fatigue calculations-ansys

16
Verity TM Weld Fatigue Method in Fe-Safe TM Using ANSYS Dr. Pingsha Dong Center for Welded Structures Research BATTELLE Columbus, OH [email protected] Abstract It is well known that stress concentration in welded joints (and notched structures) dominates fatigue behavior of welded structures. However, traditional finite element methods are not capable of consistently capturing the stress concentration effects on fatigue behavior due to their mesh-sensitivity in stress determination at welds resulted from notch stress singularity. Any use of an artificial radius is too arbitrary for the results to be reliable in fatigue design in practice. In this presentation, a robust stress analysis procedure recently developed at Battelle and extensively validated by various industries will be presented. The method is called Verity TM mesh-insensitive structural stress method which serves as a FE post-processing procedure to commercial FE packages such as ANSYS. The VerityTM method has been integrated into fe- safe TM and available from Safe Technology Ltd. The method is based on the mapping of the balanced nodal forces/moments along an arbitrary weld line available from a typical finite element run into the work-equivalent tractions (or line forces/moments). In doing so, a complex stress state due to notch effects can then be represented in the form of a simple stress state in structural mechanics in terms of through-thickness membrane and bending components at each nodal location. The resulting structural stress calculations are mesh-insensitive, regardless of element size, element type, integration order used, as long as the overall geometry of a component is reasonably represented in a finite element model. A series of simple and complex examples will be represented to demonstrate the mesh- insensitivity of the structural stress method, covering MIG seam welds, laser welds, resistance spot welds, etc. In addition to its mesh-insensitivity, the effectiveness of the structural stress parameter has been further validated by collapsing several thousands of fatigue tests available from literature into a single curve, referred to as the master S-N curve. Additional applications of the structural stress method will may also be touched upon. These include: Treatment of low cycle fatigue Treatment of multi-axial fatigue Solder fatigue in electronic packaging

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Page 1: Fatigue calculations-Ansys

VerityTM Weld Fatigue Method in Fe-SafeTM Using ANSYS Dr. Pingsha Dong

Center for Welded Structures Research BATTELLE

Columbus, OH [email protected]

Abstract

It is well known that stress concentration in welded joints (and notched structures) dominates fatigue behavior of welded structures. However, traditional finite element methods are not capable of consistently capturing the stress concentration effects on fatigue behavior due to their mesh-sensitivity in stress determination at welds resulted from notch stress singularity. Any use of an artificial radius is too arbitrary for the results to be reliable in fatigue design in practice.

In this presentation, a robust stress analysis procedure recently developed at Battelle and extensively validated by various industries will be presented. The method is called VerityTM mesh-insensitive structural stress method which serves as a FE post-processing procedure to commercial FE packages such as ANSYS. The VerityTM method has been integrated into fe-safeTM and available from Safe Technology Ltd. The method is based on the mapping of the balanced nodal forces/moments along an arbitrary weld line available from a typical finite element run into the work-equivalent tractions (or line forces/moments). In doing so, a complex stress state due to notch effects can then be represented in the form of a simple stress state in structural mechanics in terms of through-thickness membrane and bending components at each nodal location. The resulting structural stress calculations are mesh-insensitive, regardless of element size, element type, integration order used, as long as the overall geometry of a component is reasonably represented in a finite element model.

A series of simple and complex examples will be represented to demonstrate the mesh-insensitivity of the structural stress method, covering MIG seam welds, laser welds, resistance spot welds, etc. In addition to its mesh-insensitivity, the effectiveness of the structural stress parameter has been further validated by collapsing several thousands of fatigue tests available from literature into a single curve, referred to as the master S-N curve. Additional applications of the structural stress method will may also be touched upon. These include:

• Treatment of low cycle fatigue • Treatment of multi-axial fatigue • Solder fatigue in electronic packaging

Page 2: Fatigue calculations-Ansys

Introduction Fatigue design and evaluation of welded joints are typically carried out by weld classification approach in which a family (theoretically infinite) of parallel nominal stress based S-N curves are used according to joint types and loading modes [1]. Extrapolation-based hot spot stress methods offer the potential to reduce the number of the S-N curves as required in weld classification approach, which has gained an increasing popularity in offshore and marine applications [2-4]. Although extrapolation-based hot spot stress procedures have been used for tubular structures for many years, their applications in plate joints such as ship structures were only investigated during the recent past, as recently summarized Fricke [4]. As shown in Fig. 1 for a plate to I-beam joint, the hot spot stress based SCF using three extrapolation techniques demonstrate the variability showed a wide scatter band [4].

0.8

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ExperimentShell4Shell4Shell8Shell8Shell4(css)Shell4Shell8w1Solid20wSolidpw2Solid20w4Solid8w4Solid8w2Solid20w(f)

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ress

es

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es

Distance from Weld ToeExtrapolationProcedures

SCF

Attachment

Base Plate

Fig. 1: Comparison of FEA surface stress distributions using various modeling procedures and extrapolation-based hot stress SCF results at the weld toe on attachment plate [4]

One of the unique issues in using any extrapolation-based hot spot stress procedures in plate structures is that the surface stress gradients on which any extrapolation techniques are based upon are that the stress gradients are more localized in plate structures than in tubular structures, as illustrated by Dong and Hong [5], as shown in Fig. 2. In this figure, the surface stresses normal to weld (indicated by arrows) are normalized by the respective nominal bending stresses. The surface stress gradients shown become increasingly localized as the joint type changes from tube-to-tube, tube-to-plate, and plate-to-plate joints. As a result, extrapolations using 0.5t/1t, 0.5t/1.5t, or nodal value at 0.5t [1-4] yield a unity in SCF, i.e., the nominal stress. If the finite element (FE) mesh is not refined enough, or not converged yet as referred by Healy [6], the extrapolated hot spot stresses tend vary, depending on the element sizes, types, joint types, and loading mode, as illustrated in Fig. 1 [4].

Page 3: Fatigue calculations-Ansys

In all the global based stress analysis procedures (such as nominal stress, extrapolation based hot spot stresses, etc.) for fatigue evaluation purposes [1-6], the ultimate goal is to identify an appropriate stress parameter which , being able to be consistently calculated in practice, can be used to effectively correlate S-N data from various joint types and loading modes. This can be restated as both the necessary and sufficient conditions for seeking a global stress-based fatigue correlation parameter as follows:

(a) A global stress parameter must be able to be calculated consistently with a minimum mesh-sensitivity

(mesh sizes, element shapes, element types, etc.) at a fatigue prone location such as at weld toe; (b) Such a stress parameter must be demonstrated to be capable of correlating different fatigue behaviors

(such as S-N data) observed in various joint types, loading modes, etc. Obviously, nominal stress definition, if applicable for some joint configurations, satisfies the necessary conditions (a), since it can be calculated by simple formulae, i.e., without mesh-sensitivity. However, the nominal stress definition, as it is well known, does not satisfy the sufficient conditions (b), since it cannot be used to correlate S-N data from various joint types and loading modes. This is why a family of infinite number of essentially parallel S-N curves has been used with respect to the nominal stress parameter as shown in Fig. 3 [1].

The very fact that those S-N curves (Fig. 3) are essentially parallel to one another, as observed from a large mount of fatigue data, suggests the existence of a master S-N curve. A scaling parameter that correctly measures the stress concentration in various welded joint types and load modes should be able to collapse all the parallel S-N curves in Fig. 3 into a single master S-N curve. It is the purpose of this paper to present such an approach by formulating an effective global stress parameter which can be used as a basis to establish such a master S-N curve. In this context, the nodal force (always implying moments in this paper) based mesh-insensitive structural stress method (5-9) will be briefly highlighted for its consistency in stress concentration characterization as required by the necessary conditions stated above. Then, the nodal force based (referred as structural stress method throughout this paper) structural stresses are shown to posses a unique property which can be used for a rapid estimation of the stress intensity factors (K) in an arbitrary joint within fracture mechanics context. As a result, a two-stage crack growth model has been proposed and validated by a large amount of experimental data. The two-stage growth laws unifies the

t

t

t

t

Fig.2: Comparison of normalized surface stress distributions between tubular joint and plate joints: (a) tubular joint under brace tension; (b) tube-to-plate joint; (c) plate-to-plate joint; (d) normalizedsurface stress distribution (with respect to bending stress in chord for tubular joint and in base plate for tube-to-plate joint) normal to weld toe.

(c) Plate to plate T- Joint

(d) Surface Stress Distributions

F

0

1

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3

4

5

6

0.0t 1.0t 2.0t 3.0t 4.0t

Distance from Weld Toe

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ized

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face

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ess Tube to Tube

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ess Tube to Tube

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Cruciform - Tension

Tube-to-tubeTube-to-platePlate-to-plate

Tube-to-tubeTube-to-platePlate-to-plate

F

(b) Tube-to-plate T joint

(a) Tube-to tube T joint [9]

F

t = 20mm

Page 4: Fatigue calculations-Ansys

Fig. 3: A family of infinite number of fatigue S-N (or “FAT”) curves recommended by IIW for welded joints using nominal stress parameter [1]

Fig. 4: Through-thickness structural stresses definition: (a) local stresses from FE model; (b) structural stress or far-field stress ; (c) self-equilibrating stress and structural stress based estimation with respect to t1 (dashed lines)

(a)

Weld

t σx (y)τ (y)

σm σbσm σb

Weld

t τm

(b)

mσ bσ

Weld

t

(c)

t1

conventionally “short crack” anomalous crack growth with long cracks. By integrating the two-stage crack growth model, a unique scaling parameter encompassing the structural stress based stress concentration effects, loading mode effects, and thickness effects is then formulated and validated by a massive amount of historical weld fatigue S-N data from 1947 to present.

THE STRUCTURAL STRESS METHOD The essence of the new structural stress method was based on the following considerations for fatigue

evaluations of welded joints:

(a) It was postulated that stress concentration at a fatigue prone location, such as a weld toe as shown in Fig. 4a, can be represented by an equilibrium-equivalent simple stress state (as shown in Fig. 4b) and self-equilibrium stress state (as shown in Fig. 1c). The former describes a stress state corresponding to an equivalent far field stress state in fracture mechanics context [4,6], or simply, a generalized nominal stress state at the same location, while the latter can be estimated by introducing a characteristic depth t1 as shown in Fig. 1 (dashed lines), as discussed in detail in [8];

(b) Within the context of displacement-based finite element methods, the balanced nodal forces and

moments within each element automatically satisfy the equilibrium conditions at every nodal position. Therefore, the equilibrium-equivalent structural stress state in the form of membrane and bending can be calculated by using the nodal forces/moments at a location of concern.

Page 5: Fatigue calculations-Ansys

Fig. 5: The structural stress calculation procedures for an arbitrarily curved weld using shell/plate element models

y’x’

N1

N2

N3NiE1

E2

E3Ei

WeldNode at Weld Toe of Interest

x

y

zx

y

z

Shell/Plate Element Procedures

However, in order to calculate the structural stresses in terms of membrane and bending components, line forces and moments must be properly formulated by introducing work-equivalent arguments as discussed in [8-9]. As an example of such formulation for a closed weld line (i.e., two ends of an arbitrarily curve weld overlap each other, such as in a tubular joint), the nodal forces can be related to line forces along an arbitrarily curved weld as:

+

+

+

+

=

−−−−−

−−

− 1

3

2

1

1221

3322

2211

1111

1

3

2

1

.

.

3)(

60

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...6

...3

)(

06

00

063

)(6

60

63)(

.

.

nnnnn

nn

n f

fff

llll

llll

llll

llll

F

FFF

(1)

In the above equation, a closed weld line (The first node at the weld start is the same node at the weld end) is assumed, such as a tubular joint, i.e., 1FFn = and 1ff n = . The lowercase 121 ,...,, −nfff are line forces along y’. In the matrix on the left hand of Eq. (1), li (i =1, 2, …, n-1) represents the element edge length projected onto the weld toe line from ith element The corresponding line moments can be calculated in an identical manner by replacing balanced nodal forces 121 ,...,, −nFFF in local 'y direction with balanced

nodal moments 121 ,...,, −nMMM with respect to 'x in Eq (1) above, as depicted in Fig. 5. Note that nodal force Fi in Eq. (1) represent the summation of the nodal forces at node i from the adjoining weld toe elements situated on the positive side of 'y axis, as shown in Fig. 5. Before Eq. (1) can be constructed, coordinate transformation for the nodal forces and nodal moments from the global x-y-z to local x’-y’-z’ system must be performed, with x’ traveling along the weld line and y’ being perpendicular to the weld line. All these calculations have been automated as a structural stress post-processor. The linear system of equations described by Eq. (1) can be solved simultaneously to obtain line forces for all nodes along the line connecting all weld toe nodes. Substituting the corresponding nodal moments into Eq. (1), one obtains line moments in the same manner. Then, the structural stress shown in Fig. 4b at each node along the weld (such as weld toe) can be calculated as:

2

6tm

tf x'y'

bms +=+= σσσ (2)

For parabolic plate or shell elements, Eq. (1) can be formulated in an identical fashion with the relationships provided in [8]. In-plane shear can be treated in an identical manner [8].

Page 6: Fatigue calculations-Ansys

CALCULATION EXAMPLES

A tubular T-joint according to a recent round robin study on fracture assessment [5,9,10] is shown in Fig. 6a, where a detailed strain gauge measurements were also collected for deriving hot spot stress based stress concentration at the saddle positions as shown. To demonstrate the effectiveness of the present structural stress procedures, four shell element models with drastically different element sizes near the tube-to-tube weld are shown in Fig. 6b, varying approximately from 0.25tx0.25t, 0.5tx0.5t, 1tx1t, to 2tx2t. Note that the weld was not modeled at the tube-to-tube intersection in simplifying mesh generation efforts in the present mesh-sensitivity study.

Fig. 6c summarizes the structural stresses along the weld toe on the chord side obtained from the four shell models shown in Fig. 6b. Since the structural stresses along the weld possess the quarter symmetry, Fig. 6c shows only the results for a quarter of the weld length measured from the saddle point shown in Fig. 6b. The maximum structural stress concentration occurs at the saddle position. Within the angular span of 90o along the 3D curved weld from saddle to crown positions, the 2tx2t mesh represents the weld line with only three nodal positions (or about two and half linear elements) as shown by the triangle symbols in Fig. 6c. Therefore, the difference in the structural stress calculations from the 2tx2t mesh is mainly due to the geometric changes at the weld line (tube to tube intersection) resulting from the large linear element sizes used. However, the structural stress based SCF at the saddle position is still within about 5% of the fine mesh case (.25tx.25t). Excluding the 2tx2t mesh, the SCF variations in the other three models are all within the 2% of each other.

Fig. 6: Structural stress calculations for a tubular T joint investigated by Zerbst et al [10]: (a) T-joint geometry and loading conditions; (b) Four FE models with different element sizes; (c) Comparison of the current structural stress results along weld toe at chord

0.25tx0.25t 0.25tx0.25t (a) Tubular T-Joint

Hot Spot

Chord

BraceHot SpotHot Spot

Chord

Brace

(b) Four FE models with different elements sizes at weld location (t=20mm)

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0 30 60 90Angle from Saddle Point (Deg.)

SCF 2tx2t

1tx1t0.5tx0.5t0.25tx0.25t

(c) Structural stress SCF results

SaddleCrown

0.5tx5t 1tx1t 2tx2t

Saddle

0.5tx5t 1tx1t 2tx2t 2tx2t

Saddle

Page 7: Fatigue calculations-Ansys

F =122.95N

S S calculatio n

W/o weld representation

Attachment plate

Base I Beam

F =122.95N

S S calculatio n

W/o weld representation

F =122.95N

S S calculatio n

W/o weld representation

Attachment plate

Base I Beam

0.5tx0.5t 1tx1t

2tx2t

4tx4t

(a) Four FE meshes used

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Bottom Weld Toe(b) Comparison of structural stress distributions

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Top Weld Toe

Bottom Weld Toe(b) Comparison of structural stress distributions

Fig. 7: Mesh-size insensitivity demonstration for a plate to I-beam box joint used in [1] (also see Fig. 1): (a) FE models with drastically different element sizes; (b) comparison of structural stress distributions along weld toe on attachment plate.

As another example, the plate to I-beam joint shown in Fig. 1 from Fricke [1] is analyzed here using the new structural stress procedures discussed in the above. In the models shown in Fig. 7a, the box fillet weld was modeled as simple nodal connections between attachment plate edge and I beam. The weld line in this instance is considered as being open-ended. The virtual node method as discussed in [9] is automatically activated in constructing Eq. (1). Four drastically different element sizes ranging from 0.5tx0.5t to 4tx4t are used in the mesh designs, as shown in Fig. 7a. The structural stress distributions (normalized by the nominal bending stress) calculated along the weld toe on the attachment plate side are shown in Fig. 7b. It can be seen that the variation in the structural stress calculated at the weld end positions is within about 1% for all four cases. The validity of the SCF was demonstrated using models with the fillet weld being properly represented by a row of inclined shell elements [9]. Note that in all four models shown in Fig. 7a, the weld is not modeled for simplicity. As the element sizes change at the beam to attachment intersection, the geometric representation remains the same even if a 4tx4t mesh is used. This is not the case for the tubular T-joint shown in Fig. 6 discussed earlier.

MASTER S-N CURVE FORMULATION In seeking a stress-based scaling parameter to correlate the multiple S-N curves as shown in Fig. 3, it may be assumed that fracture mechanics principles are applicable, implying that crack propagation dominates fatigue lives in welded joints. The validity of such an assumption must be demonstrated by correlating a large amount of S-N test data.

Page 8: Fatigue calculations-Ansys

Structural Stress Based K Estimation

Naturally, the simple candidate fracture mechanics parameter which can be considered for the current purpose is the stress intensity factor K. However, generalized K solutions are not available for welded joints. Fortunately, the structural stress definition (Fig. 3b) is consistent with the far-field stress definition ( ∞σ ) in fracture mechanics. Therefore, the structural stress calculation process can be viewed as a stress transformation process from an actual complex joint in a structure under arbitrary loading to a simple fracture specimen, in which the complex loading and geometry effects are captured in the form of membrane and bending, as shown in Fig. 8. As a result, K for any crack size along the weld can be estimated by using the existing K solution for a simple plate fracture mechanics specimen subjected to both membrane tension and bending, by considering either an edge crack or a surface elliptical crack.

2c

a t

ta

σbσm

ta

σbσm

A general 3D Joint Geometry and Loading Mode

F

F

A Simple 2D Crack Problem

2c

a t

ta

σbσm

ta

σbσm

A general 3D Joint Geometry and Loading Mode

F

F

A Simple 2D Crack Problem

Fig. 8: Structural stress based transformation and K calculationusing a simple fracture mechanics specimen

The detailed derivations and validations can be found in [8]. For demonstration purposes, Fig. 9 shows the validation for considering an edge crack in a T fillet weld. In Fig. 9, the case corresponds to “W/O notch stress” was obtained by directly plugging the structural stress components (membrane and bending) calculated using the present structural stress method into the existing K solution for an edge notch specimen under remote tension and bending, respectively. The case “W/ notch stress” refers to the use of the self-equilibrating part of the stress state (Fig. 3c) which is analytically estimated, as discussed in [8].

Fig. 9: Validations of the structural stress K estimation for T-fillet joint using Glinka’sweight function method [8]: (a) Remote bending; (b) remote tension

0

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Far-Field Stress (Eq. 9)

Weight Function[14,15]

Notch Stress (Eqs. 8&11)

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Far-Field Stress(Eq. 9)

Weight Function[14,15]

Notch Stress (Eqs. 8&11)

aKπσ0 a

Kπσ0

(a) (b)Current solution W/O notch stressGlinka’s weight functionCurrent solution W/ notch stress

Current solution W/O notch stressGlinka’s weight functionCurrent solution W/ notch stress

(a) (b)

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Weight Function[14,15]

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Weight Function[14,15]

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aKπσ0 a

Kπσ0

(a) (b)Current solution W/O notch stressGlinka’s weight functionCurrent solution W/ notch stress

Current solution W/O notch stressGlinka’s weight functionCurrent solution W/ notch stress

0

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Weight Function[14,15]

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Weight Function[14,15]

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aKπσ0 a

Kπσ0

(a) (b)Current solution W/O notch stressGlinka’s weight functionCurrent solution W/ notch stress

Current solution W/O notch stressGlinka’s weight functionCurrent solution W/ notch stress

(a) (b)

Sym.

L

t1/2

t

h

t1/t=1h/t=1

Remote Loading: Pure Tension

Sym.

L

t1/2

t

h

t1/t=1h/t=1

Remote Loading: Pure Tension

Sym.

L

t1/2

t

h

t1/t=1h/t=1

Remote Loading: Pure Tension

Sym.

L

t1/2

t

h

t1/t=1h/t=1

Remote Loading: Pure Tension

Sym.

L

t1/2

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h

t1/t=1h/t=1

Remote Loading: Pure Tension

Sym.

L

t1/2

t

h

t1/t=1h/t=1

Remote Loading: Pure Tension

Page 9: Fatigue calculations-Ansys

It can be seen that without considering the notch stress (or self-equilibrating part of the stress state), the current solution provides an accurate K estimation for crack size a/t larger than about 0.1. With the use of the notch stress effects, K can be calculated for any given infinitesimally small a/t. The current solution in Fig. 9 is higher for small a/t than the weight function solution from Glinka (see [8] for detail). This is due to the fact that Glinka introduced a small weld toe radius in performing the finite element stress calculation to avoid the mesh-sensitivity. In the present calculations, the weld toe radius was assumed to be zero, i.e., simulating a sharp notch at the weld toe.

A Two-Stage Growth Model

The non-monotonic K behavior as a function of a/t shown in Fig. 9 is characteristic among all joint types investigated [8]. As the crack size a/t becomes smaller than about a/t ~ 0.1, the elevated K is attributed to the dominance of the notch stresses at weld toe. It can then be postulated that both the short crack and long crack growth processes may be characterized by the two distinct stages of the K behavior as a crack propagates from a/t < 0.1 to a/t > 0.1. Along this line, it can be argued that two stages of stress intensity solutions in the form of the notch-stress dominated

1.0/ <∆ taK and far-field stress dominated 1.0/ >∆ taK can be separated to characterize the full range crack growth behavior from 1.0/0 << ta (“small crack”) to 0.1≤ a/t 1≤ (“long crack”). Here, the term ∆K refers to the stress intensity factor range corresponding to remote stress range. It then follows:

[ ]1.0/21.0/1 )()( >≤ ∆×∆= tata KfKfCdNda (3)

By introducing a stress intensity magnification factor Mkn in dimensionless form and assuming a power-law form of the two stage crack growths corresponding to 1.0/1 )( ≤∆ taKf and 1.0/2 )( >∆ taKf , respectively, Eq. (3) can be re-written as:

mn

nkn KMCdN

da )()( ∆= (4)

The terms Mkn and Kn are defined below:

) and thickness throughon (basedeffects) notch local with(

bmtt

nkn K

KMσσ

= (5)

signifying the notch-induced magnification of the stress intensity factors as a/t approaches zero. The constants n represents the crack growth exponent for the first stage of the crack growth and m the conventional Paris law exponent, both of which are to be determined by experimental crack growth rate data.

The validation of the two-stage growth model is shown in Fig. 10. The crack growth data were taken from well-known short crack growth data by Tanaka and Nakai [11] and Shin and Smith [12]. Without relying on any crack closure arguments, all the so called anomalous crack growth data in Fig. 10a are collapsed into single straight data band in Fig. 10b, with a unified slope of m=3.6. Note that the first exponent in Eq. (4) was empirically determined as n=2. More detailed discussions on the notch stress formulation and the two stage growth law can be found in [13].

Page 10: Fatigue calculations-Ansys

1.E-12

1.E-10

1.E-08

1.E-06

1.E+01 1.E+02 1.E+0

delta K, MPa*m^1/2

da/d

N, m

/cyc

leSEN-0.4mm-SteelDEN-0.4mm-SSDEN-0.71mm-SSCN-s=60-SteelCN-s=76-SteelCN-s=160-Steel

SEN

DEN

CN

(Shin, 1988)

(Tanaka, 1983)

(b) Notch Structural Stress Based Two Stage Growth Model

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-01 1.E+00 1.E+01 1.E+02Kn, MPa*(m)^.5

SEN - 0.4mm Steel

CN-s=60-SteelCN-s=76-Steel

CN-s=160-SteelDEN-0.71mm-SS

DEN-0.4mm-SS

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-01 1.E+00 1.E+01 1.E+02Kn, MPa*(m)^.5

SEN - 0.4mm Steel

CN-s=60-SteelCN-s=76-Steel

CN-s=160-SteelDEN-0.71mm-SS

DEN-0.4mm-SS

delta Kn MPa*m**1/2

(b) Notch Structural Stress Based K and Two Stage Growth Model (Eq. 19)

(b) Notch Structural Stress Based Two Stage Growth Model

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-01 1.E+00 1.E+01 1.E+02Kn, MPa*(m)^.5

SEN - 0.4mm Steel

CN-s=60-SteelCN-s=76-Steel

CN-s=160-SteelDEN-0.71mm-SS

DEN-0.4mm-SS

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-01 1.E+00 1.E+01 1.E+02Kn, MPa*(m)^.5

SEN - 0.4mm Steel

CN-s=60-SteelCN-s=76-Steel

CN-s=160-SteelDEN-0.71mm-SS

DEN-0.4mm-SS

delta Kn MPa*m**1/2

(b) Notch Structural Stress Based K and Two Stage Growth Model (Eq. 19)

da/d

N*(

1/M

kn2 )

, m/c

ycle

(a)

(b)

m

1

Fig. 10: Consolidation of short crack growth data from various specimen types and notch geometries [11,12] using the current two stage growth model with n = 2: (a) da/dN versus nominal ∆K range referred as anomalous crack growth in [11,12]; (b) current two-stage growth model

Equivalent Structural Stress Parameter

The two-stage crack (Eq. 4) with two stage growth exponents being 2=n and 6.3=m ) can be integrated as

∫=

→ ∆=

faa

amn

kn KMCdaN

0 )()( (6)

As discussed in [8,13], an extensive investigation of Mkn for various joint types showed that it can be approximated by a single curve as a function of a/t for all joint types once the denominator in Eq. (5) is formulated using the mesh-insensitive structural stress.

Note that the integral in Eq. (6) is not very sensitive to the final crack size af, and therefore, can be written in a relative crack length form as:

Page 11: Fatigue calculations-Ansys

)()(1)()(

)/( 211/

0/

rItCKMC

tatdN ms

mta

tamn

kni

−−=

∆⋅⋅=∆

= ∫ σ (7)

where )(rI is a dimensionless function of bending r ( sbr σσ ∆∆= / ) after performing the following integration for a given m:

∫=

−−

=1/

0/ )()()()(

)/()(ta

tam

bmmn

kni

taf

tafr

tafM

tadrI

Then, Eq. (7) can be expressed in terms of N once the dimensionless I(r) function is known:

mmmm

ms NrItC

112

21

)(−

−−

⋅⋅⋅=∆σ (8)

Eq. (8) uniquely describes a family of an infinite number of structural stress based S-N curves ( Ns −∆σ ) as a function of thickness effects (t), and bending ratio effects r. If Eq. (8) provides a good representation of the fatigue behavior of welded joints, an equivalent structural stress parameter can be defined by normalizing the structural stress range sσ∆ with the two variables expressed in terms of t and r on the right hand side of Eq. (8):

mmm

ss

rItS 1

22

)(⋅

∆=∆ −

σ (9)

where the thickness term mmt 2/)2( − becomes unity for t =1 (unit thickness) and therefore, the thickness t can be interpreted a ratio of actual thickness t to a unit thickness, rendering the term dimensionless. With this interpretation, the equivalent sS∆ retains a stress unit. It is worth noting that the equivalent structural stress parameter described by Eq. (9) captures the stress concentration effects ( )sσ∆ , thickness effects (t), and loading mode effects (r) on fatigue behavior.

Initial Crack Size Effects

Before Eq. (9) can be used to construct a single master S-N curve for welded joints, assumptions in performing the integration and the effects of Mkn on )(rI must be quantified. It is well known that initial crack size ( tai / ) can make a significant difference in the final life prediction based on fracture mechanics as described in Eqs. (6-8).

Page 12: Fatigue calculations-Ansys

The effects of a series of assumed initial crack sizes are shown in Fig. 11 by using the edge crack based K solution. Note that )(rI is presented as mrI /1)( after considering the exponent 1/m in Eq. (11) to facilitate the comparison between Figs. 11a and 11b. Without considering the local notch effects, i.e., Mkn, different initial crack sizes tai / produce significantly different mrI /1)( curves as a function of r. Once Mkn

is considered, the dependency of the )(rI on initial crack size tai / becomes insignificant, particularly for

the two cases with small initial crack size ( tai / ), as shown in Fig.11b. The increase in mrI /1)( is about 8.5% as r increases from r = 0 (pure membrane) to r = 1 (pure bending) under load controlled conditions. This implies that with a strong notch effects in typical welded joints characterized by Mkn, the usual initial crack size effects on life predictions observed in typical fracture mechanics specimens without stress riser are significantly diminished in welded joints. Based on Fig. 11b, tai / =0.001 will be used in the rest of this paper.

Fig. 11: Comparisons of I(r) functions with and without Mkn and effects of initial ai/t(edge crack solution): (a) without considering Mkn; (b) with consideration Mkn

with considering Mkn

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

r

I(r)^

(1/m

)

ai/t=0.0001ai/t=0.001ai/t=0.01

w/o considering Mkn

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

r

I(r)^

(1/m

)

ai/t=0.0001ai/t=0.001ai/t-0.01

(a)

(b)

Fig. 12: Illustration of some representative joint types analyzed in this investigation for the development of the master S-N curve

t

Joint Gb (t=20mm)

t

Joint B(t=12.7mm), Joint B(Kihl)(6.35mm),13/10/8AW(13mm), 50/50/16AW(50mm), 50/50/16AW(DW)(50mm),100/50/16AW(100mm),100/50/16AW(QT Steel)(100mm)

t

Joint C(t=12.7mm)

t

Joint D(t=12.7mm)

t

Joint F (t=12.7mm), Joint F(Rorup)(12.5mm)

t

Bell (t=16mm)

Double Edge Gusset (90mm)

t

Joint G’ (t=12.7mm)

t

Joint-Cb(Booth)(t=38mm), Joint-Cb(Pook)(38mm)

t

Joint E (t=12.7mm)

t

t = 5-80mm

Page 13: Fatigue calculations-Ansys

S-N Data Correlation

The ultimate test for proving if the equivalent stress parameter (Eq. 9) is valid is to demonstrate if a large amount of experimental S-N data can be collapsed into a single narrow band. In doing so, a massive amount of S-N data (over 800 fatigue tests) from drastically different joint geometries, plate thicknesses, and loading modes were collected from literature and published reports, as highlighted in Fig. 12.

1.E+01

1.E+02

1.E+03

1.E+04 1.E+05 1.E+06 1.E+07

Life

Nom

inal

Stre

ss R

ange

, MPa

AT122 AT140 AT180 AT222 AT240AT280 Bell Joint G' Joint D Detail_3(Fricke)Joint F joint F(rorup) Joint-Cb(Booth) Joint-Cb(Pook) AC110AC122 AC140W AC140N AC180 AC210AC222 AC240 AC280 AC310 AC340AC380 AC422 AC440 Joint C Joint B13/10/8 AW 50/50/16 AW 50/50/16 AW (DW) 100/50/16 AW 100/50/16 AW (QT)Joint E Gurney -LW2 HHI_3 9mm-w25 9mm-w509mm-w100 9mm-w160 20mm-w25 20mm-w50 20mm-w10020mm-w160 40mm-w25 40mm-w50 40mm-w100

1.E+02

1.E+03

1.E+04

1.E+04 1.E+05 1.E+06 1.E+07

Life

Equi

vale

nt S

truct

ural

Stre

ss R

ange

, MP

a

AT122 AT140 AT180 AT222AT240 AT280 Bell Joint G'Joint D Detail_3(Fricke) Joint F joint F(rorup)Joint-Cb(Booth) Joint-Cb(Pook) AC110 AC122AC140W AC140N AC180 AC210AC222 AC240 AC280 AC310AC340 AC380 AC422 AC440Joint C Joint B 13/10/8 AW 50/50/16 AW50/50/16 AW (DW) 100/50/16 AW 100/50/16 AW (QT) Joint EGurney -LW2 HHI_3 9mm-w25 9mm-w509mm-w100 9mm-w160 20mm-w25 20mm-w5020mm-w100 20mm-w160 40mm-w25 40mm-w5040mm-w100

mmm

ss

rItS 1

22

)(⋅

∆=∆ −

σ

1.E+02

1.E+03

1.E+04

1.E+04 1.E+05 1.E+06 1.E+07

Life

Equi

vale

nt S

truct

ural

Stre

ss R

ange

, MP

a

AT122 AT140 AT180 AT222AT240 AT280 Bell Joint G'Joint D Detail_3(Fricke) Joint F joint F(rorup)Joint-Cb(Booth) Joint-Cb(Pook) AC110 AC122AC140W AC140N AC180 AC210AC222 AC240 AC280 AC310AC340 AC380 AC422 AC440Joint C Joint B 13/10/8 AW 50/50/16 AW50/50/16 AW (DW) 100/50/16 AW 100/50/16 AW (QT) Joint EGurney -LW2 HHI_3 9mm-w25 9mm-w509mm-w100 9mm-w160 20mm-w25 20mm-w5020mm-w100 20mm-w160 40mm-w25 40mm-w5040mm-w100

mmm

ss

rItS 1

22

)(⋅

∆=∆ −

σ

Fig. 13: Correlation of existing S-N data for various joint types, loading modes, and plate thicknesses: (a) nominal stress range versus life; (b) equivalent structural stress range versus life.

(a)

(b)

For each set of the specimens and fatigue tests, the structural stress calculations were performed under given loading conditions and failure criteria. The results are summarized in Fig. 13 in terms of both nominal stress range and equivalent structural stress range as given in Eq. (9). A wide scatter can be seen in Fig. 13a, which is expected. Once the equivalent structural stress range is used according to Eq. (9), the all the S-N data are collapsed into a narrower band, regardless of the diverse joint types, plate thicknesses, and load modes under which the fatigue tests were conducted over about 40 years. The same methodology has been proven with the same effectiveness in correlating tubular and pipe/vessel joints [8].

By regression analysis, the mean line of Fig. 13b can be represented in the form of:

333.31

'1

12

2 16308)(

−−

− ×==∆

NCNrIt

m

mmm

sσ (10)

Page 14: Fatigue calculations-Ansys

with the stress unit in MPa and thickness in mm and m = 3.6. In Eq. (10), '/1 m represents the negative slope of the master S-N curve in Fig. 13b. Note that although in various publications, 'mm = (=3 for steel welds) is often assumed. In this investigation, it is found that are close, but not necessarily the same. For simple fatigue test specimens, nominal stresses are often well-defined, then,

nsss SSCF ∆×=∆σ

in which ssSCF signifies the structural stress based SCF and nS∆ the typical nominal stress range definition.

For a given simple joint specimen of interest, the structural stress based SCF can be calculated using the present structural stress procedure. Then, the nominal stress based S-N behavior can be predicted by using the master S-N curve from Eq. (10) as:

'1

12

2

)( m

ss

mmm

n NSCF

rItCS−

=∆ (11)

10

100

1000

1.E+04 1.E+05 1.E+06 1.E+07Life

Nom

inal

Str

ess

Ran

ge, M

Pa

13/10/8 AW 50/50/16 AW50/50/16 AW (DW) 100/50/16 AW100/50/16 AW (QT) 13mm - Prediction50mm -Prediction 100mm -Prediction

13mm

100mm

50mm

(a) Predicted nominal stress range versus N for Data by Maddox[13]

10

100

1000

1.E+04 1.E+05 1.E+06 1.E+07Life

Nom

inal

Str

ess

Ran

ge, M

Pa

13/10/8 AW 50/50/16 AW50/50/16 AW (DW) 100/50/16 AW100/50/16 AW (QT) 13mm - Prediction50mm -Prediction 100mm -Prediction

13mm

100mm

50mm

(a) Predicted nominal stress range versus N for Data by Maddox[13]

10

100

1000

1.E+04 1.E+05 1.E+06 1.E+07Life

Nom

inal

Str

ess

Ran

ge, M

Pa

AT122 AT140AT180 AT222AT240 AT280#REF! AT140-PredictionAT180-Prediction AT222-PredictionAT240-Prediction AT280-Prediction

22mm

80mm

40mm

(a) Predicted nominal stress range versus N for Data from SR202[14]

10

100

1000

1.E+04 1.E+05 1.E+06 1.E+07Life

Nom

inal

Str

ess

Ran

ge, M

Pa

AT122 AT140AT180 AT222AT240 AT280#REF! AT140-PredictionAT180-Prediction AT222-PredictionAT240-Prediction AT280-Prediction

22mm

80mm

40mm

(a) Predicted nominal stress range versus N for Data from SR202[14]

Fig. 14: The use of the master S-N curve (mean line) for the prediction of nominal stress range versus N curves generated by Maddox [13] and SR202 [14].

Page 15: Fatigue calculations-Ansys

Both C and m’ are given in Eq. (10) based on the master S-N database generated in this investigation. Two examples are given in Fig. 14. The mean nominal stress S-N curves for various plate thicknesses are predicted using Eq. (11) for the cruciform joint tests (under remote tension loading) by Maddox [13] and T-fillet joint tests (under 3-point bending) reported in [14]. A good correlation between the test data and those predicted by the master S-N (Eq. 10) curve is evident.

Conclusion The mesh-insensitive structural stress parameter not only can be calculated consistently with the

demonstrated mesh insensitivity, but also has been shown to be an effective fatigue parameter to correlate the fatigue behavior of welded joints regardless of joint geometries, loading modes and plate thicknesses. Furthermore, the structural stress parameter can be directly related to the far-field stress in a fracture mechanics context. As a result, a rapid K estimation scheme has been demonstrated to be effective for analyzing arbitrary joints in plate structures, as well as other joint types such as tubular joints, pipe and vessel welds. With the aid of fracture mechanics principles, a master S-N curve approach has been developed by introducing an equivalent structural stress parameter, which captures three well-known factors that contribute to fatigue behavior in welded joints: (a) the stress concentration due to joint geometry; (b) the loading mode; and (c) the plate thickness. A large amount of S-N data has been collected and correlated by the equivalent structural stress parameter. The present master S-N curve approach can simplify fatigue evaluation procedures for ship structures and significantly reduce testing requirements, since the S-N data transferability in the form of the equivalent structural stress parameter has been established.

References 1. Hobbacher, A., “Fatigue Design of Welded Joints and Components: Recommendations of IIW Joint

Working Group XIII-XV, Abington Publishing, Abington, Cambridge, 1996. 2. “Fatigue strength Analysis of Offshore Steel Structures,” DNV RP-C203, May 2000. 3. “Guide for the Fatigue Assessment of Offshore Structures,” ABS, April, 2003. 4. Fricke W., “Recommended Hot-Spot Analysis Procedure for Structural Details of FPSO’s and Ships

Based on Round-Robin FE Analysis, ISOPE Proceedings, Stavanger, Noway, June 2001. 5. Dong, P. and Hong, J.K., “Analysis of Hot Spot Stress and Alternative Structural Stress Methods,”

Proceedings of 22nd International Conference on Offshore Mechanics and Arctic Engineering, June 8-13, 2003, Cancun, Mexico.

6. Healy, B.E., “A Case Study Comparison of Surface Extrapolation and Battelle Structural Stress Methodologies,” to appear in Proceedings of the 23rd International conference on Offshore Mechanics and Arctic Engineering, June 20-25, 2004, Vancouver, British Columbia, Canada.

7. Dong, P., “A Structural Stress Definition and Numerical Implementation for Fatigue Analysis of Welded Joints,” International Journal of Fatigue, 23, pp. 865-876, 2001.

8. Dong, P., Hong, J.K., Osage, D., and Prager, M., “Master S-N curve approach for welded components,” Welding Research Council Bulletin, No. 474, December, 2002, New York, New York, 10016.

9. Dong, P., “A Robust Structural Stress Method for Fatigue Analysis of Ship Structures,” Proceedings of the 22nd International Conference on Offshore Mechanics and Arctic Engineering, June 8-13, 2003, Cancun, Mexico.

10. Zerbst, U., Heerens, J., and Schwalbe, K.-H., “The fracture behavior of a welded tubular joint – an ESIS TC1.3 round-robin on failure assessment methods Part I: experimental data base and brief summary of the results,” Engineering Fracture Mechanics, 69, 2002, pp. 1093-1100.

11. Tanaka, K., and Nakai, Y., "Propagation and Non-Propagation of Short Fatigue Cracks at a Sharp Notch," Fatigue of Engineering Materials and Structures, Vol. 6, No.4, pp.315-327, 1983

12. Shin, C.S., and Smith, R.A., "Fatigue Crack Growth at Stress Concentrations- the Role of Notch Plasticity and Crack Closure," Engineering Facture Mechanics, Vol. 29, No.3, pp.301-315, 1988.

13. Dong, P., Hong, J.K., and Cao, Z., “Stresses and Stress Intensities at Notches: ‘Anomalous Crack Growth’ Revisited”, Int. J. of Fatigue, Vol. 25(9-11), pp. 811-825, 2003.

Page 16: Fatigue calculations-Ansys

14. Maddox, S.J., The Effect of Plate Thickness on the Fatigue Strength of Fillet Welded Joints, The Welding Institute, Abington Hall, Abington, Cambridge CB1 6AL, 1987.

15. SR202 of Shipbuilding Research Association of Japan, Fatigue Design and Quality Control for Offshore Structures, 1991 (in Japanese).