abaqus model for pcc slab cracking
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This article was downloaded by: [Amir Kabir University]On: 11 March 2012, At: 09:27Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Pavement EngineeringPublication details, including instructions for authors and subscription information:htt p:/ / www.t andfonline.com/ loi/ gpav20
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
To link to t his art icle: http:/ / dx.doi. org/ 10.1080/ 10298430600798994
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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
http://www.tandf.co.uk/journals
DOI: 10.1080/10298430600798994
*Corresponding author. Email: anastasios.ioannides@uc.eduEmail: jun.peng@uc.eduEmail: swindljr@email.uc.edu
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
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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
A. M. Ioannides et al.314
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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
ABAQUS model for PCC slab cracking 315
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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
A. M. Ioannides et al.316
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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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