05CHAP_5(1)

Gurson Model for Ductile Fracture

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CHAPTER FIVE

GURSON MODEL FOR DUCTILE FRACTURE

5.1. Introduction

Crack initiation and propagation is one of the common failure modes for tubular joints

subjected to tensile loads. However, simulation using finite elements based on continuum

mechanics formulation does not represent the effect of crack initiation and growth, which

violates the continuity and integrity of material and geometry.

Four numerical methods conventionally used to simulate crack initiation and propagation

include: discrete crack model, fracture mechanics, smeared crack model and continuum

damage mechanics. In the discrete crack model, crack develops only along existing

element boundaries and the crack growth depends on the mesh size and orientation

(Cofer and Will, 1992). For the fracture mechanics approach, the crack size and

orientation has to be assumed or known a-prior. In the smeared crack model, the material

stiffness decreases to zero when fracture is detected. The continuum damage mechanics


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approach introduces a damage variable to reflect the amount of damage in the structure

throughout the loading history. The constitutive relationship is modified using this

damage variable (Lemaitre 1985).

The present study adopts the void growth and nucleation approach which was established

by Gurson (1975). The Gurson model simulates the plastic yield behavior of a porous, or

void containing, material. Under plastic deformation, the material strain hardens, and

voids nucleate and grow, and subsequently lead to deformation localization and fracture.

Thomason (1990) reports that ‘all engineering metals and alloys contain inclusions and

second-phase particles, to a greater or lesser extent, and this leads to void nucleation

and growth …’.

This chapter begins with a description of the Gurson model formulation. The next section

discusses the benchmark study on the classical bar-necking problem. The subsequent

section presents the Gurson model simulation in tubular joints, followed by a sensitivity

study on the Gurson model parameters. The findings in this chapter are summarized by

Qian et al. (2005a).

5.2. Gurson Model Formulation

The yield condition in the Gurson model is modified by Tvergaard (1981) by introducing

three qi (i = 1, 2 and 3) factors as shown in Eq. 5.1.

222

1 32 2 cosh (1 ) 02

e m

yy

qfq q f

σ σΦ

σσ⎧ ⎫⎪ ⎪= + − + =⎨ ⎬⎪ ⎪⎩ ⎭

where q3 = q12 (5.1)


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in which, σe refers to the effective stress, σm denotes the hydrostatic pressure, f defines

the void volume fraction and σy is the material yield stress. The yield criterion in Eq. 5.1

becomes von Mises yield criterion when f = 0. For q1 = q2 = q3 = 1.0, the yield function

bears the same form as the original Gurson model. Tvergaard (1981) reports that q1 = 1.5,

q2 = 1.0 and q3 = q12 = 2.25 better represents materials subjected to plain-strain condition.

Instead of f in the original Gurson model, q1f represents the void volume fraction in Eq.

5.1, and magnifies the void volume fraction by a factor of q1 (Thomason, 1990).

The change of void volume arises from two sources: growth of existing voids and

nucleation of new voids (Tvergaard, 1981):

growth nucleationdf df df= + (5.2)

nucleation v pdf A dε= (5.3)

where 2

1exp22

p NNv

NN

fA

ss

ε ε

π

⎡ ⎤−⎛ ⎞⎢ ⎥= − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(5.4)

The plastic nucleation strain follows a normal distribution. The nucleated void volume

depends on the mean plastic nucleation strain, εN, standard deviation, sN, and the void

volume fraction of the nucleating particles, fN (Tvergaard, 1981). These three parameters

depend on material properties and vary with different materials. Void nucleation initiates

once yielding occurs for high strength steel FeE 690 (Arndt and Dahl 1997). fN refers to

the ratio of the void volume in the nucleating particles over the entire volume of the

material and is thus less than the total void volume fraction of the material, which is

normally less than 10% (Thomason, 1990). Otsuka et al. (1987) reports that ductile crack


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initiation occurs when the void volume fraction reaches about 4-6% for bending mode.

The Gurson model is an upper-bound theory (Gurson, 1977) and does not include the

plastic limit load failure of the inter-void matrix (Thomason, 1990).

In this chapter, the load-deformation curves for tubular joints follow the notations below.

• Curve name (G) Analysis using Gurson model

• Curve name (CT) FE model with crack-front simulation

• Curve name_f fine mesh is employed

• Curve name_m medium mesh is employed

• Curve name_c coarse mesh is employed

5.3. Benchmark Study

5.3.1. Bar-Necking Problem

The conventional bar-necking problem has been studied to verify different void growth

and nucleation models by researchers (Mahnken 1999, Needleman 1972, Tvergaard and

Needleman 1984). The FE simulation in the current study employs 2D plane-strain finite

elements. The applicability of the Gurson model in 3D continuum elements (C3D20R in

ABAQUS element library) is verified by comparing the behavior of the tensile bar

simulated using 3D elements with that of 2D axi-symmetric elements (CAX8R in

ABAQUS element library).

Figure 5.1 describes the details of the FE mesh and geometry, with the material and

geometric property obtained from ABAQUS benchmarks manual (2001). Figure 5.1(c)

indicates the Gurson model parameters used in the analysis, with qi (i =1, 2 or 3)


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parameters modified by Tvergaard (1981). Figure 5.1(c) indicates little difference

between 2D and 3D models.

(a) 3D solid model (b) 2D axi-symmetric model (c) Load-deformation curves

Fig. 5.1 3D solid model; (b) 2D axi-symmetric model; and (c) Load-deformation curves; for the bar necking problem.

5.3.2. Model Parameters qi (i = 1, 2 and 3)

The effect of void volume fraction in the Gurson’s yield function depends partly on the

definition of three qi (i = 1, 2 or 3) parameters as introduced by Tvergaard (1981). The

effect of these parameters is thus investigated using 3D tensile bar model.

Tvergaard (1981) reports that the qi parameters depend on the strain-hardening properties.

Faleskog et al. (1998) propose a relationship between strain-hardening and qi values.

According to Bessen et al. (2001), q2 = 1.15 provides the best fit for round bar and plane

strain specimens. Therefore, three values of q2 are selected: q2 = 1.0, 1.15 and 2.0. In

0.0 0.2 0.4 0.6 0.80.0

1.5

3.0

4.5

Tvergaard modelq1 = 1.5, q2 = 1.0, q3 = 2.25

P (u

nit)

δ (unit)

3D model εN=0.30 2D model sN=0.10 fN=0.04

0.5l0 = 4

R=1

Rotating axis


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addition, comparison is carried out between the Tvergaard model and the original Gurson

model. A relatively large value of q1 = 2.0 is also included for comparison.

(a) Effect of q1 (b) Effect of q2

Fig. 5.2 (a) Effect of q1; and (b) Effect of q2; on the tensile bar.

Figure 5.2 compares the effect of qi on the 3D tensile bar model. The effect of qi is

noticeable only after the strength reduction becomes significant. Increasing qi either

magnifies the effect of void volume fraction f or hydrostatic pressure p, which causes

more severe reductions in the tensile strength. In addition, variation of qi has no

observable effect on the initiation of strength reduction in the tensile bar.

5.4. Tubular Joints

5.4.1. Joints with Initial Crack

Two joint test specimens with initial crack: a T-joint with a surface crack (Zerbst et al.,

2002a) and an X-joint with a through-thickness crack (HSE, 1999), are simulated and

0.0 0.2 0.4 0.6 0.80.0

1.5

3.0

4.5

P (u

nit)

δ (unit)

q2=1.0 εN=0.30 sN=0.10 q2=1.15 fN=0.04 q2=2.0 q1=1.5 q3=2.25

0.0 0.2 0.4 0.6 0.80.0

1.5

3.0

4.5

q1=1.0 εN=0.30 sN=0.10 q1=1.5 fN=0.04

q1=2.0 q2=1.0 q3=q12

P (u

nit)

δ (unit)


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compared using the Gurson model. Both joint specimens experience brace axial loads.

Table 5.1 lists the geometric parameters of these two joints. For both joints, the initial

crack locates at the chord saddle.

Table 5.1 Geometry of tubular joints with and without initial crack

Joint Reference Loading d0(mm) β γ τ α

T-joint Zerbst et al. (2002a) Axial 298.5 0.51 7.46 1.0 10.2 Cracked

X_HSE HSE (1999) Axial 572.0 0.48 15.1 0.5 9.1

X1 Axial 407.4 1.0 25.5 1.0 17.5

X2

Sander & Yura (1986) Axial 407.4 0.35 25.5 0.83 4.5

X3 Kang et al. (1998c) OPB 169.0 0.61 11.4 0.85 16.6 Intact

K-joint Wang et al. (2000) Axial 217.4 0.65 24.8 0.75 13.9

5.4.1.1. T-Joint

Zerbst et al. (2002a) report four T-joint tests with the same joint parameters (β = 0.51, γ =

7.46, τ = 1.0, α = 10.2) and different pre-crack geometry. The crack is introduced as a

surface notch at the saddle point. The crack length, 2c, is 46.5 mm for all four joints. The

T-joint with the crack depth, a = 9.94 mm, is selected. Figure 5.3(a) shows the schematic

configuration of a T-joint, with the crack front geometry. The numerical analysis includes

four different models. Figure 5.3(b) shows crack-front modeling of the surface notch,

while Fig. 5.3(c) – (e) shows three continuous FE models which ignore the notch (i.e. an

‘intact’ model), with different mesh density, since the Gurson model depends on values

of plastic strains. The fine mesh utilizes forty elements around the quarter brace-chord

intersection curve in order to ensure the element aspect ratio to be close to 1:1:1.


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The boundary-loading condition of the T-joint assumes pin supports to the chord ends

and applies tensile loads along the brace axis. Figure 5.4 compares the load-deformation

curves of the numerical analyses and test results. In Fig. 5.4(a), the numerical analysis

includes the void growth without the void nucleation property. The initial porosity f0

imposes a critical effect on the joint strength. However, the initial porosity can be

improved during material production process. In spite of the large initial porosity (0.05)

for the crack-front model, no load reduction occurs in the Gurson model with only the

void growth process. Figure 5.4(b) compares joints with and without incorporation of the

void nucleation process. Material softening becomes significant with void nucleation

under large plastic deformations, and consequently causes a slight load reduction.

(a) T-joint configuration

(b) Crack-front modeling (c) Fine mesh (d) Medium mesh (e) Coarse mesh

Fig. 5.3 (a) T-joint configuration; (b) Crack-front modeling; (c) Fine mesh; (d) Medium mesh; and (e) Coarse mesh; for pre-cracked T-joint.

d0

t0

Surface notch

Quarter brace-to-chord

intersection

t1

d1

2c

a

Crack tip


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0.00 0.05 0.10 0.150

4

8

12

f0 = 0.05 f0 = 0.01

q1=1.5, q2=1.15

P/

f yt 02

δ /d0

Test β=0.51 γ=7.46 FE(CT) τ=1.0 α=10.18 FE(CT)(G) d0=298.5mm FE(CT)(G)

0.00 0.05 0.10 0.150

4

8

12

No nucleation fN = 0.10

q1=1.5, q2=1.15

P/f yt 02

δ /d0

Test β=0.51 γ=7.46 FE(CT) τ=1.0 α=10.18 FE(CT)(G) d0=298.5mm FE(CT)(G)

(a) f0 (b) Void nucleation

Fig. 5.4 Comparison of numerical results for the effect of: (a) f0; and (b) Void nucleation.

Figure 5.5(a) compares three fN values on the crack-front model. The Gurson model

depends significantly on the fN value. A large value of fN corresponds to a less ductile

material, and the strength reduction initiates at a very small deformation level. The

comparison between different fN values shows that the difference between fN = 0.10 and

0.20 is much more dramatic than the difference between fN = 0.04 and 0.10. Figure 5.5(a)

incorporates the load-deformation curves for the three meshes of the intact model. For all

three meshes, the Gurson parameters assume the following values: q1 = 1.5, q2 = 1.15 εN

= 0.10, sN = 0.05 and fN = 0.10. The crack-front model develops from the medium mesh

model. This implies that the mesh size further away from the crack-front is similar to the

medium mesh. Similar load-deformation characteristics are observed between the crack-

front model and the medium mesh for the same material parameter fN = 0.10. This

indicates that the mesh refinement in the crack-front may affect the local stress evaluation

around the crack front. It does not, however, impose a significant influence on the global


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joint strength computation since the very refined elements in the crack front do not form

a collapse mechanism along the joint load path. Variation of the material parameters, fN,

may exert a comparatively stronger effect on the joint strength. On the other hand, with

the same material parameters, the effect of void growth and nucleation becomes most

significant in the fine mesh which causes the largest plastic strain and most pronounced

strength reduction. Figure 5.5(b) compares the load-CTOD (Crack-Tip Opening

Displacement) curves. The difference between the numerical analysis and the test

increases as the joint deformation increase, since the current FE model does not simulate

crack propagations.

0.00 0.05 0.10 0.150

4

8

12

fN = 0.04 fN = 0.20 fN = 0.10 fN = 0.10 fN = 0.10 fN = 0.10

P/f yt 02

δ /d0

Test FE_f(G) FE(CT)(G) FE_m(G) FE(CT)(G) FE_c(G) FE(CT)(G)

0.0 0.4 0.8 1.20

1000

2000

3000

P (k

N)

CTOD (mm)

Test β=0.51 γ=7.46 FE τ=1.0 α=10.18

d0=298.5mm

(a) Effect of fN (b) CTOD

Fig. 5.5 Comparison of numerical results for: (a) Effect of fN; and (b) CTOD.

5.4.1.2. X-Joint

The HSE report (1999) describes a series of X-joint tests with and without cracks. One X-

joint with through-thickness crack at the saddle position is selected in this study. The


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crack length extends about 15% of the brace-chord intersection perimeter. The through-

thickness crack is simulated in two ways. In the normal fine FE mesh as indicated in Fig.

5.6(a), eighty elements are utilized to simulate half of the brace-chord intersection, with

the elements corresponding to the crack removed. The element aspect ratio in the brace-

chord intersection area is around 1:1:1. In the other mesh scheme, the crack-front is

simulated in the refined model.

Figure 5.6(b) compares the numerical crack-front model and test results. The tested joint

experiences strength reductions as crack propagates. Without the Gurson’s algorithm, the

FE crack-front model sustains increasing strength within the prescribed deformation.

Different fN values with q1 = 1.57, q2 = 0.97 (as suggested by Faleskog, et al., 1998) and

initial porosity equal to 0.01 are investigated. The Gurson model analysis with a smaller

fN (0.04) value terminates due to the numerical convergence at a deformation level

corresponding to the load reduction in the test. With a larger value of fN (0.20), the

analysis terminates at an even earlier deformation level due to very large deformation of

the crack-tip elements. This proves the inappropriateness of a large fN value. Table 5.2

lists the joint strength for the two joint specimens with crack-front simulations. The

strength definition follows the plastic limit load approach presented in Chapter Four. In

cases where the FE analysis terminates earlier than the plastic limit load approach, the

joint strength refers to the load level at the end of the analysis.

Figure 5.6(c) presents the results of the normal fine FE models. With the absence of the

Gurson model, the crack propagation is not captured, and the joint strength prediction can

be un-conservative if a large deformation limit defines the joint strength. Both the crack-


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front model and normal fine mesh (with the same material properties) show similar

ultimate strength levels compared to the test results. This implies that the simulation of

the crack-front singularity is not necessary in the strength analysis with the Gurson model.

(a) Crack-front and fine mesh

0.00 0.02 0.04 0.06 0.08 0.100

5

10

15

20

termination ofanalysis

β=0.48 γ=15.1τ=0.5 α=9.1 d0=572mm 15% crack

P/f yt 02

δ /d0

Test FE(CT) FE(CT)(G) fN=0.04 FE(CT)(G) fN=0.20

0.00 0.02 0.04 0.06 0.08 0.100

5

10

15

20

termination ofanalysis

β=0.48 γ=15.1τ=0.5 α=9.1 d0=572mm 15% crack

P/f yt 02

δ /d0

Test FE_f FE_f(G) fN=0.10 FE(CT)(G) fN=0.10

(b) Crack-front FE results (c) Fine mesh results

Fig. 5.6 (a) Crack-front mesh and fine mesh; (b) Crack-front FE results; and (c) Fine mesh FE results; for X_HSE joints.

Both cracked joint models demonstrate the importance of the Gurson model parameters.

Although the crack in the T-joint does not cause strength reductions, the inclusion of a

large fN (0.20) magnifies the effect of crack and leads to under-estimations of the joint


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strength. For the X-joint, a large fN (0.20) value terminates the numerical analysis at a

small deformation level with the joint strength lower than the test results, as shown in

Table 5.2. For both joints, the stress evaluation within the crack-front region differs

highly from the normal FE mesh without the crack-front simulation. However, the small

crack-front region does not form a critical failure in the joint load-path. As a result, only

marginal strength difference exists between the crack-front model and normal FE model.

Table 5.2 T- and X_HSE joint strength for varying fN with crack-front simulation

FE results ISO Other’s FE Joint fN

(kN) FE/Test (kN) ISO/test (kN) Other’s/test

Test (kN)

0.04 2246 0.94

0.10 2204 0.92 T-joint

0.20 2077 0.87

2617a

2790b

1.09

1.16 2170 0.91 2397

0.04 1885* 0.97

0.10 1908* 0.98 X_HSE

0.20 1602* 0.82

1559a

2469b

0.80

1.27 2589* 1.33 1945

* Strength taken at the end of analysis aStrength corresponding to first crack (with chord bending effect incorporated if present) bUltimate strength (with chord bending effect incorporated if present)

Table 5.2 compares the current FE predictions against ISO (2001) mean strength

formulation and FE computation from the reported study (Zerbst et al., 2002b & HSE,

1999). The ISO joint strength corresponding to crack initiation under-estimates the

X_HSE joint strength by about 20%. Over-conservatism is observed for X_HSE joint

using ISO ultimate strength formulation. On the other hand, both ISO strength predictions

over-estimate the T-joint strength. The FE prediction presented by Zerbst et al. (2002b)

shows a lower joint strength when plasticity occurs, although the joint strength does not


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indicate any sign of strength reduction. For X_HSE joint, the FE results reported by the

researchers (HSE, 1999) shows increasing joint strength with increasing joint

deformation, since crack effect is not incorporated into the FE modeling.

5.4.2. Intact Joints

Two types of intact joints (X- and K-joints) in reported studies (Sanders and Yura 1986,

Kang et al. 1998c, Wang et al. 2000) are used to verify the Gurson model. Two types of

loading conditions are studied for X-joints: the brace tensile axial loading and OPB

moment. The K-joint is loaded compressively in one brace and supported on the other

member end. Table 5.1 shows the geometrical properties for these joints.

In the selected tests (except for X2), cracks initiate in the loading history, which leads to

reductions in the joint capacity. Since the crack locates in the chord wall, the Gurson

model property is assigned to the welds and chord material. The default Gurson model

parameters are: q1 = 1.5, q2 = 1.15, q3 = 2.25; and the material parameters are: εN = 0.10,

sN = 0.05, fN = 0.10. The values for εN , sN and fN are selected within the nominal range

empirically, since they are not reported in the respective tests. Section 5.5 investigates the

effect of these parameters, through a detailed sensitivity study.

5.4.2.1. X-Joints under Axial Tension

The current study selects two of the X-joints (X1 and X2) reported by Sanders and Yura

(1986), each with two duplicated tests. Similar to the cracked T-joint, three meshes are

generated for the X-joints with different levels of refinement. In the fine mesh, element

aspect ratio is maintained around 1:1:1.


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0.00 0.02 0.04 0.06 0.08 0.100

40

80

120β=1.0 γ=25.5 α=17.5d0=407.4mm

εN=0.10, fN=0.10, sN=0.05

First crack in test B

First crack in test A

P/

f yt 02

δ /d0

X1 (test A) FE_f X1 (test B) FE_f(G)

0.00 0.04 0.08 0.120

5

10

15

20

25

εN=0.10, fN=0.10, sN=0.05

First crack in test DFirst crack in test C

P/f yt 02

δ /d0

X2 (test C) β=0.35 γ=25.5 X2 (test D) τ=0.834 α=4.49 FE_f d0=407.4mm FE_f(G)

(a) X1 (b) X2

Fig. 5.7 Comparison of test and numerical results for: (a) X1; and (b) X2.

Figure 5.7 compares the X-joint behavior with and without the Gurson’s algorithm. The

two duplicated tests for X1 fail by fracture failure. The tests stop once the crack

penetrates through the chord wall, although no sign of load reduction exists. Similar trend

among different meshes is observed as that of the T-joint. Hence, Fig. 5.7 shows only

results from fine mesh, where substantial strength reduction occurs. For both X1 and X2,

the predicted peak loads remain similar to that in the tests, although the corresponding

deformation levels are larger than that in the tests. In X1 tests, the fracture failure is

defined as the through-thickness penetration of the crack in the chord wall, which has not

mobilized the full joint capacity. In X2 tests, no sign of crack failure exists at the end of

the tests. The first crack in all four tests does not limit the joint capacity as shown in Fig.

5.7. This implies that the joints offer a significant amount of reserve strength beyond the

crack initiation. It is the propagation of crack that leads to the loss of joint capacity.


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5.4.2.2. X-Joint under OPB

For X-joint subjected to OPB moment, one side of the chord wall is under compression

and the other under tension. The possible failure modes include the instability associated

with the compression chord wall, and the fracture failure of the tensile chord wall.

Kang et al. (1998c) reported the effect of chord pre-load on the moment capacity of X-

joints. The OPB test without the presence of chord load is selected as the reference in the

current study. In Kang’s test, pure moment is applied through a four-point load

mechanism on the brace, as illustrated in Fig. 5.8(a). In the FE simulation, pure moment

is applied. The measurement of brace rotation is taken consistently within the pure

moment loaded brace corresponding to the test. The element size is refined in the brace-

to-chord intersection such that the element aspect ratio is around 1:1:1.

0.0 0.1 0.2 0.30

2

4

6

8

εN=0.10,fN=0.10,sN=0.05 Yura's

deformation limit

Mop

b/f yd

1t02

φ (radian)

Test β=0.61 γ=11.4 FE_f τ=0.85 α=16.6 FE_f(G) d0=169mm

(a) Load application in Kang’s test (b) Comparison of FE and test

Fig. 5.8 (a) Load application in Kang’s test; and (b) Comparison of FE and test; for OPB loaded X-joint.

F/2 F/2

F/2 F/2

x

Fx/2


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Figure 5.8(b) compares the test and numerical results. Without the Gurson model, the X-

joint sustains increasing load with further deformation. Although the load reduction is not

captured by the Gurson’s algorithm, the presence of voids softens the material and

reduces the joint strength when plasticity occurs and propagates in the joint. This

weakens the joint strength without introducing an abrupt load reduction. Fracture failure

occurs far beyond Yura’s deformation limit, which is a deformation level that practical

joint may not be able to achieve (Zettlemoyer, 1988).

5.4.2.3. K-Joint

The K-joint test reported by Wang, et al. (2000) is selected as the reference. No initial

crack is introduced in the joint test. Similar to the study of X- and T-joints, three FE

schemes are adopted to verify the effect of mesh density on the joint response. In the fine

mesh, four layers of elements are employed in the gap region of the chord. Sixteen

elements are utilized along the 30 mm gap length. The element aspect ratio in the gap and

the brace-to-chord intersection region is maintained around 1:1:1. Figure 5.9(a) shows the

boundary conditions of the K-joint.

Figure 5.9(b) shows the load-deformation curves for K-joints. The under-prediction of

the ultimate joint strength may be caused by the un-reported material strain-hardening. In

spite of the initiation and propagation of the crack in the K-joint, no load reduction exists

in the compression brace behavior. With the Gurson model, a slight load reduction is

observed at a relatively large deformation level, due to the reduction in the tensile brace

resistance initiated by void growth and nucleation processes. Conventional FE analysis

without the Gurson model does not indicate a reduction in the strength. The first crack


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reported in the test occurs much earlier than the ultimate strength. However, the joint

strength continues to increase with the presence of crack.

0.00 0.04 0.08 0.120

10

20

30

40

εN=0.10, fN=0.10, sN=0.05

First crack

Psin

θ/f yt 02

δ /d0

Test β=0.65 γ=24.8 FE_f τ=0.75 α=13.9 FE_f(G) d0=217.4mm

(a) Boundary condition (b) Comparison of FE and test results

Fig. 5.9 (a) Boundary condition; and (b) Comparison of FE and test results; for the K-joint

5.4.3. Discussion

Table 5.3 lists the joint strength obtained from the FE analyses (using the Gurson model

with εN = 0.10, sN = 0.05 and fN = 0.10) and the test results for the intact joints. There are

three strength definition adopted for the FE results (obtained from the fine mesh): the

plastic limit load approach; Lu’s deformation limit (Lu et al., 1994) and 15% plastic

strain (Dexter and Lee, 1999a). For the T-joint, the joint strength at the end of the test is

recorded. Among the X- and K-joints, the peak loads in the tests correspond to fracture

failure of the joint except for X2, for which, no crack develops. Table 5.3 compares the

current FE results against the ISO mean strength, which shows very conservative

At all member ends, φx = φz = 0

x

z y


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predictions for X-joints under brace tension. For X-joints under OPB and K-joint under

brace compression, over-estimation of the ISO formulation is observed.

Table 5.3 Comparison of ultimate strength for intact joints

Plastic load Lu’s deformation 15% strain ISO Joint Test

FE FE/Test FE FE/Test FE FE/Test (kN) ISO/test

X1 (kN) 2248 2154 0.96 2055 0.91 2016 0.90 1054a

2194b

0.47

0.98

X2 (kN) 397 392 0.99 388 0.98 334 0.84 200a

302b

0.50

0.76

X3 (kNm) 12.8 10.4 0.81 - - 10.9 0.85 18.2 1.42

K- (kN) 225 210 0.93 208 0.92 206 0.92 277 1.27 aStrength corresponding to first crack bUltimate strength

Table 5.4 Displacement at 15% plastic strain for three joint types

Joint Fine (δ/d0) Medium (δ/d0) Coarse (δ/d0) Coarse/Fine

X1 0.05 0.05 0.07 1.3

T-joint 0.04 0.04 0.12 3.3

K-joint 0.02 0.03 0.40 2.5

Lu’s deformation limit shows similar strength predictions as the plastic limit load

approach. Both approaches correspond to a state earlier than the fracture failure in the

tests. The plastic strain limit approach defines the joint strength corresponding to the first

attainment of 15% plastic strain at the element integration point around the brace-chord

intersection. The non-dimensional joint displacement δ/d0 corresponding to the 15%

plastic strain for different mesh schemes is tabulated in Table 5.4. The displacement in


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the coarse mesh increases up to 3.3 times that of the fine mesh, which implies significant

size dependence of the plastic-strain criterion.

Load path remains crucial in the void growth and nucleation process. Gurson’s approach

to simulate the effect of ductile fracture is based on the assumption of continuum

mechanics, with the yield condition and plastic flow rule modified by the void volume

fraction f. Stress redistribution is mobilized once material softens due to the propagation

of voids, with the adjacent material mobilized, contributing to the joint strength. It

therefore, delays the formation of the complete collapse of the tensile load path.

Consequently, load reductions predicted by the Gurson model correspond to larger

displacements as compared to those obtained from the tests for most of the joints studied.

For X-joints with high β ratio (X1), a very small portion of the chord wall between the

two braces forms the critical load path. For small β joints (X2), the chord material

between the two braces constitutes a relatively larger volume. Stress redistribution from

the highly stressed area to other regions help to release the high stress in the critical load

path, which in turn delays the effect of the void growth and nucleation. As shown by the

FE results in Fig. 5.7, there is an obvious strength reduction in X1 at around the

deformation level of δ/d0 = 0.08. However, there is only slight strength reduction for X2,

when the deformation level exceeds δ/d0 = 0.10. This shows that the effect of crack is

more significant for X-joints with high β than low β ratios, which is consistent with the

observation reported in the Health and Safety Executive (HSE) report (HSE, 1999).


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The brace-chord intersection of a T-joint, as compared to an X-joint, is under additional

tension due to bending generated by chord end reactions. This complicates the loading on

the crack front. The assumed Gurson properties for the intact joint may not be a good

representation of the real material property and this may be a possible cause for the

significant void growth and nucleation effect in the intact fine mesh with εN = 1.0, sN =

0.05 and fN = 0.10.

The gap region between the braces forms the critical load path for K-joint. The gap near

the weld toe of the tension brace experiences tensile and shear stresses. However, the

brace tension is proportioned by the brace angle θ, and the tensile stress induced by the

local bending may not be of a significant magnitude. In Fig. 5.9, there is a marginal

difference between the analysis with and without the Gurson model around the first

plateau of the load-deformation curve. However, as plasticity spreads in the joint, the

difference becomes more pronounced.

5.5. Sensitivity Study

The void nucleation material parameters: εN, sN and fN, play a significant part in ensuring

the accuracy of the analysis as observed in the T-joint study. These material properties

vary with different types of steel properties and may be affected by the manufacturing

process (Thomason, 1990). Up to now, there is no sufficient material data to quantify the

parameters such as εN, sN and fN for each type of material. In order to investigate the

effect of these material properties, a sensitivity study is conducted on three types of joints:

X- T- and K-joints, using the fine mesh for the intact models. In the study on the effect of


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each material parameter, the rest of the parameters remain as constants. As the strain

hardening parameter varies, εN, sN and fN take the values of 0.10, 0.05 and 0.10

respectively. As εN changes, sN and fN equal 0.05 and 0.10, with the reported material

strain hardening property included. For the variation of sN, εN and fN both take the value

of 0.10. With fN varied, εN and sN remain at 0.10 and 0.05.

5.5.1. Strain-Hardening

Different strain-hardening relationship causes different levels of plastic strain under the

same loading condition. The current study adopts the power hardening law described in

Eq. 5.5.

,

,

y

N

yy

y

forE

forE

σ σ σ

εσ σ σ σ

σ

⎧ ≤⎪⎪= ⎨ ⎛ ⎞⎪ >⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎩

(5.5)

The hardening relationships vary the value of N: 5, 10, 20 and ∞ (corresponding to

elastic-perfectly-plastic property). When the hardening relationship changes, the Young’s

modulus remains at 205 GPa. The joint deformation at the load reduction, as tabulated in

Table 5.5, reflects the effect of void growth and nucleation.

Similar trends in the three types of joints are observed. Strength reduction occurs at a

smaller deformation level with a larger N (lower strain-hardening). The strain-hardening

relationship for the reported X-joint test resembles that represented by N = 10. For the T-

joint, N = 10 also approximates closely the reported strain-hardening. For all three joint

types, N = 5 corresponds to an un-realistically high strain-hardening, which produces


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very high joint strength. Higher strain hardening delays the development of plastic strain,

which postpones the void growth and nucleation process.

The joints with elastic perfectly plastic property, which is normally assumed in the design,

require special consideration in the design procedure. With zero strain-hardening, the

plastic strain attains easily the nucleation strain, which causes dramatic loss of the joint

strength as demonstrated in Table 5.5. This does not reflect the real material behavior, for

which strain-hardening exists to some extent.

Table 5.5 Peak joint strength and corresponding deformation for various material properties

X1 T-joint K-joint Parameters

δ/d0 Ppeak/fyt02 δ/d0 Ppeak/fyt0

2 δ/d0 Ppeak/fyt02

5 0.095 191 0.117 17.2 0.064 38.0

10 0.051 109 0.067 12.6 0.024 30.5

20 0.029 82.2 0.043 10.8 0.018 28.7

N (εN=0.10, sN=0.05, fN=0.10)

∞ 0.011 65.0 0.032 9.53 0.015 26.1

0.0 0.097 111 0.060 11.1 0.018 26.7

0.10 0.075 103 0.049 10.9 0.017 26.8 εN

(sN=0.05, fN=0.10) 0.30 0.105 123 0.119 12.4 0.018 26.9

0.04 0.105 118 0.091 11.8 0.018 26.8

0.10 0.075 103 0.049 10.9 0.017 26.8 fN

(εN=0.10, sN=0.05) 0.20 0.031 79.6 0.033 10.0 0.018 26.7

0.05 0.075 103 0.049 10.9 0.017 26.8 sN (εN=0.10, fN=0.10) 0.10 0.084 108 0.060 11.3 0.018 26.8

Test results 0.049 105 0.103 12.5 - 28.7


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5.5.2. Effect of εN

The value of εN takes 0.0, 0.1 and 0.3 to assess the effect of εN on the Gurson model. The

case of εN = 0.0 implies immediate void nucleation once material yields. A relatively

large εN (0.3) corresponds to a very ductile material. The values of sN and fN takes 0.05

and 0.10, respectively, as εN is varied. Figure 5.10 illustrates the effect of εN on X1 and T-

joint. The ultimate load level does not vary much with the εN value, with strength

reductions first observed in both joints with εN = 0.10.

0.00 0.04 0.08 0.120

50

100

150sN=0.05fN=0.10

P/f yt 02

δ /d0

εN=0.0 (G) β=1.0 γ=25.5

εN=0.10 (G) τ=1.0 α=17.5

εN=0.30 (G) d0=407.4mm X1(test A) X1(test B)

0.00 0.04 0.08 0.120

5

10

15

sN=0.05fN=0.10

P/f yt 02

δ /d0

εN=0.0 (G) β=0.51 γ=7.46

εN=0.10 (G) τ=1.0 α=10.18

εN=0.30 (G) d0=298.5mm Test

(a) X1 (b) T-joint

Fig. 5.10 Effect of εN on Gurson model for: (a) X1; and (b) T-joint.

Table 5.5 reflects the effects of εN on three joint types via the joint peak load and the

corresponding deformation. The same trend exists for all three joints. The ultimate

strength level does not vary significantly with εN. However, the deformation level

corresponding to the peak strength becomes relatively larger for X- and T-joints with εN =


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0.30 than εN = 0.10 and 0.0. A small value of εN (0.0) does slightly reduce the joint

strength once plasticity occurs. This early initiation of void nucleation does not cause a

consequently earlier load reduction for all three joints as shown in Table 5.5 and Fig. 5.10.

For all the three joint types, the effect of void nucleation becomes more pronounced in εN

= 0.10, followed by εN = 0.0 and εN = 0.30. For X-joints, fracture failure occurs at the

deformation level around δ/d0 = 0.05 in the test. All three values of εN cause a strength

reduction at a larger deformation level. For T-joint, load reduction does not occur in the

test. The value of εΝ = 0.30 delays both the initiation of void nucleation and the

consequent load reduction, and thus shows a closer correlation with the tested joint. The

effect of the εN on the tension brace of K-joint is least prominent, due to the fact that the

primary failure mode of the K-joint is not fracture failure.

5.5.3. Effect of fN and sN

Three values of fN are selected to investigate the effect of volume fraction of the void

nucleating particles: 0.04, 0.10 and 0.20. The value of 0.04 is the nominal value

recommended by ABAQUS (2001). A very large value fN = 0.20 is included to compare

the amplified effect of void volume of the nucleating particles. The values of sN take 0.05

and 0.10. Figure 5.11 shows the effect of fN and sN on X1-joint. The joint response

exhibits a strong dependence on fN and a much lower dependence on sN.

Both Table 5.5 and Fig. 5.11 demonstrate the significant effect of fN. From Eqs. 5.3 and

5.4, the void nucleation rate is directly proportional to fN. Small fN (0.04) delays the

development of void volumes and a relatively large fN (0.2) accelerates the void


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nucleation process, causing premature strength reductions. The joint strength differs more

significantly between large fN values (0.1 and 0.2) than between smaller fN values (0.04

and 0.1). As mentioned, however, fN = 0.2 does not describe a realistic material.

0.00 0.04 0.08 0.120

40

80

120εN=0.10sN=0.05

P/f yt 02

δ /d0

fN=0.04 (G) β=1.0 γ=25.5

fN=0.10 (G) τ=1.0 α=17.5

fN=0.20 (G) d0=407.4mm X1(test A) X1(test B)

0.00 0.04 0.08 0.120

40

80

120

εN=0.10fN=0.10

P/f yt 02

δ /d0

sN=0.05 (G) β=1.0 γ=25.5

sN=0.10 (G) τ=1.0 α=17.5

test A d0=407.4mm test B

(a) fN (b) sN

Fig. 5.11 Effect of (a) fN; and (b) sN; on X1 joint.

The value of sN does not affect the joint behavior significantly. Similar joint strength

among different values of sN is observed for all joints, with slight variation in the peak

strength and the corresponding displacement. The effect of fN and sN is more pronounced

for X- and T-joints, with very little effect on the K-joint behavior.

5.6. Summary

The Gurson model reflects the fracture failure by load reductions in the load-deformation

curves of tubular joints. The accuracy of the Gurson model depends primarily on two

factors: the mesh size and the material properties. The detailed mesh refinement in the


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crack front does not form a critical load path, and thus is not necessary in joints with

initial cracks. The mesh size should remain sufficiently small near the critical load path to

compute accurately the strain value. The current study recommends more than three

layers of elements across the chord wall thickness with the element aspect ratio

maintained around 1:1:1 near the brace-chord intersection. The Gurson model parameters

qi (i = 1, 2 or 3) does not introduce an effect on the tensile bar until the post-peak state.

The void nucleation parameters (εN, sN and fN) determine the accuracy of the Gurson

model, with the effect of fN being most significant. A large value of fN (> 0.10) does not

represent the realistic material. Without material data, fN shall vary between 0.0 to 0.10.

The value of εN hardly affects the ultimate joint strength, while a large value of εN (0.30)

postpones the void nucleation process. Hence, 0.0 ≤ εN ≤ 0.10 is suggested. sN imposes

marginal effect on the joint response. A nominal range of 0.05 to 0.10 is recommended.

The effect of void growth and nucleation on the tubular joint depends also on the load

path. Among the three types of joints investigated, K-joint is the least sensitive to

variation of the nucleation parameters compared to X- and T-joints.

05CHAP_5(1)

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