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Materials and Structures ISSN 1359-5997Volume 50Number 1 Mater Struct (2017) 50:1-11DOI 10.1617/s11527-016-0901-x
Dynamic tensile test of mass concrete withShapai Dam cores
Haibo Wang, Chunlei Li, Jin Tu & DeyuLi
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ORIGINAL ARTICLE
Dynamic tensile test of mass concrete with Shapai Damcores
Haibo Wang . Chunlei Li . Jin Tu . Deyu Li
Received: 24 November 2015 / Accepted: 30 May 2016
� RILEM 2016
Abstract Static and dynamic tensile tests of dam
concrete cores were carried out to investigate the
dynamic properties as well as the tensile stress–strain
relationship under either monotonic or cyclic load.
The test specimens were prepared from the cylindrical
cores drilled from Shapai Arch Dam, which was built
with three-graded roller compacted concrete and
survived the strong shaking of Wenchuan Earthquake.
Direct tensile tests were performed on an MTS servo
controlled testing machine and the system displace-
ment was used as the control command for all tests.
The test results indicate a significant increase in
strength compared with static ones, from 2 to 47 %
under strain rate roughly from 10-4 to 10-2/s, and
more fracture energy is consumed by the concrete
under dynamic or cyclic loading than static monotonic
loading. Furthermore the static preload on the spec-
imens shows little influence on their dynamic tensile
strengths. Based on the experimental data, a simple
analytical model was proposed for entire stress–strain
relationship under both monotonic and cyclic tensile
loading, the calculated stress–strain path gives a
satisfactory approximation which can be used in
dynamic numerical analysis of concrete dams.
Keywords Mass concrete cores � Tensile properties �Cyclic loading � Dynamic strength � Stress–strain
relationship
1 Introduction
The dynamic properties of mass concrete are very
important to the analysis and review of seismic safety
of concrete dams against strong earthquakes [1–5].
Furthermore, in the investigation of the nonlinear
behavior of concrete dams, modeling of the tensile
cracking and damage process is required. Many
laboratory tests of concrete have been performed for
wide range loading rates and rate-dependent charac-
teristics of concrete have been observed [6–11]. Harris
[9] summarizes the results of a U.S. Bureau of
Reclamation research project designed to provide a
broad database of material properties of mass concrete
tested at strain rates that correspond to seismic
(dynamic) and static loading on the compressive and
splitting tensile strength. Description of rate sensitiv-
ity in concrete is normally expressed as the ratio of the
dynamic to the static value of a particular mechanical
property. The strain rate of structural responses under
earthquakes is 10-3–10-2 per second according to
Bischoff and Perry [6]. Shapai RCC arch dam of
H. Wang (&) � C. Li � J. Tu � D. Li
State Key Laboratory of Simulation and Regulation of
Water Cycle in River Basin, Earthquake Engineering
Research Center, China Institute of Water Resources and
Hydropower Research, 20 West Chegongzhuang Rd.,
Beijing 100048, China
e-mail: [email protected]
Materials and Structures (2017) 50:44
DOI 10.1617/s11527-016-0901-x
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132 m in height has been subjected to the destructive
earthquake shaking during 2008 Wenchuan Earth-
quake of magnitude 8.0, the closest fault distance to
the dam site is about 29 km. Shapai Dam survived the
strong earthquake with no visible cracking on the dam,
although many evident cracking were observed on the
RC frame structures on the top of the intake tower and
the house connected to an elevator shaft on the dam
top. The power house which is 5 km downstream was
severely damaged by the falling rocks as well [12].
No strong motion at the Shapai Dam site and
seismic response were recorded by instrument, but
from the estimation of seismological analysis, the peak
ground acceleration at the site of Shapai Dam is about
0.26 g [13]. In order to review the seismic responses of
Shapai Dam during Wenchuan earthquake and to
evaluate its ultimate capacity against seismic loading,
cylindrical cores drilled from Shapai Dam after the
earthquake were tested. In this paper, the tensile test
results of both static and dynamic loadings using the
drilled cores are summarized, which were used in the
nonlinear dynamic analysis of Shapai Arch Dam.
Direct dynamic tensile test of mass concrete of dam
is scarce. The size of the full graded specimens is very
large for mass concrete, because core diameters must
be at least twice the nominal maximum aggregate size,
and preferably, three times that size according to
ASTM C42 [14]. The nominal maximum aggregate
size is from 80 to 120 mm commonly for mass
concrete of dam. Direct tensile test is far more
complicated in preparation and installation than
splitting tensile test, especially for the specimens of
large size. But the tensile stress–strain relationship can
only be measured directly in a direct tensile test
[15–19]. And for mass concrete dams, the develop-
ment of tensile cracks is a very important concern
during earthquakes as well as routine operation.
2 Materials and methods
2.1 Material
Shapai arch dam completed in 2003 was built with
three-graded Roller Compacted Concrete. It was made
of 180–192 kg/m3 pure Portland cement, 40–50 % fly
ash and granite aggregates with nominal maximum
size of 80 mm. The total volume of the dam is
383,000 m3. The specimens for laboratory test were
vertically drilled cores and preserved in water in the
laboratory. The size of specimens is 200 mm in
diameter and 400 mm in height. Three days before the
test, the specimens were taken out and air-dried for
epoxying strain gages.
2.2 Method
There are two common ways to connect the cylindrical
concrete specimen for direct tensile test to the loading
machine. One is to embed bolts at both ends of the
specimen when molding. Another is to epoxy metal
connectors to both ends of the specimen. It is obvious
that the second way is more suitable for the specimens
prepared from the cores drilled from dams. Zheng [17]
used a similar method for direct tensile test with
specimens of square section rather than using notched
specimens [15, 16]. The most important thing for
direct tensile test of concrete is to apply a uniform
deformation on the whole specimen. Because of the
difference in elastic modulus and Poisson ratio
between concrete and metal, there will be stress
concentration near the outer surface at the ends of the
specimen if the specimen is connected to the steel
plate directly. To minimize this uneven stress distri-
bution, a pair of aluminum connectors have been used
since the elastic modulus of the aluminum is closer to
that of concrete than any other metal. And the
thickness of the epoxy-resin between the specimen
and the aluminum connector is about 10 mm in our
tests, which can release further the stress concentration
near the ends. Before being epoxied together with the
aluminum connectors, the ends of the concrete spec-
imens have been roughened to improve the bond
strength. A special epoxy-resin of high fluidity was
selected and poured into the 10 mm gaps between the
specimen and the connectors. This way can prevent air
entrainment effectively.
Figure 1 shows a specimen epoxied and ready for
tensile test, each aluminum connector is fixed to a steel
plate with 24 M16 bolts. Four auxiliary angle bars
were employed to make two steel plates be parallel
and coaxial. Besides, four auxiliary angle bars used
can prevent the specimen from undesired forces
during carry as well as installation. All the auxiliary
bars were removed just before testing. Four column
strain gages were equally spaced around the speci-
mens. Each column includes four strain gages 120 mm
long to cover the whole length of the specimens. The
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resistance of the strain gage is 350 X. Totally 16 strain
gages were used for each specimen. The maximum
strain was found at every data acquisition step during
the test and can be used to check preset points of
unload for cyclic loading test. The force and displace-
ment were measured with the load cell and Tem-
posonic digital displacement transducer of MTS
machine system.
Each steel plate was connected to the loading
system through a swivel with eight M24 bolts to
minimize any bending effects on the specimen.
The tests were performed in an MTS servo
controlled testing machine. The maximum compres-
sive capacity and tensile capacity of the machine are
15 and 8 MN, respectively. And the stiffness of the
mainframe is about 6 MN/mm. The system dis-
placement was used as the control command for all
tests.
3 Results and discussions
Totally 49 specimens have been used. Among them,
three specimens were damaged prior to tensile load-
ing. Another four specimens show big non-uniform
deformation due to either the defect of the cores or the
problems in the installation. As shown in Table 1,
monotonic tensile tests involve five sets with different
loading rates, two sets with dynamic loading plus
static preloading of either 30 or 60 % levels, and
cyclic tensile tests involve two sets with either static or
dynamic loading. Each set contains four acceptable re-
sults according to the test code for hydraulic concrete
[20].
If the specimen is equally divided into eight
sections along the axis, the occurrence of the total
amount of the specimens with their fracture location
falling each section is 2, 6, 6, 7, 13, 11, 2, 2 from top to
bottom, respectively. The smallest distance of the
fracture to the end is larger than 20 mm. This indicated
that the end preparation for the specimens was
adequate and that the epoxy adhesive had sufficient
strength to transmit uniformly the tensile stress to the
concrete until the concrete cracks, and the device and
procedure for the installation of the specimens worked
well to reach reliable test results.
3.1 Results of monotonic tensile tests
Table 2 sums up the main results of seven sets
monotonic tensile test. The maximum strain rate in
the table is calculated from one time history of the
maximum strain between 20 and 100 le. Section av-
erage strain rate is determined with the average time
history of four strain gages on the same circle between
20 and 100 le. The percentage in brackets for ZL6 and
ZL7 set test, following the value of tensile strength,
represents the actual ratio of the static preload to the
measured tensile strength of the specimen.
The eccentricity e is defined as follows for judg-
ment of the uniform loading on the specimen [20].
e ¼ e1 � e2
e1 þ e2
����
����
ð1Þ
where e1 and e2 are the maximum and minimum ones
among the column average strains of four sides,
respectively.
Fig. 1 Connection of the core specimen to the aluminum
connectors. (Color figure online)
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The e varies as the strain level changes, because
local crack may not always grow uniformly in the
specimen during tensile loading, Fig. 2 displays an
example. This is owing to the non-uniform ingredient
of concrete material. Therefore, the eccentricity
corresponding to 30 le strain on average are listed
in the table, only the results with their eccentricity less
or equal to 0.15 were taken into the statistical analysis
according to test code for hydraulic concrete [20].
The tensile strength of each specimen in the
table was calculated by the peak load divided with
the area of its cross section. Set average strength is the
simple mean value of each set.
It can be concluded from the monotonic tensile
results that (1) the dynamic tensile strength of ZL2 and
ZL3 set increased by 28 and 30 % comparing to the
static one, respectively, (2) at higher strain rate, the
average dynamic tensile strength of ZL4 and ZL5 set
increased by 47 and 37 %, respectively, (3) the static
preload had little influence on the dynamic tensile
strength as shown in the results of ZL6 and ZL7 set,
comparing with that of ZL3 set.
Due to the lack of experimental data, few accurate
models are available for the stress–strain relationship
of concrete in tension. In the paper of Gopalaratnam
[16], a simple analytical model was proposed for both
ascending part as well as descending part.
For ascending part,
r ¼ rp 1 � 1 � eep
� �A" #
ð2Þ
where A ¼ Etep=rp, Et, ep and rp are initial tangent
modulus, peak strain and peak stress, respectively.For
descending part, the relationship is given by
r ¼ rpe�k e�epð Þk ð3Þ
where k and j are constants. The crack width x in [16]
is replaced here with e-ep, which is proportional to the
crack width, if only the strain gages work. Moreover,
in numerical procedures, using e-ep is more conve-
nience although the relationship between the crack
width and strain depends on the nominal gage length in
descending part. k = 1.01 was used in [16] for the
sake of continuity at peak, however, it has little
influence in the numerical analyses. Then k = 1.0 is
assumed here and j is determined by the area under the
stress–strain curve from ep to 1200 le for every
specimen.
Figures 3 and 4 give the stress–strain curves
measured and calculated with Eqs. (2) and (3) together
for static loading ZL1 set and dynamic loading ZL2
set, respectively. The parameters used in calculation
are shown in the figures as well. For ascending part,
the measured and calculated curves fit astonishingly
well except for ZL1-1 specimen.
For descending part, the parameter j is different for
every specimen to reach a good fitting. The fracture
energy G_F given in Figs. 3 and 4 is calculated from
the area integrated between 0 and 1000 le residual
strain based on measured data, and multiplied with the
length of the strain gages used, 120 mm. On average,
the fracture energy G_F for ZL2 set is higher than for
Table 1 Loading rate of each test set and the percentage of static preload if indicated
Set no. Nominal loading speed in strain rate Specimen no.
ZL1 Static (1.0 9 10-6/s) ZL1-1, ZL1-2, ZL1-3, ZL1-4
ZL2 Dynamic (0.2 9 10-3/s)
Dynamic (1.2 9 10-3/s)
ZL2-1, ZL2-2, ZL2-3, ZL2-4
ZL3 Close to that determined by the fundamental frequency of Shapai Dam ZL3-1, ZL3-2, ZL3-3, ZL3-4
ZL4 Dynamic (5 9 10-3/s) ZL4-1, ZL4-2, ZL4-3, ZL4-4
ZL5 Dynamic (10 9 10-3/s) ZL5-2, ZL5-3, ZL5-7, ZL5-8
ZL6 30 % static preload ? dynamic (1.2 9 10-3/s) ZL6-1, ZL6-2, ZL6-3, ZL6-5
ZL7 60 % static preload ? dynamic (1.2 9 10-3/s) ZL7-1, ZL7-2, ZL7-3, ZL7-4
ZL8 Static cyclic (1.0 9 10-6/s) ZL8-1, ZL8-2, ZL8-3, ZL8-4
ZL9 Dynamic cyclic (0.2 9 10-3/s) ZL9-1, ZL9-3, ZL9-7, ZL9-8
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ZL1 set, which means that more fracture energy is
consumed under dynamic loading. The mean j values
1.34 for ZL1 set and 0.96 for ZL2 set reflect this fact.
Note, both mean j values above are from three good
fitting results of each set. And the mean value of j can
be used to defined the analytical stress–strain model
for numerical analysis.
For the tensile tests of higher strain rate, ZL3, ZL4
and ZL5 sets, the descending part of the measured
stress–strain curves fluctuated due to the inertia force
of the mass between the load cell and the tensile
specimen after sudden unloading which caused vibra-
tion of the servo-cylinder of the test system. Hence, the
installation of the load cell should be changed to
eliminate the influence of the inertia force for high
speed tests.
3.2 Results of cyclic tensile tests
System displacement of constant speed was used as
command in the cyclic tensile loading process again.
The speed of motion was determined by reference to
monotonic tensile tests. In static test, the speed was
0.001 mm/s, corresponding to a strain rate about 1 le/
s on the specimens. In dynamic test, the speed was
0.16 mm/s, corresponding to a strain rate about
Table 2 Test results of monotonic tensile
Specimen no. Maximum
strain rate (le/s)
Section average
strain rate (le/s)
Tensile strength (MPa) Eccentricity
at 30 leSpecimen Set average
ZL1-1 2.25 1.34 1.39 1.71 0.01
ZL1-2 1.45 1.22 1.66 0.09
ZL1-3 1.29 0.99 2.05 0.08
ZL1-4 1.16 0.79 1.76 0.05
ZL2-1 368.6 322.1 2.39 2.19 0.12
ZL2-2 518.3 307.9 1.76 0.15
ZL2-3 166.7 129.1 2.75 0.10
ZL2-4 212.0 182.8 1.85 0.10
ZL3-1 1368 1119 1.87 2.22 0.05
ZL3-2 1338 933 2.43 0.10
ZL3-3 1234 960.7 2.74 0.05
ZL3-4 1237 1217 1.84 0.07
ZL4-1 3491 3165 2.41 2.52 0.03
ZL4-2 3404 3178 2.68 0.07
ZL4-3 5642 4005 2.39 0.11
ZL4-6 4106 3571 2.58 0.07
ZL5-2 5638 5356 3.02 2.35 0.08
ZL5-3 11,083 9448 1.83 0.07
ZL5-7 8003 7039 2.59 0.02
ZL5-8 9402 7629 1.95 0.03
ZL6-1 1264 1066 1.68 (40 %) 2.22 (30 %) 0.11
ZL6-2 1187 1057 2.90 (23 %) 0.11
ZL6-3 1168 1099 2.40 (28 %) 0.05
ZL6-5 1692 1325 1.90 (35 %) 0.10
ZL7-1 2090 1215 2.40 (55 %) 2.28 (58 %) 0.10
ZL7-2 1623 1169 2.30 (57 %) 0.08
ZL7-3 1950 1862 2.31 (57 %) 0.06
ZL7-4 1468 1813 2.11 (63 %) 0.05
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200 le/s on the specimens. Similar to the monotonic
test, the measured strain rate may vary for different
specimens.
The preset unloading points of maximum strain
(le) for cyclic tensile test are 60, 80, 100, 150, 200,
300, 500, 700, 900, 1200, 1500, 2000 and until
fracture. If the constantly detected maximum strain
among 16 strain gage signals reaches a preset value,
the system displacement command goes to unload
immediately with the same speed of motion until the
tensile load becomes zero. Then reloading starts again
toward the next preset maximum strain point. Using
the same speed of motion for both loading and
unloading can simulate better the dynamic deforma-
tion of the structure generated by vibration at a certain
frequency. However, the drawback is that the crack
may develop quickly on the descending part of stress–
strain curves, where the crack may become unstable.
Figure 5 gives the time histories of the measured
system displacement and the maximum strain of
specimen ZL8-1 under static cyclic tensile load.
Figure 6 gives the stress–strain curves of two speci-
mens. It is can be seen in Fig. 5 that the unloading
points measured follow those preset quite well, the
whole process went very smoothly. This is true for
another five specimens as well, referring to Fig. 6. In
the post-peak softening region, the peaks of the system
displacement grew very slowly or even decreased
although the peaks of maximum strain increased from
one cycle to another. This phenomenon reflects the
fact that after the peak strength, the strains in the
section of the crack increased sharply, while the
strains on the zones away from the fracture decreased
quickly [16]. Due to a small high frequency tremor on
the system during the ZL8-1 and ZL8-2 tests, the
recorded stresses show an evident noise in Fig. 6, but
its influence is insignificant, in the subsequent tests the
tremor was minimized by the fine-tuning of the test
system.
0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.300
20
40
60
80
100
120
140
1600.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
stra
in ( μ
ε)
time (s)
Average1Average2Average3Average4
ecce
ntric
ity
Fig. 2 Time history of the average strains on four sides of
specimen ZL3-2 and the eccentricity. (Color figure online)
0 200 400 600 800 1000 12000.0
0.4
0.8
1.2
1.6
2.0
2.4
Strain (με)
Calculate3σp =2.05MPa εp= 165 με Et = 25.86 GPa κ = 1.445e-3G_F=0.135 N/mm
Calculate2σp =1.66 MPa εp= 163 με Et = 20.18 GPa κ = 1.1e-3G_F=0.138 N/mm
Stre
ss (M
Pa)
ZL1-2
ZL1-3
Fig. 3 Stress–strain
relationship for static
monotonic tensile tests.
(Color figure online)
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Using the analytical model for monotonic tensile
stress–strain relationship given in Eqs. (2) and (3), the
calculated stress–strain envelop of the measured
curves are plotted in the Fig. 6 together, with the
parameters displayed in the figure. The analytical
model developed for monotonic tension seems to be
valid for the stress–strain envelop curve subjected to
cyclic tension.
Dynamic cyclic tensile tests were not so successful
as the static ones. Figure 7 gives the stress–strain
curves of ZL9-1 and ZL9-7 specimens. Because of
high speed of motion and the intrinsic time delay of the
test machine system between the command and real
motion, the crack increased too fast to be controlled to
follow the preset unloading points soon after the peak
strength of the specimen. However, after sudden
increase in maximum strain, the cyclic loading
continued quite smoothly as preset until the maximum
strain reached 1600 le for ZL9-1 and ZL9-7. As
shown in Fig. 7, some inner loops were formed in the
stress–strain curves. These inattentive results are
helpful to define the partial unloading–reloading loops
under the stress–strain envelope.
The average tensile strength under static cyclic
loading is 1.54 MPa, 9.8 % lower than that under
monotonic loading. This reflects the effect of damage
accumulation under cyclic tension loading. The mean
value of parameter j for the results of static cyclic
tension is smaller than monotonic tension, implying
more fracture energy is consumed when subjected to
cyclic load. The variance of j for different specimens
is larger than corresponding monotonic results. How-
ever, it is more reasonable to attribute this to the big
difference in the composition near the crack section of
specimen, as shown in Fig. 8.
The average tensile strength under dynamic cyclic
loading is 2.17 MPa by four specimens of set ZL9,
41 % higher than the average tensile strength under
static cyclic loading, and almost equal to ZL2 set
under monotonic loading. Unlike the results of static
0 200 400 600 800 1000 1200 14000.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
ZL2-1
ZL2-4
ZL2-3
Strain (με)
Calculate4σp =1.85 MPa εp= 145 με Et = 26.50 GPa κ = 1.2e-3G_F=0.133 N/mm
Calculate3σp =2.75 MPa εp= 206 με Et = 29.16 GPa κ = 1.34e-3G_F=0.177 N/mm
Calculate2σp =1.76 MPa εp= 242.6 με Et = 21.65 GPa
κ = 0.7e-3G_F=0.157 N/mm
Stre
ss (M
Pa)
Calculate1σp =2.39 MPa εp= 191 με Et = 25.26 GPa κ = 0.99e-3G_F=0.225 N/mm
ZL2-2
Fig. 4 Stress–strain
relationship for dynamic
monotonic tensile tests.
(Color figure online)
0 200 400 600 800 1000 1200 14000.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0
200
400
600
800
1000
1200
1400
1600
1800
disp
(mm)
time (s)
System displacementMaxStrain
stra
in (
με)
Fig. 5 Time histories of system displacement and maximum
strain for ZL8-1. (Color figure online)
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cyclic tests, the effect of damage accumulation is not
so evident.
Now turn to the unloading and reloading loops. It
can be observed from the results of static cyclic tests
that when the specimen unloaded to zero stress from a
certain strain level, the unloading stress–strain curve is
concave from the unloading point and characterized
by high stiffness at the beginning (Fig. 6), except for
very few cycles just before the sudden fail of the strain
gages. The stiffness gradually decreased and becomes
rather flat at low stress levels and the residual strains
are reduced. When reloading is imposed from zero
stress up to a strain level higher, the reloading curve is
convex from the starting points by high stiffness
(Fig. 6). The reloading curve gradually becomes flat
and always intersects with the unloading curve at a
smaller stress than former unloading start point, and
then approaches to the stress–strain envelope curve
0 200 400 600 800 1000 1200 1400 1600 18000.0
0.4
0.8
1.2
1.6
2.0
ZL8-2
Calculate2σp =1.33 MPa εp= 207 με Et = 16.4 GPa
κ = 0.72e-3
Calculate4σp =1.49 MPa εp= 160 με Et = 16.40 GPa
κ = 1.20e-3
Strain (με)
Stre
ss (M
Pa)
ZL8-4
Fig. 6 Stress–strain
relationship for static cyclic
tensile tests. (Color
figure online)
-0.2
0.0
0.5
1.0
1.5
2.0
0 200 400 600 800 1000 1200 1400
Strain (με)
Stre
ss (M
Pa)
ZL9-1
ZL9-7
Fig. 7 Stress–strain relationship for dynamic cyclic tensile tests. (Color figure online)
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again as the strain increases. Flat turning on the
reloading path, soon after the intersecting points in
post peak tensile strength region, indicates the expan-
sion of the damage in the specimens which makes the
stress decrease.
Energy dissipation in unload-reload loops is impor-
tant for the prediction of the responses of concrete
structures under strong seismic shaking. Proper ana-
lytical expression of the shape of the unloading and
reloading loops is critical to simulate both the damage
accumulation and the energy dissipation of the mate-
rial due to cyclic loading in numerical analysis.
The shape of the unloading and reloading curves
depends on the amount of non-recoverable damage in
the concrete. Like in many models for the compressive
properties of the concrete [21], unloading strain eun is
considered herein as the parameter that defines the
unloading and reloading path and determines the
residual strain. The expression for unloading path
started from the point (eun, run) on the stress–strain
envelope is:
r ¼ run 1 � eun � eeun � ere
� �B1" #
ð4Þ
where ere is the residual strain when the specimen is
unloaded to zero stress, and exponent B1 determines
the curvature of the unloading curve. Both values are
supposed to be dependent on unloading strain eun only.
Relationship between ere and eun was obtained by
statistical regression on experimental results of ZL8
and ZL9 sets, as shown in Fig. 9. Relationship
between B1 and eun was obtained by statistical
Fig. 8 Photos of the crack surface of specimens of ZL8 set. (Color figure online)
0 200 400 600 800 1000 1200 1400 1600 18000
200
400
600
800
1000
1200
1400
1600
εre =-21.9+0.35 εun+0.00024 ε 2un ε re>0, εun<1066
εre = -319.5 + 0.885 εun, εun>=1066
Res
idua
l stra
in (
με)
Unload strain (με)
ZL8-1ZL8-2ZL8-3ZL8-4ZL9-1ZL9-4ZL9-7ZL9-8 regression
Fig. 9 Statistical regression of residual strain to unloading
strain. (Color figure online)
0 200 400 600 800 1000 1200 14000.4
0.5
0.6
0.7
0.8
0.9
1.0
ZL8-1ZL8-2ZL8-3ZL8-4 regression
Par
amet
er B
1
Unload strain (με)
B1=0.65+0.31*exp(-εun /260)
Fig. 10 Statistical regression of B1 to unloading strain. (Color
figure online)
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regression as well, as given in Fig. 10. The so-called
experimental B1 is the value by which the analytical
unloading curve defined by Eq. 4 best fits the exper-
imental result for every unloading path.
The expression for reloading path started from zero
stress point (ere, 0) up to a tensile strain level higher is
written as follows.
r ¼ rcrun
e� ere
eun � ere
� �B2
ð5Þ
rcrun=run is defined as the stress drop here and supposed
to depend on the unloading strain eun only.
Furthermore, upon the statistical analysis of the
experimental results, the strain at intersecting point
between unloading and reloading curve is about 0.99
times of eun, therefore, in Eq. 5 eun is used instead. The
statistical regression of the stress drop rcrun=run to
unloading strain eun is given in Fig. 11.
With the measured residual strain and calculated
stress drop rcrun=run, B2 was determined for every
reloading path measured so that the analytical reload-
ing curve defined by Eq. 5 best fits the experimental
result. Similar to parameter B1 for unloading path, the
relationship between B2 and eun was obtained by
statistical regression again, refer to Fig. 12.
From Fig. 7, the experimental partial unloading–
reloading loops of the stress–strain curves under the
stress–strain envelope provide us a good intimation to
use the similar analytical expressions for the unload-
ing and reloading path at any inner point (ein; rin). The
Eqs. 4 and 5 for unloading and reloading path,
respectively, are rewritten as
r ¼ rin 1 � ein � eein � ere
� �B1" #
; for unloading ð6Þ
r ¼ rin þ rcrun � rin
� � e� ein
eun � ein
� �B2
; for reloading
ð7Þ
where eun is the last unloading strain started on the
envelope, and ere, rcrun=run, B1 and B2 are determined
by the eun as above.
Figure 13 gives a calculated stress–strain path for
cyclic tensile loading by the analytical model pre-
sented in the paper. Comparing with the tests results,
the simulation is a satisfactory approximation.
0 200 400 600 800 1000 1200 14000.80
0.85
0.90
0.95
1.00
σ crun /σun= 0.89+0.11*exp(-εun /125)
ZL8-1ZL8-2ZL8-3ZL8-4ZL9-1ZL9-7ZL9-8 regression
Stre
ss d
rop
Unload strain (με)
Fig. 11 Statistical regression of the stress drop to unloading
strain. (Color figure online)
0 200 400 600 800 1000 1200 14000.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
B2=0.42+0.48*exp(-εun/380)
ZL8-1ZL8-2ZL8-3ZL8-4 regression
Par
amet
er B
2
Unload strain (με)
Fig. 12 Statistical regression of B2 to unloading strain. (Color
figure online)
0 400 800 1200 1600 20000.0
0.5
1.0
1.5
2.0
2.5
σp =2.32 MPa εp= 180 με Et = 24.9 GPa
κ = 0.9e-3
Stre
ss (M
Pa)
Strain(με)
Fig. 13 Simulation of stress–strain path under cyclic tensile
loading
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4 Conclusions
Direct tensile test of mass concrete with Shapai dam
cores under static and dynamic loads were performed.
The device and procedure presented in this paper for
the installation of tensile cylindrical specimens
worked well to reach reliable test results with high
success ratio. The tensile stress–strain relationships
were well caught with cylindrical specimens from dam
cores without notch. The tests results from the
specimens without notch can better reveal the prop-
erties of mass concrete with large size aggregates.
Dynamic direct tensile tests indicate a significant
increase in strength compared with static ones, from
28 to 47 % under strain rate roughly from 10-4 to 10-2
per second. And more fracture energy is consumed by
the concrete under dynamic or cyclic loading than
static monotonic loading. The static preload on the
specimens shows little influence on their dynamic
tensile strengths.
Based on Gopalaratnam’s proposal, a simple ana-
lytical model was developed to express the stress–
strain relationships of mass concrete under cyclic
loadings in tension, especially post peak tensile
strength. Energy dissipation in unload-reload loops
was well reflected in the present model. The model can
reproduce the complex behavior of mass concrete
under any history of uniaxial tensile cyclic loading.
Dynamic cyclic tensile tests were not so successful
as the static ones, therefore the control procedure need
to be improved further.
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