aureioannidestrb2010

7/31/2019 AureIoannidesTRB2010

1/10

12

who coded a user-defined subroutine in ABAQUS, thereby creating

a user element (UEL); an application of this UEL is described in the

2-D study of crack propagation in simply supported asphalt concrete

beams by Song et al. (11). A series of such 2-D four-noded UEL ele-

ments was inserted at the center of the beam to simulate fracture.

The corresponding case involving a three-dimensional (3-D) PCC

pavement slab on grade was first considered by Ioannides and Peng

(12), who once again used JOINTC elements from the ABAQUSlibrary. A 3-D cohesive zone UEL was also formulated by Song (10)

and was applied to a cylindrical asphalt concrete specimen.

Such efforts received a boost with the release in early 2005 of

ABAQUS version 6.5, which for the first time included a family of

cohesive elements for modeling deformation and damage in finite-

thickness adhesive layers between bonded parts. Cohesive elements

are typically connected to underlying elements with surface-based tie

constraints, so the mesh used for the cohesive layer can be independent

of the mesh used for the bonded components (13). Gaedicke and

Roesler (14) reported using these cohesive elements in their study

of crack propagation in concrete beams and slabs.

The investigation presented in this paper is a continuation of the

step-by-step development and application of fracture mechanics

tools that are based on the FCM in pavement engineering initiatedat the University of Cincinnati in the late 1990s. The main objective

is to implement the built-in cohesive elements that have recently

been added to the ABAQUS library and to compare the performance

of these elements to that reported in earlier investigations. It is hoped

that, in this manner, a contribution will be made to the ongoing effort

for more mechanistic pavement design procedures that will use

fracture mechanics concepts, thereby replacing the purely empirical

statistical transfer functions and Miners hypothesis, which are

currently in use. Validating an FE simulation of crack propagation

in a simply supported beam is considered to be a necessary precursor

to a more comprehensive analysis of slabs-on-grade required for

in situ pavement systems.

METHODOLOGY

The present study focuses on the postcracking response of simply

supported PCC beams by means of the commercial, general purpose

FE program ABAQUS (Standard) version 6.7-1 (15). The geometric

and material properties of the beams considered in this paper are

shown in Table 1. To begin with, however, a linear elastic analysis

using 2-D and 3-D elements is described, and the results are compared

with available closed-form solutions. This initial step is considered

essential in ensuring the robustness of the proposed FE formulation.

Simulation of Crack Propagationin Concrete Beams with Cohesive

Elements in ABAQUSTemesgen W. Aure and Anastasios M. Ioannides

This paper discusses the simulation of crack propagation in concrete

beam specimens with a finite element package, ABAQUS, version 6.7-1.

Special-purpose cohesive elements are used to model the fracture process

by means of the fictitious crack model. Two- as well as three-dimensional

finite element discretizations are carried out. Parameters influencing the

responses, such as mesh fineness, cohesive zone width, type of softening

curve, and analysis technique, are studied. The responses are then com-

pared with previous experimental and numerical investigations conducted

by various independent researchers, and it is shown that cohesive elements

can be used in modeling crack propagation as required in pavement

engineering.

The development of a mechanisticempirical approach for the

design of pavement systems has received increased attention recently,

reigniting the debate over the use of statisticalempirical transfer

functions, whose experimental verification is questionable, at best

(1, 2). Following an exhaustive examination of various fracture

mechanics options offered as replacements to Miners hypothesis

(3), Hillerborg et al.s fictitious crack model (FCM) was found to be

the most promising for studying crack propagation in portland

cement concrete (PCC) pavements (4), and a step-by-step

approach was outlined for its implementation (5). Accordingly,

Ioannides and Sengupta (6) formulated a two-dimensional (2-D)

numerical procedure to simulate crack propagation on the basis of

the FCM for a simply supported beam. The response of the beam

over the elastic region was analyzed with the commercial finite ele-

ment (FE) software GTSTRUDL (7), while its fracture behavior

was studied with a specially coded FORTRAN program, called

CRACKIT. To facilitate the generalized application of the con-

cepts implicit in the GTSTRUDLCRACKIT combination, Ioan-

nides et al. (8) subsequently implemented their approach by using

the general purpose FE package ABAQUS (9). They reported that the

applicability of the built-in fracture analysis capabilities of

ABAQUS was too limited for pavement engineering, especially

because the FCM was not used. Consequently, in their 2-D study of

simply supported beams, the investigators employed a nonlinear

spring element from the ABAQUS library, JOINTC, to model the

fracture zone. An alternative approach was developed by Song (10),

Department of Civil and Environmental Engineering, University of Cincinnati

(ML-0071), P.O. Box 210071, Cincinnati, OH 45221-0071. Corresponding author:

T. W. Aure, [email protected].

Transportation Research Record: Journal of the Transportation Research Board,

No. 2154, Transportation Research Board of the National Academies, Washington,

D.C., 2010, pp. 1221.

DOI: 10.3141/2154-02


2/10

Upon the successful conclusion of the linear elastic analysis, the

simulation of crack propagation can be carried out, by implementing

the built-in cohesive elements of ABAQUS, on the basis of the FCM

for fracture. In all analyses, Elements CPS4 and C3D27 are used

for 2-D and 3-D discretizations of the intact material, respectively,

while the cohesive fracture zone is simulated with COH2D4 (2-D)

and COH3D8 (3-D) elements. Program runs reported here capture

the effects of the numerical analysis technique, mesh fineness,

cohesive-zone width, and softening-curve type. The resulting simula-

tion is finally used to reproduce numerical and experimental studies

conducted by other independent researchers, thereby adding to its

credibility.

ANALYSIS OF LINEAR ELASTIC RESPONSE

This section reports on testing of the robustness of the proposed

FE formulation through simulation of the linear elastic response of

Beam A (Table 1). This beam was originally used by Sengupta (16)

and was later adopted by Ioannides et al. (8), whose results are,

therefore, available for comparison, along with published closed-

form solutions. Both concentrated and uniformly distributed loads

are considered.

Beam A was assumed to be simply supported on rollers, requiring

2 degrees of freedom (vertical displacement) to be fixed at the two

support nodes. Because of symmetry, half of the beam was meshed

uniformly with 64 48 CPS4 and 40 30 5 C3D27 elements, forthe 2-D and 3-D discretizations, respectively. All elements used were

nearly square, which thereby eliminated any significant aspect ratio

effects. Once the mesh for the half of the beam on the right-hand side

was defined, a mirror image was created along the plane of symmetry

and surface-based tie constraints used to connect surfaces on either

side of the symmetry plane.

The simulations first considered a concentrated load of 33.7 kips,

applied at the midspan. For the 2-D model, the load was applied on

the top node at the midspan. To avoid localized effects in the 3-D

idealization, a small loaded area 0.4 1.5 in. (two elements alongthe symmetry line) was defined at the center of the beam, on which a

pressure of 56.17 ksi was applied. The same beam was also subjected

to a uniformly distributed load of 10 kips/in. This load was applied

at a pressure of 6.667 ksi (10 kips/in. divided by 1.5 in.) over the

top-surface elements.

Simulation results were compared with closed-form solutions.

For the beam subjected to concentrated load, the midspan deflection

was compared with that reported by Timoshenko and Goodier (17)

as accounting for the shear deformation, with which it was found to

agree within 2.3% and 0.3%, for the 2-D and 3-D discretizations,

respectively. For the uniformly distributed load, results were com-

Aure and Ioannides 13

pared with values computed in accordance with various theories, as

summarized by Shames and Dym (18). It was found that the FE

idealization exhibited near-perfect agreement with the theory of

elasticity accounting for Poissons ratio effect. Moreover, the FE

solution provided an excellent approximation (98.86% and 98.66%

for 2-D and 3-D, respectively) to the beam with a shear deformation

solution. It was, therefore, concluded that the numerical approach

implemented in this study was robust as far as the linear elastic aspects

of the analysis were concerned and that it might be considered a good

candidate for investigating the more-demanding fracture mechanics

issues of crack propagation in simply supported beams.

SIMULATION OF CRACK PROPAGATION

BY USING BUILT-IN COHESIVE ELEMENTS

This section outlines the FE formulation of the cohesive elements

recently introduced in ABAQUS and their implementation in accor-

dance with the FCM for investigating the postfracture response of

simply supported concrete beams. The sensitivity of the FE solution

to a variety of aspects of the numerical procedure employed is also

examined.

Idealization of Cohesive Zone

Among the three classes of cohesive elements that are available in

ABAQUS, those based on the so-called tractionseparation formu-

lation are the most suitable for use in crack propagation studies that

use the FCM. Accordingly, the load-response process can be sub-

divided into three stages: precrack, initiation of crack, and postpeak

(or softening) behavior.

Precrack Behavior

During the precrack stage, the material along the beam centerline

is considered to experience a very small but finite separation and

the cohesive-element response is governed by the following elastic

straindisplacement relations (15):

where and w are the nominal strain and elastic separation, respec-

tively, in the normal (n) and two shear directions (s and t) and T0 is

n

s

t

n

s

t

T

w

w

w

=

1

0

( )1

TABLE 1 Geometry and Material Pro pert ies of Beams Studied

Youngs Tensile FractureSpan Length Depth Width Modulus Strength EnergyS L h b E f t GF

Beam Source (in.) (in.) (in.) (in.) (ksi) (ksi) (lbf/in.)

A Sengupta (16) 16.0 16.0 6.0 1.5 4,000 0.463 0.431

B Liu (19) 12.0 12.0 3.0 1.0 5,405 0.463 0.431

C Roesler et al. (20) 39.4 43.3 9.8 3.2 4,641 0.602 0.954

NOTE: 1 lbf= 4.444 N; 1 in. = 25.4 mm; 1 ksi = 6.89 MPa.


3/10

initial width of the cohesive zone. The elastic stress components can

then be computed from Equation 2:

where Kis a nominal stiffness (also referred to as penalty stiffness)

and tis the nominal stress, in the normal and two shear directions,

respectively. If the shear and normal components are uncoupled,

Equation 2 will reduce to

For a 2-D model, only the first two rows and columns of Equation 3

are used.

Selection of the initial width of the cohesive zone and of the penalty

stiffness, K, is largely based on prior experience with using thesoftware, yet it can influence the solution convergence significantly.

ABAQUS (15) recommends computing the penalty stiffness from

Knn =E/T0, whereEis theYoungs modulus oftheintact(or uncracked)material. Similarly, it may be assumed that Kss = Ktt = G/T0, whereG is the corresponding shear modulus of the intact material.

Initiation of Crack and Postpeak Behavior

Crack initiation refers to the beginning of the degradation of the

material. In PCC crack propagation studies, it is often assumed that

the crack initiates when the stress reaches the tensile strength, ft, ofthe material (6, 19, 20). Once the crack initiates, material damage

evolves on the basis of a predefined softening law. Park et al. (21)

provided a comprehensive list of softening options proposed for

concrete. First, Hillerborg et al. (4) used a linear softening curve.

Today, it is more common to employ a bilinear curve characterized

by three points, as shown in Figure 1: (a) the crack initiation point,

defined by traction stressft and separation wcr ; (b) the kink point,

t

t

t

K

K

K

n

s

t

nn

ss

tt

=

0 0

0 0

0 0

n

s

t

( )3

t

t

t

K K K

K K

K

n

s

t

nn ns nt

ss st

tt

=

symm

n

s

t

( )2

14 Transportation Research Record 2154

at which the separationtraction stress pair is wk, ft; and (c) thecritical (or maximum) separation, wf, for which the traction stress

is zero.

The crack evolution for any kind of softening is facilitated

by using a dimensionless damage variable defined as shown in

Equation 4 (14, 15):

where ts is the traction stress for separation w, along the softeningcurve, and ti is the traction stress that would have correspondedto w had the precrack stiffness endured, as explained in Figure 1. For

linear softening, Equation 4 yields

For bilinear softening, the location of the kink point is obviously

important.

ABAQUS incorporates only linear and exponential softening

curves. Other kinds of curves (including the bilinear one), however,may be specified by the user in a tabular form.

Sensitivity Study for Proposed Discretization

Previous investigators have examined in detail the effect of several

variables influencing numerical solutions analogous to that proposed

in the present paper. Thus, Song et al. (11) studied the effect of total

fracture energy, GF (defined as the area under the bilinear curve

shown in Figure 1), of tensile strength, ft, and of cohesive zone mesh,on the fracture of asphalt concrete beams. Furthermore, Park (22)

examined the sensitivity of the solution to the initial fracture energy,

Gf (defined as the area under the first two limbs of the softening

curve shown in Figure 1), as well as of the location of the kink point,

for PCC specimens. The latter was also investigated by Gaedicke

and Roesler (14) by using built-in cohesive elements.

In this section, the sensitivity of the proposed FE discretization to

the analysis technique, mesh size, cohesive zone width for PCC, and

type of softening curve is investigated. The beams are idealized by

Dw w w

w w ww w w

f

f

f=( )( )

cr

cr

cr( )5

D

t

t w w w

s

if=

1 4cr ( )

Separation, w

Tractions, t

(wi, ti)

Knn

wfw1wcr

(wk, ft)

ft (wi, ts)

ti

wi

FIGURE 1 Bilinear softening curve.


4/10

using 3-D elements, C3D27 for the intact material and COH3D8 for

the cohesive zone, respectively.

Effect of Analysis Technique

For the purposes of this investigation, there are two analysis options

in ABAQUS: the general (or default, which uses a NewtonRaphson

method) and the modified Riks procedures (15). The latter is par-ticularly suited for potentially unstable problems that occasionally

exhibit negative stiffness values or present convergence difficulties

(e.g., buckling andsnap-backoftheloaddisplacementcurve). Because

cohesive elements involve softening that results from progressive

material damage, they may also experience such numerical problems.

A possible solution when using the NewtonRaphson approach is to

apply viscous regularization (15), but a preferable alternative is to

employ the modified Riks procedure. For example, Song et al. (11, 23)

encountered divergence problems when using the NewtonRaphson

method, whereas the modified Riks approach produced convergence.

For their part, Yang and Proverbs (24) studied the efficacy of various

solution strategies for fracture and concluded that arc lengthbased

solvers (such as that used in the modified Riks method) would cap-

ture the softening behavior in the snap-back type of loaddisplacement

relations.

To study the effect of the two analysis options in ABAQUS,

Beam A in Table 1 was considered in an unnotched configuration.

The cohesive zone width was set to 0.001 in. The penalty stiffnesses,

Knn, Kss, and Ktt, were computed as noted earlier. Linear softening

was used for simplicity.

For the NewtonRaphson method, the loading parameters that

need to be specified are initial time increment (ITI), time period of

the step (TPS), minimum time increment (MnTI). and maximum time

increment (MxTI). These were set to 6.E3, 1.0, 1.E9, and 3.E2,

respectively, on the basis of previous work by Ioannides et al. (8).

To alleviate convergence problems, viscous regularization is used


with viscosity set to 1E06, selected on the basis of several pre-

liminary trials. The maximum number of time increments is set to 200.

In contrast, the modified Riks approach requires four parameters to

be defined: the initial increment in the arc length along the static

equilibrium path, lin; the total arc length scale factor, lperiod; and theminimum and maximum arc length increments, lmin and lmax, respec-

tively. A convenient way to assign numerical values for these param-

eters is to retain those specified above: 6.E3, 1.0, 1.E9, 3.E2,

respectively. Moreover, the terminal increment is similarly set to 200.

The results are plotted in Figure 2 as crack mouth opening displace-

ment versus load (CMOD-P) and load line displacement versus load

(LLD-P) curves. It is clear that the trends captured by the two methods

are significantly different; those in accordance with the modified

Riks approach are considered more realistic because they reflect

the snap-back behavior expected in the softening stage. It is recom-

mended, therefore, that the modified Riks procedure be adopted for

use with built-in cohesive elements.

Effect of Mesh Fineness

Beam B in Table 1 is considered for this set of runs, which involves

both multiple mesh configurations for the intact region of the beam

and a constant mesh for the cohesive zone. Linear softening was

used for simplicity. The coarse mesh consisted of 24 6 2 elementsin the length, depth, and width directions, whereas the fine mesh

had 60 15 8 elements and the median mesh had 36 9 4 sub-divisions. The cohesive zone mesh was set to be 10 times as fine as

the median mesh in the depth and width directions and had one ele-

ment in the length direction. In view of space limitations, no graph

of results is presented here because the effect of fineness was

surprisingly insignificant for the meshes adopted, especially for the

CMOD response. Inasmuch as there were differences between the

coarser and the finer meshes for the LLD response, it was observed

that a finer mesh generally resulted in a slightly lower load before

1.80

1.60

1.40

1.20

1.00

0.80

Load(

kips)

0.60

0.40

0.20

0.000.000 0.001 0.002

CMOD/2 or LLD (in.)

CMOD

CMOD

LLD

LLD

0.003 0.004 0.005

NR

Riks

NR

Riks

FIGURE 2 Effect of analysis technique (NewtonRaphson versus modified Riks)on Beam A.


5/10

the peak and a slightly higher load after the peak, but the differences

did not exceed 5%.

Effect of Cohesive-Zone Width

To investigate the influence of the cohesive-zone width, three widths

were considered: T0 = 1.0, 0.01, and 0.001 in. Three-dimensional FE

discretization of Beam B with linear softening was carried out.Figure 3 shows that, as the cohesive zone width increased, the elastic

deformation of the cohesive zone increased, contributing much to

the elastic fracture energy until damage began. Once damage started,

the responses were not sensitive to the cohesive-zone width. An

increase in cohesive-zone width decreased the peak load that could

be supported by the beam. The 0.01- and 0.001-in. widths, however,

yielded almost the same result, which led to the decision to use

0.001 in. in all subsequent sections.

Effect of Softening Curve

In the cases considered above, linear softening was used for its

simplicity. A comparison between linear and bilinear curves employed

in conjunction with Beam B indicated that linear softening wouldoverpredict the peak load by about 11%. After the peak, however, the

bilinear curve would eventually give a higher load than the linear

one. The areas under CMOD-P curves for both softening models

seemed to be approximately the same because the areas under both

softening curves were also assumed to be the same.

The sensitivity of the proposed fracture formulation to various

parameters has been investigated. It can be concluded that careful

attention should be given to the selection of the type of solver, the

width of the cohesive zone, and the type of softening curve in model-

ing crack propagation by using cohesive elements. Once the effect

of each parameter involved in the cohesive zone FE analysis is

understood, the approach is then applied to reproducing numerical


and experimental results obtained by other researchers, as described

in the following sections.

COMPARISON WITH PREVIOUS

NUMERICAL MODELS

In this section, comparison is made of simulation results obtained

from this study with those from numerical studies conducted by otherindependent researchers. The purpose is to validate the proposed FE

procedure by contributing evidence confirming its credibility.

GTSTRUDLCRACKIT by Sengupta

In the earliest University of Cincinnati effort to simulate crack propa-

gation in simply supported concrete beams, Sengupta (16) developed

a combination approach that employed a commercial software FE

package, GTSTRUDL, for the elastic response and the corresponding

flexibility matrix, in tandem with a specially coded FORTRAN com-

puter program named CRACKIT for the ensuing fracture behavior, in

accordance with a bilinear softening law and the FCM. To illustrate

his approach, Sengupta (16) reproduced a beam that had first beenstudied by Liu (19) and discretized it with four-node plane stress

elements of size 0.2 0.5 in. Lius beam is Beam B in Table 1 and isconsidered in the present study with a 3-D idealization by using C3D27

elements of size 0.2 0.2 0.2 in. Results from the 2-D CRACKITapproach by Sengupta and the 3-D COH3D8 procedure employed

for the cohesive fracture zone in this study are shown in Figure 4, in

which the LLD-P curves appear to agree better than the CMOD-P

curves. The difference between the LLD-P curves can be explained

by the difference in the intact region elements used (2-D four-node

element versus 3-D 27-node element), whereas the CMOD-P dis-

crepancy is mainly due to the implicit assumption in CRACKIT that

the notch remains undeformed until the crack begins to propagate.

0.45

0.40

0.35

0.30

0.25

0.20Load(

kips)

0.15

0.10

0.05

0.000.000 0.001 0.002

CMOD/2 (in.)

t=1.000 in.

t=0.010 in.

t=0.001 in.

0.003 0.004 0.005

FIGURE 3 Effect of cohesive zone width (Beam B).


6/10

ABAQUSJOINTC by Ioannides et al.

To model crack propagation in concrete beams with ABAQUS,

Ioannides et al. (8) used CPS4 elements for the intact region and a

JOINTC nonlinear spring element for the fracture zone, prescribing

the same bilinear FCM curve as Sengupta (16). Beam A in Table 1

was considered and discretized with a coarse FE mesh consisting

of elements of size 1 1 in. The notch-to-depth ratio was 13. Anidentical mesh pattern and element type were used in the present


2-D study, in which each intact zone element was a CPS4 and the

cohesive zone was discretized by using COH2D4 instead of JOINTC

elements.

The results are shown in Figure 5. The cohesive model over-

predicts the load for a given CMOD in the postpeak stage. In general,

however, the two models are in good agreement for the particular

mesh considered. Nonetheless, the wavy curves in Figure 5 suggest

some convergence difficulties, which may easily be overcome when

the mesh is refined. To complicate matters, however, mesh refinement

0.18

0.16

0.14

0.12

0.10

0.08Load(

kips)

0.06

0.04

0.02

0.000.000 0.001 0.002

CMOD/2 or LLD (in.)

CMOD

CMOD

LLD

LLD

2D CRACKIT

3D COH3D8

2D CRACKIT

3D COH3D8

0.003 0.0040.001 0.002 0.003

FIGURE 4 Comparison of 2-D CRACKIT with 3-D cohesive elements (Beam B).

CMOD

CMODLLD

LLD

2D JOINTC

2D COH3D82D JOINTC

2D COH3D8

0.70

0.40

0.50

0.60

Load(

kips)

0.30

0.20

0.10

0.000.000 0.001 0.002

CMOD/2 or LLD (in.)

0.003 0.004

FIGURE 5 Comparison between 2-D JOINTC and 2-D cohesive elements (Beam A).


7/10

is also found to increase the discrepancy between JOINTC and the

cohesive elements. Further research is required to identify the source

of this phenomenon, but the preliminary postulate is that it is related

to differences in the assumptions of the two elements in regard to

precrack behavior.

ABAQUS2-D UEL by Roesler et al.

Roesler et al. (20) employed a UEL for the fracture zone and a

four-node plane stress element for the intact material in a mesh that

was significantly finer near the cohesive zone than farther away. The

geometry and material properties of their beam are shown in Table 1

in relation to Beam C. The following parameters were also adopted:

T0 = 0.04 in.,E= 4.6 Mpsi, Gf= 0.323 lbf/in., GF= 0.954 lbf/in., and = 0.25.

To reproduce in the present study the results of the 2-D analysis

presented by Roesler et al. (20), Beam C was meshed uniformly

with 0.2- 0.2-in. elements in both directions for the intact material,whereas the mesh of the cohesive zone was made 5 times as fine.

The penalty stiffnesses for the givenEand T0 values were computed

as Knn = 118 Mpsi/in. and Kss = 51 Mpsi/in.

The CMOD-P curve obtained in the present study along withthat presented by Roesler et al. (2007) is shown in Figure 6. As can

be seen, good agreement is obtained between the two numerical

simulations. The small difference in the elastic region may be due

to the respective approaches followed in establishing the penalty

stiffnesses.

ABAQUS2-D COH2D4 by Gaedicke and Roesler

Gaedicke and Roesler (14) were the first to use the built-in 2-D

ABAQUS cohesive element COH2D4 to model the fracture process


of Beam C for a variety of kink point locations. Their mesh was

similar to that used by Roesler et al. (20). Their CMOD-P curve for

= 0.25 is plotted in Figure 7, along with the corresponding resultsfrom the present study. The peak load predictions differ by about 7%;

Gaedicke and Roesler had reported that their model underpredicted

the peak load by 12% and 7% with respect to the average and min-

imum experimental peak load, respectively (14). This may again

be attributed to penalty stiffness differences.

From the comparisons with previously reported results, it maybe concluded that the proposed use of built-in ABAQUS cohesive

elements is effective in simulating PCC fracture. Comparison with

experimental measurements reported by other researchers appears

in the following section.

COMPARISONS WITH EXPERIMENTAL RESULTS

In this section, FE simulations conducted by means of the proposed

procedure that implements cohesive elements in ABAQUS are

compared with experimental measurements reported by various

independent researchers. The simulations employ 3-D discretizations

with bilinear softening curves in all cases.

Experimental Results by Liu

Liu (19) tested notched-beam specimens under center-point loading.

The pertinent geometry and average material properties reported

are shown in Table 1, as Beam B. A comparison of the CMOD-P

and LLD-P curves is shown in Figure 8. Good agreement is obtained

between the numerical solution in the present study and Lius

experimental results, especially for the CMOD-P curve. The small

difference in the elastic portion of the LLD-P was explained by Liu

as the result of support settlement, which can cause the measured

1.60

1.40

1.20

1.00

0.80

Load(

kips)

0.60

0.40

0.20

0.000.000 0.005 0.010

CMOD (in.)

Roesler et al. (2007)

This study

2D UEL

2D COH2D4

0.015 0.020

FIGURE 6 Comparison of Roesler et al. (20) numerical results with current studysresults (Beam C).


8/10


1.60

1.40

1.20

1.00

0.80

Load(

kips)

0.60

0.40

0.20

0.000.000 0.005 0.010

CMOD (in.)

Gaedicke and Roesler (2009)

This study

2D COH2D4

2D COH2D4

0.015 0.020

FIGURE 7 Comparison of Gaedicke and Roesler (14 ) numerical results with currentstudys results (Beam C).

0.18

0.16

0.14

0.12

0.10

0.08Load(

kips)

0.06

0.04

0.02

0.000 0.002

CMOD or LLD (in.)

CMOD

CMOD

LLD

LLD

Experimental

3D COH3D8

Experimental

3D COH3D8

0.003 0.0050.0040.001

FIGURE 8 Comparison of Liu s (19) experimental results with current studys results(Beam B).


9/10

load-point deflection [to be] larger than the actual one. As a result,

the predicted curves are stiffer in comparison with the measured

ones. A more sophisticated testing set-up is needed to overcome this

problem (19).

Experimental Results by Roesler et al.

Beam C of Table 1 is considered here. Good agreement is obtained

between the experimental results by Roesler et al. (20) and the FE

simulation conducted in the present study, as shown in Figure 9. In

the elastic region, the idealization underpredicts the load for a given

CMOD. This can be explained by the use of a low-penalty stiffness.

The numerical procedure reproduced the peak load well. The postpeak

behavior is accurately reproduced up to CMOD of 0.0063 in. If one

keeps in mind the variability in the experimental results from replicate

specimens reported by Roesler et al. (20), it can be concluded that

the numerical simulation has reasonably captured the fracture process.

The results shown in Figures 8 and 9 indicate the potential use

of cohesive elements available in ABAQUS to simulate crack

propagation in concrete on the basis of the FCM. The findings from

this study affirm the potential of the proposed numerical procedurewhen extended to PCC slabs-on-grade in the near future.

CONCLUSION

This study focused on the use of 2-D and 3-D cohesive elements

that have recently become available in the commercial FE software

package ABAQUS for studying crack propagation in simply sup-

ported concrete beams. With reliance on Hillerborgs FCM, the input

parameters needed for cohesive elements that are based on traction

separation could be specified. These elements were then inserted at


the cohesive fracture zone with the top and bottom faces of the

elements tied to beam elements at the right and left sides of the crack

plane. Analyses examined the effect of solution techniques, mesh

size, and thickness of cohesive zone.

Comparison of this study with other numerical studies was

presented. Good agreement was found with Senguptas GTSTRUDL

CRACKIT combination. The small discrepancy observed can be

ascribed to an expedient assumption implicit in CRACKIT. Theproposed FE formulation also gave good agreement with the UEL

created by Park (22). Results from the present study were also com-

pared with experimental data reported by different researchers and

good agreement was again found.

The main advantages of cohesive elements over other numerical

models presented can be summarized as (a) capability to be used in

3-D FE analysis, (b) ability to use different kind of softening curve

assumptions, and (c) possibility to use other failure criteria. The use of

cohesive elements, however, is computationally extremely intensive

and requires significant computer resources.

Use of cohesive elements that are based on tractionseparation in

tracking material damage introduces nonlinearity to the system, which

results in convergence problems, especially if the NewtonRaphson

solution algorithm is used. To avoid this problem, the modified Riksmethod (or the NewtonRaphson method with viscous regularization)

can be used.

From the findings in this study, it is anticipated that cohesive

elements that are based on tractionseparation will be used in

large problems with PCC pavement slabs rest ing on layered foun-

dations. It is hoped that the use of fracture mechanics concepts

will eventually lead to the definition of more reliable and realistic

failure criteria, ones addressing the current weaknesses of statistical

empirical transfer functions commonly employed in pavement

design guides.

1.60

1.40

1.20

1.00

0.80

Load(

kips)

0.60

0.40

0.20

0.00

0.000 0.005 0.010CMOD (in.)

Roesler et al. (2007)

This study

Experimental

3D COH3D8

0.015 0.020

FIGURE 9 Comparison of the Roesler et al. (20) experimental results with currentstudys results (Beam C).


10/10

ACKNOWLEDGMENT

This work was supported in part by an allocation of computing time

from the Ohio Supercomputer Center.

REFERENCES

1. Ioannides, A. M. Pavement Fatigue Concepts: A Historical Review.In Proc., 6th International Conference on Concrete Pavements, Vol. 3,Indianapolis, Ind., 1997, pp. 147159.

2. Khazanovich, L., and D. Tompkins. Singularities in Concrete PavementAnalysis. In Proc., Workshop on Fracture Mechanics for Concrete Pave-ments: Theory to Practice, International Society for Concrete Pavements,Copper Mountain, Colo., 2005, pp. 4958.

3. Miner, M. A. Cumulative Damage in Fatigue. Transactions of theAmerican Society of Mechanical Engineers, Vol. 67, Sept. 1945,pp. A-159A-164.

4. Hillerborg, A., M. Modeer, and P. E. Petersson. Analysis of Crack For-mation and Crack Growth in Concrete by Means of Fracture Mechanicsand Finite Elements. Cement and Concrete Research, Vol. 6, No. 6, 1976,pp. 773782.

5. Ioannides, A. M. Fracture Mechanics in Pavement Engineering: TheSpecimen-Size Effect. In Transportation Research Record 1568, TRB,National Research Council, Washington, D.C., 1997, pp. 1016.

6. Ioannides, A. M., and S. Sengupta. Crack Propagation in Portland CementConcrete Beams: Implications for Pavement Design. In Transporta-tion Research Record: Journal of the Transportation Research Board,

No. 1853, Transportation Research Board of the National Academies,Washington, D.C., 2003, pp. 110117.

7. GTSTRUDL: Finite Element Computer Software System for StructuralAnalysis and Design. Users Manual.Georgia Tech Research Corporation,Georgia Institute of Technology, Atlanta, 1993.

8. Ioannides, A. M., J. Peng, and J. R. Swindler. Simulation of Concrete Frac-ture Using ABAQUS. In Proc., 8th International Conference on ConcretePavements, Vol. 3, Colorado Springs, Colo., Aug. 2005, pp. 11381154.

9. ABAQUS, v. 6.4-1. Abaqus, Inc., Providence, R.I., 2003.10. Song, S. H. Fracture of Asphalt Concrete: A Cohesive Zone Modeling

Approach Considering Viscoelastic Effects.PhD dissertation. Universityof Illinois at UrbanaChampaign, 2006.

11. Song, S. H., G. H. Paulino, and W. G. Buttlar. A Bilinear CohesiveZone Model Tailored for Fracture of Asphalt Concrete Considering


Viscoelastic Bulk Material.Journal of Engineering Fracture Mechanics,Vol. 73, No. 18, 2006, pp. 28292848.

12. Ioannides, A. M., and J. Peng. Finite Element Simulation of Crack

Growth in Concrete Slabs: Implications for Pavement Design. In Proc.,

5th International Workshop on Fundamental Modeling of Concrete Pave-

ments, Istanbul, Turkey, International Society for Concrete Pavements,

April 2004.

13. What Is New in ABAQUS 6.5? Abaqus, Inc., Providence, R.I., 2004.

www.pdfgeni.com. Accessed July 24, 2009.

14. Gaedicke, C., and J. R. Roesler. Fracture-Based Method to Determinethe Flexural Load Capacity of Concrete Slabs. FAA-05-C-AT-UIUC.

University of Illinois at UrbanaChampaign, 2009.

15. ABAQUS Analysis Users Manual, v. 6.7-1. Dassault Systmes, Prov-

idence, R.I., 2007.

16. Sengupta, S. Finite Element Simulation of Crack Growth in PCC Beams:

Implication for Concrete Pavement Design. MS thesis. University of

Cincinnati, Ohio, 1998.

17. Timoshenko, S. P., and J. N. Goodier.Theory of Elasticity. McGraw-Hill,

New York, 1970.

18. Shames, I. H., and C. L. Dym.Energy and Finite Element Methods in

Structural Mechanics. Hemisphere Publishing, New York, 1985.

19. Liu, P. Time-Dependent Fracture of Concrete. PhD dissertation. Ohio

State University, Columbus, 1994.

20. Roesler, J., G. H. Paulino, K. Park, and C. Gaedicke. Concrete Fracture

Prediction Using Bilinear Softening. Cement and Concrete Composites,

Vol. 9, No. 4, 2007, pp. 300312.

21. Park, K., G. H. Paulino, and J. R. Roesler. Determination of the Kink Pointin the Bilinear Softening Model for Concrete.Journal of Engineering

Fracture Mechanics, Vol. 75, No. 13, 2008, pp. 38063818.

22. Park, K.Concrete Fracture Mechanics and Size Effect Using a Specialized

Cohesive Zone Model. MS thesis. University of Illinois at Urbana

Champaign, 2005.

23. Song, S. H., G. H. Paulino, and W. G. Buttlar. Simulation of Crack

Propagation in Asphalt Concrete Using an Intrinsic Cohesive Zone

Model. Journal of Engineering Mechanics, Vol. 132, No. 11, 2006,

pp. 12151223.

24. Yang, Z. J., and D. Proverbs. A Comparative Study of Numerical Solutions

to Non-Linear Discrete Crack Modelling of Concrete Beams Involving

Sharp Snap-Back.Journal of Engineering Fracture Mechanics, Vol. 71,

No. 1, 2004, pp. 81105.

The Rigid Pavement Design Committee peer-reviewed this paper.

aureioannidestrb2010

Documents