ABAQUS Model for PCC Slab Cracking

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ABAQUS model for PCC slab crackingAnastasios M. Ioannides

a, Jun Peng

a& James R. Swindler Jr.

a

aDepartment of Civi l and Environmental Engineering, Universit y of Cincinnati (ML-0071),

Box 210071, Cincinnat i, OH, 45221 0071, USA

Available onl ine: 24 Nov 2006

To cite this art icle: Anastasios M. Ioannides, Jun Peng & James R. Swindler Jr. (2006): ABAQUS model for PCC slab crackingInternat ional Journal of Pavement Engineering, 7:4, 311-321

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ABAQUS model for PCC slab cracking

ANASTASIOS M. IOANNIDES*, JUN PENG and JAMES R. SWINDLER Jr.

Department of Civil and Environmental Engineering, University of Cincinnati (ML-0071), PO Box 210071, Cincinnati, OH 45221 0071, USA

(Received 13 September 2005; revised 5 May 2006)

To contribute towards the development of improved failure criteria for pavement systems that couldpotentially replace Miners hypothesis in future pavement design guides, Hillerborgs Fictitious CrackModel can be used to simulate crack propagation in concrete pavement slabs, thereby dispensing withthe need to conduct time consuming and expensive physical experiments in the laboratory and the field.Commercial finite element program ABAQUS is used for slabs assumed to rest on a dense liquidfoundation, and to be loaded by an edge load. Both notched and unnotched slabs are considered, and theeffects of various loading parameters, notch size, size of the loaded area, slab thickness and slab size areexamined. A comparison is made between displacement and loading-controlled testing of the slabs.

Keywords: Concrete pavement fracture; Fictitious crack model; ABAQUS; Finite element analysis

1. Introduction

The majority of current pavement analysis and design

procedures have two primary features: (a) With respect to

the prediction of behavior from initial loading until shortly

before failure, current methods are based on the theory of

linear elasticity; (b) With respect to the prediction of

performance, distress, and failure, current methods resort

to rather simple, mostly empirical and phenomenological

concepts, such as Miners cumulative linear fatigue

hypothesis (Miner 1945). The conventional approach to

pavement design is commonly a two-stage one: first, a

critical primary response is calculated, which is

subsequently passed into a statistical/empirical algorithm

that converts it into a measure of performance. A cursory

review of existing analytical and design procedures for

pavements might lead to the impression that these two

aspects are decoupled, when in fact they are closely

interrelated. It should be appreciated that expediency is

the only justification for such practices, pending the

development of more reliable and rational (mechanistic)

alternatives. It is often the case, meanwhile, that the choice

of a particular empirical and phenomenological perfor-

mance criterion is by far the most overriding consideration

in any design exercise. Consequently, derivation of

improved performance relationships, preferably ones that

recover their interrelationship with the primary response

calculation process, is an on-going objective of pavement

researchers. The study reported herein is intended as a

contribution to this effort, which seeks to develop models

that are implementable in sophisticated finite element

codes and allow parametric studies and predictions of

structural behavior. More specifically, this paper focuses

on the application of the ABAQUS/STANDARD finite

element software (Hibbitt et al. 1994) in tracking crack

propagation in Portland Cement Concrete (PCC) pave-

ment slabs, subject to the usual restrictive assumptions of

Westergaard (1926), namely: (a) full contact (no

temperature differential); (b) single slab (no load transfer);

(c) single placed layer (no subbase); (d) semi-infinite

foundation (no rigid bottom); and (e) one tire-print.

Fracture mechanics, particularly the Fictitious Crack

Model (FCM) proposed by Hillerborg et al. (1976) to

simulate crack propagation, can be an important tool

toward a better understanding of crack formation and

propagation in pavements, which in turn can provide us

with models capable of capturing these phenomena.

2. Context of investigations at the University of

Cincinnati

The context for the present study is provided by a long-

term, step-by-step effort that started in the late 1990s, with

a historical review of the major research activities

concerning the development of fatigue cracking in both

International Journal of Pavement Engineering

ISSN 1029-8436 print/ISSN 1477-268X online q 2006 Taylor & Francis

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DOI: 10.1080/10298430600798994

*Corresponding author. Email: [email protected]: [email protected]: [email protected]

International Journal of Pavement Engineering, Vol. 7, No. 4, December 2006, 311321
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PCC and bituminous pavements (Ioannides 1997b).

Efforts spanning almost a century were examined, with

the objective of identifying the sequence of events that

have led to the formulation of current approaches used to

account for fatigue in pavement design codes. In

particular, the roots of Miners cumulative linear fatigue

hypothesis were traced, and its advantages and limitations

were discussed. It was shown that the foundations for thiscrucial aspect of current design procedures were

surprisingly feeble, and the desirability of enhanced

mechanistic approaches to fatigue cracking prediction was

established. The scarcity of suitable candidates to replace

Miner was highlighted, and fracture mechanics was

proposed as a promising realm to explore.

An exhaustive examination of the advantages and

limitations of a variety of fracture mechanics options for

pavement engineering was then embarked on (Ioannides

1997a). Early pavement fracture mechanics efforts were

primarily associated with Paris Law (Paris and Erdogan

1963), a phenomenological construct that was eventually

shown to offer few breakthroughs compared to Minershypothesis. Of primary historical importance are the

studies by Majidzadeh et al. (1971) at Ohio State

University (OSU), and of Prof. Robert L. Lytton (Lytton

and Shanmugham 1982) at Texas A&M University (TX

A&M). The major shortcomings of these efforts, which

probably account for the lack of progress achieved, were

identified as their acceptance of the validity of Linear

Elastic Fracture Mechanics (LEFM) (Broek 1986) as

applied to (bituminous) concrete mixtures, and of Paris

law for explaining pavement fatigue.

Having established the inability of Paris Law to

address the fundamental weakness of current pavement

design procedures satisfactorily, a number of alternativeapproaches were then examined. Noteworthy among these

were investigations conducted by Prof. Heshmat A. Aglan

(Aglan and Figueroa 1993) at Tuskegee University, whose

more mechanistic flavor was clearly discernible.

Advanced concepts of thermodynamics and viscoelasti-

city were employed for the development of the Modified

Crack Layer Model for the characterization of the near-

failure behavior of bituminous mixes. Another very

promising investigation was conducted by Prof. Yeou-

Shang. Jenq (Jenq and Perng 1991) of OSU, who applied

to asphalt pavements the FCM introduced by Swedish

investigator Arne Hillerborg. Such research establishes the

fact that intercontinental collaboration is invaluable inthe development of effective procedures to account for the

fatigue cracking and fracture phenomena in pavements. Its

limitations notwithstanding (Bazant 2002), the FCM was

also identified as a most promising tool in this effort, and

additional pavement applications thereof were sought.

Of particular interest were found to be the contributions

of two Danish investigators, Hans H. Bache and Ib

Vinding, who applied Hillerborgs FCM to concrete

pavement engineering (Bache and Vinding 1990). They

also suggested a number of similitude considerations that

flow naturally from the application of this model, and that

highlight the significance of the application of the

principles of dimensional analysis. Their work validated

some earlier observations made by pavement engineers

concerning the relative size of beam specimens compared

to that of in situ pavement slabs. The specimen size effect

(Bazant and Planas 1998) was thus found to be at the heart

of the concrete fracture problem, and its resolution to be

essential before unraveling the complex phenomenon ofpavement fatigue cracking.

An opportunity to overcome the specimen size

limitation was identified in the work of Russian

investigator Vyacheslav D. Kharlab (Kharlab 1995,

personal communication) of St. Petersburg State Univer-

sity of Architecture and Civil Engineering (SPSUACE).

Kharlabs approach was shown to have similarities to

Hillerborgs FCM, but also to be in contradiction to it in

some respects. These two proposals were selected in this

study as the most promising tools available for PCC

pavement fracture mechanics applications at this time.

To begin with, finite element analysis was used in

simulating crack propagation in PCC beams (Ioannidesand Sengupta 2003). Experimental data (Liu 1994)

pertaining to the load vs. deflection (P 2 d) and the load

vs. crack mouth opening displacement (P-CMOD)

behavior of simply supported PCC beams subjected to a

point load at mid-span were successfully reproduced in

this way, past the elastic limit to failure. Finite element

package GTSTRUDL (1993) was used to generate the

flexibility matrix pertaining to the linear elastic aspects of

structural response, whereas fracture behavior in accord-

ance with the FCM was examined using CRACKIT, a

FORTRAN computer program coded during the course of

the study. The GTSTRUDL/ CRACKIT combination was

then used to generate numerical analysis data for differentbeam sizes, and these data were interpreted in dimension-

less format. These beam test results were subsequently

confirmed by implementing the FCM in the commercial

package ABAQUS (Ioannides and Peng 2004). The

validity of the ABAQUS simulation for beams was

checked through comparison with results obtained using

CRACKIT, and from other independent laboratory and

analytical investigations. The methods adopted for the

analysis of pavement slabs in the present paper are an

extension of those applied to beams, thereby affirming the

suitability of the step-by-step approach adopted in this

project.

3. Fracture analysis of slabs using JOINTC elements

A slab is assumed to be resting on a dense liquid

foundation loaded by a single, square (or rectangular)

edge load. The dimensions of the slab are selected to

correspond roughly to those in an actual concrete

pavement, and to lend themselves for a series of finite

element runs without undue demands on memory and

other computer resources, as follows: length, L 6.10 m

(240 in.), width, W 3.05 m (120 in.) and thickness,

A. M. Ioannides et al.312
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h 0.152 m (6in.) (see also table 1). Additional

advantages of the slab dimensions selected include the

elimination of slab-size effects by retaining a value above

5 for the ratio of slab size or width to the radius of relativestiffness of the slab-subgrade system, and the adoption of

a rectangular rather than of a square slab. Both notched

and unnotched slabs are considered, with the notch (when

present) specified to extend both through the slab

thickness (vertically), and along the slab symmetry line

(horizontally). Consequently, the crack is assumed to

follow the symmetry line, as it propagates from the bottom

up. The notch is described by two ratios, of notch-to-slab

thickness, (az/h), and of notch-to-slab width, (ay/W), in

the vertical and horizontal directions, respectively. The

values of these ratios are limited by the number of

elements used in the two directions, since only notches

spanning an entire element are considered. Figure 1illustrates the geometry of the slab and the definition of

notches in the z- and y-directions.

In cross section, the slab is subdivided into three layers,

a value that allows meaningful consideration of crack

propagation through the thickness without undue penalty

in terms of execution time. In discussing the use of three

dimensional finite element analysis to slabs on grade,

Ioannides and Donnelly (1988) had found that even two

layers are adequate for linear elastic analysis. A more

detailed examination of through-the-slab thickness crack-

ing will probably require a finer subdivision. In plan view,

the subdivision is into 40 elements in the x-direction and

10 elements in the y-direction; each element is, therefore,

152 305 mm (6 12 in.). The element adopted in

modeling the slab is the C3D27R, which is described as anisoparametric, 3D, 27-node, reduced integration element.

For the purpose of simulating crack propagation, a series

of JOINTC elements is used to connect each pair of nodes

on either side of the symmetry plane, which thereby serves

as the potential fracture plane.

There are three main classes of spring-type elements

available in ABAQUS/STANDARD that allow the user to

define explicitly the desired fracture process: SPRING,

ITS (tube support elements), and JOINTC (flexible joint

element). The stiffness (force per relative displacement)

for all these is defined using the *SPRING option.

SPRING elements include three distinct types: SPRING1,

SPRING2, and SPRINGA, and all three can be linear ornonlinear. SPRING1 is a spring between a node and the

subgrade, acting in a fixed direction and is not useful at

this time. SPRING2 and SPRINGA connect two nodes,

but whereas the first acts in a fixed direction, the line of

action of the second can rotate, as might be necessary in

large displacement analyses. Moreover, SPRINGA

requires that the two nodes it connects be separated by a

finite distance, whereas SPRING2 can connect two nodes

that occupy the same geometrical location. ITS elements

are not relevant to this work. JOINTC elements consist of

translational and rotational springs and connect two nodes

that are essentially at the same geometric location, thereby

dispensing with the need to define a spring length.Ioannides et al. (2005) report that when ABAQUS/STA-

NDARD is used, SPRINGA appears to be sensitive to the

choice of spring length, for which no rational method of

determination could be devised. Analyses were conducted

to compare the behavior of SPRING2 and JOINTC

elements, and to justify the choice of the latter in this

study. It was observed that these two element types give

Figure 1. Slab geometry and Notch definition

Table 1. Baseline pavement system considered

Slab geometry Slab properties

Length 240 in. Modulus 4 MpsiWidth 120 in. Poissons ratio 0.15Thickness 6 in. Tensile strength 463psiNumber of slablayers 3 (or 2)

Fracture energy 4.31 1024 kips/in.

Foundation characteristics Applied load

WINKLER Dense l iquid Edge loadingSubgrade modulus 200psi/in. 12 12in.

Pressure 100psi

Met ric c onversio ns Westergaa rd re sp onses

1in. 25.4 mm Maximum bending stress 768psi1 lb 4.44822 N Maximum deflection 40 mils1psi 6.89476kPa1 psi/in. 0.27145MN/m

3

ABAQUS model for PCC slab cracking 313
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very similar results and reinforce the credibility of the

model using the JOINTC element. The latter was chosen

because the plot of the deflected shape produced by

ABAQUS/POST is more attractive. There is no other

compelling reason to pick one over the other.In this study, a bilinear closing pressure vs. crack

opening displacement (s2 w) curve, required by the

FCM, is assumed, for the definition of which three points

are needed, depending primarily on the tensile strength of

the concrete, ft. The first point is, of course, (ft, 0), while

the two other define the location of the intersection point

(fI, wI), and the value of the critical crack opening at (0,

wc). Following the proposals by Petersson (1981) and

Gustafsson (1985), wc is set to 3.6 Gf/ft, in which Gf is the

concrete fracture energy, while the knee of the bilinear

curve is located at 1/3 ft and 2/9 wc. The experimental

beam FCM bilinear curve employed by Liu (1994) is

retained here, as shown in figure 2. For this curve, criticalCMOD, wc 0.085154 mm (0.0033525 in.); intersection

point, (wI, fI) (0.0189 mm, 0.1064988 MPa) or

(0.000745 in., 0.1544233 ksi); for comparison purposes,

a second, linear FCM curve with critical CMOD,

wc 0.047262 mm (0.0018607 in.) was also considered;

both these curves correspond to Gf 75.4N/m

(0.431 lb/in.). The nonlinear response of the JOINTC

elements is defined in accordance with this curve,

converted into a cohesive force vs. crack mouth opening

displacement relationship, on the basis of energy

equivalence and moment balance considerations (Shah

et al. 1995).

Because of the inability of ABAQUS to accommodatetruly unnotched slabs, a fictitious notch extending half an

element in each of the two directions of interest is

introduced in such cases. This operation is compensated

by doubling the stiffness of the two neighboring JOINTC

elements.

The load is applied at the middle of the long edge of the

slab, and the size of the loaded area considered is

305 305 mm (12 12 in.) in most cases. The maxi-

mum bending stress predicted by Westergaard (1948)

under a pressure of 689 kPa (0.100ksi) is 5292kPa

(0.768 ksi) and the corresponding maximum vertical

deflection is 1.01 mm (39.6 mils). This guarantees that

the slab will crack under a reasonable pressure, producing

simultaneously a measurable deflection. For the sake of

additional comparisons, loaded areas of 305 610 mm

(12 24 in.) and 610 305 mm (24 12 in.) are also

examined. Fracture in slabs is simulated the only two

available options with regard to load application, i.e.,

(vertical) displacement control (maintaining a maximumdisplacement) and loading control (maintaining a

maximum load).

By default, under displacement control, the displace-

ment is applied as a RAMP function, starting at 0 and

reaching a relative maximum value of 1.0 at the end of

the load step. The absolute maximum displacement is set

to a value of 25.4 mm (1.0 in.) downward at each of the

nodes defining the loaded elements. In most cases, two

fully loaded elements were considered, and the

displacement was fixed at 18 nodes. Under loading

control conditions, the load may be applied as a

uniformly distributed load or as a series of concentrated

nodal loads. The latter is particularly useful in casesinvolving partially loaded elements. For the distributed

load, the applied pressure will increase linearly from 0 at

time 0 to a relative maximum value of 1.0 at time 1.0.

The absolute value of the maximum pressure is typically

fixed between 1380 kPa (0.2 ksi) to 34,450 kPa (5ksi).

These choices correspond to approximately 0.5 to 10

times the tensile strength of the material (ft 3192 kPa

or 0.4633 ksi), depending on the particular stage of the

fracture process one is interested in.

It is freely admitted that several of these choices are

rather unrealistic when in situ pavements are considered,

and that the mesh idealization described is not ideal.

They were considered, however, expedient and quiteadequate for the purposes of this preliminary analytical

investigation, which aims primarily at verifying the most

significant aspects of the formulation, and at delineating

the most prominent trends to be expected. Occasional

tests with much finer meshes verify this assertion. Figure

3 displays the stress contours for a typical case

examined. It is noted that the analyses presented are

not aimed at calculating the stress or displacement levels

per se, but only to create a robust numerical model.

Increasing the mesh fineness would improve the

accuracy of the solution, but would not serve identify

any weaknesses of the numerical model, while at the

same time it would inhibit the efficiency of the study,increasing execution times and resource expenditures.

Consistent with this approach, results obtained are

presented primarily in the form of Tables, which lend

themselves better to verifying their veracity by those

wishing to adopt the approach proposed. A practical

interpretation for the purposes of modifying existing

pavement design procedures is pursued only to the

limited extent possible at this time. It is anticipated that

future research will explore the agreement between the

finite element results and in situ measurements, pending

the performance of pertinent experiments.

Figure 2. Bilinear s 2 w curve for PCC, per FCM

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4. Discussion of results

4.1 Effect of loading parameters

A series of runs with the smallest notch considered in this

study was performed. The notch has a depth of one layer

(50.8 mm or 2 in.) and a width of one row of elements

(305 mm or 12 in.); it is formed by removing two JOINTC

elements in each of the vertical and horizontal directions.

Loading control is employed, but the comments and

observations below apply to displacement control, as well.

Load application requires at least two parameters: the

initial time increment (ITI), and the time period of the step

(TPS). Two additional parameters may also be specified:

the minimum time increment (MnTI), and the maximum

time increment (MxTI), for which there is no default

value, i.e. it is left unbounded. The TPS is set at 1.0, forconvenience, ensuring that all time values are also

prescribed in relative terms. In earlier linear elastic

analyses (Ioannides et al. 2005), it was found that MxTI

has no influence on the results, as might have been

expected. Changing the maximum displacement or

velocity leads to proportional changes in the maximum

reactive force as predicted by linear elasticity. The

following procedure for selecting these parameters was

formulated: it is desired to have about 30 time increments

during the entire loading process, each increment lasting

3 1022 relative units of time or 3.33% of the total time;

this defines the MxTI. The ITI is set to about 1/5th of this

value (6 1023); the MnTI is chosen as approximately

1/5th of the ITI (1 1023).

In particular, the following responses are monitored in

table 2: the first peak stress occurring at the horizontal

crack tip, stip; the applied pressure when stip is achieved,

pc; the load line displacement (LLD), i.e. the vertical

displacement at the center edge node at the top surface of

the slab; and the crack mouth opening displacement

(CMOD), which is twice the value of the horizontal (x-

direction) displacement along the loaded edge of the

center edge node at the bottom surface of the slab. Table 2

also gives the number of layers (NL) into which the152.4 mm (6 in.) slab was subdivided, the specified

maximum pressure to be applied, pmax (set to a value

many times greater than ft, to ensure cracking occurs), the

number of loading increments (NINC) sustained, and

the execution time (CPU) consumed. It is observed that the

choice of loading parameters can influence the maximum

responses calculated by up to about 25%. This reflects

Table 2. Effect of loading parameters.

NL ITI MxTI pmax (ksi) NINC CPU (min) LLD (mils) CMOD/2 (mils) stip (ksi) pc (psi)

2 5.00 1025

N/S 50 37 56 37.1 1.0 0.609 82.9

2 2.50 1025 N/S 50 44 66 42.7 13.1 0.768 94.92 2.00 102

5N/S 50 46 66 33.9 0.9 0.557 75.9

2 1.50 1025

N/S 50 34 46 33.8 1.1 0.630 85.72 1.50 102

5N/S 25 49 76 43.7 1.4 0.787 96.9

2 1.50 1025

N/S 10 41 61 39.1 1.2 0.642 87.43 5.00 1025 N/S 50 44 152 36.7 0.7 0.579 82.93 2.50 1025 N/S 50 43 144 42.0 0.8 0.667 94.93 1.50 1025 N/S 50 48 151 37.9 0.8 0.600 85.73 1.50 10

25N/S 25 45 156 42.9 0.9 0.683 96.9

3 1.50 1025

N/S 10 39 132 38.7 0.8 0.612 87.43 1.50 10

25N/S 5 38 91 43.6 1.0 0.694 98.5

3 5.00 1023

2.00 1022

5 54 112 38.7 0.8 0.613 87.53 2.00 102

32.00 102

25 59 120 40.4 0.8 0.641 91.3

3 5.00 1023

2.00 1022

2 53 121 40.4 0.8 0.641 91.3

Note: Runs employ loading control; slabs notched: (ay/W) 10%; (az/h) 33.3%; N/S: not specified; MnTI 1.00 1025.

Figure 3. Stress contours for typical case considered

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the sensitivity of the maximum responses to the particular

instant in time at which they are sampled, which in effect

means to the particular applied pressure under which they

occur. This follows from the fact that time elapsed is

linearly related to the applied pressure. Additional

investigations are necessary to establish the procedures

needed to produce accurate as well as precise predictions

of the maximum responses desired. In contrast, the

distribution of the responses does not appear to be nearly

as sensitive to the input loading parameters, as illustrated

in figure 4, in which the various curves cluster over oneanother.

It is noted that the peak stress, stip, in table 2 assumes

values well in excess of ft. A possible explanation for this

phenomenon is provided by the stress gradient criteria of

strength (SGCS) for quasi-brittle materials, first proposed

by Kharlab and Minin (1989) and later elaborated by

Kharlab (1989, 1990). The formulation of the SGCS is

based on the experimental observation that local strength

of the material is higher where elastic stress distribution is

more non-uniform. Kharlabs SGCS are based on the

hypothesis that a material may not fail under theoretically

predicted high (or even infinite) stresses, if these are

sufficiently localized in nature. Stated in another way, thishypothesis recognizes the significance of the stress

gradient, a factor recognized in the West, as well (Siemes

1982). Additional discussion of the incorporation of SGCS

in fracture mechanics analyses has been presented by

Khazanovich and Ioannides (1993).

4.2 Influence of Notch size

To investigate the influence of the two aforementioned

notch ratios on slab response, a series of runs were

performed using loading control, with the input par-

ameters specified above. In each case, the JOINTC

elements were removed from an additional row of

elements or an additional layer, simulating increased

notch sizes. The unnotched case is modeled using a

fictitious notch of half an element row or (ay/W) 5%,

and half an element layer or (az/h) 16.7%, and

compensating for these changes by doubling the stiffness

of the first JOINTC elements at every point of the force-

displacement curve defined by the FCM. Four different

responses are tracked, namely, the maximum stress at the

node located at the notch tip, stip, occurring when thecrack begins to propagate; the applied pressure, pc,

corresponding to the development of stip; the horizontal

displacement at the bottom of the slab at the middle of the

loaded edge, directly below the center of the applied load,

corresponding to the development ofstip, and being equal

to one-half the CMOD; and the vertical displacement at

the top of the slab at the middle of the loaded edge, at the

center of the applied load, corresponding to the

development of stip, and being equal to the LLD. In

each case, a pair of these parameters is recorded,

pertaining to the two directions of crack propagation: at

the notch front in the vertical, z, or at the notch back in the

horizontal, y, directions.In general, the crack propagates in each direction at a

different time and load level, but in each instance when the

crack opening exceeds wc, as indicated by the results in

tables 3 and 4. It is observed that the crack propagates first

in the vertical direction and then horizontally, indicating

that the vertical notch size ratio, (az/h), is more significant.

This is not unexpected, since the primary contributor to

the stiffness of the slab is its thickness; consequently even

a small reduction in thickness leads to pronounced

deterioration in performance. The maximum stip is much

smaller for the notched cases than for the unnotched case.

Figure 4. Effect on loading parameters on stress distribution

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In general, as the notch size increases, stip, LLD, and pcdecrease, whereas CMOD increases; trends contrary to

this may occur at large notch sizes, presumably because

the notch in such cases is beyond the limits of the loaded

area, which is 305 305 mm (12 12 in.), i.e. extends

only one element-row width on either side of the slab

center-line. In some cases, the crack does not even

propagate at all in the y-direction by the end of the

specified load step (maximum applied pressure of

1378 kPa or 0.2 ksi). The latter was defined by considering

the response of the unnotched beam, in a manner that

would apply to all cases analyzed and result in efficient

utilization of available computer resources. The maximumbending stress predicted by Westergaard (1948) noted

earlier is already almost twice the specified tensile

strength of the material.

A similar series of runs was also conducted using

displacement control, with maximum applied displacement

arbitrarily set at 25.4 mm (1 in.). The variation of the CMOD

is tracked as a function of the applied load, RF3, calculated

as the sum of vertical reaction forces under loaded area. The

pressure to be applied in not specified in this case, but for the

cases considered it was as high as 6000 kPa (0.900ksi). It is

observed in figure 5 that, in general, the fracture process

consists of three stages: an initial quasi-linear elastic region,

extending to about 178 kN (40 kips) or pressure of 3824 kPa

(0.555 ksi); a main fracture region, extending to about

267 kN (60 kips) or pressure of 5739 kPa (0.833 ksi); and a

collapse region, which is also quasi-linear at loads aboveabout 267kN (60 kips). The linearity of the latter region can

be ascribed to the more significant role that the Winkler

foundation, itself linearly elastic, plays in this region. The

Table 3. Influence of notch Size (at notch Front).

Unnotched in both directions

(ay/W) (%) stip (ksi) % CMOD/2 (mils) % LLD (mils) % pc (psi) %

0 0.796 100.00 1.00 100 66.2 100.00 153.1 100.00(az/h) 33.3%

10 0.693 87.06 1.01 101 50.1 75.68 113.1 73.8720 0.708 88.94 1.19 119 47.5 71.75 105.1 68.65

30 0.689 86.56 2.11 211 45.9 69.34 101.1 66.0450 0.694 87.19 1.18 118 45.9 69.34 101.1 66.0470 0.694 87.19 1.15 115 45.9 69.34 101.1 66.04100 0.694 87.19 1.18 118 46.0 69.49 101.1 66.04

(az/h) 66.7%

10 0.412 51.76 2.23 223 41.0 61.93 85.1 55.5820 0.353 44.35 2.94 294 42.7 64.50 85.1 55.5830 0.350 43.97 4.47 447 56.6 85.50 113.1 73.8750 0.395 49.62 9.22 922 101.6 153.47 189.1 123.5170 0.393 49.37 9.39 939 100.0 151.06 185.1 120.90100 0.393 49.37 9.49 949 102.4 154.68 189.1 123.51

Note: Runs employ loading control: pmax 0.2ksi; slab dimensions 120 240 6 in.; ITI 5 1023; MnTI 1 1025; MxTI 2 1022; Columns marked

% provide the results of the preceding columns as a ratio of the corresponding unnotched case response.

Table 4. Influence of notch Size (at notch back)

Unnotched in both directions

(ay /W) (%) stip (ksi) % CMOD/2 (mils) % LLD (mils) % pc (psi) %

0 0.874 100.00 1.00 100 66.2 100.00 153.1 100.00

(az/h) 33.3%

10 0.833 95.31 1.14 114 51.9 78.40 117.1 76.49

20 0.605 69.22 2.41 241 59.9 90.48 129.1 84.3230 0.553 63.27 6.40 640 72.4 109.37 145.1 94.7750 0.348 39.82 9.08 908 105.3 159.06 200.0 130.6370 0.065 7.44 9.10 910 105.6 159.52 200.0 130.63100 No crack tip

(az/h) 66.7%

10 0.313 35.81 1.57 157 32.6 49.24 69.1 45.1320 0.316 36.16 2.74 274 40.5 61.18 81.1 52.9730 0.307 35.13 4.26 426 54.4 82.18 105.1 68.6550 0.306 35.01 9.86 986 104.9 158.46 193.1 126.1370 0.199 22.77 8.72 872 93.4 141.09 173.1 113.06100 No crack tip

Note: Runs employ loading control;pmax 0.2 ksi; slab dimensions 120 240 6 in.; ITI 5 1023; MnTI 1 102

5; MxTI 2 102

2; Columns marked%

provide the results of the preceding columns as a ratio of the corresponding unnotched case response. The value given is the stress observed at the last step increment; no maximum was observed.

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notch size appears to have a significant impact on the

responses obtained during the first two stages of loading.Thus, the initial slope of the curves decreases as the notch

size increases. Similarly, the CMOD corresponding to any

given load level increases as the notch size increases, while

the maximum load sustained before entering the second

stage decreases as notch size increases. Responses are much

more sensitive to increases in notch size in the vertical than

in the horizontal direction. The pattern exhibited by the

unnotched curve suggests that themodeling of this case may

call for some additional refinement.

On the basis of these observations, the value of half the

CMOD is tracked as the notch size increases, at two

different levels for half the total applied force, between

169 kN (38 kips) and 178kN (40 kips) and between 267 kN(60 kips) and 285 kN (64 kips), respectively. The LLD is not

tracked in table 5 since this is controlled directly. At the

lower level of applied force, the CMOD increases as the

notch size increases, whereas at the higher level, the CMOD

does not changemuch as notch size increases.This indicates

that whereas at the low load level the slab is primarily

responsible for carrying the load, at the higher load level the

subgrade is carrying a bigger portion of the load.

4.3 Effect of loaded area size

A few cases were selected for an investigation of the effect

of the size of loaded area, and additional runs were

performed using loaded areas of 305 610mm

(12 24 in.) and 610 305mm (24 12 in.). The ITI

was 1.5 1025, the MnTI was the default (1 1025)

and the MxTI was not specified. The response tracked in

table 6 is the applied pressure, pc, corresponding to thedevelopment of the maximum stress at the notch tip,

which occurs when the crack first begins to propagate. It is

observed that pc first decreases as the notch size increases,

as expected, but then increases indicating that the notch tip

is now beyond the limits of the loaded area. These

observations also explain the fact that, in general, pc-

values for a 610 305 mm (24 12 in.) area are higher

than thos e obtained us ing the 305 610mm

(12 24 in.) area, since the horizontal notch extends

into the slab away from the edge, i.e. in the y-direction.

4.4 Effect of slab thickness

To investigate the effect of slab thickness on the fracture

process, a run was conducted using a thickness of 229 mm

(9 in.) and the results are compared to those from an

identical run using the standard thickness of 152 mm

(6 in.). Displacement control was used in both cases, and

the notch size was one slab layer by one element row. The

impact of the notch is more pronounced on the thicker

slab, which exhibits a lower initial slope, as well as a lower

sustained load prior to the main fracture region (figure 6).

At any given applied load, the thicker slab shows a higher

CMOD. This observation can be explained by recallingthat the FCM model is defined in terms of cohesive stress

vs. crack opening near the crack tip; consequently, a larger

CMOD is required to produce the same crack opening near

the tip if the thickness of each slab layer increases. It is

also observed in table 7 that the thicker slab requires fewer

load increments to complete the fracture process, and that

the load steps for it are larger than those applied to the

thinner slab. This is probably due to the internal manner in

which ABAQUS determines the appropriate load step at

every stage. In the initial few stages of loading, the thicker

slab allows the load step to increase faster, and once large

steps begin to be taken, they are continued to the end of the

fracture process, thereby resulting in fewer load steps. Incontrast, the thinner slab dictates a slower increase in the

load step magnitude, and leads to a greater total number of

steps required to complete the entire process. Thus, at any

particular load level, the thicker slab exhibits a larger

CMOD than the thinner slab. Finally, the thicker slab

requires a total load of 1620 kN (364 kips) to complete the

fracture process, compared to only 1190 kN (267 kips) for

the thinner slab. It is apparent that these observations are

influenced to a great extent by the use of displacement

control and the selection of the three loading parameters,

ITI, MnTI, and MxTI.

Figure 5. The three stages of the loading-and-fracture process. (RF3/2:sum of vertical reactio

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ABAQUS Model for PCC Slab Cracking