Nuclear Engineering and Design 250 (2012) 173–183

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Nuclear Engineering and Design

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Tensile strength of concrete under static and intermediate strain rates:

Correlated results from different testing methods
Shengxing Wu, Xudong Chen ∗ , Jikai Zhou
College of Civil and Transportation Engineering, Hohai University, Nanjing, Jiangsu 210098, China

h i g h l i g h t s

I Tensile strength of concrete increases with increase in strain rate.

I Strain rate sensitivity of tensile strength of concrete depends on test method.
I High stressed volume method can correlate results from various test methods.

a r t i c l e i n f o a b s t r a c t

Article history: This paper presents a comparative experiment and analysis of three different methods (direct tension,
Received 23 November 2011 splitting tension and four-point loading flexural tests) for determination of the tensile strength of concrete
Received in revised form 27 April 2012 under low and intermediate strain rates. In addition, the objective of this investigation is to analyze the
Accepted 5 May 2012
suitability of the high stressed volume approach and Weibull effective volume method to the correlation
of the results of different tensile tests of concrete. The test results show that the strain rate sensitivity
Keywords:
of tensile strength depends on the type of test, splitting tensile strength of concrete is more sensitive to
Concrete
an increase in the strain rate than flexural and direct tensile strength. The high stressed volume method
Tensile strength
Strain rate
could be used to obtain a tensile strength value of concrete, free from the influence of the characteristics
Direct tension of tests and specimens. However, the Weibull effective volume method is an inadequate method for
Splitting tension describing failure of concrete specimens determined by different testing methods.
Four-point loading © 2012 Elsevier B.V. All rights reserved.

1. Introduction A lot of efforts have been dedicated to the research in the field
of dynamic properties of concrete. More emphasis has been placed
It has long been known that concrete has a low tensile strength on the compressive behavior, for which more data is available, and
compared to its compressive strength. Since concrete is inherently less on the tensile response, because the behavior in compression
weak in tension, it has been used as compressive member mate- is more easily measured (Yan and Lin, 2006). Results of loading
rial in most concrete structures (Raphel, 1984; Nonaka et al., 1995; tests have confirmed an increase in compressive or tensile strength
Lorefice et al., 2008). However, even though static tensile loads on of concrete subjected to dynamic loading. Some comprehensive
concrete members are avoided, it is difficult to isolate concrete review papers of more recent works of this topic can found in sev-
members from dynamic tensile stresses. The propagation of ten- eral surveys (Soroushian et al., 1986; Fu et al., 1991a,b; Bischoff and
sile stress waves in structural members is generated by explosives, Perry, 1991; Malvar and Ross, 1998; Li et al., 1993; Lu and Li, 2010;
impingement of projectiles, earthquakes, and so on (Georgin and Ficker, 2011). From the literature review (Goldsmith et al., 1966;
Reynouard, 2003). In fact, in the Great Hanshin-Awaji Earthquake, Birkimer and Lindemann, 1971; Zielinski et al., 1981; Zielinski and
some uncommon fractures and damages of concrete structures Reinhardt, 1982; Oh, 1987; Ross et al., 1995; Brara et al., 2001;
were observed which might have been caused by the propagation Cadoni et al., 2001; Klepaczko and Brara, 2001; Brara and Klepaczko,
of stress waves and/or interface of tensile stress waves (Nonaka 2006; Lin et al., 2007; Yan and Lin, 2008; Asprone et al., 2009; Zhang
et al., 1995). et al., 2009; Snozzi et al., 2011), it can be found that the dynamic ten-
sile strength of concrete has not been studied extensively yet and
also the data on the rate effect is mostly in the regime of high strain
∗ Corresponding author. Tel.: +86 25 83786551; fax: +86 25 83786986; mobile:
rates (above 1 s−1 ). The research on intermediate and quasi-static
+86 1360149968.
strain rates (10−6 to 10−2 ) has been rarely performed.
E-mail addresses: sxwu@hhu.edu.cn (S. Wu), cxdong1985@hotmail.com There are three main methods for measuring the tensile strength
(X. Chen), jkzhou@hotmail.com (J. Zhou). of concrete: (1) splitting tensile test; (2) beam flexural test; and (3)

0029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.nucengdes.2012.05.004
174 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183

direct tension test (McNeely and Lash, 1963; Rashid et al., 2002). flexural tensile strength is generally 35% higher than splitting ten-
The first two test methods are indirect in the sense that the tensile sile strength. According to the above literature review, it follows
stresses are applied indirectly to the specimens. In the beam flexu- that, unless a better knowledge of the factors that produce such a
ral test, the specimens are subjected to center-point or third-point wide range of tensile strength results is acquired, any attempt to
loading until the beam fails by flexure. The theoretical maximum apply the test results to actual structures, whose dimensions and
tensile stress in the bottom fiber, known as the modulus of rup- field of stresses differ markedly from those of the specimens, is
ture, is calculated by assuming a linear distribution of flexural questionable.
stress across the section at failure. As is to be expected, the flexural The present paper deals with the effect of strain rates on the
tensile strength determined in this way is generally significantly tensile strength of concrete, as given by different test methods.
higher than that predicted from the direct pull tests. This proba- Such knowledge is required for the design of concrete structures.
bly arises because of the nonlinear stress–strain characteristics of Early investigators (Oh, 1990; Zhou et al., 2010, 2011) showed that
the concrete in tension zone. Wright (1952) has shown that if the flexural strength to increase with an increase of the strain rate by
compressive stress distribution is linear and the tensile stress dis- a logarithmic relationship at low and intermediate strain rates.
tribution parabolic, the maximum tensile stress is 0.735 times the At high strain rates, power and exponential relationships have
conventional modulus of rupture value. In the split-tensile test, a been proposed (Wittmann et al., 1987; Rossi and Parant, 2008).
concrete cube or cylinder is laid horizontally between the loading Similar observations were observed for splitting tensile strength
platens of the testing machine and compressed along two opposite (Brühilert and Wittmann, 1990; Tedesco and Ross, 1993; Gomez
generators until the specimen splits across the vertical diamet- et al., 2001) and direct tension strength (Tinic and Brühilert, 1985;
ric plane. A theoretical basis for the test has been postulated by Körmeling and Reinhardt, 1987). However, there are disagreements
Thaulow (1957) and Davies and Bose (1968). It is believed, how- on the strain rate sensitivity of tensile strength by different test-
ever, that neither of these two test methods yields the true tensile ing method. Reinhardt et al. (1990) suggested that the strain rate
strength. Theoretically, the direct tension should yield a tensile effect on bending seemed to be almost the same as in uniaxial ten-
strength closer to the true strength of the concrete under pure sile testing, this was concluded from comparing test results from
uniaxial tension. Several variations of the direct tension test have other researchers assuming a constant Young’s modulus. Suaris
been developed. The following modes of gripping the specimens for and Shah’s (1983) results showed that the four point flexural tests
direct tension test have been adopted by many researchers. They was a little more sensitive to strain rate than uniaxial tensile
are by means of (1) rings on truncated cones (Elvery and Haroun, loading, they attribute this difference to the vectorial behavior of
1968), (2) embedded steel bars (Evans and Marathe, 1968; Xie and micro-cracks which are oriented differently for the various types
Liu, 1989; Swaddiwudhipong et al., 2003; Sahamitmongkol and of loading. While Oh’s (1987) results showed that direct tensile
Kishi, 2011), (3) gluing (Wang et al., 1990; Mohamed and Li, 1994; strength appears to be more sensitive to strain rate than flexural
Li and Shah, 1994; Li and Ansari, 2000; Zheng et al., 2001; Paul, strength. However, no study was found in which all three tests for
2008) and (4) lateral gripping (Petersson, 1981; Gopalaratnam and evaluating the dynamic tensile strength of concrete were analyzed
Shah, 1985). Each method suffers from different setbacks. Based and compared.
on three-dimensional finite element analysis, Zheng et al. (2001) This paper presents a comparative analysis of three different test
suggested that the most reliable method of applying direct ten- methods (direct tension, splitting tension and four-point flexural
sion without inducing secondary stresses is by gluing steel plates tests) for determination of the tensile strength of concrete under
to the ends of the concrete specimen and pulling the steel plates so intermediate strain rates. In addition, the objective of this inves-
that the tension is transmitted to the concrete in the form of pure tigation is to analyze the suitability of the high stressed volume
tension. approach and Weibull effective volume method to the correlation
The flexural and splitting tensile tests are much cheaper, sim- of the results of different tensile tests of concrete.
pler and quicker to carry out because the samples are smaller,
and the set up time for the tests is much less. However, it is
2. Experimental procedure
known that flexural and splitting tensile strength are different
from direct tension samples. In fact, the tensile strength of con-
2.1. Materials and mix proportion
crete was found to be highly sensitive to the testing techniques
and the size and shape of the specimens used. This fact has pre-
The mix proportion of concrete used is given in Table 1, where
vented a deep understanding of the behavior of concrete under
Type 42.5R Portland cement was used in all mixes. The mixes con-
tension and, therefore, has made the practical application of the
tained fly ash (ASTM Class A) to save cement and reduce the heat of
tensile strength values obtained from tests unreliable. Usually, only
hydration for practical application. The crushed granite rock with
the compressive strength of concrete is measured for the purpose
maximum aggregate size 40 mm was used as coarse aggregate. The
of quality control as construction proceeds; the tensile strength is
maximum sand grain size was 4 mm. The specific gravities of the
estimated from the compressive strength using empirical correla-
fine and coarse aggregates were 2.40 and 2.58, respectively. The
tion equations (Narrow and Ullberg, 1963; Bhanja and Sengupta,
coarse aggregate and sand were air dried prior to mixing.
2005; Choi and Yuan, 2005). An appraisal of the various testing
Following casting, the specimens were covered with a plastic
techniques has been given by Price (1951) and the results indi-
membrane to prevent the moisture from evaporating. Specimens
cate that, of the tensile tests, the splitting tensile method predicts
were demolded after 24 h and moist cured for 6 months.
a tensile strength between the values obtained from the other two
methods. Raphel (1984), after examining a large number of tensile
test results, assumed that the direct tensile strength is about 10% 2.2. Test specimens and methods
of its compressive strength; splitting tensile strength is about the
same as the direct tensile strength; and flexural strength is about In this investigation the specimens were wet cured until the
15% of compressive strength. Popovics (1982, 1998) and Khaliq time of test, to minimize the possible occurrence of shrinkage
and Kodur (2011) concluded that the splitting tensile strength is cracks. As shown in Fig. 1, three types of test specimens were
usually 5–12% greater than the direct tensile strength, whereas made for four-point flexure, splitting-tensile and direct tension
it is 40–50% lower than the flexure tensile strength. Zheng et al. tests. Details of the tests are given in Table 2 together with dif-
(2001), reviewing works of several researchers, have reported that ferent loading rates were also presented. All tests were carried out
 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183 175

Table 1
Details of mix proportions (in kg per cubic meter of concrete).

Cement Fly ash Water Coarse aggregate Fine aggregate Superplasticizer Air-entraining
admixture

5–20 mm 20–40 mm

134 57 86 548 390 534 1.43 0.0105

Fig. 1. Concrete specimens in testing machine: (a) four-point loading; (b) splitting tension; and (c) direct tension.

in a MTS test machine (500 kN). Strains were determined by surface easily recognized. At high loading rates, the load–time curves for
electrical resistance gauges. the concrete beams exhibited unsteady behavior. However, overall,
For flexural tensile tests, the machine was set to apply load at it was considered that the load was acceptable. For splitting tensile
the four points over 450 mm span. The beams were tested with the tests, the loading regime was similar to that of flexural tensile tests.
trowelled face to one side and with hardboard packing strips under For direct tensile tests, bonded steel end plates were used to
the points of load and support to distribute some of the load. Typical apply rapid tensile load to the specimen as shown in Fig. 1(c). The
load–time curves for concrete beams tested at a low and a high load- steel end plate had a cross-sectional dimension of 68 × 68 mm and
ing rate are shown in Fig. 2(a) and (b), respectively. These illustrate a thickness of 35 mm. An epoxy adhesive was used to glue the steel
that the linearity of loading was acceptable and that failure was end plate to the specimen. When applying the epoxy adhesive, the

Table 2
Tensile strength as measured by various methods of tests.

Method of test Specimen size (mm) Loading rate Number of

specimens

Four-point loading 150 × 150 × 550 0.25 kN/s 4

2.5 kN/s 3
25 kN/s 3
450 kN/s 4

Splitting tension 150 × 150 × 150 0.25 kN/s 3

2.5 kN/s 3
25 kN/s 3
250 kN/s 3

Direct tension 160 × 68 dia. 0.02 mm/s 5

0.2 mm/s 5
2.0 mm/s 5
20 mm/s 5
176 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183

Fig. 3. Concrete specimens after failure (four-point loading).

distributed between the load carrier clamps suggesting that the

loading during the test was as intended. Fig. 4 shows the concrete
specimens after failure under direct tension tests. As it is typically
observed in these types of experiments (Snozzi et al., 2011), a single
crack develops approximately in the specimen, induced by uniform
tensile stresses acting along the specimen cross section.
Fig. 5 shows the fractured surfaces of concrete specimens after
split tension tests. It can be seen from Fig. 5 that the fractured sur-
faces of the specimens became more and more flattened with the
increasing strain rate; and an increasing number of aggregates were
broken along the fractured surface. As a consequence of shrink-
age effects, micro-cracks exist in the unloaded concrete mainly at
the interface boundaries between the matrix and the aggregates
(Scrivener and Nemati, 1996; Bisschop and van Mier, 2002; De Sa
et al., 2008). Upon loading, high stresses occur at the tip of these
micro-cracks (Hsu et al., 1963; Nemati et al., 1998). As a result of
tensile or compressive strain, these high stresses are relieved by
the growth of hair cracks in the matrix and bond cracks at the
interfaces between the matrix and the aggregates. The material
Fig. 2. Typical load time curves for concrete beams: (a) loading rate 0.25 kN/s and
(b) loading rate 450 kN/s.

bonded surface between the steel end plate and the specimen was
subjected to a compressive stress of 0.5 MPa. The epoxy adhesive
was generally controlled to be approximately 0.5–1.0 mm thick.
The assembly was cured for more than two days for the epoxy
adhesive to develop strength. The upper and lower sides of the spec-
imens were rigidly attached to a firmly load cell and to a moveable
loading frame, respectively, through bolts in the uniaxial loading
apparatus, thereby restraining the end rotations of the specimen.
Tensile loads were applied to the specimens by moving the loading
frame downward at the specified rates under displacement control.
Four loading rates of 0.02, 0.2, 2.0 and 20 mm/s were adopted.

3. Test results and discussion

3.1. Failure modes

The crack pattern and failure modes of concrete specimens

under both static and dynamic loading are presented in Figs. 3–5,
respectively. Fig. 3 shows the flexural failure mode of concrete
beams. The failure location of all concrete beams is at the mid-span,
no effect of strain rate was observed. Thus it would appear that for
concrete beams subjected to flexure, strain rates will not affect the
failure mode. The failure locations of the beams tested were evenly Fig. 4. Concrete specimens after failure (direct tension).
 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183 177

Fig. 5. Typical failure surface of concrete after split tension tests.

is, therefore, weakened. The stress-deformation relationship may Contrary to the behavior at low strain rates, the resulting stored
be described in two parts. In the first part, which is ascending, strain energy remains reversible almost until the maximum stress
the energy gained during loading is not lost upon loading, i.e. the is reached; this is where energy release starts. For delayed crack
energy is reversible. However, in the second, descending part of the propagation the cracks occur spontaneously, i.e. without signifi-
stress deformation relationship, a portion of the energy is lost due cant formation of micro-cracks. As a consequence, there is a great
to crack-formation and is, therefore, irreversible. Both reversible amount of energy released, the cracks are no longer stable and
and irreversible energy occur at every point of the descending part they propagate unhindered and relatively quickly. The failure then
of the relationship. Under increasing strain and decreasing stresses occurs according to a relatively direct path through the matrix and
these portions decrease. the aggregates themselves (Fig. 6b).
Generally, at low strain rates in tests, the formation of micro-
cracks is shown after the maximum stress is attained (Ross et al., 3.2. Tensile strength
1989). Furthermore, the cracks in the matrix are prevented by
the aggregates from growing further. Therefore, they are initially The tensile strength results are shown in Fig. 7. The strain rate
stable, i.e. more energy needs to be supplied for their continued was calculated in the linear part of the strain–time curves in tests
propagation. The zone around the micro-cracks remains capable where strain gages were used. Fig. 7 indicates that with every
of carrying load, but in a steadily decreasing magnitude. This con-
tinues until a critical crack width is reached (Shah, 1990). After
this point, the crack is unstable. Finally, the failure occurs at a
low reversible energy. The fracture surface grows relatively slowly
according to the path of least resistance through the matrix and the
interfaces around the aggregates, i.e. matrix-failure occurs (Fig. 6a).

Fig. 6. Description of failure (schematic), (a) failure characteristics at static loading

rates and (b) failure characteristics at dynamic loading rates. Fig. 7. Tensile strength versus strain rate for concrete.
178 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183

order of magnitude increase in strain rate, the tensile strength of the tensile strength of concrete is related in some way to the rate of
concrete increases. This trend is nearly linear increase over four change of stress across the section. One could speculate that failure
orders of magnitude. On the basis of the results presented here, it might be associated with the size and number of localized regions
is clear that concrete is a strain rate sensitive material. Concrete of weakness, i.e., the probability that localized regions of weak-
materials showed higher tensile strength with increase in strain ness occurring within the critical area is in proportion to the area
rate. Several explanations can be suggested to account for these stressed to near failure. The variation of tensile strength among
trends. One explanation may be based on fracture mechanics con- different testing methods observed in this study agrees with the
cepts (Weerheijm and Van Doormaal, 2007; Zhang et al., 2010). The results reported by other investigators (Price, 1951; Raphel, 1984;
phenomenon of strain rate sensitivity can be explained by com- Brühilert and Wittmann, 1990; Khaliq and Kodur, 2011). Ho and
bining the classical Griffith theory with the concept of sub-critical Peng (2011) suggested that, in making concrete test specimens,
crack growth. According to Griffith theory, failure in brittle mate- planes of weakness may occur, due to bleeding, at the undersur-
rials occurs when a flaw exceeds the critical flaw size for a given faces of the coarse aggregate. In the direct tension test critical stress
stress. If the load is applied very slowly, the sub-critical flaws have is normal to these planes of weakness, whereas in the flexure and
time to grow and thus the failure occurs at a lower value of load. splitting tests the critical stresses are parallel to these planes. He
However, if the load is applied at a very high rate, there is little thought this may be the main reason for the lower strength in the
or no time available for the growth of the sub-critical flaws, and a direct tension tests. Statistical considerations, based on the “weak-
higher load can be reached by the structural element before failure est link” theory, may also be pertinent.
occurs. Zhang et al. reported that pre-peak crack growth is reduced The type of aggregate used for concrete is known to affect the
at high rates of loading. influence of strain rate on the dynamic tensile strength (Li et al.,
Under dynamic tensile loading conditions much energy is intro- 2004). Aggregate that exhibits a good bond, and which minimizes
duced into the specimen in a short time, and cracks are forced to the differences in stiffness with the surrounding mortar matrix, will
develop along shorter paths of higher resistance through stronger have a good loading resistance. Tests by Harris et al. (2000) showed
matrix zones and hard aggregate particles. It has been observed concrete with stiffer aggregates was less rate-sensitive, but they
from Fig. 5 that cracks usually passed under aggregate particles did not consider the effect of surface texture (roughness) between
in statically loaded concrete, but in the case of dynamic loading the difference aggregates. A smaller size of aggregate should also
the fracture surfaces comprised many fractured aggregate particles. improve the tensile strength of concrete under dynamic loading
The very rapidly increasing overall tensile stress causes extensive (Cadoni et al., 2001). In further investigation, the influence of aggre-
micro-cracking in other areas, since relaxation cannot occur in the gate type (stiffness, surface texture, size, shape and strength) on the
extremely short time of fracture. Also, crack branching can occur strain rate tensile behavior of concrete will be carried out.
due to interactions between the rapidly moving crack front and
the aggregate particles or other in-homogeneities. So the reasons 3.3. Dynamic increase factor (DIF)
for the higher tensile strength of concrete under dynamic loading
can be distinguished: (a) cracks develop through hard aggregate; The effect of strain rate on the compressive or tensile strength of
and (b) extensive microcracking take place in the whole volume of concrete-like materials is typically denoted as a dynamic increase
stressed material. factor (DIF) (i.e. the ratio of dynamic to static strength) versus strain
Although many researchers associate the strain rate effects to rate (or the logarithm of strain-rate). The use of normalized DIF
inertia effects related to the material cracks, Rossi and Toutlemonde reduces the influence of the material strength on the DIF formu-
(1996) suggested that the free water in concrete has an impor- lae. For this investigation, the DIF versus logarithm of strain rate
tant influence named as “Stefan-effect” on the strain rate effects is plotted in Fig. 8. As shown in Fig. 8, the tested results from the
of strength. The Stefan-effect is the phenomenon that occurs when three test methods, namely four-point loading, splitting tension
a viscous liquid is trapped between two plates that are rapidly sep- and direct tension, shows that splitting tensile strength is most
arated; causing a reaction force on the plates that is proportional sensitive, while the rate sensitivity of direct tensile and flexural
to the velocity of separation. Forquin and Erzar (2010) investigated strength is almost same.
the effect of relatively long cure in somewhat humid environment Comparing direct tensile strength from this investigation and
on the rate effects, without checking if the free water in concrete compressive strength data from other literatures (Harsh et al.,
exhibits any influence. The drying process and sufficiently precise 1990; Grote et al., 2001; Li et al., 2009) at given strain rate (Fig. 9),
test-tube conservation enabled to compare the rate effects on a dry it appears that the tensile strength is more sensitive to strain rate
and wet concrete. The obtained results by Forquin and Erzar (2010) effects at lower strain rates than the compressive strength. The
showed that in the range of rate where the evolution of the resis-
tance is slow (until 1 s−1 ), the content in free water appear as the
main parameter inducing the rate effects.
From Fig. 7 it appears that the specimens tested in a four-point
loading arrangement exhibit a high strength than those tested in
a direct or split tension tests. The differences between the tensile
strength in these three testing methods tend to exist under differ-
ent strain rates. The differences between splitting tensile strength
and direct tensile strength are small compared the corresponding
differences between the flexural and the direct tensile strength.
The results of this study, however, appear to indicate that con-
crete will resist failure more effectively, and therefore show greater
strength, under a loading condition where a relatively small area
of the section is stressed to a point. Lower strengths were obtained
in direct tension, that is when the entire cross section is stressed
in tension to near failure, than when progressively smaller areas
of the cross section were stressed to near failure by the splitting
tension and four-point loading. There seems to be little doubt that Fig. 8. Dynamic increase factor (DIF) versus strain rate for concrete.
 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183 179

Fig. 9. Comparison between tensile and compressive dynamic increase factor.

same results have been observed by others (Ross et al., 1989; Grote
et al., 2001; Li et al., 2009). Cusatis (2011) suggested that in the
strain rate range (10−7 to 10−1 s−1 ) the dynamic increase factor
(DIF) data includes the effect of inertia and it should be considered Fig. 10. The contour of the high stressed volume (HSV) for various tests.
a structural property, rather than a material property. An impor-
tant consequence of Cusatis’s (2011) results is that the common
one. Fracture will be caused by a crack starting in the tensioned
practice of calibrating constitutive models by directly fitting DIF
zone and spreading rapidly through the whole section of the speci-
experimental data is not justified and the fitting should be based
men. That crack will start in the critical zone, where tensile stresses
on the actual dynamic simulation of the tests in the experiments.
are higher, and where the most severe defect, flaw or micro-crack
Both macroscopic compressive and tensile behaviors of concrete
exists. The HSV is, by definition, the volume of material subjected
are the result of the same physical phenomena, which are meso-
to a tensile stress amounting to 95% or more of the tensile strength
scale fracturing and shearing (and there is no softening postulated
in the specimen. As a matter of fact, this also constitutes an arbi-
meso-scale compression) (Wittmann et al., 1987). The difference
trary definition of the critical region and therefore, it is assumed
between tensile and compressive DIF can be explained as follows,
that the crack leading to fracture will develop somewhere within
according to the discussion above one can write:
HSV.
= ft∗ + ftin
dyn dyn
−fc = −fcin + fcin ; ft (1) According to the concepts stated above it is assumed that the
crack leading to fracture will start in some point of HSV. There-
dyn dyn
where fc , ft are the peaks of the nominal stresses at the spec- fore, the greater the HSV, of the specimen in testing, the greater
imen; fc , ftin are the (structural
in dependent) contributions due to probability a more severe flaw or micro-crack in that zone that
inertia effects. In an ideal situation in which exact the same test set- leads more easily to failure. In other words, the greater the HSV
up and the same specimen is used for tension and compression, the of the specimen, the smaller the maximum tensile stress at the
accelerations in tension and compression have the same magnitude moment of failure is. For descriptive purposes, the shapes adopted
but opposite sign. Consequently, one can write fcin = −ftin = −f in . By by the HSV contour for different test methods are outlined in
exploiting this observation, dividing Eq. (1) by static strength, one Fig. 10.
obtains The HSV values were computed using the following formulae,
dyn
fc f in ft
dyn
f in derived from the stress distributions in the specimens assuming
DIFc = = DIF∗ +  ; DIFt = = DIF∗ + (2) linear-elastic behavior, their derivation can be found in:
f c fc ft ft
D2 L
where DIF* is the DIF without the effect on inertia and representa- HSV(direct tension) = (3)
tive of the intrinsic material behavior. The assumption that intrinsic 4
DIF* is the same for tension and compression is motivated by the 14bhl
HSV(four-point loading prismatic beam) = (4)
fact that macroscopic tensile failure and macroscopic unconfined 1600
compressive failure are both the effect of meso-scale tensile frac-
HSV(split tension) = 0.0475 c 3 (5)
turing and shearing. By comparing DIFc and DIFt it is evident that
the tensile DIFt is bigger than the compressive DIFc . where D is the diameter of cylindrical specimens (mm), L is the
length of specimens (mm), b is the width of prismatic beams (mm),
4. Comparison of tensile strength determined by various h is the height of prismatic beams (mm), l is the span length of
testing methods prismatic beams (mm), and c is the width of cubic specimens for
split tension tests.
4.1. Highly stressed volume (HSV) method Mihashi and Izumi (1977) used the thermodynamic approaches
to predict the loading rate influence on the strength of concrete.
An alternative approach using the concept of highly stressed vol- They combined a thermodynamic approach to some extent with
ume was first proposed by Kuguel (1961) and subsequently used by fracture mechanics. They state that the fracture of concrete may be
Torrent (1977), Torrent and Brooks (1985), Lin and Lee (1998), and caused by series of local failure processes in the phase a crack of
Gaenser (2008). The “highly stressed volume” approach is based on hydration products of cement and interfaces between cement and
the fact that in the fracture of brittle materials, it is not necessary aggregate. As soon as a failure criterion is satisfied in one part of
to analyze what happens in the whole volume of the specimen, but the phases a crack is initiated. Extension of cracks and coalescence
only in its most critical area, that is to say in the most highly stressed of cracks cause fracture. The concrete system consists of a group of
180 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183

Fig. 11. Model of concrete with linked elements (Mihashi and Izumi, 1977).

elements linked in series, Fig. 11. Each element contains a circular 4.2. Weibull effective volume (WEV) method
crack the length of which depends on the pore sizes of hardened
cement paste. The distribution of the material defects and the char- The Weibull distribution is a statistical method based on the
acteristic properties of each element are statistically equal over the weakest link model and is widely applied in brittle materials where
material. The relationship between strength increase and relative failure occurs by crack growth from a single, critical flaw. The
strain rate ε̇d /ε̇s is expressed as Eqs. (6) and (7). Analysis of the Weibull model has historically been applied to brittle materials to
results for HSV dependency revealed the following: describe the scatter in strengths as well as strength variations due
to specimen size (Danzer, 1992). In concrete, micro-cracks develop
 ε̇ ˛ and become connected link, causing larger cracks at multiple sites
d
DIF = (6)
ε̇s and a progressive failure mode.
Weibull strength theory assumes that the strength of a brittle
1 material is controlled by flaws of similar shape and orientation and
˛= (7)
(0.512HSV/ ln(HSV)) + (5087.27/HSV ) − 29.952 in which the critical flaws are statistically distributed. As such, it
predicts an increase in strength with a decrease in specimen vol-
Fig. 12 shows that Eqs. (6) and (7) are reasonably representative ume, as probability dictates that on average the criticality of the
for the results of this investigation. Consequently, Eqs. (6) and (7) flaw found will be less with decreasing specimen size.
may be used to estimate the approximate strength of any size of the A two parameter model was used by Bullock (1974) and
specimen or test methods for any strain rate, provided the strength Cattell and Kibble (2001) in correcting four-point flexure and
is known for one strain rate. This finding implies that when the tensile-coupon data for unidirectional graphite–epoxy composite.
relative tensile strength is considered, the influence of the stress Excellent agreements were obtained between theory and experi-
distribution in the specimen is small compared with the influence ment. The Weibull model has been applied previously to ceramic
of size, as quantified by HSV. materials (Danzer, 1992). For a two parameter Weibull model, the
probability of survival, P(s), for a specimen under stress field, , and
with volume, V, is
   m 

P(s) = exp − dV (8)
v
0

where  0 is the characteristic stress and m is the Weibull modulus.

The Weibull modulus defines the shape of the statistical distri-
bution. Higher values of the modulus, m, indicate less scatter in
the data and greater confidence in the reliability of the material.
The characteristic strength is a normalizing parameter that relates
strength for a given probability of survival to the effective volume
under load.
In order to compare different tensile tests, it is necessary to take
into account the differing stress field present in the specimens for
each test type, as outlined below.
In direct tension tests, the stress field is constant over the entire
Fig. 12. Comparison of experimental and predicted results by HSV. gauge length. For a given Weibull distribution, assuming equal
 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183 181

Fig. 13. The concept of Weibull effective volume (WSV) method.

probability of failure, the stresses for specimens of different vol- Similarly, the WEV of the specimens in splitting tension test,
umes V1 and V2 can be compared using Eq. (9): VEs , was calculated from the dimensions of the specimens (Table 2),

1
 V 1/m and suing the appropriate equation suggested by Neergaard et al.
2 (1997):
= (9)
2 V1
where  1 (P, V1 ) and  2 (P, V2 ) are the ultimate strengths for a given VEs = c 3 (0.21209 m−0.48338 ) (12)
failure probability P and for the stressed volumes V1 and V2 .
Fig. 14 shows the predicted results by WEV method. The val-
As an example of the application of Eq. (9), the four-point flexu-
ues obtained from WEV method are smaller than the test results.
ral strength ( f ) may be compared with the direct tensile strength
This trend indicates that the WEV method is not appropriate for the
( d ):
description of tensile strength of concrete determined by different
 1/m test methods. This result is in agreement with other researchers’
f VEd
= (10) conclusion (Calard and Lamon, 2002). Bazant et al. (1991) sug-
d VEf
gested that, Weibull theory cannot be applied to the failure of
where VEf is the effective volume of the four-point loading test concrete structures unless the effect of stable macroscopic crack
specimen and VEd is the effective volume of the direct tension growth on the stress distribution is taken into account. Concrete
specimen. Many specimens or components, such as flexural loaded normally undergoes pronounced inelastic deformation with large
specimens have stress gradients and VEf < V. Sometimes, the rela- macroscopic stable crack growth prior to reaching the failure load
tionship between the two is expressed as VEf = KV, where K is called (maximum load). This inevitable engenders stress distributions,
the loading factor (Cattell and Kibble, 2001; Quinn, 2003). V is the such that the stress distribution at incipient failure is very differ-
total volume within the outer loading point. The concept of Weibull ent from the elastic stress distribution, which has commonly been
effective volume (WSV) is shown in Fig. 13. As shown in Fig. 13, assumed in previous studies of the statistical strength theory. The
only a part of the total volume, V, of the flexure specimen on the existence of macroscopic crack growth is also documented by the
left is in tension. Only a small fraction of this region (depicted by load-deflection diagram, in which the start of macroscopic crack
the shaded area) is exposed to large tension stresses. VE is the vol- growth is manifested by a significant reduction of slope.
ume of a hypothetical tensile specimen, which, when subjected to
the stress  max , has the same probability of fracture as the flexure
specimen stressed at  max . In other words, a flexural specimen of
volume V is equivalent to a tensile strength of size VE . How much
of the flexure specimen volume should be counted depends on the
Weibull modulus. For concrete specimens loaded in tension, m is
taken as 9 (Xu and He, 1990).
In flexural specimens the stress field is variable. In accordance
with the beam theory the stress distribution for four-point flexure
can be assumed to vary linearly through the thickness. The surface
stress varies linearly along the length from zero at the supports,
to maximum at the loading nose. The stress distribution across the
width of the specimen is assumed to be constant. Integrating Eq.
(8) over the half thickness of the specimen in tension give an WEV
of

m+3
VEf = (bhl) (11)
(m + 1)2 Fig. 14. Comparison of experimental and predicted results by HSV.
182 S. Wu et al. / Nuclear Engineering and Design 250 (2012) 173–183

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