ANCOLD · RC frame structures on the top of the intake tower and the house connected to an elevator...

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: wanghb@iwhr.com

Materials and Structures (2017) 50:44

DOI 10.1617/s11527-016-0901-x

Author's personal copy

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

44 Page 2 of 11 Materials and Structures (2017) 50:44

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

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

��

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)

Materials and Structures (2017) 50:44 Page 3 of 11 44

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" #

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

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

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

1600.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

in ( μ

time (s)

Average1Average2Average3Average4

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

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

Fig. 3 Stress–strain

relationship for static

monotonic tensile tests.

(Color figure online)

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

Strain (με)

Calculate2σp =1.76 MPa εp= 242.6 με Et = 21.65 GPa

κ = 0.7e-3G_F=0.157 N/mm

relationship for dynamic

monotonic tensile tests.

(Color figure online)

0 200 400 600 800 1000 1200 14000.00

time (s)

System displacementMaxStrain

Fig. 5 Time histories of system displacement and maximum

strain for ZL8-1. (Color figure online)

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

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)

relationship for static cyclic

tensile tests. (Color

figure online)

0 200 400 600 800 1000 1200 1400

Strain (με)

Fig. 7 Stress–strain relationship for dynamic cyclic tensile tests. (Color figure online)

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" #

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

εre =-21.9+0.35 εun+0.00024 ε 2un ε re>0, εun<1066

εre = -319.5 + 0.885 εun, εun>=1066

l stra

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

ZL8-1ZL8-2ZL8-3ZL8-4 regression

B1=0.65+0.31*exp(-εun /260)

Fig. 10 Statistical regression of B1 to unloading strain. (Color

figure online)

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

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

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

σ crun /σun= 0.89+0.11*exp(-εun /125)

ZL8-1ZL8-2ZL8-3ZL8-4ZL9-1ZL9-7ZL9-8 regression

Fig. 11 Statistical regression of the stress drop to unloading

strain. (Color figure online)

0 200 400 600 800 1000 1200 14000.3

B2=0.42+0.48*exp(-εun/380)

ZL8-1ZL8-2ZL8-3ZL8-4 regression

Fig. 12 Statistical regression of B2 to unloading strain. (Color

figure online)

0 400 800 1200 1600 20000.0

σp =2.32 MPa εp= 180 με Et = 24.9 GPa

κ = 0.9e-3

Strain(με)

Fig. 13 Simulation of stress–strain path under cyclic tensile

loading

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.

References

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compacted concrete dams; Appendix E, tensile strength of

roller compacted concrete. Engineer pamphlet 1110-2-12,

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2. U.S. Army Corps of Engineers (1995) Gravity dam design,

EM 1110-2-2200

3. U.S. Army Corps of Engineers (2003) Arch dam design, EM

1110-2-2201, 2003

4. U.S. Bureau of Reclamation (1974) Design criteria for

concrete arch and gravity dams

5. U.S. Bureau of Reclamation (1992) Concrete manual, part

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electr Eng 28(5):92–96 (in Chinese)13. Zhang C, Yu Y, Chen H (2014) Study on earthquake ground

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