal stresses or hot spot stresses) for welded structures, a relevant stress parameter must satisfy the two basic requirements: (a) mesh-size insensitivity in finite element solutions; (b) ability to differentiate stress concentration effects in different joint types (e.g., butt joints versus T-fillet cruciform joints) in welded structures. In the following, such a stress parameter is presented and the corresponding finite element procedures using both solid and shell element models are given. The validation of such a structural stress parameter is demonstrated by reprocessing a series of existing S-N data for joint types listed in Fig. 1(a).
2. Structural stress definition and formulation As discussed in Dong [10] and Dong et al. [11], a structural stress definition that follows elementary structural mechanics theory can be developed with following considerations:
Fig. 1.
Stress concentration behavior in welded joints.
(a) It can be postulated that for a given local throughthickness stress distribution as shown in Fig. 2(a) obtained from a finite element model, there exists a corresponding simple structural stress distribution as shown in Fig. 2(b), in the form of membrane and bending components that are equilibrium-equivalent to the local stress distributions in Fig. 2(a). (b) The structural stress distribution must satisfy equilibrium conditions within the context of elementary structural mechanics theory at both the hypothetical crack plane [e.g., at weld toe in Fig. 2(a)] and a nearby reference plane, on which local stress distributions are known a priori from typical finite element solutions. The uniqueness of such a structural stress solution can be argued by considering the fact that the compatibility conditions of the corresponding finite element solutions are maintained at this location in such a calculation. (c) While local stresses near a notch are mesh-size sensitive due to the asymptotic singularity behavior as a notch position is approached, the imposition of the equilibrium conditions in the context of elementary structural mechanics with respect to this regime
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a membrane component (sm) and bending component (sb), consistent with elementary structural mechanics definition: ss⫽sm⫹sb.
(1)
The normal structural stress (ss) is defined at a location of interest such as Section A–A at the weld toe in Fig. 2(b) with a plate thickness of t. In the above, the transverse shear (tm) of the structural stress components [to be calculated based on local transverse stress distribution from Fig. 2(a)] is not considered in the structural stress definition in the present discussions. In practice, the transverse shear component can play an important role in controlling crack propagation path if the remote loading imposes a significant transverse shear at the weld toe. A second reference plane can be defined along Section B–B in Fig. 3(a), along which both local normal and shear stresses can be directly obtained from a finite elements solution. The distance, d, represents the distance 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 finite element model. By imposing equilibrium conditions between Sections A–A and B–B, the structural stress components sb and sm must satisfy the following conditions:
冕 t
1 sm⫽ sx(y)·dy t
(2)
0
Fig. 2. Structural stresses definition for through-thickness fatigue crack. (a) Local through-thickness normal and shear stress at weld toe, (b) Structural stress definition at weld toe.
冕 t
0
should eliminate or minimize the mesh-size sensitivity in the structural stress calculations. This is due to the fact that the local stress concentration close to a notch is dominated by self-equilibrating stress distribution, as discussed by Niemi [9].
冕 t
t2 t2 sm· ⫹sb· ⫽ sx(y)·y·dy⫹d txy(y)·dy. 2 6
(3)
0
Eq. (2) represents the force balances in x direction, evaluated along B–B and Eq. (3) represents moment bal-
Along this line, the following typical situations are considered: 2.1. Solid model with monotonic through-thick stress distributions As shown in Fig. 2(a), the stress distribution at the Tfillet weld toe is assumed to exhibit a monotonic through-thickness distribution with the peak stress occurring at the weld toe. It should be noted that in typical finite element based stress analysis, the stress values within some distance from the weld toe can change significantly as the finite element mesh design changes (e.g., [9]), referred to as mesh-size sensitivity in this paper. The corresponding statically equivalent structural stress distribution is illustrated in Fig. 2, in the form of
Fig. 3. Structural stresses calculation procedure for through-thickness fatigue crack.
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ances with respect to Section A–A at y =0. The integral term on the right-hand side of Eq. (3) represents the transverse shear force as an important component of the structural stress definition. It is then follows that if element size (d) is small or transverse shear is negligible, the integral representations of sb and sm 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 finite depth is used as a fatigue failure criterion (e.g. [9]), the corresponding structural stress can be then defined in a similar manner to that in Fig. 3. In Fig. 4, the depth of the fatigue crack
at failure is assumed to be t1. Without losing generality, the structural stress procedures [10,11] can be effectively demonstrated using the example in Fig. 4. Note that for convenience, the local y coordinate is defined as shown in Fig. 4(a), different from that in Fig. 3, At a horizontal cross section of depth t1 from the top surface, both normal stress (sy) and shear stress (tyx) are present in general. By imposing equilibrium conditions between Sections A–A and B–B, as well as the horizontal cross section in between, it can be shown that the structural stress components (sb and sm) must satisfy the following equations:
冕
冕
t1
d
1 1 sm⫽ sx(y)·dy⫹ tyx(x)·dx t1 t1
(4)
0
0
冕
冕
t1
t1
t21 t21 sm· ⫹sb· ⫽ sx(y)·y·dy⫹d txy(y)·dy 2 6 0
(5)
0
冕 d
⫹ sy(x)·x·dx. 0
Additionally, by considering the bottom element (spanning t–t1) between Sections A–A and B–B, it can be shown that sm⫺sb⫽sm⬘⫹sb⬘
冕
t⫺t1
1 sm⬘⫽ t−t1
(6)
冕 d
1 sx(y)·dy⫺ t (x)·dx. t−t1 yx
0
(7)
0
As in Eqs. (2) and (3), the integrals in the above can be accurately evaluated using finite element solutions at cross sections away from the geometric discontinuity. The structural stress components sb and sm, including sb⬘ and sm⬘ can then be solved. 2.3. Solid model with non-monotonic throughthickness stress distributions
Fig. 4. Structural stresses definition for partial thickness (t1) fatigue crack weld toe. (a) Local normal stress distribution, (b) Structural stress definition.
In thick section joints and some joint configurations, non-monotonic through-thickness or in-plane distributions may develop, as shown in Fig. 5(a). The corresponding structural stress definition can be consistently defined in a similar manner as that in Fig. 4. Note that the parameter t1 can be determined based on the position at which the transverse shear stress changes direction, if there is no specified crack depth as a failure criterion. Eqs. (4, 5) and (7) can be directly used for structural stress calculations except a minor modification of Eq. (6) as follows: sm⫺sb⫽sm⬘⫺sb⬘.
(8)
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Fig. 5. Structural stresses definition for non-monotonic throughthickness stress distribution.
A special case in this category is that the joint configuration and loading are symmetric with respect the neutral axis of the horizontal member in Fig. 5 such as Joints B and C in Fig. 1(a). In this case, the shear stress on the cross section along the symmetry line is zero as shown in Fig. 6. Thus, Eqs. (4) and (5) can be directly used to calculate the structural stress components sb and sm, by substituting t1=t/2 and tyx=0. 2.4. Shell/plate element models Shell or plate element models of complex structures are widely used for performing stress analyses for fatigue evaluation. The underlying principles of structural stress calculations are the same between the
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Fig. 6. Structural stresses procedure for non-monotonic throughthickness stress distribution: (a) symmetry with respect to plate midthickness (t1=t/2); (b) structural stress definition.
shell/plate and solid element models, while special consideration will be given to shell/plate structural theory and its finite element implementation. It should be noted that shell/plate element solutions at geometric discontinuities (e.g., weld toe) converge only to the solutions described by the corresponding shell/plate theory. Consequently, the local stresses at a weld toe in actual structures are forced to obey the shell/plate theory used in the finite element model and are not the structural stresses sought in Eq. (1), even though a linear throughthickness stress distribution is maintained. Two general methods for structural stress calculations are presented below:
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2.4.1. Using stresses and stress resultants As shown in Fig. 7, stresses and nodal quantities from shell or plate element models are often defined in a global coordinate system (x, y, z), depending on the finite element codes used. Given the definition of the structural stress components in Eq. (1), it is the local coordinate system (x⬘, y⬘, z⬘) that is convenient for calculating the structural stresses with respect to a weld, with local x⬘ and y⬘ being perpendicular and parallel to the weld direction, respectively. Consistent with the solid element model approach (e.g., see Fig. 3), three components of the stress resultants (sectional forces and moments), i.e., fx⬘, fz⬘, and my⬘, at Section B–B in Fig. 7 can be used to calculate the structural stress components at Section A–A: fx⬘ 6(my⬘+d·fz⬘) . ss⫽sm⫹sb⫽ ⫹ t t2
(9)
In the above, a finite element formulation with six degrees of freedom at each node is assumed, i.e., six components of generalized forces at each node (three translational and three rotational). If stresses in the global coordinate system (x, y, z) are used, they must be transformed to the local coordinate system (x⬘, y⬘, z⬘) before Eqs. (2) and (3) can be used for the structural stress calculations. 2.4.2. Using element nodal forces In some applications, the reference section B–B in Fig. 7 may not be available. This situation arises if welds are rather close to each other or load transfer at a weld of interest is very localized. If the element sectional forces and moments (with respect to the reference element in Fig. 7) at Section A–A are available from a finite element solution, the equilibrium requirements
described by Eq. (9) are automatically satisfied within the accuracy of the finite element solutions. In view of typical finite element procedures in commercial codes, a general structural stress calculation procedure is presented below: With respect to the global coordinate system (x, y, z), the element stiffness matrix {Ke} can be obtained either directly from a finite element solution or formulated separately afterwards. The nodal displacements at a node within the reference element are typically described in the form of: {u}Ti ⫽{uxi, uii, uzi, qxi, qyi, qzi} where uxi, uyi, uzi represent the three translational displacements in x, y, and z directions at node i and qxi, qyi, qzi three rotational displacements, respectively. The subscript i takes 1, 2, …, n, with n being the number of the nodes in the element. The element nodal force vector, {Fe}Ti ⫽{Fxi, Fyi, Fzi, Mxi, Myi, Mzi, …}, i⫽1,2,…,n can be obtained by: {Fe}⫽{Ke}{u}.
(10)
The element nodal forces in the local coordinate system (x⬘, y⬘, z⬘) can then be computed as {Fe⬘}⫽{T}{Fe}
(11)
where the matrix {T} is the coordinate transformation matrix built up of directional cosines of angles formed between the two sets of axes in Fig. 7. Once the element nodal forces are obtained for the nodal positions along Section A–A in Fig. 7, the corresponding sectional forces and moments (fx⬘, fz⬘, and my⬘) can be calculated using appropriate shape functions. Then, Eq. (9) simply becomes at Section A–A: fx⬘ 6my⬘ ss⫽sm⫹sb⫽ ⫹ 2 . t t
(12)
Note that transverse shear effects in Eq. (9) are already taken into account in the finite element solution in this instance.
3. Numerical examples
Fig. 7. Structural stress procedures for a shell/plate element adjacent to a weld.
The structural stress procedures presented above [e.g., Eqs. (2–5, 9) and (12)] can be implemented as post-processing procedures to the finite element results for a structure using commercially available finite element codes. In what follows, a series of numerical examples will be presented to demonstrate the effectiveness of the structural stress procedures. Some typical joint details discussed in the open literature are considered here.
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3.1. Plate lap fillet weld A typical lap fillet joint is shown in Fig. 8(a). A plane strain eight-node solid element model is used as shown in Fig. 8(b) and (c), illustrating two representative meshes with drastically different element sizes at the weld toes, among those used for mesh-size sensitivity investigation. Since large element sizes are to be investigated, parabolic elements with reduced integration are used for this joint, due to shear lock considerations if a strong bending action is present [11]. As the element size (a/b) varies from a/b=0.16t/0.1t to a/b=2t/1t, the structural stresses calculated at the weld toe according Eqs. (2) and (3) remains essentially identical. Fig. 8(d) summarizes the structural stress based SCF values calculated with different element mesh sizes. The SCF values were calculated using the structural stresses calculated using the present methods normalized
871
with respect to remote nominal stress, i.e., F/A, where F is remote loading and A the area of loaded member. It is important to note that once the mesh-size insensitivity is ensured, the structural stress should serve as an intrinsic stress parameter for a given geometry and boundary conditions, regardless of numerical procedures used. Then, it is natural to expect a similar structural stress value from a shell/plate element model for a geometrically similar joint. Indeed, this is the case, as evidenced in Fig. 9. The structural stress values obtained using the two shell element procedures, i.e., Eqs. (9) and (12), give the same results. The slightly higher structural stress values than those obtained by solid element models reflect, to a large extent, the simple representation of the fillet weld geometry in the shell model, as shown in Fig. 9(a). A proper definition of the shell element thickness in the weld area should further improve the shell element results. Attempts were not made here to optimize some of the detailed modeling issues. 3.2. Double plate lap fillet weld The double plate lap fillet weld shown in Fig. 10 serves as a special case of the proposed structural stress definition for non-monotonic through-thickness stress distributions, as shown in Fig. 5. The calculation procedures are illustrated in Fig. 6 by imposing the symmetry conditions at the base plate mid-thickness, as
Fig. 8. Structural stress and mesh insensivity–single plate lap joint: (a) model definition; (b) a representative FE model with fine mesh (0.16t/0.1t) at weld toe; (c) a representative FE model with coarse mesh (0.8t/t) at weld toe; and (d) structural stress SCF calculated from six FE models.
Fig. 9. Comparison of structural stresses calculated from both solid and shell models as a function of element size–single plate lap joint: (a) model definition; (b) structural stress results.
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Fig. 10. Structural stress and mesh insensivity–double plate lap fillet weld: (a) model definition; (b) structural stresses calculated from four FE models with various mesh-sizes at weld toe.
shown in Fig. 10(a). For this case, shell or plate element models cannot be not directly used. Instead, plane-strain solid element models (same as those in Fig. 8) are used for demonstrating the mesh-size insensitivity in the structural stress calculations. The structural stress results based on Eqs. (4) and (5) are shown in Fig. 10(b). The mesh-size insensitivity can be demonstrated from very small element size up to a/b=0.4t/0.5t. Unlike the single plate lap weld, the mesh-insensitivity of the structural stress results will be limited to less than 1/2t since a larger element size will not be able to capture the localized stress distribution in this joint type. 3.3. Rectangular hollow section joint Fig. 11(a) shows a typical rectangular hollow section (RHS) joint under the given loading of 1000 N. Fournode linear shell elements were used with a full integration scheme. The procedure discussed earlier [Eq. (9)] is used for calculating the structural stresses at the weld toe on the vertical member. The results are summarized in Fig. 11(b). The local stress values directly obtained from the shell element model are also shown for comparison purposes. Again, the mesh-size insensitivity of the current structural stress procedures is evident. It should be noted that the curved geometry of the
Fig. 11. Structural stress at weld toe for RHS joint (shell element model). (a) Stress distribution - Shell Element model, (b) comparison of local stress and structural stresses.
RHS limits the increase in element sizes along the weld direction without losing a proper representation of the actual geometry, particularly at corner positions. The element size represents the variations in the direction perpendicular to the weld.
4. Implications and discussions 4.1. Properties of the structural stresses As discussed in the previous sections, the structural stress parameter presented in this paper demonstrates that it can be robustly computed from typical finite element models used in practice. The mesh-size insensitivity in the structural stress calculations can be generally maintained using rather coarse mesh (up to multiple thicknesses) for monotonic through thickness stress distributions (Fig. 3) and up to 1/2t for non-monotonic through-thickness distributions such as those shown in Figs. 5 and 6. In former, other considerations in final finite element mesh design should be given in practice since interactions of welds and a proper representation of structural geometry should be considered. In the latter, the element size limitation can be removed if the structural stress based SCF can be used in conjunction with the structural stresses calculated from shell/plate element
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models (nominal structural stresses in this context) in actual applications. By its very definition, the structural stress provides a global stress measure at a location of interest such as weld toe, which satisfies equilibrium conditions in the context of elementary structural mechanics. Consequently, the structural stress parameter is solely determined by joint geometric parameters, such as joint type, plate thickness (or reference fatigue crack depth), and weld shape under a given load transfer mode. Once it is determined in a mesh-insensitive manner, the structural stress parameter can be viewed as an equivalent structural stress state with respect to a hypothetical crack orientation such as a weld toe crack in a simple geometry. Both the stress concentration and load mode effects in the actual structure are fully contained in the structural stress parameter. It should also be noted that the average transverse shear stress [e.g., Eq. (3)] should also be viewed as a component of the structural stress parameter. The effects of the transverse shear component of the structural stress parameter are currently under investigation [14] for joints loaded with significant transverse shear loading. 4.2. Correlation between structural stresses and S-N data Structural stress calculations were performed for a series of selected weld details upon which well-documented S-N fatigue data can be found from a database at Battelle. These joint configurations are shown in Fig. 1(a) [Joints (A), (B), (C), (E), and (F)]. For comparison purposes, nominal stress and hot-spot stress extrapolation procedure based on 0.4t and 1.0t positions [1–4] were also used with the same finite element models, in addition to the structural stress procedures discussed in the above. A relatively small element size (about t/6) at the weld toe were used in all these calculations due to the hot spot stress extrapolation requirements, even though the present structural stress procedures are capable of generating the same results with much coarser models. Fig. 12 summarizes the results processed using the three stress definitions. To facilitate an one-to-one comparison, all plots in Fig. 12 are plotted with same length scale and ranges both for the abscissa and ordinate. As shown in Fig. 12(a), the nominal stress range versus life plot shows that each of the joint types essentially retains its own S-N data trend, as expected. The hot spot stress based plot in Fig. 12(b) did not provide any noticeable improvement. Note that some of the data points such as those for Joint E and Joint F have a lower nominal stress value (F/A) than 100 MPa and are outside of the plot in Fig. 12(a), but having higher hot spot stress and structural stress values than 100 MPa in Fig. 12(b) and (c). As discussed earlier, due to the highly localized stress
Fig. 12. Comparison of S-N data representations using various stress parameters published in the literature: (a) nominal stresses; (b) IIW hot spot stresses (0.4t/1.0t); (c) structural stresses.
concentration behavior in most of the joints (see Fig. 1), the stress extrapolations from positions at 0.4t and 1t from weld toe yield essentially the nominal stresses for almost of all the joint types, except Joint (F). However, the structural stress based plot in Fig. 12(c) for the same S-N data shows that the structural stress parameter provides more effective consolidation of the S-N data from the different joint types studied. The structural stress based SCFs (sm and sb values corresponding to unit remote tension) are listed in the parenthesis following each legend in Fig. 12(c). The results suggest that the structural stress parameter can be used to establish the transferability of the S-N data among the joint types. Further results will be reported separately [14] for more
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detailed considerations of various other joint types and corresponding S-N data. 4.3. Effects of bending and membrane components With a S-N curve approach, fatigue lives (in cycles) of welded joints are described as a function of either nominal or hot spot stress ranges. In the present approach as shown in Fig. 12(c), structural stress ranges are used as a stress parameter to relate to fatigue lives of the welded joints under remote tension. For all the joints analyzed in Fig. 12(c) under remote tension, the bending content (sb) of the structural stresses at weld toe varies from 0.214 to in Joint A to 0.522 to Joint F, normalized by remote tension stress. It is conceivable that the peak surface stress range parameter (sm+sb), although it can be calculated in a mesh-insensitive manner, may not be adequate to consolidate S-N data from drastically different remote loading modes, such as pure remote tension versus pure remote bending. In such situations, the corresponding structural stress components sm and sb will be drastically different as well. The effects of the membrane and bending effects on S-N behavior can be inferred from fracture mechanics considerations since stress intensity factor provides an effective one-parameter characterization of cracking driving force under linear elastic conditions. A large amount of published work demonstrates that crack growth rate can be related to the stress intensity factor (⌬K) range in a form of the Paris law. Once a structural stress is available for a joint geometry in terms of sm and sb in Eq. (1), the calculated structural stress transforms the actual geometry and loading effects for a given weld detail on to a simple geometry as shown in Fig. 13, where a represents a fatigue crack length and tr stands for a reference length corresponding to the final crack depth at failure. For such a simple crack geometry in Fig. 13, stress intensity factor solutions are readily available in the open literature. For the hypothetical crack plane in Fig. 13, the two structural stress components serve as an equivalent remote loading with respect to the hypothetical crack face spanning from a=0 to a=tr. Then, the corresponding stress intensity factor solution a for welded joint with structural stress components of sm and sb can be directly constructed
Fig. 13. Fracture mechanics interpretation of the structural stresses as fatigue propagation parameter.
using existing solutions such as those given by Tada et al., [15]. It can be shown that the Mode I stress intensity factor range for the crack geometry in Fig. 13 can be expressed as a function of the ranges of the structural stress components using superposition principle:
冋 冉冊
⌬K⫽⌬Km⫹⌬Kb⫽冑tr ⌬smfm
冉 冊册
a a ⫹⌬sbfb tr tr
(13)
The fm(a/tr) and fb(a/tr) are dimensionless functions of a/tr for the membrane and bending components, expressed as follows [15]:
冉冊 冋
冉冊 冉
冉冊 冋
冉
fm
fb
冪2tan2t
冊册
a a pa ⫽ 0.752⫹2.02 ⫹0.37 1⫺sin tr tr 2tr
pa r
pa cos 2tr
冪2tan2t .
冊册
pa a ⫽ 0.923⫹0.199 1⫺sin tr 2tr
4
3
(14)
pa r
pa cos 2tr
(15)
For a given combination of ⌬sm and ⌬sb, fm(a/tr) and fb(a/tr) are functions of a/tr. It should be pointed out that the K solutions using Eqs. (13)–(15) are strictly valid for joint types without symmetry with respect to the midplane of the horizontal member in Fig. 1(a), such as Joints D, G, and G⬘. For joints with symmetry, such as Joints A–C, E, and F, the K solutions using Eqs. (13)– (15) should be viewed as an approximation. The adequacy of the approximation for the current applications can be justified from the fact that the structural stress calculations have already taken into account of the stress gradient effects in the symmetrical joints (A–C, E, and F). Similar assumptions have been used in various early publications, e.g., [1,2]. A more refined approach by incorporating more appropriate functions fm(a/tr) and fb(a/tr) in place of Eqs. (14) and (15) is given by Dong et al. [16] for joints with horizontal symmetry. In typical fatigue testing of welded joints for S-N data generation, crack size information such as a/tr is often not available. For S-N data interpretation purposes, as a reasonable approximation, it may be assumed that a crack size at the beginning of a test is infinitesimally small (a/tr苲0) and reaches about a/tr苲1 (complete separation) at the final failure. (It is recognized that in most of fatigue tests, a final failure often occurs well before a/tr=1. More detailed considerations are discussed in [16].) Then, an effective ⌬K can be defined by considering an averaged stress intensity factor integrating over a=0苲tr. The average stress intensity factor range ⌬K can be obtained by integrating the expressions in Eqs. (14) and (15) over a/tr=0 to 1 as:
P. Dong / International Journal of Fatigue 23 (2001) 865–876 a/tr⫽1
⌬K⫽冑tr
冕冋 冉冊
a/tr⫽0
⌬smfm
875
冉 冊册 冉 冊
a a a ⫹⌬sbfb d . tr tr tr
The following simple expression can then be obtained: ⌬K⫽冑tr(33.17⌬sm⫹11.87⌬sb).
(16)
For a given combination of ⌬sm and ⌬sb, Eq. (16) provides the stress intensity factor based measurement of their relative contributions to the total ⌬K. Note that Eq. (16) is rather general since the structural stress components already contain the geometry and three dimensional loading effects from an actual welded joint, as discussed earlier. Naturally, if ⌬K serves as an effective parameter for crack propagation behavior, a single ⌬K⫺N log–log plot should be expected regardless of the relative magnitude of the membrane and bending components. However, in a log–log based S-N plot, if ⌬sm and ⌬sb are significantly different from those in Fig. 12(c), a slope change is anticipated since Eq. (1) provides only a peak structural stress measure at the surface with an equal weight on both components. To substantiate this augment, two sets of S-N data for Joints G and G⬘ by Huther et al. [12] and Maddox [13] were processed using both Eqs. (1) and (16). Joint G⬘ is the same as Joint G in geometry, but under horizontal tension instead of pure bending. The structural stress based SCFs for the bend component (sb) for Joints G and G⬘ are 1.206 and 0.383, respectively, as indicated in Fig. 14(a), drastically different from each other. The structural stress based S-N plot is shown in Fig. 14(a). It can be seen that the two sets of SN data exhibit two distinct bands separating membranedominant structural stress from the bending-dominant. However, if Eq. (16) is used, Fig. 14(b) shows that the fatigue data from the two drastically different membrane and bending combinations fall into essentially the same band. Fig. 14(c) summarizes the stress intensity factor ranges versus cycles at failure for all joints [Joints A though G in Fig. 1(a) and G⬘]. The results in Fig. 14(c) clearly suggest that the S-N data for Joints A through G⬘ can be effectively consolidated into a single band, even though both the joint types and loading are significantly different from one another. In calculating the structural intensity factor ranges for each of the joint types, Eq. (16) was used throughout the process once the meshinsensitive structural stresses are obtained for all joint types and loading mode. This can also be viewed as a validation of the structural stress based transformation process shown in Fig. 13 and its potential usefulness for general applications. Further discussions on the use of the relation in Eq. (13) or Eq. (16) and more refined definition of ⌬K parameters for consolidating existing S-N data be reported separately [16], in which structural stresses at notches
Fig. 14. Effects of bending and membarane components on S-N curve slope: (a) structural stress range; (b) stress intensity factor range; (c) stress intensity factor range for joints A–G⬘.
and detailed K solutions were also used [16]. It should be noted that the structural stress based ⌬K⫺N interpretation for fatigue behavior has been successfully used for consolidating S-N data of spot welds under drastically different loading conditions (e.g., [17,18]), where ⌬K, however, can be expressed in a simple form without explicit crack size dependency, as initially proposed by Zhang [17]. A recent independent evaluation on its effectiveness can be found in Gao et al. [19].
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5. Summary 1. A structural stress definition has been presented as an effective measure of the stress state at welded joints. The structural stress parameter in terms of both membrane and bending components is consistent with elementary structural mechanics theory and can be readily calculated in a mesh-insensitive manner from conventional finite element models. Its applications in characterizing weld toe cracking for a series of joint types were demonstrated. 2. The structural stress based interpretation of existing S-N data considered here suggests that transferability of S-N data among different joint types and loading conditions can be established. 3. The relative bending to membrane ratio of the structural stress components calculated at weld toe can provide additional insight on the S-N curve behavior, particularly for those joints tested under drastically different remote loading conditions. A simplified stress intensity factor range parameter (an integrated average) proposed in this study has demonstrated its potential in correlating such S-N curve behaviors in terms of the membrane and bending effects. 4. The structural stress procedure can be viewed as a transformation process that maps a complex stress state at a joint situated in an actual structure subjected to complex loading mode onto a simple structural stress state represented by membrane and bending components. Consequently, stress intensity factors can be readily calculated using existing solutions for a simple crack problem for various welded joints once the structural stress components become available.
[3] [4] [5]
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[7]
[8]
[9] [10]
[11] [12]
[13]
[14] [15] [16]
[17] [18]
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on fatigue design of welded joints and components. Welding in the World 1997;39(5):272–8. Code of practice for fatigue design and assessment of steel structures. BS7608, British Standards Institution, 1993. Design of steel structures—Part 1-1. ENV 1993-1-1. Eurcode 3, European Committee for Standardization, Brussels, 1992. Radaj D. Review of fatigue strength assessment of non-welded and welded structures based on local parameters. International Journal of Fatigue 1996;18(3):153–70. Lawrence FV, Mattos RJ, Higashida Y, Burk JD. Estimating the fatigue crack initiation life of welds. ASTM STP 1978;648:134–58. Stress determination for fatigue analysis of welded components. IIS/IIW 1221-93. Abington, Cambridge: Abington Publishing, 1993. Niemi E. Recommendations concerning stress determination for fatigue analysis of welded components. IIW-1458-92/XV-79792, 1992. Niemi E, Tanskanen P. Hot spot stress determination for welded edge gussets. IIW XIII-1781-99. 1999. Dong P. A robust structural stress parameter for characterizing fatigue behavior of welded joints. SAE Technical Paper Series: No. 2001-01-0086, Fatigue and Failure of Spot Welds and Weld Joints, March, 2001. Dong P, Zhang J, Hong JK. Structural stress calculation scheme. Battelle’s Patent Application (pending), 2000. Huther I, Gorski S, Lieurade HP. Longitudinal non-loaded welded joints geometric stress approach. Welding in the World 1999;43(3):20–6. Maddox SJ. Influence of tensile residual stresses on the fatigue behavior of welded joints in steel. Residual stress effects in fatigue. ASTM STP 1982;776:63–96. Dong P, Hong JK. A master fatigue damage curve approach for welded joints. Under preparation, 2001. Tada H, Paris P, Irwin G. The stress analysis of cracks handbook. 2nd ed. St Louis, MO: Paris Productions, 1985. Dong P, Hong JK, Cao Z. A mesh insensitive structural stress procedure for fatigue evaluation of welded structures. International Institute of Welding (IIW), IIW Doc. XIII-1902-01/XV1089-01, July, 2001. Zhang S. Stress intensities at spot welds. International Journal of Fracture 1997;88:167–85. Zhang J, Dong P, Gao Y. Evaluation of stress intensity factor based predictive technique for fatigue life of resistance spot welds. SAE Technical Paper Series: No. 2001-01-0830, Fatigue and Failure of Spot Welds and Weld Joints, March, 2001. Gao Y, Chucas D, Lewis C, McGregor IJ. Review of CAE fatigue analysis techniques for spot-welded high strength steel automotive structures. SAE Technical Paper Seires: No. 2001-01-0835, Fatigue and Failure of Spot Welds and Weld Joints, March, 2001.