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FINITE ELEMENT SIMULATION OF DUCTILE FRACTURE IN REDUCED

SECTION PULL-PLATES USING MICROMECHANICS-BASED FRACTURE

MODELS

By A. M. Kanvinde1, Associate Member and G.G Deierlein

2, Fellow

ABSTRACT

Micromechanics-based models that capture interactions of stress and strain provide

accurate criteria to predict ductile fracture in finite element simulations of structural steel

components. Two such models – the Void Growth Model and the Stress Modified

Critical Strain Model are applied to a series of twelve pull-plate tests that represent

reduced (or net) section conditions in bolted and reduced beam section connections. Two

steel varieties, A572 Grade 50 and a high-performance Grade 70 bridge steel are

investigated. The models are observed to predict fracture much more accurately than

basic longitudinal strain criteria, by capturing stress-strain interactions that lead to

fracture. The flat stress and strain gradients in these pull plates allow the use of relatively

coarse finite element meshes providing economy of computation while capturing

fundamental material behavior and offering insights into localized ductile fracture effects.

1 Assistant Professor, Dept. of Civil and Environmental Engineering, University of California, Davis, CA

2 Professor, Dept. of Civil & Environmental Engineering, Stanford University, Stanford, CA

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INTRODUCTION

Ductile fracture is an important mode of failure in steel structures, often controlling the

ultimate strength and ductility of connections and members. For earthquake engineering,

fracture resistance is of primary concern in the design of ductile beam-column

connections, bracing, and other regions of the structural system that are designed to yield

and absorb energy through inelastic deformations. Even in routine “elastic” design, there

are many situations, such as the net-section rupture checks for bolted connections, where

there are implicit assumptions for minimum ductility and the steel’s ability to

inelastically redistribute stress and strain concentrations. Owing to the complexity in

modeling ductile fracture, development of guidelines and standards for fracture resistant

design has relied heavily on empirical test data to characterize fracture in structural steel

members and connections.

Traditional fracture mechanics offers models that work well for highly constrained sharp

cracks, but have limited ability to model ductile fracture, which is characterized by large-

scale yielding with or without crack-initiating flaws. Modern finite element methods

provide the ability to accurately characterize inelastic stresses and strains; but,

fundamental criteria to establish ductile fracture limit states under triaxial stresses and

strains are not well established. This paper seeks to address these issues through

modeling techniques derived from the fundamental micromechanics of ductile fracture

and can be applied through detailed finite element simulations. The models are first

described and then applied to simple pull-plate test specimens which simulate ductile

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fracture in situations that are representative of effective net-section conditions in bolted

connections and reduced sections in members and connections.

Widespread fractures to steel moment frame connections in the Northridge Earthquake

drew attention to the vulnerability of modern building structures to fracture. Most of the

fractures that occurred in the Northridge earthquake were fairly brittle in nature, and were

caused by a combination of low-toughness materials and large initial flaws. Studies

conducted since the Northridge Earthquake demonstrated that traditional fracture

mechanics can reliably evaluate brittle fractures caused by these conditions (e.g., Chi et

al, 2000). On the other hand, these traditional fracture criteria, such as stress intensity, J-

integral energy release rate, or Crack Tip Opening Displacement, are subject to two key

assumptions that limit their applicability to more ductile connection details (see Anderson

1995). First, traditional fracture mechanics methods require the presence of a real or

presumed sharp crack, or stress singularity. Second, the fracture metrics are based on

assumption of a highly-constrained crack tip with very localized (if any) yielding. These

two limitations invalidate the use of traditional fracture mechanics to assess ductile

fracture that governs the strength and ductility of a wide range of commonly encountered

situations in structural steel bolted connections and ductile (“post-Northridge”) welded

connections and other situations with geometrical concentrations of stresses and strains.

Enabled by the capabilities of modern computational methods to accurately simulate

localized inelastic stresses and strains, micromechanics-based models aim to evaluate

ductile fracture through principles derived from fundamental fracture phenomena as

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represented in continuum models of stresses and plastic strains. As such, these models are

limited by fewer assumptions than traditional fracture indices and can be applied to flaw-

free conditions subjected to large-scale yielding. Much of the pioneering development of

these models was done from the early 1970’s through mid-1980’s, with the primary

applications being mechanical components, such as pressure vessels and nuclear

containment vessels (e.g., see Rousselier, 1987, Panontin, 1995). Validation and

applications of the models to low-carbon structural steels and civil/structural engineering

applications has been very limited.

This paper describes one part of a comprehensive study by the authors (Kanvinde and

Deierlein, 2004) that includes development, calibration, verification and application of

micromechanical models for the prediction of ductile crack initiation in structural steel

details. The focus in this paper is on twelve pull-plate type tests intended to represent

ductile fracture conditions in the net-section region of bolted connections and

components with similar stress and strain concentrations. Three pull plate geometries are

evaluated, two of which represent the net-section of a bolted connection (Figs. 1a and

1b). The third (Fig. 1c) mimics a smoother “dogbone” reduction, such as has been used to

create a fuse in seismically designed Reduced Beam Section (RBS) details. The

specimens are fabricated from two varieties of low-carbon structural steels – one is A572

Grade 50 ( Yσ = 345 MPa) structural steel, and the other is a high performance bridge steel

HPS70W ( Yσ = 480 MPa). The A572 Grade 50 steel is a common grade used in building

construction, and the HPS70W steel is a new high-performance bridge steel that provides

greater strength with comparable toughness and weldability to lower-strength steels.

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Tensile coupon and notched-bar tests for material and micromechanical model calibration

conducted in the parent study (Kanvinde and Deierlein 2004) are briefly summarized and

used as input to finite element analyses to predict fracture in the pull plate tests.

The paper begins with a brief introduction to a widely accepted mechanism for ductile

crack initiation, which provides the basis for micromechanics-based models. The two

models considered herein are the Void Growth Model (VGM) and the Stress Modified

Critical Strain (SMCS) model. Complementary finite element simulations for all the tests

are then presented along with the methodology for predicting ductile fracture in these

situations using the VGM and the SMCS models. After introducing the experiments and

simulations, model based predictions of failure are compared to experimental

observations. The efficacy of the models relative to empirical ultimate strain measures

and the relative performance of two grades of steel (A572 and HPS70W) are compared

and contrasted.

Given the current state of research in the micromechanical simulation of ductile fracture,

e.g., the development of SMCS and VGM models and their application to structural

steels, this paper serves to bridge the gap between small material-scale experiments and

large-scale structural tests by implementing and validating these models for modest size

experiments representative of situations found in structural components. It aims to

present a verified methodology for fracture prediction by inelastic finite element analysis

in flaw-free structural components with large scale yielding.

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DUCTILE FRACTURE MECHANISM AND MICROMECHANICAL MODELS

Mild low-carbon steel, used in structural engineering applications, typically exhibits

ductile fracture accompanied by plastic deformation. As shown schematically in Fig. 2,

ductile fracture is characterized by micro-void nucleation, growth and coalescence

(Anderson 1995). Void nucleation occurs around secondary particles or inclusions such

as carbides in the steel matrix. When sufficient stress is applied to the interfacial bonds

between these particles and the matrix, these bonds break, and a void nucleates around

the secondary particle (Argon et al, 1975). Void growth and coalescence occur after

nucleation, when plastic strains and hydrostatic stresses cause the voids to grow. Initially,

the voids initiate and grow independently, until neighboring voids grow to the point that

they interact and plastic strain concentrates along planes between the voids. At this point

local necking instabilities cause the voids to grow suddenly and form a macroscopic

fracture surface. In common structural steels, void growth and coalescence are assumed

to govern the fracture process, such that macroscopic fracture initiation can be predicted

through the void growth rate as a function of the locally evolving stress and strain fields.

The Void Growth Model (VGM) and the Stress Modified Critical Strain (SMCS) are both

based on the concept of tracking microvoid growth and coalescence. The models both

models assume that void growth is the defining step in the fracture process, and they do

not explicitly model void nucleation and coalescence. Other models, besides SMCS and

VGM, have been proposed, which similarly represent the processes of void growth and

coalescence (e.g., Pardoen and Hutchinson, 2000, Benzerga et al, 2004, Hancock and

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Brown, 1983, Johnson and Cook, 1985). The features and relative merits of these

alternative models are discussed in Chi et al. (2006) and Kanvinde et al, (in review). The

simplifying assumptions of the SMCS and VGM are judged to be appropriate for

structural engineering applications, considering the inherent variability and lack of

detailed material data in typical applications and the otherwise highly empirical and

approximate models that are commonly used to evaluate fracture.

STRESS-STRAIN CRITERION FOR DUCTILE FRACTURE INITIATION

According to Rice and Tracey’s analytical derivations (1969) for a single spherical void

in an elastic perfectly plastic material, the void growth rate is dependent on two

quantities: (1) the equivalent plastic strain pε and (2) the stress triaxiality /m eT σ σ= .

The equivalent plastic strain quantifies the deformation in the material, while the

triaxiality is a convenient measure of the ratio between the hydrostatic (dilational) stress

mσ to the von Mises (distortional) stress eσ . Mathematically, the void growth rate can be

expressed as,

( )exp 1.5 . p

dRC T d

Rε= (1)

where R is the instantaneous void diameter, C is a constant determined by Rice and

Tracey (1969) to be equal to 0.283 for a single void in an elastic-perfectly plastic

continuum, and the other terms are as defined previously. After a series of mathematical

simplifications, (1) can be reduced to the form.

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( )0

0

ln

exp 1.5 .

p

p

R

RVGI T d

C

ε

ε

= = ∫ (2)

where VGI, the Void Growth Index, is a normalized measure of the void growth

“demand” that can be determined as a function of the stress triaxiality and plastic strain

histories.

Void Growth Model (VGM): To predict fracture, the void growth demand can be

compared to a void growth “capacity”, criticalVGI , which is assumed to be a material

property based on the notion of a critical void size. The criticalVGI of the material can be

determined using notched bar tensile tests (see Fig. 3), in a similar manner to the way that

other material properties are routinely measured. The notched bar is identical to the

standard round tension coupon, but with the addition of a circumferential notch whose

radius is varied to control the ratio of stress triaxiality to plastic strain (Kanvinde et al.

2004, Panontin et al. 1995). As described later, data from notched bars provide the key

micro-mechanical fracture parameters for the two steels investigated in the pull-plate

tests. Knowing the value of criticalVGI from the notched bar test, fracture is predicted in

structural assemblies when the calculated void growth demand exceeds the capacity, i.e.,

criticalVGI VGI> . This inequality and calculation of VGI per (2) comprise the essential

components of the Void Growth Model (VGM).

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Stress Modified Critical Strain (SMCS) Model: According to (2), the VGM explicitly

integrates the triaxiality with respect to plastic strain, which implies that both quantities

vary during loading. However, in many realistic situations, the stress triaxiality (which is

primarily geometry dependent) remains fairly constant during loading while the plastic

strain increases. This observation, as first noted by Hancock and Mackenzie (1976),

forms the basis of the Stress Modified Critical Strain (SMCS) model, which is derived

from setting the triaxiality in (2) to be a constant. This leads to the following equation for

a critical value of plastic strain , εpcritical

, that is inversely related to the exponential of

stress triaxiality,

)5.1exp(. Tcritical

p −= αε (3)

where α is a material dependent material property. Like the criticalVGI parameter in the

VGM, the α parameter is determined from the notched tensile bar tests.

The SMCS criterion is defined as the difference between the critical plastic strain and the

calculated equivalent plastic strain, i.e.,

critical

ppSMCS εε −= (4)

where fracture is predicted to occur when SMCS = 0. Based on the assumption that the

triaxiality (T) does not change appreciably with respect to increasing plastic strain εp, the

critical plastic strain in (3) and (4) depends only on the instantaneous level of triaxiality

and ignores history effects. Thus, the SMCS is simpler to evaluate than the VGM since it

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does not require integration over the loading history. For many. In addition, by

considering only the final value of triaxiality, it places a larger weight on the latter stages

of loading (when the voids have grown), rather than the initial stages, as compared to the

VGM. Thus, the SMCS may implicitly model void coalescence better than the VGM,

leading to better predictions in some cases (Panontin et al, 1995). On the other hand,

where the triaxiality often changes appreciably during loading, such as in situations with

large geometry changes in ductile materials or during reverse cyclic loading, the VGM

would likely be more accurate.

Length Scale: In addition to evaluating the VGM or SMCS criteria given by (2) and (4),

respectively, one must consider the critical volume of material over which these criteria

are satisfied to form a macro crack. Referring back to Fig. 2, the critical volume is

physically related to the sampling and coalescence of voids, which are spaced finite

distances apart. According to the proposed models, ductile fracture initiation is assumed

to occur when the VGM or the SMCS reach their critical condition over a characteristic

volume. In two dimensions, this volume is represented by a characteristic length – l*.

As reported by Kanvinde and Deierlein (2004), for typical mild steels, the critical l* is on

the order of 0.1 mm.

In concept, the fracture models require accurate assessment of the stress and strain field

within the length scale (about l* = 0.1 mm for mild steels), which is computationally

challenging in geometries with sharp cracks and steep stress/strain gradients ahead of the

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crack tip. For the assumed flaw-free pull-plate specimens in this study, the stress/strain

gradients in the critical region are extremely flat (relative to l* = 0.1mm), and

consequently, the fracture evaluation is insensitive to the length scale parameter. Apart

from relieving the need to assess the length-scale, this insensitivity to the length-scale

permits the use of relatively coarse finite-element meshes (compared to the micro-scale

fracture processes) and thereby maintains reasonable computational demands.

STEEL MATERIAL PROPERTIES

Properties of the A572 Grade 50 and HPS70W steels, as determined from standard

material tests and notched tension bar tests (Fig. 3), are summarized in Table 1. Data

from the standard round tension tests provide basic information on the yield strengths,

ultimate strengths and uniaxial ductility, where the latter is measured in terms of axial

strain and diameter of the necked cross section at fracture. Referring to Table 1, both

steels have similar yield ratios (Fy/Fu) of about 0.85 and the diameter reductions (do/df)

of about 1.5 for the A572 steel and 1.95 for the HPS70W steel suggest that HPS70W is

more ductile. The ultimate uniaxial strains at fracture (back-calculated from the diameter

of the fractured cross-section of the tension coupon) are 0.81 for the A572 steel and 1.28

for the HPS70W steel. Similarly, the Charpy V-Notch data indicate that the HPS70W has

higher toughness, with almost twice the absorbed impact energy as that A572 steel. Both

steels exhibit ductile upper shelf behavior at room temperature, with toughness values on

the order of 5 to 10 times the value of CVN = 27 Joules (20 ft-lbs) that is typically

associated with the toughness close to the transition temperature. The chemical analyses

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reveal that both steels have low carbon content as compared to the maximum allowable

content specified by ASTM. The A572 steel contains 0.07% Carbon versus the specified

ASTM limit of 0.23%; and the HPS70W contains 0.08% Carbon as compared to the

specified limit of 0.11%. The HPS70W has a larger carbon equivalent than the A572

steel (Ceq = 0.51 versus 0.35), due to higher percent weights of Chromium, Molybdenum,

and Nickel, which presumably give the HPS70W steel its larger strength and weldability.

The HPS70W steel also has finer grain sizes, which contribute to its larger ductility and

toughness.

The notched bar tests provide data to determine the critical plastic strain parameters (α

and criticalVGI ) summarized in Table 2 and defined according to (2) and (3). The

percentages shown in parentheses for these parameters (in Table 1) represent the

statistical coefficients of variation in the measured data. The fracture strain parameters

(α and criticalVGI ) suggest that the high-performance HPS70W steel has two to three

times the ductility of the A572 Grade 50 steel, which follows the trend observed in

uniaxial fracture strain and the CVN toughness data. The authors have noted a similar

correlation between CVN upper-shelf toughness and the fracture indices, α and criticalVGI ,

with data from other steel samples (see Kanvinde and Deierlein 2004), which suggests

how the micromechanical model parameters could be inferred from the CVN data, or vice

versa.

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PULL PLATE EXPERIMENTS

The pull-plate specimens, shown in Figs. 1a and 1b, are configured to demonstrate

application of the micro-mechanical models to simulate ductile fracture initiation in the

net section regions of bolted connections. Typical design provisions (e.g., AISC 2005)

are based on the assumption that mild steels have sufficient ductility such that the

ultimate steel strength, Fu, can be assumed to act over the effective net section of

material. In many cases, the effective net section is easily calculated as equal to the actual

net section. However, where there are large stress and strain gradients, such as in steel

angle members that are connected through one leg, the effective net section calculation is

not as straightforward and necessitates modification by an empirical coefficient. Bolted

connection design provisions generally do not distinguish between cases where the loads

are introduced away from the net section (Fig. 1a) or through bolt bearing in the net

section region (Fig. 1b). The extent to which these and other factors affect the ductility

demand (and, hence the effective net section) are examples of the practical research

questions that can be investigated using the micro-mechanical fracture criteria.

The dog bone shaped pull plate specimen in Fig. 1c represents a more general

configuration where a reduction in member area leads to net section type behavior in a

larger region of a member or connection. This might, for example, represent a simple

idealization of the Reduced Beam Section (RBS) concept has been used in welded

moment frames to move regions of large inelastic strains out away from the column face.

As noted by Kuwamura et al. (1997), Uriz and Mahin (2004), among others, the initiation

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of small ductile tears can quickly grow and propagate under reverse cyclic loading,

sometimes involving transition from ductile cracking into brittle cleavage fracture. Thus

the tests and analyses of the specimen in Fig. 1c are intended to illustrate how the fracture

models can be applied through finite element simulations to help quantify the inelastic

deformation capacity of such fuse details.

Pull plate – Net Section Specimens (BH): The pull plate specimen shown in Fig. 1a is

referred to as the bolt-hole (BH) configuration, where the specimens are cut out of a

50.8mm X 25.4 X 152.4mm (2” X 1” X 6”) plate. The holes in the end (thicker) region of

the plate are for attachment to the loading fixture. The central 76 mm (3”) of the plate is

milled down to a thickness of 9.5mm (0.375”), and two 12.5 mm diameter bolt holes are

drilled in the center region. The area of the critical net section is equal to 240 mm2.

Monotonic tension is applied in a displacement-controlled 250 kN capacity MTS servo-

hydraulic testing machine. Two displacement transducers are attached to either side of

the specimen to monitor the elongation over the central 76 mm gage length, where the

average displacement measurements are used for comparison with the finite element

analyses. Four such tests are conducted, two for each variety of steel.

Load-displacement curves for the four BH tests are shown in Fig. 4a. The load climbs

linearly until yielding initiates in the net section. At the peak point the net section

ligaments start to neck, and the load starts dropping. After this point, the deformations

localize in the ligament and the plastic strain increases rapidly until a critical value of

plastic strain is reached and the material ruptures. The fracture was observed to initiate at

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the outside edge of the bolt hole, causing the outer ligament to fracture first and the load

to drop suddenly. Further straining causes the middle ligament to fail, resulting in the

fracture state shown in Fig. 5. For purposes of this study, the initial fracture of the edge

ligament is considered the point of failure and the corresponding displacement Test

failure∆ is

reported in Table 2.

The A572 BH specimens reached a maximum load of 141 kN, which is about 6% larger

than the computed ultimate strength of 142 kN, equal to the product of the measured

material tensile strength Fu = 590 MPa and the net section area. The difference is likely

due to the triaxial constraint at the net section, which elevates the effective ultimate

strength beyond that of the uniaxial tension tests. The HPS70W BH specimen reaches

maximum loads of 182 kN, about 7% larger than the calculated value of 170 kN (based

on Fu = 690 MPa). The failure displacements are comparable for all four tests, on the

order of 3.5 to 4.0 mm (refer to Table 2). Note that while the failure displacements are

similar, the earlier peak and sharper drop-off in the load-deformation curves (Fig. 4a) for

the HPS70W specimens suggest that these undergo more severe necking and plastic

strain prior to fracture. Differences of this sort in the localized strain demands are

detected through the finite element analyses presented later.

Pull plate – Net Section Specimens with Bolt Bearing (BB): The specimens shown in

Fig. 1b are similar to the previous set (Fig. 1a), except for the introduction of loading

through the bolt bearing. Referred to as the bolt bearing (BB) tests, the load is applied by

means of 12.7 mm dowel pins passing through the bolt holes. Other than this,

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displacement measurements, load application, and steel types are identical to the BH

specimens.

Similar to the BH tests, fracture initiation in the BB tests occurs in the edge ligament, and

the failure pattern is almost identical to that in BH. Referring to Fig. 4a and 4b and Table

2, the overall load displacement curves and failure elongations are comparable for the BB

and BH specimens. On average, the failure elongations are about 8% larger for the BB

specimens, which presumably is due to additional bearing deformations caused around

the loading pins.

Pull plate – Reduced Beam Section (RBS): As shown in Fig. 1c, the overall geometry

of the RBS specimen is similar to that of the BH or BB specimens, but instead of bolt

holes, the central part of the plate is narrowed down to a width of 25.4 mm for the A572

and 23.0 mm for the HPS70W by means of two circular cuts on either side. The cut

radius is 19 mm for the A572 specimens and 12.7 mm for the HPS70W specimens. The

difference in geometry for the A572 and HPS70W steels was an unintentional result of

machining difficulties encountered with the HPS70W steel, which arose due to the higher

hardness of this steel. The instrumentation and loading fixtures for the RBS test are

identical to those in the BH tests.

The load displacement curves for the RBS specimens are shown in Fig. 4c and follow the

same general trends as the BH and BB tests, although some differences are evident due to

the difference in radius between the A572 steel (r=19.0 mm) and HPS70W steel (r=12.7

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mm). In particular, the smaller radius in the HPS70W specimen results in a steeper drop

off and less displacement ductility than the A572 specimen. There is also a relatively

large variation (± 9%) in maximum load between the two HPS70W tests, which is

attributed in part to dimensional tolerances (± 4%) and variation in the material

properties. The maximum strength of 156 kN in the A572 specimen was about 10%

higher than the 142 kN predicted based on the nominal tensile strength (Fu = 490 MPa);

this difference is comparable to the trends noted previously for the other specimen types.

One of the two HPS70W specimens reached a load of 165 kN, which is also about 9%

larger than the calculated strength of 151 kN (Fu = 690 MPa); the second specimen

reached a peak load that is just about equal to the calculated value.

In all RBS specimens, the fracture propagates quickly across the critical section,

suggesting that the ductile initiation takes place nearly simultaneously over a large area,

after which the entire section fails suddenly through a mixture of ductile tearing and some

brittle fracture. This is evident from examination of the fracture surfaces which have the

appearance of ductile tearing dimples, smooth shear lips, as well as shiny cleavage facets.

While the combined fracture mechanism behavior is complicated, the presence of a large

dimpled region (indicative of ductile fracture) over a large portion of the center of the

specimens suggests that the failures initiated by ductile fracture. As described later, this

trend is supported by the finite element analyses, which show the SMCS and VGM

criteria to be at critical values over this same region. The fracture elongations are on the

order of 6.3 mm for the A572 steel and 5.4 mm for the HPS70W steel. The larger

deformations for the A572 specimen are attributed to the fact that it has a larger radius

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cut than the HPS70W specimen, as well as a smaller net section. Otherwise, one would

expect the HPS70W to exhibit larger failure deformations since the basic material tests

(Table 1) indicate it to be the more ductile material. Overall, the RBS specimens exhibit

about 50% more elongation ductility than the BH and BB specimens, due to the fact that

they have a larger yielded region.

FINITE ELEMENT SIMULATIONS OF FRACTURE INITIATION

Three dimensional finite element models of the pull-plate geometries were created and

analyzed using ABAQUS/CAE 6.2. The analyses were run using large deformation

theory and isotropic incremental Mises plasticity models, which were calibrated to stress-

strain data from uniaxial material tests. The critical regions were modeled with three-

dimensional 20 node hexahedral elements with reduced integration.

FEM Analyses – BH and BB Configurations: The BH configuration has three planes of

symmetry, so it is sufficient to construct a FEM model of one-eighth of the complete

specimen. The boundary conditions are fixed according to the requirements of symmetry,

and the model is loaded in displacement control. Displacements are monitored at the

locations where the displacement transducers are attached to the test specimen. The finite

element mesh has approximately 1000 hexahedral elements, where the smallest elements

(in the regions of fracture) are on the order of 1 mm – roughly a tenth of the ligament

width. This element size is large, compared to the characteristic ductile fracture length of

l* = 0.1 mm, however, a detailed mesh sensitivity analysis verified that the stress/strain

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gradients in this specimen are sufficiently flat to achieve accurate solutions with this

mesh density.

The BB configuration has only two axes of symmetry, so a one-quarter model with

approximately 3500 hexahedral and tetrahedral elements is used. Otherwise, the model is

similar to that for the BH configuration. The deformed mesh of the BB model, shown in

Fig. 6, indicates the relatively coarse mesh and the predicted fracture location in the outer

ligament.

To predict fracture initiation, stress and strain contours are monitored over the highly

strained net section region of the specimens. The critical VGM and SMCS criteria,

described by (2) and (4), are first breached at the rim of the bolt hole near the outer edge

of the specimen. The gradients are flat enough so that the stress and strain fields need to

be monitored only at this one location. As shown in Fig. 7, the plot of the SMCS versus

applied displacement indicates the load increment when the fracture criterion is triggered,

i.e., when SMCS > 0. The corresponding displacement Analysis

failure∆ is retained as the failure

displacement and reported in Table 2. A similar procedure is followed for the VGM

model (graph not shown), where the VGI is tracked to predict failure displacement.

Figure 8 shows a load-displacement curve from the analysis of the BH A572 specimen,

which is overlaid on the measured test data. The close agreement in the load-

displacement curve, which accounts for large-deformation necking, and the predicted

fracture point is encouraging. Further comparisons of the measured and calculated

failure displacements in Table 2 and Fig. 9 indicate that the agreement is good (within

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about 8%) for the A572 specimens. The discrepancies are larger for the HPS70W

specimens, where the analysis overestimates the fracture displacement by 13% to 24%

for the SMCS model and 26% to 41% for the VGM model. Further discussion regarding

the relative accuracy is provided later.

Referring to Fig. 7, the HPS70W specimens with the larger α values, achieve a larger

Analysis

failure∆ than the A572 specimens. In this case, the difference between the SMCS fracture

indices, α = 2.9 versus 1.2 for HPS70W and A572 steels, respectively, results in a 33%

increase in Analysis

failure∆ for the HPS70W steel. The amplitudes of the initial values of SMCS

(close to ∆ = 0) represent the plastic strain capacity of the specimens, which is a function

of the triaxiality present under elastic conditions. The accelerated rate of increase for the

HPS70W specimens beyond about 2-1/2 mm displacement is associated with the

significant necking at this point. The plots in Fig. 7 further demonstrate that the FEM

analyses reveal practically no difference between the BH and BB specimens. This is not

unexpected, given the similarities in the test data, but the FEM analyses further confirm

that the similarities extend from the global behavior, as observed in the experiment, to the

local behavior evident from the analysis. The insights that such models offer into

localized effects are likely one of the most important features of these models. Similar

trends between the BH and BB configurations and the A572 versus HPS70W steels are

observed using the VGM fracture criterion.

FEM Analyses – RBS Configuration: The RBS model has three planes of symmetry,

and so a one-eighth FEM model is sufficient for analysis. The mesh has about 1500

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hexahedral elements with the minimum element size on the order of 1 mm, similar to the

BH and BB configurations. The critical SMCS and VGM criteria are both observed to be

satisfied over a large portion of the net section, which is consistent with the sudden

failure that rapidly propagated across the section.

The calculated failure displacements, reported in Table 2, match the measured response

fairly well. For the A572 specimens, the SMCS and VGM predictions are both within 5%

of the measured values. For the HPS70W specimens, the maximum displacements

determined using the SMCS and VGM models exceed the measured displacements by

13% and 25%, respectively. The larger discrepancies for the HPS70W steel might be

attributed to the fact that fracture surfaces for both these tests showed a combination of

cleavage and ductile fracture, though the evidence suggests that the fracture was initiated

by ductile tearing.

Summary Comparison of SMCS and VGM Predictions: Data from all of the pull-

plate tests and analyses are summarized graphically in Fig. 9. Results for the A572 steel

tests (the solid markers) demonstrate that both sets of model predictions (SMCS in Fig.

9a and VGM in Fig. 9b) were accurate within 7%. On the other hand, larger discrepancies

were observed for the HPS70W specimens, where the SMCS model predictions were

within about 25% of the measured values and the VGM predictions had errors up to 42%.

In general, both models tended to overestimate the failure deformations.

22

Uniaxial Strain Index: Another comparison relevant to the accuracy of the fracture

models is to the basic parameter of longitudinal plastic strain as a fracture criterion. As

shown in Table 1, the peak true strains at fracture in the uniaxial tension tests are equal εu

= 0.81 for the A572 steel and εu = 1.28 for the HPS70W steel. For comparison, the

uniaxial strains calculated from the finite element analyses and corresponding to the

measured failure displacement are summarized in the last column of Table 2. Failure

strains for the A572 BH and BB specimens range from 0.66 to 0.79, which are 19% and

2% less, respectively, than εu = 0.81 from the tension coupon. Peak strains for the A572

RBS specimens are 0.45, which is about 45% less than εu = 0.81. Failure strains for the

HPS70W BH and BB specimens range from 1.03 to 1.22, which are 10% to 24% less

than the εu = 1.28 from the uniaxial test; and values for the HPS70W RBS specimens of

0.63 to 0.71 are to 47% to 53% less. The rather dramatic differences in the failure strains

demonstrate the significance of triaxiality on the critical plastic strain in the material.

Were one to base the failure predictions on the uniaxial strains, the errors would be

considerably larger than those observed with the SMCS and VGM models. Thus, these

comparisons demonstrate the improvement in ductile fracture predictions made using the

micromechanical models.

CONCLUSIONS

Application of the VGM and the SMCS models to predict ductile fracture in steel

connection details are examined through a series of twelve pull-plate experiments and

complementary finite element analyses of bolted connection (BH and BB) and Reduced

Beam Section (RBS) type details. Two structural steels, A572 Grade 50 and a high

23

performance bridge steel HPS70W, are investigated. Four tests are conducted for each of

the BH, BB and RBS specimens. In the BH and BB specimens, fracture initiates and

propagates in the outer ligament, followed by rupture of the inner ligament. The fracture

initiates after the nominal tensile strength has been reached, which is consistent with

current design procedures to apply the ultimate stress (Fu) over the net section. The RBS

tests exhibit ductile fractures that occur nearly simultaneously over a large part of the

reduced cross-sectional area.

The VGM and the SMCS micromechanics-based models are observed to be reasonably

accurate tools to predict ductile crack initiation. For the A572 steel, the calculated

fracture displacement is predicted within about 7% of the measured values. Discrepancies

are larger for the HPS70W steel, where the SMCS and VGM models over predicted the

displacement at fracture by up to 27% and 42%, respectively. The larger differences for

the HPS70W steel are attributed to (1) mixed-mechanism fracture (combining void

growth based fracture, with other mechanisms such as cleavage) and (2) larger variations

in the material properties. Further testing is suggested to clarify these issues.

Comparing between the two models, the SMCS is more accurate than the VGM. For the

monotonic loading without significant geometry changes during loading, the SMCS

model does better since it is based on the instantaneous values of triaxiality to predict

failure. Consequently the stresses at failure exercise larger influence on the prediction of

failure using the SMCS model than they would in the VGM. In previous comparisons of

the models, Panontin et al (1995) observed a similar effect. However, in cases where the

24

triaxiality change is large, this effect might be compensated by the ability of the VGM

model to explicitly integrate triaxialities, and the VGM might predict fracture more

accurately (Kanvinde and Deierlein, 2004). Owing to the relatively modest changes in

geometry in all of the specimens discussed in the study, the differences between the

SMCS and VGM predictions are not too dramatic. Both the the micro-mechanical

models are significantly more accurate than the simplified longitudinal strain parameter,

which does not account for variations in triaxial stresses and their influence on ductile

fracture.

To summarize, the micromechanics-based models are found to be useful tools to predict

fracture in structural components. Many structural component situations are flaw free and

have flat stress-strain gradients, which obviates the need for a detailed calibration of the

length scale parameter for these models. Consequently, all the calibration for these

models can be done using relatively inexpensive notched bar tests, as opposed to more

elaborate fracture tests. Moreover, a recent study by the authors (Kanvinde and Deierlein,

2004), indicates a strong correlation between the CVN upper shelf energy values and the

α and criticalVGI parameters, thus suggesting how the standard Charpy toughness tests can

be used to determine model parameters for the SMCS and VGM fracture criteria.

Another important observation is that the absence of sharp cracks or flaws, the stress and

strain contours in critical fracture regions are often sufficiently gradual to permit

application of the micro-mechanical fracture models using relatively course finite

element meshes. Thus, the paper demonstrates the abilities of the new micromechanics-

25

based models to predict ductile fracture under various situations of relevance to structural

and earthquake engineering encouraging their application for analysis as well as design.

While the paper applies these models to medium scale tests with reasonable success, the

use of these models for real-scale structural components will require accurate modeling

of other phenomena such as local buckling and weld fracture that may ultimately be

responsible for fracture. Moreover, the for seismic performance assessment, modeling of

the earthquake-induced ultra-low cycle fatigue behavior of materials is necessary. Future

work in modeling all aspects of behavior is needed to fully utilize the benefits offered by

the micromechanical fracture models.

ACKNOWLEDGEMENTS

This paper is based upon research supported by the National Science Foundation under

the US Japan Cooperative Research for Urban Earthquake Disaster Mitigation initiative

(Grant No. CMS 9988902). Additional support was provided by the Steel Structures

Development Center of the Nippon Steel Corporation (Futtsu, Japan), and by donations

of steel material from the Garry Steel Company (Oakland, CA) and the ATLSS

Engineering Research Center (Bethlehem, PA).

NOMENCLATURE

eA : Effective area of net section (total area minus area of holes)

c : Void growth rate proportionality constant

*l : Characteristic length scale

26

nP : Nominal load capacity of tension member

r* : Radius of notch in notched tensile test

R : Microvoid radius

0R : Initial microvoid radius

e

mTσ

σ= : Triaxiality

criticalVGI : Toughness index for VGM

α : Toughness index for SMCS

Analysis

failure∆ : Analytically predicted failure displacement

Test

faillure∆ : Experimentally observed failure displacement

uniaxial

fractureε : Longitudinal fracture strain in uniaxial tension test

1

failureε : Longitudinal plastic strain at failure

2.

3

ij ij

p p pε ε ε=

: Equivalent plastic strain

eσ : Effective or von Mises stress

mσ : Mean or Hydrostatic stress

uσ : Ultimate tensile strength

Yσ : Yield stress

27

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28

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29

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30

TABLES

Table 1 – Material Parameters for A572 and HPS70W steels

Notched Tension

Bar Test Data

Charpy

Impact Standard Tension Bar Test Chemical and

Grain Size Tests

Steel

criticalVGI

α CVN

(Joules)

Fy

(MPa)

Fu

(MPa)

uniaxialfractureε

True

do/df Grain

(mm)

%C %Ceq2

A572 Gr.

50

1.1

(18%)

1.2

(15%)

146 420 490 0.81 1.50 0.019 0.07 0.35

HPS70W 3.2

(7%)

2.9

(7%)

278 590 690 1.28 1.95 0.007 0.08 0.51

Notes:

1. Percentage values in parenthesis represent the coefficient of variation on the fracture parameters.

2. Carbon Equivalent: 6 5 15

eq

Mn Cr Mo V Ni CuC C

+ + += + + +

3. True Strains at fracture are calculated from the ratio of the fractured specimen diameter to the

original diameter, i.e.

2

ln

=

f

ouniaxial

fractured

dε

Table 2 – Summary of pull-plate test data and analytical fracture predictions

Steel Test Pfailure

(kN)

Test

failure∆

(mm)

Analysis

failure∆

SMCS

(mm)

Analysis

failure∆

VGM

(mm)

1

failureε

BH1 141 3.6 0.73

BH2 142 3.5

3.5 3.3

0.66

BB1 142 3.9 0.79

BB2 142 3.7

3.5 3.5

0.75

RBS1 140 6.3 0.45

A572

RBS2 140 6.3

6.6

6.5

0.45

BH1 146 3.7 1.03

BH2 150 3.9

4.7 5.2

1.15

BB1 146 4.0 1.18

BB2 155 4.2

4.7 5.8

1.22

RBS1 113 5.6 0.71

HPS70W

RBS2 135 5.2

6.1 6.8

0.63

31

FIGURES

Fig. 1 – Pull-plate specimens (a) bolt-hole net section – BH, (b)

bolt-hole with bearing – BB, (c) reduced beam section – RBS

(b)

All dimensions in mm

(a) 12.7 dia typ

25.4 , 23.0 50.8

(c)

152.4

76.2

25.4 9.5

r =19.0,12.7

32

Fig. 2 – Mechanism of void nucleation growth and coalescence

(Anderson, 1995)

33

Fig. 3 – Notched Tensile Bar (a) Test setup (b) diagram of notched section

(a)

12.7

6.4

r* = 3.2,6.3,12.7

All dimensions

in mm

(b)

34

Fig. 4 – Load displacement curves for the different specimens

(a) BH (b) BB and (c) RBS

0

100

200

0 2 4 6 8

Displacement (mm)

Loa

d (k

N)

0

100

200

0 2 4 6 8

Displacement (mm)

Loa

d (k

N)

A572

HPS70W

(b)

0

100

200

0 2 4 6 8

Displacement (mm)

Loa

d (k

N)

(c)

A572

HPS70W

A572

HPS70W

(a)

35

Fig. 5 – Fractured BH specimen (A572 Steel)

36

Fig. 6 – FEM model of BB tests showing fracture location

Minimum element

size ? 1mm

Face loaded in

displacement control

Bottom half of the hole

constrained in the

longitudinal direction

Fracture Location

Plane of

symmetry

37

Fig. 7 – Evolution of the SMCS in BH and BB situations

38

Fig. 8 – Comparison of analysis versus test results for BH specimens of A572 Steel

0

100

200

0 2 4 6

Displacement (mm)

Loa

d (k

N)

Analysis

failure∆

(VGM,SMCS)

ABAQUS

Analysis

Test

Test

failure∆

39

Fig. 9 – Comparison between tests and analyses for all tests based on (a)

SMCS and (b) VGM

0

2

4

6

8

0 2 4 6 8

∆failure from Experiment (mm)

∆fa

ilure

fro

m S

MC

S P

red

icti

on (

mm

)

A572 BH

HPS70W BH

A572 BB

HPS70W BB

A572 RBS

HPS70W RBS

0

2

4

6

8

0 2 4 6 8

∆failure from Experiment (mm)

∆fa

ilu

re f

rom

VG

M P

red

icti

on (

mm

)

A572 BH

HPS 70W BH

A572 BB

HPS70W BB

A572 RBS

HPS70W RBS

1.25Analysis Testfailure failure∆ = ∆

1.25Analysis Testfailure failure∆ = ∆

0.75Analysis Testfailure failure∆ = ∆ 0.75Analysis Test

failure failure∆ = ∆

Analysis Testfailure failure∆ = ∆

Analysis Testfailure failure∆ = ∆

(a) (b)