MODE II FRACTURE PARAMETERS FOR VARIOUS SIZES OF BEAMS IN PLAIN CONCRETE

Darsigunta Seshaiah Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 11, (Part - 5) November 2015, pp.13-25
www.ijera.com 13 | P a g e
MODE II FRACTURE PARAMETERS FOR VARIOUS SIZES
OF BEAMS IN PLAIN CONCRETE
Darsigunta Seshaiah
Structural Engineering Studying M.Tech In Chinthalapudi Engineering College Roll No: 137r1d8703
ABSTRACT
Blended aggregate in concrete and arriving at the structural properties of blended aggregate concrete is a thrust
area. Pumice is very light and porous igneous rock that is formed during volcanic eruptions.Cinder is a waste
material obtained from steel manufacturing units. Shear strength is a property of major significance for wide
range of civil engineering materials and structures. Shear and punching shear failures particularly in deep beams,
in corbels and in concrete flat slabs are considered to be more critical and catastrophic than other types of
failures. This area has received greater attention in recent years. For investigating shear type of failures, from the
literature it is found that double central notched (DCN) specimen geometry proposed by Prakash Desai and
V.Bhaskar Desai is supposed the best suited geometry. In this present experimental investigation an attempt is
made to study the Mode-II fracture property of light weight blended aggregate cement concrete combining both
the pumice and cinder in different proportions, and making use of DCN test specimen geometry . By blending
the pumice and cinder in different percentages of 0, 25, 50, 75 and 100 by volumeof concrete, a blended light
weight aggregate concrete is prepared. By using this the property such as in plane shear strength is studied.
Finally an analysis is carried out regarding Mode-II fracture properties of blended concrete. It is concluded that
the Ultimate load in Mode-II is found to decrease continuously with the percentage increase in Pumice aggregate
content. It is also observed that the ultimate stress in Mode II is found to increase continuously with percentage
increase in cinder aggregate content.
Key Words: Cinder, light weight aggregate, Mode II fracture, shear strength, Pumice.
I. Introduction
Fracture mechanics is the science of studying the
behavior of progressive crack extension in structures
subjected to an applied load. Fracture mechanics is
all about cracks, stress fields around cracks, stress
intensity factors at cracks, failures due to cracks,
growth rates of cracks, etc. This website covers all of
these topics, beginning with some historical
perspective for motivation.
The story begins way back at the turn of the 20th
century with analytical solutions for stresses at holes
in 1898, then at ellipses in 1913. Fracture mechanics
research is considered to have officially begun in
1920 with Griffith’s energy based analysis of cracks.
Nevertheless, similar landmark contributions
remained few and far between until World War II,
when several structures that should have never failed,
did. It was determined that the cause of these failures
was cracks. This led to a rapid expansion of research
into the areas of fracture mechanics and fatigue
growth.
In 1983, the National Bureau Of Standards (now
the National Institute for science and Technology)
and Battelle Memorial Institute estimated the costs
for failure due to fracture to be $119 billion dollars
per year in1982. The dollars are important, but the
cost of many failures in human life and injury is
infinitely more.
Failure have occurred for many reasons,
including uncertainties in the loading or environment,
defects in the materials, inadequacies in design, and
deficiencies in construction or maintenance. Design
against fracture has a technology of its own, and this
is a very active area of current research. This module
will provide an introduction to an important aspect of
this field, since without an understanding of fracture
the methods in stress analysis discussed previously
would be little use. The module on the dislocation
basis of yield shows how the strength of structural
metals particularly steel can be increased to very high
levels by manipulating the microstructure so as to
inhibit dislocation motion. Unfortunately, this renders
the material increasingly brittle, so that cracks can
form and propagate catastrophically with very little
warning.
Fracture mechanics goes along with the
recognition that real structures contain discontinuities
which has originates in 1921 by Griffith and was for
a long time applied only to metallic structures and
ceramics. Concrete structures, on the other hand,
have so far been successfully designed and built
without any use of fracture mechanics, even though
their failure process involves crack propagation.
RESEARCH ARTICLE OPEN ACCESS

II. HISTORICAL ASPECTS OF
FRACTURE MECHANICS
For a long time, we always had some idea about
the role of a crack or a notch. While cutting a tree, we
would make a notch with an axe at its trunk and then
pull it down with a rope. While breaking a stick we
would make a small notch with a knife before
bending it. Good swords were made by folding a thin
metal sheet at the center line and then hammering it
to make it thin again so that it could be further
folded. Thus, a sword would have many layers. If a
crack develops in one of the layers, it is not likely to
move to another, thus making the sword very tough.
In 19th
century and early part of 20th
century, the
entire industry was obsessed with production. Even
the failure theories were developed quite late: Tresca
in (1864) and Mises (1913). However, World War II
accelerated the industrial production at a very rapid
rate, due to unusually high demands of war. Within 6
years of the war, the know-how of aircraft making
improved dramatically. Also, the ships, which were
earlier made by joining plates together through the
process of riveting, were changed to welded frames.
Many cargo ships, known as liberty ships, were
rolled out from American docks within a short span.
However, soon there were complications regarding
welded structures. Many of these failed in the cold
temperatures of the North Atlantic Ocean. As a
matter of fact some of these broke up into two parts.
However, ships made by riveting plates did not
display such failures. If a crack nucleated and grew in
a plate, it would only slit that plate into two parts; the
crack would not grow into another plate. A welded
structure is a large single continuous part and
therefore, if the crack becomes critical, it will run
through the entire hull of the ship.
Fig 1.1: The tower bridge in London, completed
in1894
III. FRACTURE MECHANICS
Fracture mechanics is the science of describing
how a crack initiates and propagates under applied
loads in many engineering materials like ceramics,
rocks, glasses and concretes. Fracture mechanics is
generally applied in the field of earth sciences such as
petroleum engineering, geological engineering,
mining engineering and civil engineering.
Concrete is a stone like material obtained by
permitting a mixture of cement, fine aggregate and
gravel or other aggregate and water to harden in
forms of desires shape of the structure. Concrete has
become a popular material in civil engineering for
several reasons, such as the low cost of the aggregate,
the accessibility of the needed materials and its high
compressive strength. On the other hand concrete is a
relatively brittle material with low tensile strength
compared to the compressive strength.
IV. LINEAR ELASTIC FRACTURE
MECHANICS (LEFM)
Linear Elastic Fracture Mechanics (LEFM) first
assumes that the material is isotropic and linear
elastic. Based on the assumption, the stress field near
the crack tip is calculated using the theory of
elasticity. When the stresses near the crack tip exceed
the material fracture toughness, the crack will grow.
In Linear Elastic Fracture Mechanics, most formulas
are derived for either plane stresses or plane strains,
associated with the three basic modes of loadings on
a cracked body: opening, sliding, and tearing.
Griffith was the first to develop a method of
analysis for the description of fracture in brittle
materials. Griffith found that, due to small flaws and
cracks, stress concentrations arise under loading,
which explains why the theoretical strength is higher
than the
Fig 1.1: Arbitrary body with an internal crack of
length a subjected to an arbitrary force F
By superposition, the potential energy of the
body is given by the fracture process in concrete is
given in equation

?=? e +? F +? k +? c
? e = the elastic energy content in the body
? F = the potential of the external forces
? k = the total kinetic energy in the system
? c = the fracture potential
The fracture potential is the energy that
dissipates during crack growth. By assuming that
crack growth is only dependent on the crack length,
a, the equilibrium equation can be stated, by
requiring that the potential energy of the system
equals to zero.
Griffith introduced a parameter, the energy
release rate, G. R is the fracture resistance of the
material, which is assumed to be constant in LEFM.
The total potential energy of a system increases when
a crack is formed because a new surface is created,
thus increasing the fracture potential. However, the
formation of a crack consumes an amount of energy,
G, in the form of surface energy and frictional
energy. If the energy release rate is larger than the
energy required forming a crack.
Ucracked – Uuncracked = - (πa2
σ2
)/E + 4aγ
Where Ucracked and Uuncracked are elastic energies of the
plate, σ is the nominal normal stress and γ is the
elastic surface energy of the plate.
Fig.1.2 (a) Plate with crack 2a (b) Load-
Displacement diagram
V. MODES OF FAILURE
(a) Mode I (Tension, Opening)
(b) Mode II (In-Plane Shear, Sliding)
(c) Mode III (Out-Of-Plane Shear, Tearing)
Fig 1.3: Modes of failure
Mode I (Tension, Opening): Mode I fracture is the
condition in which the crack plane is perpendicular to
the direction of the applied load.
Mode II (In-Plane Shear, Sliding): Mode II fracture
is the condition in which the crack plane is parallel to
the direction of the applied load.
Mode III (Out-Of-Plane Shear, Tearing): Mode III
fracture corresponds to a tearing mode and is only
relevant in three dimensions.
An inclined crack front in a component can be
modeled as a superposition of the three basic modes
and then, the effect of loading by each mode can be
analyzed separately. Mode I usually play a dominant
role in many engineering applications and is
considered to be the most dangerous. However, in
certain applications, components fail through the
dominant roles played by Mode II or Mode III. Any
of these modes of fracture are typically assumed to
initiate at the point of greatest stress concentration,
which is commonly assumed to be at a flaw in the
material. As shown in Figure 5, this flaw can be
internal of length 2a or external of length a (Callister
2005).
Fig. 1.4 Internal and external flaw geometry
Mode I and Mode II fracture is also referred to as
an opening and in-plane shear mode, respectively. In
practical calculations, only the first order term of
equation is included. This is because, that for  ? ,
the first order term approaches infinity while the
higher order terms are constant or zero. Because the
stress tends towards infinity when  ? . A stress
criterion as a failure criterion is not appropriate. For
this reason, Irwin derived a relationship between the
SIF and the release rate, G,
K= G*E

The fracture criterion can thereby be written as
K=Kc
It should be noted that the global energy balance
criteria by cite bib Griffith is equivalent to the local
stress criteria by Irwin. Moreover, Kc is also referred
to as the fracture toughness of the material, and is
regarded as a constant in LEFM. The Griffith (1921)/
Irwin (1958) theory assumes that the stresses in
vicinity of the crack tip tend to infinity. This
contravenes the principle of linear elasticity, relating
small strains to stresses through Hooke’s law. In the
fracture process zone, abbreviated FPZ, ahead of the
crack tip, plastic deformation of the material occurs.
Specifically for concrete de-bonding of aggregate
from the cement matrix and micro cracking occurs.
Moreover, cracks coalesce, branch and deflect in
FPZ. To describe this highly non-linear phenomenon,
non-linear fracture mechanics, abbreviated NLFM,
must be adopted.
VI. STRESS INTENSITY FACTOR
Since in LEFM the state of stress near crack tip
is singular, it is not possible to evaluate the stress
value around crack tip accurately. In order to more
accurately evaluate or predict the state of stress near
crack tip Irwin and his colleagues developed a
relation to calculate the amount of energy available
for fracture in terms of asymptotic stress and
displacement fields around a crack front in a linear
elastic solid. This asymptotic relation is
 ij = (f ij)
Since  ij are the Cauchy stresses, r is t he
distance from the crack tip, ? is the angle with respect
to the plane of the crack geometry and loading
conditions. The quantity K was called stress intensity
factor. Since functions f ij are dimensionless, the
stress intensity factor has the units of N/mm2
. This
formation is not valid for the areas very close to
crack tip i.e., inside process region. This concept is a
theoretical construct applicable to elastic materials
and is useful for providing a failure criterion for
brittle materials.
Stress intensity factor can be defined for
different modes of loading (I, II, and III) and in such
cases is referred to as KI , KII, and KIII . Stress
intensity factors are related to energy release rate
according to following equation for 2D problems
G= (K2
I /E*) + (K2
II /E*)
Where E* = E (young’s modulus) for plane stress and
E* = E / (1- µ2
) for plane strain
µ is the Poisson’s ratio
Stress and displacement equations for the centre
cracked body are similar for other modes. For mode
II in plane strain and far field stress 12= with KII =
 a. Stress intensity factor especially in mode II,
can be used as a crack propagation criterion. There is
a critical value for stress intensity factor, required to
propagate the crack. This critical value determined
for mode II loading in plane strain is referred to
critical fracture toughness (K IIc).
Mode II fractures initiation and propagation
plays an important role under certain loading
conditions in rock fracture mechanics. Under pure
tensile, pure shear, tension- and compression-shear
loading, the maximum Mode I stress intensity
factor, , is always larger than the maximum
Mode II stress intensity factor, . For brittle
materials, Mode I fracture toughness, KIC, is usually
smaller than Mode II fracture toughness, KIIC.
Therefore, reaches KIC before reac
hes KIIC, which inevitably leads to Mode I fracture.
Due to inexistence of Mode II fracture under pure
shear, tension- and compression-shear loading,
classical mixed mode fracture criteria can only
predict Mode I fracture but not Mode II fracture. A
new mixed mode fracture criterion has been
established for predicting Mode I or Mode II fracture
of brittle materials. It is based on the examination of
Mode I and Mode II stress intensity factors on the
arbitrary plane θ,KI(θ) and KII(θ), varying
with θ(−180°⩽θ⩽+180°), no matter what kind of
loading condition is applied. Mode II fracture occurs
when and
at θIIC. The validity of the new
criterion is demonstrated by experimental results of
shear-box testing.
VII. Size Effects In Non- Linear Elastic
Fracture Mechanics
Consider a uniformly stressed panel and suppose
that fracture propagates via the formation of a crack
band of thickness hf. The load required to propagate
the band follows from energy balance equation, i.e.,
energy available is equal to the fracture energy (the
energy required for band extension). To that end,
assume that due to presence of crack band the strain
energy in the band and cross-hatched area drops from
σ2 N/2E to zero (this region is called the stress relief
zone). Next, consider a geometrically similar panel. It
is usually the case, that the larger the panel, the large
the crack band and consequently the larger the cross-
hatched area = in a larger structure, more energy is
released in a strip by the same extension of the crack
band. It is usually assumed that the edges of the
specimen are fixed during the crack advance
(displacement control), and so the external work is
zero. The condition balancing the total energy
released from the stress relief zone and the fracture
energy needed to advance the crack by ∆a reads
b(hf∆a + 2ka0∆a)σN 2 /2E = Gfb∆a .
Following Baˇzant we further denote
Bf 0 = Gfhf /E = const and
D0 = hfD / 2 ka0 = const

where f 0 is the tensile strength and D/a0 = const due
to geometrical similarity. Combining the above
expressions together with Eq. (85) gives the Baˇzant
size effect equation in the form
σNu =Bf 0(1 + D/D0)
Note that both Bf 0 and D0 depend on the
fracture properties of the material and on the
geometry of the structure, but not on the structure
size. Also not that equation is approximate, valid
only within a range of about 1:20 for most structures.
The elastic stress and displacement field
surrounding a non linear process zone correspond to
a certain crack length a. In this case one can write
a= a0+c
where a0 is the initial length on the notch or crack.
Fig. 1.5 Stress Elongation Curve
The fracture energy is a measure of the materials
fracture toughness. It is important to note that, in
addition to the fracture energy, this model requires
that one must know the ultimate strength, and the
shape of the fracture curve (Shah et al. 1995). Some
state that since the fracture energy is a material
property, it should be invariant with beam size
(Muralidhara et al. 2011). However, because of the
size effect the value of GF may decrease with
increasing size (Bazant and Planas 1998).
The length of the fracture process zone is termed
the characteristic length and using this analysis can
be estimated using the following relationship, where
E is Young’s modulus, Gf is the fracture energy, and
ft is the ultimate strength.
lch = (EGf )/(ft
2
)
On an ending note, it has been pointed out that,
in its formulation, Fictitious Crack Model can be
applied to un-cracked as well as cracked structures
(Gustafsson and Hillerborg 1985). This gives it an
advantage over other fracture models that are only
applicable for post crack behavior.
VIII. MODELING FRACTURE CURVE:
In the fictitious crack model, as mentioned,
it is important to know the shape of fracture curve
(stress-crack opening displacement curve) (Shah et
al. 1995). Knowing the shape of the curve offers a
description to the behavior of the energy dissipation
process. Linear (Figure 1.7.2 a), bi-linear (Figure
1.7.2 b), exponential (Figure1.7.2 c), and power
curves (Figure 1.7.2 d) are just a few of the common
shapes used to describe the nature of this process
(Shah et al. 1995). Each of these curves starts with
the ultimate tensile stress ft at a crack opening of
zero, and are terminated (zero stress) at some critical
crack opening wc.
Fig: 1.6 Constitutive models
IX. FRACTURE BEHAVIOR
The quasi-brittle behavior of concrete can be best
explained by the following five stages (Shah et al.
1995) as depicted graphically in figure 1.8 and with
the use of figure1.8
Elastic:
The material exhibits elastic behaviour until the
proportional elastic limit (PEL) is reached. The PEL
in concrete is typically assumed to be the point of
first crack (Shah et al. 1995).
Damage Localization:
The micro-cracks will localize forming a micro-
crack, which occurs at the point of initial crack
localization. At which point the material undergoes
stable crack growth (crack propagates only when load
increases) and a softening behavior occurs (Shah et
al. 1995).
Unstable Crack Growth:
Once the ultimate strength is reached at a critical
crack length the crack will undergo unstable growth
(crack propagates even though load decreases) (Shah
et al. 1995).
Fig.1.7 Stages of quasi-brittle behaviour

Fig. 1.7 Fracture process zone: stage II (a) and
stage III (b)
For analysis purposes it is commonly assumed
that a crack propagates in a linear fashion. However,
concrete is a composite material so cracks tend to
propagate along non-linear or chaotic crack paths due
to the heterogeneity of the material. This can be
associated with several toughening mechanisms that
occur within the Fracture Process Zone (FPZ) as
pointed out by Shah et al. (1995).
Micro crack Shielding:
Randomly oriented micro cracks occur at flaws
ahead of the crack tip. The micro-cracking is caused
by the high stress concentration near the crack tip
(Shah et al. 1995). The formation of micro-cracks
releases energy, which increases the amount of
energy required to form unstable cracks (Anderson
2005).
Crack Deflection:
This occurs when inclusion (i.e., aggregates or
fibers) is strong enough to divert the path of least
resistance around the inclusion (Shah et al.1995).
X. FRACTURE MECHANICS OF
CONCRETE
Concrete is a heterogeneous anisotropic non
linear inelastic composite material, which is full of
flaws that may initiate crack growth when the
concrete is subjected to stress. Failure of concrete
typically involves growth of large cracking zones and
the formation of large cracks before the maximum
load is reached. This fact and several properties of
concrete, points toward the use of fracture mechanics.
Furthermore, the tensile strength of concrete is
neglected in most serviceability and limit state
calculations. Neglecting the tensile strength of
concrete makes it difficult to interpet the effect of
cracking in concrete.
Two basic types of structural failure may be
stated: brittle and plastic. Plastic failure occurs in
materials with a long yield plateau and the structure
develops plastic hinges. For materials with a lack of
yield plateau, the fracture is brittle, which implies the
existence of softening. During softening the failure
zone propagates throughout the structure, so the
failure is propagating.
In 1983 Wittmann suggested to differentiate
between three different levels of cracking in concrete.
The levels are categorized as follows:
1. Micro cracks that can only be observed by an
electron microscope.
2. Meso cracks that can be observed using a
conventional microscope.
3. Macro cracks that visible to the naked eye.
Micro cracks occur on the level of the hydrated
cement, where cracks form in the cement paste. Meso
cracks form in the bond between aggregates and the
cement paste. Finally, macro cracks form in the
mortar between the aggregates.
XI. THE FRACTURE PROCESS IN
COMPRESSION
The compressive stress strain curve for concrete
can be divided into four regions, see fig 1.3.1. The
figure describes four different states of compressive
cracking.
Initial cracks on the micro-level, caused by
shrinkage, swelling, and bleeding, are observed in the
cement paste prior to loading. For loads of
approximately 0-30% of the ultimate load the stress
strain curve is approximately linear and no growth of
the initial cracks is observed. Between approximately
30-50% of the ultimate load a growth in bonding
cracks between the cement paste and aggregates is
observed. The curve is divided into four regions for
different stages of cracking:
1. Pre existing bond cracks extend only slightly
under load.
2. Slow growth of bond cracks.
3. Additional growth of bond cracks plus slow
growth of matrix cracks.
4. Rapid growth of matrix cracks.
Fig: 1.8 The compressive stress- strain curve for
concrete
XII. THE FRACTURE PROCESS IN
TENSION
The tensile strength of concrete is much like the
compressive strength, dependent on the strength of
each link in the cracking process, i.e., micro cracks in
the cement paste, meso cracks in the bond and macro
cracks in the mortar. Consider a concrete rod under
pure tensile loading. The fracture process initiates
with crack growth of existing micro cracks at
approximately 80% of the ultimate tensile load. This
is followed by formation of new cracks and a halt in

formation of others due to stress redistribution and
the presence of aggregates in the crack path. These
cracks are uniformly distributed throughout the
concrete beam. When the ultimate tensile load is
reaches, a localized fracture zone will form in which
a macro crack that splits the specimen in two will
form. The fracture zone develops in the weakest part
of the specimen.
XIII. TEST SETUP & TEST
PROCEDURE
After 28days of curing the samples were taken
out from the curing tank and kept for dry. Then notch
is provided at one-third length of the beam with a/w
i.e., notch depth to specimen depth ratio of 0.3, 0.4,
0.5, 0.6. After this the sample was coated with white
wash. One day later the sample was kept for testing.
The notched beam specimen was kept on the
supports of testing machine as shown in figure.
Uniformly distributed load was applied over the
central one third part between the notches and square
cross section steel supports were provided at bottom
along the outer edges of the notches as shown in
figure, so that the central portion could get
punched/sheared through along the notches on the
application of loading. When performing a test, a
gradually increased load is applied to the notched
beam until a stress level is reached which results in
crack propagation. With the increase of load
propagation of these cracks in more or less vertical
direction along with the formation of new cracks at
the bottom of one or both the notches was observed.
Finally the specimens failed by shearing along the
notches.
Fig. 1.9 testing setup of notched beam specimen
The beam specimens were tested on the
Universal Testing Machine of capacity 1000 Tons.
All the beam specimens were tested under the four
point bending test with the displacement rate control.
A photograph of the Universal Testing Machine is
shown in Fig 4.1. The Load and Displacement was
measured by using the Universal Testing Machine.
XIV. SIZE EFFECT METHOD (SEM):
It may be noted that, the structural size effect is
the most important manifestation of fracture
phenomena. Geometrically similar specimens of
concrete as a quasi-brittle material, exhibit a
pronounced size effect on their failure loads.
Therefore, it is worthwhile to relate the size effect
behavior to the fracture properties of materials. SEM
has been developed according to effective elastic
crack model originally proposed by Bazant and
Pfeiffer. This method has been included in RILEM
TC-89. In this method, major fracture parameters are
determined using three-point bending test on
geometrically similar notched beams with varying
sizes. Based on this method, nominal strength of
geometrically similar concrete specimen can be
described by size effect law as:
σN = Bft /√(1+B);
Where β = d/d0
In which σN = nominal stress at the maximum
load Pu (σN =cnPu/bd.); b = thickness of the
specimen; d = size or characteristic dimension of the
specimen (e.g., beam depth in bending test); cn =
coefficient introduced for convenience; B and d0 are
two parameters determined experimentally; and ft is a
measure of material strength, which can be taken as
the tensile strength of the concrete. The theory is
based upon three distinct types of fracture regimes,
which are linear elastic fracture mechanics (LEFM),
nonlinear fracture mechanics (NLFM) and strength
theory, as illustrated in Fig. 14.
Fig. 1.10 The generalized size effect law
Therefore, Bazant and Kazemi concluded that if
the peak loads of geometrically similar specimen
with varying sizes are extrapolated to a specimen of
infinite size, the obtained fracture energy is unique
and is consequently independent of the specimen size
and thus, LEFM assumptions can be used. Therefore,
using linear regression of the peak loads of
geometrically similar specimens of varying sizes, and
according to size effect law Eq., the coefficients B
and d0 can be determined as:
Y = AX + C
In which: X = d, Y = (ft/σN) 2
, B = 1/√C and d0 =
C/A the slope A and intercept C in the plot of Y vs. X
can be used to determine the two parameters B and d0
in the given equation. Based on the concept of
LEFM, Bazant and Kazemi concluded that in failure

of structures of infinite size, the two major fracture
parameters, fracture energy (Gf) and effective length
of process zone (Cf) can be determined through:
Gf = g(α)/AE *ft
2
And Cf = g(α)/g’
(αo) *C/A
Where E is the modulus of elasticity of concrete, A is
the angular coefficient of regression line; C is the y-
intercept of the regression line, g(α) is the non-
dimensional energy release rate and g’(α0) is the
derivative of g(α) with respect to the relative initial
crack length (α0 = a0/d). These functions (g(α) and
g’(α) ) are geometry dependent and obtained
according to LEFM . g(α0) , for some cases, can be
found in RILEM-TC 89 recommendation. In SEM,
other fracture parameters of interest including
fracture toughness (KIC) can be determined as:
XV. DETERMINATION OF FRACTURE
PARAMETERS:
The Size Effect Law (SEL) which is applied to
geometrically similar specimens of different sizes
takes the form as shown below:
σN = Bft /√(1+B);
;Where β = d/d0 (1)
The fracture energy (Gf) was obtained from the
slope of the Regression line (A) and the elastic
modulus (E) as given below:
Gf = g(α)/AE *ft
2
Where ft = direct tensile strength, in absence of
direct tensile test the value ft is taken as 0.665 times
the split tensile strength of concrete.
The effective length of Fracture Process Zone
(Cf) was calculated by:
Cf = g(α)/g’
(αo) *C/A
Where
for
beams having geometry of S/D = 4. (5)
The stress intensity factor (KI) was calculated by: KI
= σN × f (α) × √πa0
Fig 1.11: Specimen geometry for the four – point
bending beams
XVI. TESTING PROGRAM and
MATERIALS
The main ingredients used were cement, fine
aggregate, coarse aggregate, water, and steel plates.
Cement
Ordinary Portland cement of 53 grade
conforming to IS: 12269-1987 was used for the
study.
Fine Aggregate
River sand passing through 4.75 mm sieve and
conforming to grading zone II of IS: 383-1970 was
used as the fine aggregate.
Coarse Aggregate
Crushed granite stone with a maximum size of
20 mm was used as the coarse aggregate.
Steel plates
Steel plates of 2 mm mean thickness and 22.5,
30, 37.5, 45, 60, 75, 90, 120, 150, 180mm in width
were used at a/w ratios of 0.3, 0.4, 0.5, 0.6.
Water
Potable water supplied by the college was used
in the work
Tamping Rod
Tamping rod was used for compacting the test
specimens, beams.
Curing:
The specimens were removed from the moulds
after 24 hours of casting and the specimens were
placed in water for curing.
Preparing of Notch:
The notch was prepared with steel plates with
different a/w ratio sizes.
XVII. MIX PROPORTIONING
The normal strength concrete mix M30 was
proportioned as per Indian Standard for a target mean

strength 30MPa. After various trial mixes, the
optimum mix proportion was selected as
0.45:1:1.562:2.902 with cement content of 405.81
kg/m3
. The different constituents in the order of
water: cement: fine aggregate: coarse aggregate were
proportioned as 183.0:406.81:635.4:1180.56 for
making 1m3
of mix. The steel plates were prepared
with a/w ratios. The plates were placed at one-third
of the length of the beam.
Fig 1.12universal testing machine
Fig.1.13 steel plate for notch
Fig. 1.14 wooden mould for large size beams
Fig.1.15 wooden mould for medium size beams
Fig .1.16 wooden mould for small size beams
XVIII. CALCULATIONS
The beam specimens were tested on the
Universal Testing Machine under displacement rate
control. All the beam specimens were tested under
the three point bending under the displacement rate
control. A photograph of the test setup is shown in
Fig 4.0. To understand the fracture behavior of plain
concrete beams the following graphs were drawn,
Load Vs Mid span deflection (Fig 5.0, 5.1, 5.2). The
normal and shear stress and stress intensity factor and
fracture energy of the beams subjected to three point
bending with eccentric notch calculated by using the

eq.s (5.1 to 5.8) from reference (1) and reported in
Table 5.0 and in table 5.1. From the graphs and
Tables it was observed that, for mixed-mode failure
of concrete, It was found that the stress intensity
factor and fracture energy increases with the
increasing of beam sizes and decreasing the failure
stresses with increasing the beam sizes. The
brittleness of the beam increases with increase the
size of the beam.
For the calculation of the stress intensity
factor the following formulae can be used for mode –
II
GII = (1-µ2
) K2
II
E
K2
II = GII * E
(1-µ2
)
Where
GII = W FS
Aeff
W FS = work of fracture in mode II, calculates as
the area under the load – slip curve.
Aeff = effective area of cross section of the
specimen.
µ = poisons ratio for concrete = 0.3
E = young’s modulus of the concrete
Table 1.1 Failure stresses
Graphs for Load vs Displacement FOR SMALL
SIZE BEAMS
GRAPH FOR 0.3(NOTCH SIZE: 22.5MM)
S.
No
Size
of the
beam
a/w
ratio
Fracturee
nergy
(G)N/mm
Stress
intensity
factor(K)
N/mm 1.5
1 1000*
300*7
5
0.3 27.32 901.31
0.4 28.33 917.85
0.5 25.75 866.78
0.6 24.21 849.03
2 500*1
50*75
0.3 6.87 424.51
0.4 12.76 600.81
0.5 8.67 490.94
0.6 7.23 465.43
3 250*7
5*75
0.3 14.72 649.02
0.4 12.60 600.87
0.5 12.60 615.71
0.6 10.47 559.32
Table 1.2: Fracture Energy and Stress intensity
factors
GRAPHS FOR 0.4(NOTCH SIZE: 30MM
GRAPHS FOR 0.5(NOTCH SIZE: 37.5MM)
GRAPHS FOR 0.6(NOTCH SIZE: 45MM)
FOR MEDIUM SIZE BEAMS
S.
No
Size of the
beam
a/w
ratio
Ultimate
load(KN)
Shearstress
( N )n/mm2
1 1000*300
*75
0.3 209.55 13.3
0.4 184.20 13.6
0.5 180.10 16.0
0.6 165.80 18.42
2 500*150*
75
0.3 55.60 7.06
0.4 53.50 7.97
0.5 52.30 9.29
0.6 42.30 9.4
3 250*75*7
5
0.3 40.80 10.36
0.4 37.50 11.11
0.5 35.80 12.70
0.6 32.70 14.53

GRAPHS FOR 0.6(NOTCH SIZE: 90MM) FOR
LARGE SIZE BEAMS
Fig:1.17 Graphs for load vs displacement for all
sizes of beams
Load Vs shear stresses for all a/w ratio of all sizes
of beam
Graph for 0.3 a/w ratio for all sizes of beams

Fig 1.18 Graphs for load vs shear stress of a/w
ratio of all sizes of beams
Graphs for Fracture energy vs a/w ratio of all
sizes of beams
Graph for G vs a/w ratio of small size beams
Graph for G vs a/w ratio of medium size beams
Graph for G vs a/w ratio for large size beams
Fig 1.19 Graphs for G vs a/w ratio for all sizes of
beams
Graph for stress intensity factor vs a/w ratio of all
sizes of beams
Graph for K vs a/w ratio for small size beams
Graph for K vs a/w ratio for medium size beams
Graph for K vs a/w ratio for large size beams
XIX. CONCLUSIONS
Based on the tests on twenty four double
centered notched concrete beam specimens, the
following conclusions have been drawn
1. From tests we observe that the failure stresses
increases with increase in beam sizes.
2. Fracture toughness or stress intensity factor
increases up to certain point and then decreases
with increase in notch depth for all sizes of
beams.
3. Fracture energy or energy release rate increases
up to certain point and then decreases with
increase in notch depth for all sizes of beams.
4. For constant notch to depth ratio of beam size
increases with decrease in stress intensity factor.
5. For constant notch to depth ratio of beam size
increases with decrease in fracture energy.
6. It is observed that the load increases with
decrease in notch size.
7. It is observed that the load decreases with
increase in failure stresses.
8. For constant notch to depth ratio of beam sizes
the failure stresses decreases.
REFERENCES
[1.] Agarwal, B.D. and Giare, G.S., “Fracture
toughness of short-fiber composites in
Modes-I and II”, Engineering Fracture
Mechanics, Vol. 15, No. 1, 1981, pp.219-
230.
[2.] Bazant , Z.,p, and Pfeiffer, P.A., “Shear
fracture tests of concrete”, materials and
structures (RKLEM) , 1984, vol. 19, pp.111-
121.
[3.] Watkins, J. and Liu, K.L.W., “A Finite
Element Study of Short Beam Test
Specimens under Mode-II loading”, The
International Journal of Cement Composites
and Light Weight Concrete, Vol.7, No.1,
Feb.1985, pp.39-47.

[4.] Bazant , Z.,p, and Pfeiffer, P.A., “Tests on
shear fracture and strain softening in
concrete”, proceedings of second
symposium on interaction of Non-nuclear
Munition with structures Florida, USA,
April 1985, pp. 254-264.
[5.] Davies, J., Yim, C.W.A and Morgan, T.G.,
“Determination of Fracture parameters of
punch through shear specimens”, The
International Journal of Cement Composites
and Light weight Concrete, Vol. 9, No. 1,
Feb. 1987, pp. 33-41.
[6.] BhaskarDesai . V, “Some studies on Mode -
II fracture and stress – strain behavior in
shear of cementitious materials”, Ph.D
thesis, Indian Institute of Science,
Banglore”.
[7.] PrakashDesayi, Raghu Prasad .B.K, and
Bhaskar Desai . V, “Experimental
determination of KIIc from compliance and
fracture energy”, proceedings national
seminar on Aerostructures, organized by
IIT, Kanpur, India, 29-30, Dec, 1993, pp.
33-34.
[8.] Prakashdesayi, B.K.Raghu Prasad and
V.Bhaskar Desai, “Mode – II fracture of
cementitious materials- part – I : Studies on
specimens of some new geometries”,
Journal of Structural Engineering, Vol.26,
No.1, April 1999, pp.11-18.
cementitious materials- part – II: Fracture
toughness of cement paste, mortar, concrete
and no-fines concrete. Journal of structural
engg Vol. 26, No. 1, April 1999, pp. 19-27.
cementtiotus materials- part – III: Studies on
shear strength and slip of cement paste,
mortar, concrete and no-fines concrete.
Journal of structural engg Vol. 26, No.2,
July 1999, pp. 91-97.
V.Bhaskar Desai, conducted Mode-II
fracture of cementitious materials- part-IV:
Fracture toughness, shear strength and slip
of fibre reinforced cement mortar and
concrete. Journal of structural engg. Vol. 26,
No. 4, Jan 2000, pp. 267-273.

MODE II FRACTURE PARAMETERS FOR VARIOUS SIZES OF BEAMS IN PLAIN CONCRETE

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