COMPLETE TRIAXIAL STRESS-STRAIN CURVES OF

HIGH-STRENGTH CONCRETE
By D. C. Candappa,1 J. G. Sanjayan,2 and S. Setunge3

ABSTRACT: The axial-stress–axial-strain and axial-stress–lateral-strain behavior of concrete under active lateral
confinement is presented. The uniaxial strengths investigated are 41.9 MPa, 60.6 MPa, 73.1 MPa, and 103.3
MPa. The confining pressures (␴3) used are 4 MPa, 8 MPa, and 12 MPa. Details of an economical lateral-strain-
measuring device used in this investigation to produce accurate and repeatable measurements are presented. For
low levels of confinement, the constant in the Mohr-Coulomb failure criterion (k) is shown to be closer to five
than the traditional value of four. The axial strain at peak stress is shown to have a strong linear relationship
with the level of confinement. Parameter values are suggested for Ottosen’s constitutive model based on nonlinear
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elasticity. It was discovered that descending portion of the nonlinearity index (␤) versus the secant value of
Poisson’s ratio (␯a) was independent of the uniaxial strength and the level of confinement. A simple model is
proposed for the descending portion of the ␤ versus ␯a curves.

INTRODUCTION data. This research aims to increase the almost nonexistent

database of complete stress-strain curves for high-strength
Although the compressive strength of concrete ( f ⬘) c is the concrete under active lateral confinement.
key property employed in the design of reinforced and pre- A previous paper by the writers (Candappa et al. 1999) pub-
stressed concrete elements, a number of practical design situ- lished complete stress-strain curves for 60 MPa and 100 MPa
ations require the prediction of concrete behavior under mul- concretes. This paper includes previously unpublished stress-
tiaxial stress states. For example, spiral columns, containment strain curves for 40 MPa and 75 MPa concretes.
vessels and shells of revolution (Ansari and Li 1998).
For quite some time now, researchers have been insisting EXPERIMENTAL PROGRAM
that knowledge of the complete (ascending and descending)
stress-strain behavior of concrete is essential for accurate con- Triaxial Cell
stitutive modeling (Ahmad and Shah 1982; Imran and Panta- The triaxial cell used in the experiments is shown in Fig. 1.
zopoulou 1996; Chin et al. 1997; Candappa et al. 1998). While The main component is a pressure cell that accommodates a
the ascending portion of the axial-stress–axial-strain curve 100 mm ⫻ 200 mm cylindrical specimen. The required con-
provides key material parameters such as the Young’s modu- fining pressure was applied using oil through a flexible poly-
lus, the descending or softening portion gives an indication of urethane membrane. As shown in Fig. 1, a bleed hole and an
the ductility of concrete (Attard and Setunge 1996). It is also oil inlet was provided in the middle of the cell body. The
important to know the complete axial-stress–lateral-strain be- compressive load was applied to the specimen using two
havior. This knowledge is needed to understand the volumetric spherically seated cylindrical loading blocks that were de-
behavior of concrete and hence to predict the level of passive signed to fit either end of the cell with a clearance of 1.3 mm.
confinement provided by lateral reinforcements (Candappa et The specimen to be tested was enclosed in a membrane that
al. 1996). had the confining pressure applied to it. The membrane was
However, since about the 1980s to present, researchers have made out of flexible polyurethane and had a thickness of 2
been lamenting the fact that there is a severe lack of such mm. The required confining pressure was applied using an
complete stress-strain data [e.g., van Mier (1986), Smith et al. Enerpac hydraulic hand pump. Also, a pressure gauge was
(1989), Dahl (1992a), Imran and Pantazopoulou (1996) and connected in series in order to measure the pressure being
Candappa et al. (1999)]. In the case of high-strength concrete applied. The pressure gauge used was a PDCR 610 (Druck
(strengths over 50 MPa), this lack of stress-strain data becomes Ltd., Leicester, U.K.) and it required a 10-V exciter.
even more pronounced. This is because most of the experi-
mental work to date has involved the testing of normal Measurement of Axial Displacement
strength concrete (Ansari and Li 1998).
Linear variable displacement transducers (LVDTs) were
RESEARCH SIGNIFICANCE used to measure the longitudinal or axial deformation. As
LVDTs measure the overall displacement, adopting this
As high-strength concrete is currently being used exten- method requiring accounting for the flexibility of the loading
sively in many parts of the world, there is an urgent need to components. A calibration was done using an aluminum cyl-
obtain experimental data of its ‘‘complete’’ constitutive behav- inder that had a yield stress higher than the compressive
ior, in order to calibrate and/or validate the many constitutive stresses encountered in the testing of concrete. The ratio be-
models, most of which are based on normal-strength concrete tween the LVDT readings and the strain gauges fitted onto the
1
aluminum cylinders was used as the correction factor for the
PhD Student, Dept. of Civ. Engrg., Monash Univ., Clayton Campus,
Victoria 3168, Australia.
LVDT readings.
2
Sr. Lect., Dept. of Civ. Engrg., Monash Univ., Clayton Campus, Vic- Further, a number of concrete specimens were fitted with
toria 3168, Australia. two vertical strain gauges to verify the correction procedure.
3
Sr. Lect., Monash Univ., Churchill, Victoria 3842, Australia. The corrected LVDT readings matched the average strain
Note. Associate Editor: Nemkumar Banthia. Discussion open until No- gauge values quite closely (until approximately peak stress,
vember 1, 2001. To extend the closing date one month, a written request when the strain gauges failed). The lines of fit between the
must be filed with the ASCE Manager of Journals. The manuscript for
this paper was submitted for review and possible publication on July 13,
LVDT readings and strain gauge readings of the concrete spec-
1999; revised June 9, 2000. This paper is part of the Journal of Materials imens typically produced slopes between 0.98 and 0.97 (a ex-
in Civil Engineering, Vol. 13, No. 3, May/June, 2001. 䉷ASCE, ISSN act match would produce a slope of 1.0) with R2 values of
0899-1561/01/0003-0209–0215/$8.00 ⫹ $.50 per page. Paper No. 21396. 0.999.
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FIG. 2. Lateral Expansion Measuring System

hence any movement of the arms were recorded into a data

logger.
The accuracy of the lateral strain measurements obtained by
the new device was verified using strain gauges (until the
strain gauges failed, which was usually around the peak
strength). Except at small strains, when the device needed to
overcome friction between the studs and the piano wire, there
was good agreement between the average strain gauge read-
ings and the readings of the new apparatus. The repeatability
of the device was checked by carrying out two tests at each
FIG. 1. Triaxial Cell uniaxial strength/confining pressure combination and found to
give repeatable measurements.
Measurement of Lateral Displacement
Details of Concrete Mixes
Strain gauges of length 70 mm were used to measure the
The 60-MPa and 100-MPa strengths basically represent the
radial or lateral strain. However, as expected, preliminary tests
lower and upper ends of the high-strength concrete spectrum.
showed that the strain gauges malfunctioned during the de-
scending portion of the stress-strain curve due to the large
TABLE 1. Uniaxial Strengths
magnitude of the postpeak lateral strains and cracking of the
concrete specimen. Therefore, a dual measuring system was Concrete type U40 U60 U75 U100
incorporated to measure the lateral deformation. The strain 28-day strength (MPa) 32.6 49.7 62.0 86.4
gauges were used to measure the ascending portion of the Strength at testing (MPa) 41.9 60.6 73.1 103.3
stress-strain curve and a special device, shown in Fig. 2, was Age at testing (days) 216 56 405 90
built to measure the large lateral strains along the descending
portion of the curves. To ensure that the flexible membrane
TABLE 2. Concrete Mix Proportions
surrounding the concrete specimen did not influence lateral
strain measurements, ‘‘direct contact’’ with the specimen was Mix type U40 U60 U75 U100
achieved by drilling 10 holes half way up the membrane and Cement (kg/m3) 520 360 400 550
fitting the holes with studs. A sealant was used to ensure that Water (kg/m3) 330 180 180 165
no fluid reached the specimen. A groove was cut on the outside Coarse aggregate (kg/m3) 2550 1130 1310 1340
of the studs and a thin wire (piano wire) was placed in these Fine aggregate (kg/m3) 1370 830 630 520
grooves with the ends connected to the two arms of the clip Superplasticizer (L/m3) — — 2.2 8.7
W/C ratio 0.63 0.5 0.45 0.3
gauge as shown in Fig. 2. The arms of the clip gauge were Target slump (mm) 150 100 100 150
fitted with four strain gauges forming a Wheatstone bridge and
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J. Mater. Civ. Eng. 2001.13:209-215.

 The 40-MPa set was included to aid comparisons with normal- EXPERIMENTAL RESULTS
strength concretes. The concrete specimens were cured for 28
days in standard bath conditions (23⬚C) and allowed to air dry The complete axial and lateral strains obtained for concrete
for at least four weeks to stabilize the strength. This condi- under compressive axial stresses are shown in Figs. 3–6. Re-
tioning process has an effect of increasing the uniaxial strength sults of uniaxial tests are also presented marked with 0 MPa
of the concrete. Table 1 gives the 28-day strengths and the lateral confinement on the figures. These tests were performed
strength/age at the time of testing. The mix proportions used without the use of triaxial apparatus as no confining pressure
are given in Table 2. The concrete batches were named U40, was required.
U60, U75, and U100, with ‘‘U’’ representing uniaxial and the
numbers representing the target test strengths. The type of MOHR-COULOMB FAILURE CRITERION
coarse aggregate used was crushed basalt with a maximum-
The Mohr-Coulomb failure criterion is the classical failure
size aggregate of 14 mm. The type of superplasticizer used
criterion used in many applications, and assumes only two
was Rheobuild 1000 (MBT PTY Ltd., Melbourne, Australia).
different failure modes: the sliding failure and the separation
Specimen Preparation failure. This failure criterion is still used extensively due to its
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simplicity and relatively good accuracy. It is usually written

The size of the cylindrical specimens were 98 mm ⫻ 200 in the form:
mm (98-mm diameter specimens were used rather than the
standard 100 mm to enable ease of removal of tested samples). ␴1P = f ⬘c ⫹ k␴3 (1)
In order to achieve a diameter of 98 mm, a 1-mm-thick sheet
of plastic sleeve was inserted in the mold. The specimens were where ␴1P = peak axial stress; f ⬘c = uniaxial strength ( f ⬘c > 0);
demolded 24 h after casting and were bath cured at 23⬚C. The ␴3 = lateral confining pressure (␴3 > 0); and k = a constant,
cylinders were taken out at 28 days and ground at both ends usually assumed to be 4.
before testing. To prevent membrane damage, the specimen Eq. (1) may be normalized as shown
pores were filled with casting plaster. ␴1p ␴3
=1⫹k (2)
Testing Procedure f c⬘ f ⬘c
The confining pressures used were 4 MPa, 8 MPa, and 12 The constant k can only be found using triaxial tests. The
MPa. Two specimens were tested at each confining pressure value of k commonly used is 4. However, it was found that
for each strength. Two lateral strain gauges and two longitu- the best-fit value of k for the peak stresses obtained from the
dinal gauges were attached on the middle third of the speci- test results was 5.3. (Fig. 7). It should be noted that only low
men. The longitudinal gauges were placed diametrically op- confinements were investigated. The low confinement (␴3/f ⬘c <
posite each other as were the lateral gauges. 0.2) results of Dahl (1992b), Xie et al. (1995) and Attard and
Once the triaxial cell was placed in the stiff, servocontrolled Setunge (1996) are shown in Fig. 8. Again, k = 5.3 better
compression testing machine (Amsler 5,000 kN), a small ver- predicted the observed peak stress values. Therefore, for low
tical load was applied, to ensure that the cell was secure, be- confinement levels, it appears that a value of k = 5 (a rounded
fore applying the confining pressure. The testing machine was off value of 5.3) is more suitable than the traditional value of
then set to compressive displacement control at the rate of 2.5 k = 4.
mm vertical displacement every 10 min. The tests were con- However, this increased value of k cannot be extended to
tinued until the postpeak behavior of the axial-stress–axial- higher levels of confinement. Dahl (1992b) found that even
strain curves was well defined, thus giving an indication of the lower of k = 4 commenced overpredicting the peak stresses
the ductility of the specimen. once the level of confinement exceeded 0.5. Also, for high

FIG. 3. Stress Strain Curves for U40

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FIG. 4. Stress Strain Curves for U60

FIG. 5. Stress Strain Curves for U75

levels of confinement (up to 1.0), Ansari and Li (1998) found was R2 = 0.95. In (3), f c⬘ = uniaxial strength; ␴3 = confining
the best fit value of k to be as low as 2.6. pressure; ε uc = axial strain at peak stress in uniaxial compres-
sion; and ε u1 = axial strain at peak stress in triaxial compres-
AXIAL STRAIN AT PEAK STRESS sion. Using (3), providing a level of confinement of 0.1 will
result in the failure strain increasing by threefold.
Fig. 9 shows the axial strain at peak stress ratio plotted A similar plot was produced for high-strength concrete un-
against the level of confinement (confining pressure/uniaxial der triaxial compression by Ansari and Li (1998). They ob-
strength). It shows that the axial strain at peak stress increases tained
linearly with the level of confinement, regardless of the uni-
axial strength of the concrete. The correlation for the relation-
ship
εu1
ε uc
= 1 ⫹ 15.15 冉冊
␴3
f ⬘c
(4)

冉冊
with a slightly lower correlation of R2 = 0.93. Ansari and Li
ε u1 ␴3 (1998) predicted that providing a level of confinement of 0.1
= 1 ⫹ 20 (3)
ε uc f ⬘c will produce a 2.5-fold increase in the failure strain, which is
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FIG. 6. Stress Strain Curves for U100

Young’s modulus and Poisson’s ratio are changed appropri-

ately. The Ottosen (1979) model was given prominence when
it was adopted by CEB (Comité Euro-International du Béton,
Paris) to be included in the CEB Model Code 1990 [cited in
Dahl (1992a)]. The basic model is described by
␴1 ⫺ ␯ a(␴2 ⫹ ␴3)
ε1 = (5a)
ES
␴2 ⫺ ␯ a(␴1 ⫹ ␴3)
ε2 = (5b)
ES
␴3 ⫺ ␯ a(␴1 ⫹ ␴2)
ε3 = (5c)
ES
FIG. 7. Mohr-Coulomb Failure Criterion (Writers’ Triaxial Test Re- where ε1 = axial strain and ε2 = ε3 = lateral strain (for triaxial
sults) tests); ␴1 = axial stress and ␴2 = ␴3 = confining pressure; ␯ a
= secant value of Poisson’s ratio; and ES = secant Young’s
modulus.
Ottosen (1979) proposed the following model for the vari-
ation of ␯ a with the axial stress, until failure:
␯ a = ␯ ia when ␤ ⱕ ␤1 (6a)

␯ a = ␯ af ⫺ (␯ af ⫺ ␯ ai ) 冑 冉
1⫺
␤ ⫺ ␤1
1 ⫺ ␤1
冊
2

when ␤ > ␤1 (6b)

FIG. 8. Mohr-Coulomb Failure Criterion (Previous Triaxial Test Re-

sults)

15% lower than what was observed in the current research

project.

PARAMETER VALUES FOR OTTOSEN’S

CONSTITUTIVE MODEL
Ottosen (1979) proposed a constitutive model for concrete
based on nonlinear elasticity, where the secant value of the FIG. 9. Axial Strain at Peak Stress versus Level of Confinement

JOURNAL OF MATERIALS IN CIVIL ENGINEERING / MAY/JUNE 2001 / 213

J. Mater. Civ. Eng. 2001.13:209-215.

 where ␤ = nonlinearity index, defined as the actual stress state
in relation to the failure stress (␴1/␴1p). The variable ␤1 is the
stress point at which ␯ a deviates from a constant value. The
symbol ␯ ai is the initial Poisson’s ratio and ␯ af is the secant
value of Poisson’s ratio at peak stress. Eq. (6) and its symbols
are shown in Fig. 10. Ottosen (1979) lamented the fact that
only little was known about the increase of ␯ a in the postpeak
region. Although Dahl (1992a) stated that postpeak behavior
is indeed important, his experimental work did not extend into
the postpeak region. However, as this research included post-
peak behavior, it was possible to plot complete ␤ versus ␯ a
curves for the 40-MPa, 60-MPa, 75-MPa, and 100-MPa con-
cretes. Figs. 11 and 12 show the complete ␤ versus ␯ a curves
for 40-MPa and 100-MPa concretes. The curves for the other
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concretes (60 MPa and 75 MPa) are also of similar shape and FIG. 13. Descending Portions of ␤ versus ␯ a Graphs
therefore not shown here.
It was found that ␤1 = 0.7 for the 40 MPa concrete but was For all strengths, ␯ af was found to be approximately 0.5,
approximately 0.8 for the 60-MPa, 75-MPa, and 100-MPa which was the same value proposed by Dahl (1992a) for tri-
concretes. This compared well with Ottosen’s (1979) proposed axial tests. The value of 0.36 suggested by Ottosen (1979)
value of 0.8 and Dahl’s (1992a) value of 0.6. appears to be somewhat low.
With regards to ␯ ai , Dahl (1992a) suggested a constant value
of 0.15 for triaxial tests. However, Dahl did state that increas-
ing the concrete strength will result in an increase in ␯ ai . Ot-
tosen (1979) used different ␯ ai values but did not state any
reasons for doing so. In this research, it was found that the
following equation closely matched (R2 = 0.995) the ␯ ai values
for the triaxial tests:

␯ ai = 8 ⫻ 10⫺6( f ⬘)
c
2
⫹ 0.0002 f ⬘c ⫹ 0.138 (7)

Descending Portion of ␤ versus ␯ a Graph

When modeling the descending portion, it was discovered
that the descending curves were approximately the same, re-
FIG. 10. Ottosen’s Model for Variation of ␤ versus ␯ a gardless of the uniaxial strength of the concrete or the level
of lateral confinement being applied. It was found that (8)
modeled all the descending curves with a reasonable degree
of accuracy (R2 = 0.82). The modeling was restricted to a
maximum ␯ a value of 1

␤D = ⫺0.5(␯ a)2 ⫹ 0.45␯ a ⫹ 0.9 (8)

where ␤D = Beta descending portion.

The descending portions of all the ␤ versus ␯ a curves, to-
gether with the predicted descending portion, is shown in Fig.
13. In Figs. 11 and 12, the descending portions were predicted
by incrementally increasing the value of ␯ a (by 0.02) and
hence predicting the value of ␤ using (8).
The predicted complete ␤ versus ␯ a curves, which incor-
porates the proposed values for ␯ ai , ␯ af , ␤1, and ␤D are also
FIG. 11. ␤ versus ␯ a (U40)
shown in Figs. 11 and 12. Figs. 11 and 12 show that the com-
plete ␤ versus ␯ a curves were very closely predicted.

CONCLUSIONS

• This paper described an innovative and economical lat-

eral-strain-measuring device developed at Monash Uni-
versity. The device measures the lateral strains in concrete
under active triaxial confinement, which traditionally have
been difficult to measure. The device was observed to
produce accurate and repeatable measurements.
• The constant in the Mohr-Coulomb failure criterion (k)
was shown to significantly exceed the usual value of 4.
The results indicate that a k value of 5 would better pre-
dict the peak stresses at low confinements (less than 0.2).
• The axial strain at peak stress was shown to have a strong
FIG. 12. ␤ versus ␯ a (U100) linear relationship with the level of confinement.
214 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING / MAY/JUNE 2001

J. Mater. Civ. Eng. 2001.13:209-215.

 • Parameter values were suggested for Ottosen’s (1979) Imran, I., and Pantazopoulou, S. J. (1996). ‘‘Experimental study of plain
constitutive model based on nonlinear elasticity. concrete under triaxial stress.’’ ACI Mat. J., 93(6), 589–601.
Ottosen, N. S. (1979). ‘‘Constitutive model for short-time loading of con-
• For low levels of confinement, the descending portion of crete.’’ J. Engrg. Mech. Div., ASCE, 105(2), 127–141.
the nonlinearity index (␤) versus the secant value of Pois- Smith, S. S., William, K. J., Gerstle, K. H., and Sture, S. (1989). ‘‘Con-
son’s ratio (␯ a) was shown to be independent of the uni- crete over the top, or: Is there life after peak?’’ ACI Mat. J., 86(5),
axial strength and the level of confinement. A simple 491–497.
model was proposed for the descending portion. van Mier, J. G. M. (1986). ‘‘Fracture of concrete under complex stress.’’
• Currently at Monash University, the results of this re- HERON, 31(3), 1–90.
Xie, J., Elwi, A. E., and MacGregor, J. G. (1995). ‘‘Mechanical properties
search project are being used to develop a constitutive of three high-strength concretes containing silica fume.’’ ACI Mat. J.,
model for concrete based on Ottosen’s (1979) nonlinear 92(2), 135–145.
elastic model. This model will be the focus of a future
publication. NOTATION
The following symbols are used in this paper:
REFERENCES
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ES = secant value of Young’s modulus;

Ahmad, S. H., and Shah, S. P. (1982). ‘‘Stress-strain curves of concrete
f ⬘c = uniaxial compressive strength of concrete ( f ⬘c > 0);
by spiral reinforcement.’’ ACI J., 79(46), 484–490. k = constant in Mohr-Coulomb failure criterion;
Ansari, F., and Li, Q. (1998). ‘‘High-strength concrete subjected to triaxial ␤ = nonlinearity index;
compression.’’ ACI Mat. J., 95(6), 747–755. ␤D = nonlinearity index, descending portion;
Attard, M. M., and Setunge, S. (1996). ‘‘Stress-strain relationship of con- ε uc = axial strain at peak stress in uniaxial compression;
fined and unconfined concrete.’’ ACI Mat. J., 93(5), 432–442. ε u1 = axial strain at peak stress in triaxial compression;
Candappa, D. P., Sanjayan, J. G., and Setunge, S. (1996). ‘‘Behavior of ε1, ε2, ε3 = principal strains (compression positive);
high performance concrete under lateral confinement.’’ Proc., 21st ␯ = Poisson’s ratio;
Conf. on Our World in Concrete and Struct., CI-Premier, Singapore, ␯a = secant value of Poisson’s ratio;
77–83. ␯ ai , ␯ af , ␤1 = parameters in Eq. (6);
Candappa, D. P., Setunge, S., and Sanjayan, J. G. (1998). ‘‘Experimental
␴1, ␴2, ␴3 = principal stresses (␴1 ⱖ ␴2 ⱖ ␴3, compression pos-
stress-strain curves for high strength concrete under triaxial confine-
ment.’’ Proc., Int. Conf. on High Perf. High Strength Concrete, Curtin
itive); and
University of Technology, Perth, Australia, 285–297. ␴1p = peak axial stress.
Candappa, D. P., Setunge, S., and Sanjayan, J. G. (1999). ‘‘Stress versus
strain relationship of high strength concrete under lateral confinement.’’ Subscripts
Cement and Concrete Res., 29(12), 1977–1982. a = apparent;
Chin, M. S., Mansur, M. A., and Wee, T. H. (1997). ‘‘Effects of shape, c = compression;
size, and casting direction of specimens on stress-strain curves of high- D = descending;
strength concrete.’’ ACI Mat. J., 94(3), 209–219.
f = final;
Dahl, K. K. B. (1992a). ‘‘A constitutive model for normal and high
strength concrete.’’ Project 5, Rep. 5.7, American Concrete Institute, i = initial;
Detroit. p = peak;
Dahl, K. K. B. (1992b). ‘‘A failure criterion for normal and high strength s = secant; and
concrete.’’ Project 5, Rep. 5.6, American Concrete Institute, Detroit. u = ultimate.

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