12

who coded a user-defined subroutine in ABAQUS, thereby creatinga user element (UEL); an application of this UEL is described in the2-D study of crack propagation in simply supported asphalt concretebeams 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) PCCpavement 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 ofABAQUS version 6.5, which for the first time included “a family ofcohesive elements for modeling deformation and damage in finite-thickness adhesive layers between bonded parts. Cohesive elementsare typically connected to underlying elements with surface-based tieconstraints, so the mesh used for the cohesive layer can be independentof the mesh used for the bonded components” (13). Gaedicke andRoesler (14) reported using these cohesive elements in their studyof crack propagation in concrete beams and slabs.

The investigation presented in this paper is a continuation of thestep-by-step development and application of fracture mechanicstools that are based on the FCM in pavement engineering initiatedat the University of Cincinnati in the late 1990s. The main objectiveis to implement the built-in cohesive elements that have recentlybeen added to the ABAQUS library and to compare the performanceof these elements to that reported in earlier investigations. It is hopedthat, in this manner, a contribution will be made to the ongoing effortfor more mechanistic pavement design procedures that will usefracture mechanics concepts, thereby replacing the purely empirical–statistical transfer functions and Miner’s hypothesis, which arecurrently in use. Validating an FE simulation of crack propagationin a simply supported beam is considered to be a necessary precursorto a more comprehensive analysis of slabs-on-grade required forin situ pavement systems.

METHODOLOGY

The present study focuses on the postcracking response of simplysupported PCC beams by means of the commercial, general purposeFE program ABAQUS (Standard) version 6.7-1 (15). The geometricand material properties of the beams considered in this paper areshown in Table 1. To begin with, however, a linear elastic analysisusing 2-D and 3-D elements is described, and the results are comparedwith available closed-form solutions. This initial step is consideredessential in ensuring the robustness of the proposed FE formulation.

Simulation of Crack Propagation in Concrete Beams with CohesiveElements in ABAQUS

Temesgen W. Aure and Anastasios M. Ioannides

This paper discusses the simulation of crack propagation in concretebeam specimens with a finite element package, ABAQUS, version 6.7-1.Special-purpose cohesive elements are used to model the fracture processby means of the fictitious crack model. Two- as well as three-dimensionalfinite element discretizations are carried out. Parameters influencing theresponses, such as mesh fineness, cohesive zone width, type of softeningcurve, and analysis technique, are studied. The responses are then com-pared with previous experimental and numerical investigations conductedby various independent researchers, and it is shown that cohesive elementscan be used in modeling crack propagation as required in pavementengineering.

The development of a mechanistic–empirical approach for thedesign of pavement systems has received increased attention recently,reigniting the debate over the use of statistical–empirical transferfunctions, whose experimental verification is questionable, at best(1, 2). Following an exhaustive examination of various fracturemechanics options offered as replacements to Miner’s hypothesis(3), Hillerborg et al.’s fictitious crack model (FCM) was found to bethe most promising for studying crack propagation in portlandcement concrete (PCC) pavements (4), and a step-by-stepapproach 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 ofthe FCM for a simply supported beam. The response of the beamover the elastic region was analyzed with the commercial finite ele-ment (FE) software GTSTRUDL (7 ), while its fracture behaviorwas studied with a specially coded FORTRAN program, calledCRACKIT. To facilitate the generalized application of the con-cepts implicit in the GTSTRUDL–CRACKIT combination, Ioan-nides et al. (8) subsequently implemented their approach by usingthe general purpose FE package ABAQUS (9). They reported that theapplicability of the built-in fracture analysis capabilities ofABAQUS was too limited for pavement engineering, especiallybecause the FCM was not used. Consequently, in their 2-D study ofsimply supported beams, the investigators employed a nonlinearspring element from the ABAQUS library, JOINTC, to model thefracture 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, auretw@mail.uc.edu.

Transportation Research Record: Journal of the Transportation Research Board,No. 2154, Transportation Research Board of the National Academies, Washington,D.C., 2010, pp. 12–21.DOI: 10.3141/2154-02

Upon the successful conclusion of the linear elastic analysis, thesimulation of crack propagation can be carried out, by implementingthe built-in cohesive elements of ABAQUS, on the basis of the FCMfor fracture. In all analyses, Elements CPS4 and C3D27 are usedfor 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 capturethe 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 studiesconducted by other independent researchers, thereby adding to itscredibility.

ANALYSIS OF LINEAR ELASTIC RESPONSE

This section reports on testing of the robustness of the proposedFE formulation through simulation of the linear elastic response ofBeam 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 loadsare considered.

Beam A was assumed to be simply supported on rollers, requiring2 degrees of freedom (vertical displacement) to be fixed at the twosupport nodes. Because of symmetry, half of the beam was mesheduniformly with 64 × 48 CPS4 and 40 × 30 × 5 C3D27 elements, forthe 2-D and 3-D discretizations, respectively. All elements used werenearly square, which thereby eliminated any significant aspect ratioeffects. Once the mesh for the half of the beam on the right-hand sidewas defined, a mirror image was created along the plane of symmetryand surface-based tie constraints used to connect surfaces on eitherside 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 onthe top node at the midspan. To avoid localized effects in the 3-Didealization, a small loaded area 0.4 × 1.5 in. (two elements alongthe symmetry line) was defined at the center of the beam, on which apressure of 56.17 ksi was applied. The same beam was also subjectedto a uniformly distributed load of 10 kips/in. This load was appliedat a pressure of 6.667 ksi (10 kips/in. divided by 1.5 in.) over thetop-surface elements.

Simulation results were compared with closed-form solutions.For the beam subjected to concentrated load, the midspan deflectionwas compared with that reported by Timoshenko and Goodier (17 )as accounting for the shear deformation, with which it was found toagree 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, assummarized by Shames and Dym (18). It was found that the FEidealization exhibited near-perfect agreement with the theory ofelasticity accounting for Poisson’s ratio effect. Moreover, the FEsolution provided an excellent approximation (98.86% and 98.66%for 2-D and 3-D, respectively) to the beam with a shear deformationsolution. It was, therefore, concluded that the numerical approachimplemented in this study was robust as far as the linear elastic aspectsof the analysis were concerned and that it might be considered a goodcandidate for investigating the more-demanding fracture mechanicsissues 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 elementsrecently introduced in ABAQUS and their implementation in accor-dance with the FCM for investigating the postfracture response ofsimply supported concrete beams. The sensitivity of the FE solutionto a variety of aspects of the numerical procedure employed is alsoexamined.

Idealization of Cohesive Zone

Among the three classes of cohesive elements that are available inABAQUS, those based on the so-called traction–separation formu-lation are the most suitable for use in crack propagation studies thatuse 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 centerlineis considered to experience a very small but finite separation andthe cohesive-element response is governed by the following elasticstrain–displacement 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 Properties of Beams Studied

Young’s 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.

initial width of the cohesive zone. The elastic stress components canthen be computed from Equation 2:

where K is a nominal stiffness (also referred to as penalty stiffness)and t is 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 3are used.

Selection of the initial width of the cohesive zone and of the penaltystiffness, 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 fromKnn = E/T0, where E is the Young’s modulus of the intact (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 thematerial. In PCC crack propagation studies, it is often assumed thatthe crack initiates when the stress reaches the tensile strength, f ′t, ofthe material (6, 19, 20). Once the crack initiates, material damageevolves on the basis of a predefined softening law. Park et al. (21)provided a comprehensive list of softening options proposed forconcrete. First, Hillerborg et al. (4) used a linear softening curve.Today, it is more common to employ a bilinear curve characterizedby three points, as shown in Figure 1: (a) the crack initiation point,defined by traction stress f ′t 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 separation–traction stress pair is wk , ψ f ′t; and (c) thecritical (or maximum) separation, wf , for which the traction stressis zero.

The crack evolution for any kind of softening is facilitated by using a dimensionless damage variable defined as shown inEquation 4 (14, 15):

where t ′s is the traction stress for separation w, along the softeningcurve, and t ′i is the traction stress that would have correspondedto w had the precrack stiffness endured, as explained in Figure 1. Forlinear softening, Equation 4 yields

For bilinear softening, the location of the kink point is obviouslyimportant.

ABAQUS incorporates only linear and exponential softeningcurves. 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 severalvariables influencing numerical solutions analogous to that proposedin the present paper. Thus, Song et al. (11) studied the effect of totalfracture energy, GF (defined as the area under the bilinear curveshown in Figure 1), of tensile strength, f ′t, 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 softeningcurve shown in Figure 1), as well as of the location of the kink point,for PCC specimens. The latter was also investigated by Gaedickeand Roesler (14) by using built-in cohesive elements.

In this section, the sensitivity of the proposed FE discretization tothe analysis technique, mesh size, cohesive zone width for PCC, andtype of softening curve is investigated. The beams are idealized by

Dw w w

w w ww w wf

ff=

−( )−( ) ≤ ≤cr

crcr ( )5

Dt

tw w ws

if= − ′

′≤ ≤1 4cr ( )

Separation, w

Tractions, t

(wi, t´i)

Knn

wfw1wcr

(wk, ψf´t)

f´t (wi, t´s)

t´i

wi

FIGURE 1 Bilinear softening curve.

using 3-D elements, C3D27 for the intact material and COH3D8 forthe cohesive zone, respectively.

Effect of Analysis Technique

For the purposes of this investigation, there are two analysis optionsin ABAQUS: the general (or default, which uses a Newton–Raphsonmethod) and the modified Riks procedures (15). The latter is par-ticularly suited for potentially unstable problems that occasionallyexhibit negative stiffness values or present convergence difficulties(e.g., buckling andsnap-backof theload–displacementcurve). Becausecohesive elements involve softening that results from progressivematerial damage, they may also experience such numerical problems.A possible solution when using the Newton–Raphson approach is toapply “viscous regularization” (15), but a preferable alternative is toemploy the modified Riks procedure. For example, Song et al. (11, 23)encountered divergence problems when using the Newton–Raphsonmethod, whereas the modified Riks approach produced convergence.For their part, Yang and Proverbs (24) studied the efficacy of varioussolution strategies for fracture and concluded that arc length–basedsolvers (such as that used in the modified Riks method) would cap-ture the softening behavior in the snap-back type of load–displacementrelations.

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 softeningwas used for simplicity.

For the Newton–Raphson method, the loading parameters thatneed to be specified are initial time increment (ITI), time period ofthe step (TPS), minimum time increment (MnTI). and maximum timeincrement (MxTI). These were set to 6.E–3, 1.0, 1.E–9, and 3.E–2,respectively, on the basis of previous work by Ioannides et al. (8).To alleviate convergence problems, viscous regularization is used

Aure and Ioannides 15

with viscosity set to 1E–06, 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 tobe defined: the initial increment in the arc length along the staticequilibrium path, Δlin; 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.E–3, 1.0, 1.E–9, 3.E–2,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 methodsare significantly different; those in accordance with the modifiedRiks approach are considered more realistic because they reflectthe snap-back behavior expected in the softening stage. It is recom-mended, therefore, that the modified Riks procedure be adopted foruse with built-in cohesive elements.

Effect of Mesh Fineness

Beam B in Table 1 is considered for this set of runs, which involvesboth multiple mesh configurations for the intact region of the beamand a constant mesh for the cohesive zone. Linear softening wasused for simplicity. The coarse mesh consisted of 24 × 6 × 2 elementsin the length, depth, and width directions, whereas the fine meshhad 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 asthe median mesh in the depth and width directions and had one ele-ment in the length direction. In view of space limitations, no graphof results is presented here because the effect of fineness was surprisingly insignificant for the meshes adopted, especially for theCMOD response. Inasmuch as there were differences between thecoarser and the finer meshes for the LLD response, it was observedthat a finer mesh generally resulted in a slightly lower load before

1.80

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0.000.000 0.001 0.002

CMOD/2 or LLD (in.)

CMODCMODLLDLLD

0.003 0.004 0.005

NRRiksNRRiks

FIGURE 2 Effect of analysis technique (Newton–Raphson versus modified Riks) on Beam A.

the peak and a slightly higher load after the peak, but the differencesdid not exceed 5%.

Effect of Cohesive-Zone Width

To investigate the influence of the cohesive-zone width, three widthswere considered: T0 = 1.0, 0.01, and 0.001 in. Three-dimensional FEdiscretization of Beam B with linear softening was carried out.Figure 3 shows that, as the cohesive zone width increased, the elasticdeformation of the cohesive zone increased, contributing much tothe elastic fracture energy until damage began. Once damage started,the responses were not sensitive to the cohesive-zone width. Anincrease in cohesive-zone width decreased the peak load that couldbe supported by the beam. The 0.01- and 0.001-in. widths, however,yielded almost the same result, which led to the decision to use0.001 in. in all subsequent sections.

Effect of Softening Curve

In the cases considered above, linear softening was used for itssimplicity. A comparison between linear and bilinear curves employedin conjunction with Beam B indicated that linear softening wouldoverpredict the peak load by about 11%. After the peak, however, thebilinear curve would eventually give a higher load than the linearone. The areas under CMOD-P curves for both softening modelsseemed to be approximately the same because the areas under bothsoftening curves were also assumed to be the same.

The sensitivity of the proposed fracture formulation to variousparameters has been investigated. It can be concluded that carefulattention should be given to the selection of the type of solver, thewidth of the cohesive zone, and the type of softening curve in model-ing crack propagation by using cohesive elements. Once the effectof each parameter involved in the cohesive zone FE analysis isunderstood, the approach is then applied to reproducing numerical

16 Transportation Research Record 2154

and experimental results obtained by other researchers, as describedin the following sections.

COMPARISON WITH PREVIOUS NUMERICAL MODELS

In this section, comparison is made of simulation results obtainedfrom this study with those from numerical studies conducted by otherindependent researchers. The purpose is to validate the proposed FEprocedure by contributing evidence confirming its credibility.

GTSTRUDL–CRACKIT by Sengupta

In the earliest University of Cincinnati effort to simulate crack propa-gation in simply supported concrete beams, Sengupta (16) developeda combination approach that employed a commercial software FEpackage, GTSTRUDL, for the elastic response and the correspondingflexibility matrix, in tandem with a specially coded FORTRAN com-puter program named CRACKIT for the ensuing fracture behavior, inaccordance with a bilinear softening law and the FCM. To illustratehis approach, Sengupta (16) reproduced a beam that had first beenstudied by Liu (19) and discretized it with four-node plane stresselements of size 0.2 × 0.5 in. Liu’s beam is Beam B in Table 1 and isconsidered in the present study with a 3-D idealization by using C3D27elements of size 0.2 × 0.2 × 0.2 in. Results from the 2-D CRACKITapproach by Sengupta and the 3-D COH3D8 procedure employedfor the cohesive fracture zone in this study are shown in Figure 4, inwhich the LLD-P curves appear to agree better than the CMOD-Pcurves. The difference between the LLD-P curves can be explainedby the difference in the intact region elements used (2-D four-nodeelement versus 3-D 27-node element), whereas the CMOD-P dis-crepancy is mainly due to the implicit assumption in CRACKIT thatthe notch remains undeformed until the crack begins to propagate.

0.45

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0.35

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0.25

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s)

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).

ABAQUS–JOINTC 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 aJOINTC nonlinear spring element for the fracture zone, prescribingthe same bilinear FCM curve as Sengupta (16). Beam A in Table 1was considered and discretized with a coarse FE mesh consistingof elements of size 1 × 1 in. The notch-to-depth ratio was 1⁄3. Anidentical mesh pattern and element type were used in the present

Aure and Ioannides 17

2-D study, in which each intact zone element was a CPS4 and thecohesive zone was discretized by using COH2D4 instead of JOINTCelements.

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 particularmesh considered. Nonetheless, the wavy curves in Figure 5 suggestsome convergence difficulties, which may easily be overcome whenthe mesh is refined. To complicate matters, however, mesh refinement

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0.000.000 0.001 0.002

CMOD/2 or LLD (in.)

CMOD

CMODLLD

LLD

2D CRACKIT

3D COH3D82D 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

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0.50

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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).

is also found to increase the discrepancy between JOINTC and thecohesive elements. Further research is required to identify the sourceof this phenomenon, but the preliminary postulate is that it is relatedto differences in the assumptions of the two elements in regard toprecrack behavior.

ABAQUS–2-D UEL by Roesler et al.

Roesler et al. (20) employed a UEL for the fracture zone and afour-node plane stress element for the intact material in a mesh thatwas significantly finer near the cohesive zone than farther away. Thegeometry and material properties of their beam are shown in Table 1in 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., andψ = 0.25.

To reproduce in the present study the results of the 2-D analysispresented by Roesler et al. (20), Beam C was meshed uniformlywith 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 given E and T0 values were computedas 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 canbe seen, good agreement is obtained between the two numericalsimulations. The small difference in the elastic region may be dueto the respective approaches followed in establishing the penaltystiffnesses.

ABAQUS–2-D COH2D4 by Gaedicke and Roesler

Gaedicke and Roesler (14) were the first to use the built-in 2-DABAQUS cohesive element COH2D4 to model the fracture process

18 Transportation Research Record 2154

of Beam C for a variety of kink point locations. Their mesh wassimilar to that used by Roesler et al. (20). Their CMOD-P curve forψ = 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 “underpredictedthe peak load by 12% and 7% with respect to the average and min-imum experimental peak load, respectively” (14). This may againbe attributed to penalty stiffness differences.

From the comparisons with previously reported results, it maybe concluded that the proposed use of built-in ABAQUS cohesiveelements is effective in simulating PCC fracture. Comparison withexperimental measurements reported by other researchers appearsin the following section.

COMPARISONS WITH EXPERIMENTAL RESULTS

In this section, FE simulations conducted by means of the proposedprocedure that implements cohesive elements in ABAQUS arecompared with experimental measurements reported by variousindependent researchers. The simulations employ 3-D discretizationswith 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 reportedare shown in Table 1, as Beam B. A comparison of the CMOD-Pand LLD-P curves is shown in Figure 8. Good agreement is obtainedbetween the numerical solution in the present study and Liu’sexperimental results, especially for the CMOD-P curve. The smalldifference in the elastic portion of the LLD-P was explained by Liuas “the result of support settlement, which can cause the measured

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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 study’sresults (Beam C).

Aure and Ioannides 19

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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 currentstudy’s results (Beam C).

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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 study’s results(Beam B).

load-point deflection [to be] larger than the actual one. As a result,the predicted curves are stiffer in comparison with the measuredones. A more sophisticated testing set-up is needed to overcome thisproblem” (19).

Experimental Results by Roesler et al.

Beam C of Table 1 is considered here. Good agreement is obtainedbetween the experimental results by Roesler et al. (20) and the FEsimulation conducted in the present study, as shown in Figure 9. Inthe elastic region, the idealization underpredicts the load for a givenCMOD. This can be explained by the use of a low-penalty stiffness.The numerical procedure reproduced the peak load well. The postpeakbehavior is accurately reproduced up to CMOD of 0.0063 in. If onekeeps in mind the variability in the experimental results from replicatespecimens reported by Roesler et al. (20), it can be concluded thatthe numerical simulation has reasonably captured the fracture process.

The results shown in Figures 8 and 9 indicate the potential useof cohesive elements available in ABAQUS to simulate crackpropagation in concrete on the basis of the FCM. The findings fromthis 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 elementsthat have recently become available in the commercial FE softwarepackage ABAQUS for studying crack propagation in simply sup-ported concrete beams. With reliance on Hillerborg’s FCM, the inputparameters needed for cohesive elements that are based on traction–separation could be specified. These elements were then inserted at

20 Transportation Research Record 2154

the cohesive fracture zone with the top and bottom faces of theelements tied to beam elements at the right and left sides of the crackplane. Analyses examined the effect of solution techniques, meshsize, and thickness of cohesive zone.

Comparison of this study with other numerical studies was presented. Good agreement was found with Sengupta’s GTSTRUDL–CRACKIT combination. The small discrepancy observed can beascribed to an expedient assumption implicit in CRACKIT. Theproposed FE formulation also gave good agreement with the UELcreated by Park (22). Results from the present study were also com-pared with experimental data reported by different researchers andgood agreement was again found.

The main advantages of cohesive elements over other numericalmodels presented can be summarized as (a) capability to be used in3-D FE analysis, (b) ability to use different kind of softening curveassumptions, and (c) possibility to use other failure criteria. The use ofcohesive elements, however, is computationally extremely intensiveand requires significant computer resources.

Use of cohesive elements that are based on traction–separation intracking material damage introduces nonlinearity to the system, whichresults in convergence problems, especially if the Newton–Raphsonsolution algorithm is used. To avoid this problem, the modified Riksmethod (or the Newton–Raphson method with viscous regularization)can be used.

From the findings in this study, it is anticipated that cohesiveelements that are based on traction–separation will be used inlarge problems with PCC pavement slabs resting on layered foun-dations. It is hoped that the use of fracture mechanics conceptswill eventually lead to the definition of more reliable and realisticfailure criteria, ones addressing the current weaknesses of statistical–empirical transfer functions commonly employed in pavementdesign guides.

1.60

1.40

1.20

1.00

0.80

Lo

ad (

kip

s)

0.60

0.40

0.20

0.000.000 0.005 0.010

CMOD (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 currentstudy’s results (Beam C).

ACKNOWLEDGMENT

This work was supported in part by an allocation of computing timefrom the Ohio Supercomputer Center.

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The Rigid Pavement Design Committee peer-reviewed this paper.