ACI MATERIALS JOURNAL
TECHNICAL PAPER
Title no. 93-M67
Experimental Study of Plain Concrete under Triaxial Stress
by I. Imran and S. J. Pantazopoulou
The mechanical behavior of plain concrete under triaxial stress
states was investigated experimentally to establish the sensitivity of
the mechanical properties to an array of physical and experimental variables, and to enhance the available database of such
tests for the benefit of improved understanding and modeling of the
constitutive behavior of the material. The test program was particularly focused on the deformation aspects of the response and the
process of damage buildup, and consisted of 130 different tests.
Parameters of the experimental program were the porosity of
concrete, the moisture content at the time of testing, and the load
path used in the tests. The most important response index was the
history of damage accumulated in the microstructure, which was
measured in terms of volumetric expansion and plastic deformation of the material. A simple strain-based model was formulated
to represent the experimental trends. It was shown that the expansive strain of the cross section supporting the compressive strut
best organized the experimental data. This result indicates that
expansion due to damage is responsible for the stiffness degradation and the softening of resistance that is observed in concrete
with increasing deformation levels; hence, this is one of the most
important state variables characterizing the constitutive behavior
of the material.
Keywords: damage; deformation; moisture content (of aggregate, hardened
concrete; triaxial stresses.
INTRODUCTION
The response of concrete to triaxial stress is governed by
several material, geometric, and load variables. This parametric complexity is the reason why it is not easy to
adequately describe the mechanical problem in mathematical terms. The greatest challenge to constitutive modeling
of concrete stems from the fact that the array of significant
parameters has not yet been completely isolated, whereas
the available database of tests that can be used as a point of
reference for such developments has a number of limitations: 1) attention in the past was primarily focused on
strength issues, as evidenced by the established practice of
expressing the triaxial response in stress coordinates (e.g.,
yield and failure envelopes), whereas very few complete
experimental records of the deformation/damage characteristics have been reported in the literature; 2) many of the
parameters considered in the experimental studies were not
entirely independent, to the extent that interpretation of the
experimental trends is often tainted by the effects of interaction of these variables.
Recent research results have illustrated that damage in
concrete due to microcracking is manifest by volumetric
expansion of the material (Van Mier, 1986; Smith, 1987;
ACI Structural Journal/November-December 1996
Smith et al., 1989; Imran, 1994; Pantazopoulou and Mills,
1995). The rate of volume change (i.e., volume increment
per unit of initial volume) represents volumetric strain,
which is also defined by the trace (or first invariant) of the
strain tensor. Partial or total restraint against expansion,
which is usually imposed through boundary conditions, has
a profound influence on the internal stress state of the material (Pantazopoulou, 1995; Kosaka, 1984). It has been
proposed that the residual stiffness and strength of concrete
under arbitrary stress can be entirely quantified from the
state of damage (described by strain measures) and from the
kinematic restraints imposed by the boundary conditions
(Pantazopoulou, 1995). Implementation of this concept in
constitutive modeling of concrete relies on the availability of
an extensive database of test results supplemented by
complete deformation records. It was already mentioned that
such records are rare in the literature (Smith, 1987; Smith et
al., 1989; Van Mier, 1986; Imran, 1994). One contributing
factor for the scarcity of data has been the technical difficulty
in obtaining a complete set of credible deformation measurements; usually, the standard triaxial test is conducted by
encasing the specimen in a triaxial testing device, with practically no physical access to it but through the loading mechanism. To date, the most commonly used database that is
complete with regards to deformation characteristics has
been developed at the University of Colorado (Scavuzzo et
al., 1983; Smith, 1987; Smith et al., 1989); the effect of the
load path was the main parameter investigated in that study.
However, the value of the water-cement ratio (w/c) used was
unconventionally high (0.82), and hence, although trends for
the respective responses of other concretes could be extrapolated from the Colorado study, quantification of the actual
magnitudes of the various effects for different values of w/c
is not yet possible.
It is also well established in the field of materials research
that the influence of water held in the pore structure of
concrete (i.e., the moisture content, which depends on the
relative humidity of the surrounding conditions) has a
noticeable influence on the strength properties of concrete
(Mills, 1966; Rehbinder et al., 1948). The influence of moisture
on deformability is not well documented in the literature, but
ACI Structural Journal, V. 93, No. 6, November-December 1996.
Received Feb. 23, 1995, and reviewed under Institute publication policies. Copyright © 1996, American Concrete Institute. All rights reserved, including the making
of copies unless permission is obtained from the copyright proprietors. Pertinent
discussion will be published in the September-October 1997 ACI Structural Journal if
received by June 1, 1997.
589
I. Imran is a lecturer in the Department of Civil Engineering, Institut Teknologi Bandung, Indonesia, where he received his BSc degree in civil engineering. He received his
MASc and PhD degrees from the University of Toronto, Canada. His research interests
include constitutive modeling of concrete and other cementitious materials.
ACI member S. J. Pantazopoulou is an associate professor of civil engineering at the
University of Toronto. She received her degree in civil engineering from the National
Technical University of Athens, Greece, and MSc and PhD degrees from the University of California at Berkeley. She is a member and secretary of ACI Committee 341,
Earthquake Resisting Bridges, and a member of ACI Committees 368, Earthquake
Resisting Elements and Systems, and 442, Lateral Forces. Her research interests
include the mechanics of concrete and concrete structures.
it is known that creep in concrete is a phenomenon entirely
linked to the presence of water (Mills, 1966). However,
despite this obvious influence on mechanical properties,
moisture content is more often than not neglected in constitutive models of the mechanical behavior of concrete, even
in the case of the most elaborate formulations. Again, this
practice has been propagated in the mathematical models by
the lack of experimental quantification of the moisture effect
and its sensitivity to the simultaneous presence and intensity
of triaxial stress conditions in concrete.
The experimental work presented in this paper was motivated
by the list of limitations of the existing database of test
results outlined in the preceding. A parametric experimental
study consisting of a series of 130 triaxial tests was undertaken at the University of Toronto, aiming to explore and
quantify the influence of a number of significant variables on
the triaxial behavior of concrete for the benefit of improved
understanding and modeling of the material. The experimental program and the results of the study are outlined in
the following sections.
RESEARCH SIGNIFICANCE
The response of concrete to mechanical load is characterized
by progressive damage of its microstructure, a process manifest
by cracking and volumetric expansion of the material mass.
Partial restraint against this expansive tendency provided by
Fig. 1—Triaxial testing device.
590
confining mechanisms generally gives rise to a state of
triaxial stress, the intensity of which depends on the damage
characteristics. Development of reliable constitutive models
to describe this stress state requires an extensive database of
triaxial test results supplemented by complete deformation
records. In this paper, the results of an experimental parametric study of the constitutive properties of concrete is
presented, exploring the relationship between stress and
several physical and experimental variables.
EXPERIMENTAL PROGRAM
The tests were done on cylindrical specimens using a
triaxial cell that was originally designed for testing the
mechanical properties of rock (Hoek and Franklin, 1968).
The essential features of the test apparatus are illustrated in Fig.
1. The cell was hydraulically operated and consisted of a
stainless steel chamber bound in the interior by a cylindrical
membrane that hosted the concrete specimen. The
membrane was made of flexible urethane rubber, and rested
on the end caps of the cell. Its function was to shield the
specimen from fluid penetration into its pore structure.
When the chamber was filled with pressurized oil, the specimen was the structural fuse of the loading setup, at least in
the case of low and normal-strength concrete. The cell and
the jacket used in this study were designed to withstand
hydraulic pressures of up to 70 MPa, and accepted only Nxcore-sized specimens (i.e., specimens with a diameter of
54 mm [2.125 in.] and height of 108 mm [4.25 in.]).
While encased in the triaxial cell, the specimen was loaded
axially under displacement control using an MTS servocontrolled loading frame. Two 88-mm-high hardened steel
rams were inserted between the specimen ends and the frame
to facilitate load transfer.
Specimen preparation and instrumentation
The dimensions of the specimens used in the study were
prescribed by the geometry of the available cell. To insure
statistical homogeneity of the specimens, larger concrete
blocks were cast (150 x 250 x 250 mm), from which cylindrical cores of 54 mm diameter were extracted. The cores
were then cut to lengths of 115 mm (4.3 in.) using a concrete
saw to remove irregularities and soft concrete from the top
surfaces. Both end faces of each core were ground (to within
0.001 in.) with a surface grinding machine so that they would
be exactly orthogonal to the longitudinal axis of the cylinders. Additional control cylinders of 150 x 300 and 100 x
200-mm size were cast for assessing the magnitude of possible
size effects from the uniaxial test data.
Three different batches of concrete were cast, with w/c of
0.4, 0.55, and 0.75, respectively. The complete mix proportions
are given in Table 1. The maximum aggregate size used was
10 mm.
The specimens were moist-cured at 23 ± 2 C at 100
percent relative humidity. The 28-day wet uniaxial compressive strengths (i.e., fc′ at 100 percent relative humidity) of
Batches 1, 2, and 3 were 48.1, 38.3, and 19.9 MPa, respectively;
however, at the time of testing concrete was approximately
3.5 months old, and the corresponding wet uniaxial compressive strengths were increased to 64.7, 43.5, and 21.2 MPa
because of continuous moist-curing of the specimens. The
prolonged supply of water enabled continuation of the
hydration process; the solid mass product partially filled in
the existing pores, thereby causing an effective reduction in
porosity and increase in strength. This effect was more
ACI Structural Journal /November-December 1996
pronounced for concrete with smaller initial porosity (for w/
c of 0.4, the strength increase over the 28-day value reached
35 percent, whereas for w/c of 0.75 it was only 6.5 percent).
Axial deformation was measured independently during
the tests by a system of external LVDTs. To evaluate the
volumetric strain εv , which was of primary interest in the
study, the lateral strain εlat of the cylinders was measured
using 60-mm-long strain gages with high-strain capacity
(nominal strain limit of 10 to 20 percent) that were glued
directly on the surface of the cylinders at midheight and
encased within the membrane of the triaxial cell during
testing. (Long-gage sensors were selected because of their
ability to average the lateral strain measurements rather than
sense a localized effect. Note that the gage length provided
was about one-third of the perimeter of the specimen.) Due
to axisymmetry, εv was calculated from the algebraic sum ε3 +
2εlat (the convention used here is tension positive; ε3 represents the strain in the axial direction of the cylinder).
Description of parametric dimensions of study
It is axiomatic that the mechanical properties of concrete
are influenced by the characteristics of its components,
namely cement paste, pores, and aggregates. Of these, pores
are known to have the most profound influence on strength,
acting as weakening agents in the microstructure of concrete,
since they have no contribution towards interparticle
bonding but nevertheless occupy space in the material mass.
Note that increasing the value of w/c causes a corresponding
increase in the effective void ratio and a reduction in nominal
strength. Furthermore, the pore structure of concrete enables
penetration of fluids into its mass, the degree and rate of
penetration being dependent on the molecular size of the
fluid and the size and morphology of the pores. The motion
of fluids in and out from the pores, which is recognized
macroscopically by the change in moisture content, is known
to influence the uniaxial response of concrete. This phenomenon has been primarily attributed to weakening the intensity
of the Van der Waals attractions between gel particles due to
dilation of the cement gel by adsorbed water (Mills, 1966;
Rehbinder et al., 1948).
Another characteristic of concrete is path dependency, i.e.,
its mechanical behavior is influenced by way and type of
loading. It has been proposed that the material is sensitive to
the kinematic restraint against volumetric expansion that
boundary conditions provide, and that failure is marked by
uncontrolled growth of volumetric strain (Pantazopoulou,
1995). It is likely that path sensitivity manifests different
damage processes that result from different rates of volumetric expansion. Kinematic restraint is a possible means of
slowing down the phase of volume expansion, thereby
delaying the onset of failure.
Based on these considerations, the following emerged as
the essential parametric dimensions of the study: w/c, moisture content, level of confining stress, and load path. The
ranges of values considered for these experimental variables
are given in the following section. Two or more specimens
were tested for each combination of values of these variables
to determine repeatability of the experimental results.
Consistent with the previous discussion, performance was
evaluated in light of the recorded deformation histories.
1. w/c and moisture content—It was mentioned already that
three different values for w/c were considered, i.e., 0.4, 0.55,
and 0.75. To study the effect of moisture, two groups of specimens were tested, with the following moisture conditions:
ACI Materials Journal/November-December 1996
a) Dry—This condition was achieved by exposing the
specimens to a temperature of 60 C for a duration of 72 hr
prior to testing. (Higher temperatures were not used to avoid
possible breakdown of weak compounds in the gel. Therefore, a small amount of residual water, denoted henceforth as
ωdry , remained in the specimen at the time of testing.)
b) Saturated—Achieved by submerging the specimens in
water for a duration of 72 hr prior to testing.
At the time of testing, the adsorbed water ωsat (mass ratio),
water content of the dry specimens ωdry (mass ratio), and
total void volume νvoid measured based on ASTM C 642-82
were as follows: For w/c of 0.4, νvoid = 13.6, ωsat = 5.85, and
ωdry = 2.25 percent; for w/c of 0.55, νvoid = 14.3, ωsat = 6.25,
and ωdry = 1.87 percent; and for w/c of 0.75, νvoid = 16, ωsat
= 7.15, and ωdry = 1.15 percent. The theoretical values for the
total porosity of the three mixes were 25, 30, and 32 percent,
respectively, whereas the volume fraction of capillary pores
was calculated as 10, 13, and 15 percent; the latter set of values
approximates well the void ratios measured experimentally.
[Theoretical expressions were based on w/c, the rate of hydration of the paste α∞, and the volumetric fraction of paste in
concrete (1-Va) (Pantazopoulou and Mills, 1995).]
2. Level of confining stress—Seven levels of confining
pressure σlat were planned in the study. Normalized with
respect to the corresponding uniaxial compressive strength
of wet concrete (fc′ ), these were 0, 0.05, 0.10, 0.20, 0.40,
0.70, and 1.0. Concrete specimens with w/c of 0.4 were only
subjected to a maximum confining pressure ratio of 0.80 due
to the limitation in the capacity of the triaxial test equipment.
3. Load path—Four different load paths were considered;
these are given schematically in Fig. 2. All paths had a
common first phase during which the confining pressure was
gradually increased to a specified level while the specimen
was unrestrained in the axial direction. Beyond that stage,
one of the following load programs was used:
Type a—Axial compressive stress was gradually applied
under displacement control, while the level of confining
pressure was maintained [Fig. 2(a)].
Type b—The confining stress was increased further in a
stepwise manner, alternating with a stepwise increase of the
axial compressive stress [Fig. 2(b)].
Type c—The confining pressure was reduced in a stepwise
manner, alternating with a stepwise increase of the axial
compressive stress [Fig. 2(c)].
Type d—The axial compressive stress was cycled while
maintaining the level of confining stress at a specified level
[Fig. 2(d)].
DISCUSSION OF TEST RESULTS
The experimental responses of wet specimens subjected
to Load Path a are plotted in Fig. 3 for the three values of
w/c. The abscissa in all plots is the axial strain in the direction
of the applied load (Axis 3 in the reference coordinate
system). The axial stress and volumetric strain histories are
Table 1
Mix
Water, Cement Gravel, Sand, Density,
w/c kg/m3 , kg/m3 kg/m3 kg/m3 kg/m3
a/c*
Va†
Batch 1 0.40
185
462
830
870
2398
3.68
0.61
Batch 2 0.55
185
336
830
976
2404
5.38
0.65
Batch 3 0.75
185
247
830
1052
2348
7.62
0.69
*
Aggregate-cement ratio.
†
Total aggregate volume ratio.
591
identified in pairs by the level of lateral confining stress
applied during each test, which is marked next to the
response curves. The elastic properties of wet specimens
(initial stiffness E and Poisson’s ratio ν), obtained by averaging the values from all relevant uniaxial tests, were
32,900, 29,570, and 21,250 MPa, and 0.28, 0.22, and 0.21
for w/c of 0.4, 0.55, and 0.75, respectively.
A consistent relationship between the evolution of the
volumetric strain history and the characteristics of the
compressive stress-strain curve is evident in the plots. Note
that in ideally elastic conditions, εv is contractive throughout
the response under uniaxial stress and is given by (1 – 2ν)ε3
where ν is the initial Poisson’s ratio of the material and ε3 is
the axial compressive strain (Fig. 4). However, during the
actual response of concrete, the εv-ε3 relationship deviates
from linearity and eventually becomes expansive beyond the
point ε30 in the axis of axial strains. The descending branch
Fig. 2—Load paths used in test program.
of the stress-strain curve is associated with this phase of
volumetric growth. For any stress state, the coordinate bound
between the εv-ε3 curve and a 45 deg line in the contractive
quadrant represents the strain εA3 (=ε1 + ε2) of the cross
section supporting the load at an axial strain of ε3. This coordinate εA3, referred to in this work as area strain, increases at
a precipitous rate that is clearly related to the rate of softening of the stress-strain curve. The value of ε30 is of great
significance in this regard, because from the geometry of the
graph in Fig. 4 it can be easily deduced that ε30 also represents
the critical point at which the area strain is exactly equal in
magnitude (but of opposite sign) to the imposed axial strain.
It therefore separates the region of volume contraction to
volume expansion, and also defines the strain limit beyond
which the strength of concrete deteriorates from its peak
value (Pantazopoulou and Mills, 1995).
From Fig. 3 it is evident that in the case of axially
compressed confined concrete the region of volumetric
contraction is extended, and so is the prepeak range of the
stress-strain envelope. Volumetric expansion, when it
occurs, does so at a much slower rate than in the uniaxial
case where area growth is free of lateral restraint. Note that
for the same level of axial strain ε3, the area strain εA3 of the
principal compressive strut becomes smaller for higher
intensity of confining stress; reduction in εA3 is also associated with improved ductility in the postpeak regime of the
stress-strain response. Therefore, it can be concluded that the
presence of confinement imposes a kinematic restraint
against area growth of the compressive strut supporting the
axial load. Under uniaxial stress, the ratio of axial stress to
the corresponding axial strain is the secant stiffness associated with that strain Es. For axially symmetric confined stress
conditions, the secant stiffness is given by Es = σ3mod/ε3, where
the axial stress σ3mod accounts for the restraining of axial
expansion by the testing equipment [σ3mod = σ3 – 2νσlat /(1 –
ν)], obtained by assuming elastic initial conditions). Increase
Fig. 3—Triaxial axial stress-axial strain and volumetric strain-axial strain plots for all Type a tests on saturated specimens.
592
ACI Materials Journal/November-December 1996
in the magnitude of the axial deformation generally causes a
reduction of Es from the initial value Eo, which corresponds
to the unloaded, undamaged state. Fig. 5 plots the normalized values of the secant stiffness for all data points obtained
from the Type a triaxial tests done on wet specimens with w/c
of 0.4; note that the data are plotted with respect to the two
alternative strain variables ε3,εA3. The plot implies that axial
strain ε3 is not the ideal index for describing the variation of
Es since the same value of Es can be obtained for different
values of ε3; rather, it appears that there is a one-to-one relationship between the rate of reduction of Es and the amount
of area strain (i.e., damage due to cracking) that develops in
the cross section of the compressed strut that carries the axial
load. It was found that the pattern plotted in Fig. 5 is repeatable for all the tests in the database.
Failure patterns and transition point from brittle to
ductile response
Failure modes observed during the tests revealed that
concrete specimens subjected to low levels of lateral
confinement (i.e., from 5 to 20 percent of fc′ ) experienced
macrocracking similar to that observed in uniaxial (unconfined) tests. However, the restraining action provided by the
confining system delayed the softening and degradation of
load resistance, causing the failure pattern to be less brittle
than in the unconfined case. At high levels of confinement
(i.e., σlat greater than 40 percent fc′ ), almost no strength
degradation was observed past the peak load, the response
being ductile and resembling plastic flow (Fig. 3). No
visible macrocracks could be seen in these tests; however,
bulging was notable due to development of area strain. This
failure pattern is most likely related to the collapse and
compaction of the pore structure in the material, a concept
encouraged by the degree of dispersion of light observed
when examining the surface of such specimens in the laboratory.
Based on these experiments, it appears that transition from
brittle to ductile-type failure occurred at hydrostatic stress
levels in excess of 1.1fc′ (i.e., σlat > λfc′ , with λ being a function
of the initial porosity of the material and therefore varying
with w/c). Hence, for w/c of 0.75, transition occurred at
confining stresses in excess of 20 to 40 percent fc′ , while for
the higher strength materials this was observed to occur in
the range of confinement levels greater than 40 to 60 percent
fc′ . The maximum volumetric contraction achieved by the
specimens also increased with increasing confining pressure;
the corresponding coordinate of the transition point in terms
of volumetric contraction strain was in the range of 0.7 to 1
and 0.6 to 1.3 percent for wet and dry concretes, respectively.
where f cc
′ is the axial compressive strength of concrete
confined by the lateral stress σlat and fc′ is the uniaxial compressive strength of concrete. Observed strengths of the test specimens for all concrete mixes tested in this study are compared
with the estimates from Eq. (1) in Fig. 6. The correlation is
deemed satisfactory for the weaker specimens; however, the
compressive strengths of the stronger ones have been overestimated by Eq. (1), particularly so for specimens subjected to high
levels of confinement. This difference may be partly because
the original database used in deriving Eq. (1) only contained
triaxial test data of dry concrete specimens with low uniaxial
compressive strength (i.e, fc′ in the range of 5 to 25 MPa).
Fig. 6(b) plots the relationship between the observed axial
compressive strain at peak stress ε′cc (normalized with respect
to axial strain at peak uniaxial stress ε′c ), and the peak stress
f cc
′ (normalized with respect to uniaxial compressive strength
′ relationship
f c′ ). It is evident from the figure that the ε′cc - f cc
is basically linear. Earlier proposals for this relationship
were (Richard et al., 1928)
Fig. 4—Geometry of volumetric strain plots for axially
loaded concrete subjected to different levels of confinement.
Effect of confinement on characteristic
mechanical properties
Table 2 summarizes the observed peak stress f cc
′ along
with the corresponding axial strain at peak stress ε cc
′ , and at
zero volumetric strain ε30 from the triaxial tests of both wet
and dry concrete specimens with w/c of 0.75, 0.55, and 0.4,
respectively (averaged values). The anticipated axial
strengths of the test specimens were computed using the
familiar empirical formula that was originally proposed by
Richart et al. (1928)
f cc
′ = f c′ + 4.1σ lat
ACI Materials Journal/November-December 1996
(1)
Fig. 5—Degradation of secant stiffness plotted as function
of (a) axial strains, (b) area strains
593
f cc
′
ε′cc
= β 1 ------- – β 2 where β 1 = 5, β 2 = 0.8
-------f c′
ε′c
(2)
This result is plotted by the dashed line in Fig. 6(b); it is
evident that Eq. (2) underestimates most of the recorded
strain data. Correlation is improved for β1 = 6 and β2 = 0.83
[shown by the solid line in Fig. 6(b)]. Eq. (1) and (2) facilitate simultaneous evaluation of the increase in strength and
deformation capacity of concrete resulting from active
confining pressures. It was noted earlier that the axial strain
ε30 corresponding to zero volumetric strain (εv = 0) marks the
onset of strength degradation. To evaluate the available
deformability of concrete, the observed values of ε30 are
correlated with those of ε′cc in Fig. 7. The diagonal in the
Table 2—Summary of Type a test results
Dry specimens
w/c
0.4
0.55
0.75
σlat,
MPa
f ′cc,
MPa
Saturated specimens
0
ε ′cc,
ε3 ,
percent percent
f ′cc,
MPa
ε ′cc ,
ε3 0,
percent percent
0
73.35
–0.325
–0.31
64.69
–0.297
–0.277
3.2
96.13
–0.495
–0.445
80.85
–0.455
–0.51
6.4
108.7
–0.65
–0.66
96.8
–0.61
–0.8
12.8
125.6
–1.045
–1.1
113.51
–1.125
–1.335
25.6
38.4
168.6
203.95
–2.025
–3.105
–2.465
–4.525
153.87
190.59
–2.235
–3.495
–2.585
–3.11
51.2
240.5
–4.09
–3.92
230.48
–5.03
–5.39
0
47.4
–0.28
–0.2725
43.11
–0.25
–0.295
2.15
57.65
–0.43
–0.395
45.95
–0.43
–0.405
4.3
67.3
–0.69
–0.585
53.51
–0.65
–0.63
8.6
83.59
–1.46
–1.305
72.98
–1.66
–1.86
17.2
118.07
–2.53
–2.47
107
–2.81
–3.58
30.1
161.06
–3.6
–2.015
149.34
–4.23
–5.41
43
204.7
–4.73
–5.95
184.2
–5.02
–8.5
0
28.62
–0.26
–0.24
21.17
–0.22
–0.26
1.05
33.6
–0.47
–0.315
25.85
–0.36
–0.43
2.1
36.37
–0.675
–0.405
28.48
–0.66
–0.51
4.2
48.08
–1.385
–0.81
37.99
–1.62
–1.2
8.4
65.15
–2.375
–2.34
55.23
–2.96
–2.32
14.7
92.26
–3.425
–3.57
79.4
–3.96
–3.93
21
114.5
–4.46
–3.8
102.6
–5.05
–5.34
figure is the equal value line (i.e., ε′cc /ε30 = 1); most of the
experimental data points are clustered around this line,
suggesting that the axial compressive strain at the onset of
expansive volumetric growth is approximately equal to the
axial compressive strain corresponding to peak strength.
This demonstrates that peak stress and therefore strength of
concrete specimens is essentially limited by the initiation of
volumetric expansion. Based on the geometry of the volumetric plot, it also follows that at the point of zero volumetric
strain the area strain εA30 of the cross section supporting the
main compressive strut is equal in absolute magnitude to the
axial compressive strain, i.e., |εA30| = |ε′cc |. Hence, Eq. (2)
also describes the limiting area growth of the main compressive strut at which strength deterioration is imminent.
Effect of water content on mechanical behavior
of concrete
To study the sensitivity of the mechanical behavior of
concrete to changes in the amount of water residing in its
pore structure, test results obtained from otherwise identical
groups of wet and dry specimens were evaluated collectively
(Fig. 8). Fig. 8(a) compares the uniaxial responses of wet and
dry specimens with w/c of 0.75; it is evident that the wet
specimens are weaker than the corresponding dry ones.
Microscopically, this difference has been attributed to the
fact that when the wet specimen was loaded uniaxially,
adsorbed water in the pores developed a considerable
amount of pressure (referred to as pore pressure) due to the
initial volumetric contraction of the specimens (Akroyd,
1961); this water pressure acted against the concrete microstructure from inside the pores and therefore weakened the
effectiveness of the specimens in sustaining the axial load.
To explain the weakening effect of water on strength, Mills
(1966) suggested that pore water that is adsorbed in the gel
develops a swelling pressure (also referred to as disjoining
pressure) that has to be equilibrated by interparticle forces
(or Van der Waals forces). This process reduces the effectiveness of the Van der Waals attractions, which are responsible for the cohesive bond properties of concrete material in
supporting the external loads. Following a similar model for
intermolecular binding, Rehbinder et al. (1948) attributed
the strength reduction observed in wet concrete to the
shielding effect of cohesion forces acting between opposite
solid surfaces due to penetration of water; shielding effectively
Fig. 6—Summary of experimental results: (a) experimental strength values versus strength estimates from Eq. (1); (b) axial
strain versus axial stress at peak.
594
ACI Materials Journal/November-December 1996
reduces the molecular force of cohesion in the material. It
was shown by experiment that the affinity of liquid molecules (which depends on the types of liquid used) to the solid
surfaces affects the magnitude of strength reduction of the
solids; hence, higher forces of attraction of the liquid molecule to the solid surface cause a greater loss in the effective
cohesive strength of the solid.
1. Effective lateral pressure—From these physical interpretations of the weakening influence of water, it follows
that the effective lateral stress that was experienced by the
saturated specimen during the triaxial tests was actually less
than the nominal pressure applied. By extending the model
of Terzaghi and Peck (1967), which was originally developed
Fig. 7—Axial strain at zero volumetric change versus axial
strain at peak stress.
for porous soils, it is proposed that the effective lateral stress
σlat
′ of concrete was
σ′lat = σ lat – pv void
(3)
where σlat is the applied lateral stress, νvoid is the effective
porosity of the material, and p is the pore pressure build up
due to the initial volumetric contraction of the material.
According to this model, the effective lateral stress that is
actually mobilized in the specimen under a pure uniaxial test
is equal to –pνvoid (tensile). This explains why saturated
specimens exhibit lower uniaxial strength than the otherwise
identical dry specimens. The strength difference between
wet and dry specimens is also apparent in the test results of
confined concrete. Fig. 8(b) displays stress-strain plots of
concrete with w/c of 0.75 subjected to confining stress equal
to the uniaxial strength of wet concrete, i.e., 21 MPa.
However, based on Eq. (3), the effective confining stress in
the wet specimen is actually less, so that Eq. (1) would
consistently predict a lower strength for the wet specimen,
provided that σ lat
′ rather than σlat is considered in the calculations. To estimate the value of σ lat
′ from Eq. (3), the pore
water pressure p was taken equal to the applied lateral pressure; as effective porosity, νvoid of the wet specimens,
measured values were used. It was stated earlier that the dry
specimens contained some residual water; for these specimens, the effective porosity used in Eq. (3) was approximated based on
ω dry
ν dry = ----------ν
ω sat void
(4)
Fig. 8—Comparison of responses of otherwise identical saturated and dry specimens: (a) unconfined; and (b) confined.
ACI Materials Journal/November-December 1996
595
where ωsat and ωdry are the measured mass water contents of
dry and wet specimens respectively, and νvoid is the effective
porosity of the wet specimens. The corrected values for
strength calculated using Eq. (1) in combination with Eq. (3)
are compared in Fig. 9(a) with the experimental values for all
the specimens. It is evident from the figure that the correlation
of Eq. (1) with the test results is improved further when the
influence of adsorbed water on strength is considered in the
analytical model.
2. Compressive meridian—Based on the overall database
of tests, it was observed that the weakening influence of
water content on the triaxial strength of concrete increased
with confining stress, and was more pronounced for
concretes with higher w/c. Note that porosity increases with
w/c, thereby enabling the more porous concrete to adsorb
larger quantities of water. The strength data (i.e all Type a
tests) were also analyzed in hydrostatic and deviatoric stress
coordinates [Fig. 9(b)]; here, the abscissa represents the
hydrostatic stress invariant at peak stress I1, whereas the
ordinate is the corresponding deviatoric component √J2
(Chen and Han, 1988). Both coordinates were normalized by
the corresponding (i.e., wet or dry) uniaxial compressive
strength of concrete. When using the corrected values for the
lateral pressure [by accounting for the weakening influence
of water through Eq. (3)], the experimental points were organized in a linear pattern, suggesting a linear relationship
Fig. 9—Corrected strength values: (a) estimates of Eq. (1)
and (3) versus experimental values; (b) experimentally
derived compressive meridian of failure envelope.
596
between √J2 and I1 that was independent of the concrete type
(strength) and condition (wet or dry)
I
0.3  ----1- + 1 =
f ′

c
J
1
--- – --------23 f c′
(5)
Note that the pattern of the experimental data points
became more scattered and nonlinear when the water influence on lateral pressure was neglected in computing the
values of √J2 and I1 from the measured strength values,
suggesting a lower rate of increase of the deviatoric
strength with confining pressure in the wet than in the dry
specimens. It is common practice in the field of concrete
plasticity to model the profile of the compressive meridian
of three-dimensional failure envelopes by nonlinear
expressions; many such models are available in the literature (summarized in Chen and Han, 1988, and Chen and
Saleeb, 1994). The magnitude of this nonlinearity,which is
not evident in the trends of Fig. 9, could have been
enhanced by the different specimen geometries (e.g.,
cubes) and testing conditions used in some of the reference
experimental studies. It is also likely that nonlinearity has
been primarily propagated in the analytical models because
of the current experimental practice of neglecting the influence
of adsorbed water on mechanical properties. Note that most
specimens contain various amounts of water during testing
that are more often than not neglected in the evaluation of
the stress states and are seldom reported in experimental
studies of this type published in the available literature.
3. Effect of adsorbed water on deformation measures—
Volumetric strain plots of uniaxial tests on otherwise identical wet and dry specimens did not differ substantially; a
relatively slower rate of expansion was observed in wet
specimens at the later stages of the test [Fig. 8(a)]. However,
the observed differences in volumetric strain histories
increased with the magnitude of confining pressure (Imran,
1994), with the wet specimens experiencing a significantly
larger amount of lateral contraction during early stages of
loading than the dry specimens [e.g., Fig. 8(b)]. This implies
that the pore water pressure might have induced considerable
damage to the internal structure of wet concrete during the
application of confining stress (i.e., during the first phase of
the typical test; this is illustrated by the initial vertical
segment in the εv-ε3 plot of wet specimens). The large
amount of initial contraction of the wet confined specimens
suggests early compaction and softening of concrete, which
is consistent with the observation that, during the second
phase of the typical test, increased amounts of volume
contraction occurred in the wet specimens as compared to
their dry counterparts. For completeness, to evaluate the
strain envelope at peak strength in terms of strain invariants
(J ′2 and I ′1), the following points were considered: a) the first
strain invariant I ′1 is the volumetric strain εv; b) at peak
stress, the volumetric strain is approximately zero (Fig. 4, 7);
c) because of axisymmetry (ε1 = ε2), the deviatoric strain
invariant J ′2 = (ε3 - ε1)2/3, which at peak strength (εv = 0,
hence ε1 = -0.5ε3) reduces to J ′2 = 3ε32/4. Hence, at peak
strength the points corresponding to the strain envelope
have coordinates of (0, ε3√3/2) in the I ′1-√J ′2 system of axes.
Characteristics of cyclic response
A total of 21 dry concrete specimens from the three
different concrete mixes were tested under cyclic triaxial test
ACI Materials Journal/November-December 1996
conditions to study the characteristics of the cyclic compressive
behavior of concrete. In each test, unloading and reloading
of axial load was carried out while maintaining the level of
specified confining stress constant [Fig. 2(d)]. Typical
recorded responses from this test series are plotted in Fig. 10
(solid line) along with the corresponding monotonic test
results (dashed line). It can be observed from the stress-strain
curve in that figure that the reloading path of the cyclic
response always joins the monotonic stress envelope, almost
at the level of previously attained maximum stress; this
observation is also valid in cyclic tests of confined concrete.
The similarity of monotonic and cyclic stress-strain envelopes indicates that the specimens subjected to unloading
and reloading cycles experience very little or no strength
degradation due to cycling, which suggests that the strength
criteria for concrete are basically path-independent.
However, the cyclic stress-strain curves exhibit hysteresis
during unloading and reloading cycles, especially past the
peak stress level of concrete (at the postpeak segment of the
stress-strain response). The magnitude of this hysteresis is a
measure of the amount of energy dissipation due to crack
formation during the loading cycles. Hysteretic behavior is
also observed in the plots of the corresponding volumetric
strain response of the specimens; evidently, the average
slope of the hysteretic loops (line connecting the turning
points) has a horizontal trend, particularly so in the postpeak
regime. Thus, it can be said that during the cycling process
the volumetric strain is more or less constant. This implies
that during unloading, the change of area strain in the material is approximately equal to the decrease in axial strain.
(Note that this result is consistent with observations reported
by Van Mier [1986] obtained from cyclically loaded
concrete cubes under similar lateral stress conditions, i.e.,
σ1 = σ2.) In addition, the size of the hysteretic loops in the
εv -ε3 plots progressively increased with higher levels of
applied confining stress. While decreasing the axial load,
the applied confining stress was kept constant, thereby
squeezing the specimen. This resulted in a reduction of area
strain and promoted further reversal of the axial strain. Note
that this additional reversal of axial strain is the reason why
at high confining pressure the specimen experienced a
decrease in volume during the initial stages of the reloading
cycle, which represents compaction of the material in the
axial direction. Thus, for the same level of axial strain at
initiation of unloading, specimens subjected to higher
confining stress experience greater reversal of strain in the axial
direction, and hence they undergo a larger volume contraction
during the reloading stage, as illustrated by the larger size
hysteretic loops.
Furthermore, it appears that for the same level of axial
deformation, specimens that were subjected to cyclic loading
experienced more overall volumetric expansion than specimens subjected to monotonic loading. It has been stated
earlier that volumetric expansion is a reliable measure of
internal damage. The results of this comparison suggest that
the cyclic nature of the load promotes damage built up at
faster rates than monotonic loading, and hence the path
dependence in the behavior of concrete is primarily with
respect to the deformations.
Degradation of elastic modulus
The slope of the unloading loops in the stress-strain curve,
which represents the residual elastic stiffness of concrete in
the axial direction, decreases as the magnitude of induced
axial deformation at which unloading begins is increased.
This is attributed to the continuous growth of cracks within
the concrete specimens with increasing level of imposed
deformation. The crack formation caused an increase in the
Fig. 10—Sample results of cyclic tests.
ACI Materials Journal/November-December 1996
597
apparent porosity of the specimens (i.e., crack-induced
voids) that was orthotropically oriented and primarily
affected the area normal to the compressive load. It has been
established from early works (Powers, 1958) that the larger
the effective void ratio of concrete, the smaller its elastic
modulus. It is for this reason why cracking, as well as natural
porosity, are considered to both have a parallel weakening
influence on the elastic modulus of concrete.
To study the pattern of stiffness degradation in the axial
direction, the average slope of the hysteretic loops in the
axial stress-axial strain diagram (i.e., the line connecting the
turning points at the ends of a loop), normalized with respect
to the initial stiffness of undamaged concrete, was plotted for
all tests in Fig. 11 against the area strain of the specimen
cross section carrying the axial load. It is evident from the
figure that, again, the area strain variable organizes successfully the data for both confined and unconfined tests, thereby
suggesting that degradation of the elastic stiffness is a manifestation and direct consequence of expansion (area strain)
due to cracking of the cross section of the compressive strut.
The experimental trend of Fig. 11 is described mathematically as follows
E
1
------ = --------------ε
E0
1 + ----Aα
(6)
where α is a normalizing constant; based on the available
tests, the value of α used in plotting the solid line in Fig. 11
was taken as 0.05. From this correlation it can be concluded
that the degradation of the elastic stiffness due to cyclic loads
is primarily a function of geometrical change (in this case
area strain) and is insensitive to the type of concrete considered. Furthermore, using Eq. (6), it is possible to evaluate the
magnitude of axial strain ε3R, at which complete removal of
axial stress is achieved (the point where the unloading loop
intersects the axis of strains [Fig. 12]). Note that the stress at
the point of initiation of unloading from the envelope σ3E is
given in terms of the accumulated area strain εA3E and the
corresponding axial strain ε3E (Fig. 12) by the following relationship (Pantazopoulou and Mills, 1995)
E
E
σ3
Eε 3
= ------------------E
ε A3
1 + ----------β
(7)
where β is a material constant (it was shown in the previous
reference that β = Vp3νcap /3, where Vp is the volumetric fraction
of paste in the concrete mix and νcap is the capillary porosity
of the paste; for the three concrete mixes considered in this
study, the values of β were 0.006, 0.0035, and 0.002 for w/c
of 0.4, 0.55, and 0.75, respectively). From Eq. (6) and (7),
the axial strain at complete removal of stress ε3R is given as
a fraction of the strain at the envelope ε3E (at the initiation of
unloading) as follows (Fig. 12)
R
E
ε A3
ε3
1 1
- --- – ---------- = ------------------E
E
ε A3 β α
ε3
1 + ----------β
Fig. 11—Degradation of unloading stiffness with increasing
area strain of compressive strut.
Fig. 12—Residual stress values after cycling of axial load:
comparison of results of Eq. (8) with uniaxial model (Karsan and Jirsa, 1969).
598
(8)
Eq. (8) is an empirical summary of the available cyclic
test results, stating that the strain ratio ε3R/ε3E and hence the
slope of the unloading loops (defined as the ratio of envelope stress over the elastic strain ε3E-ε3R) decays with
increasing amount of area strain in the cross section of the
compressive strut. Note that for the same level of envelope
strain ε3E, concrete specimens with higher confining pressures generally experienced lower amounts of expansion
strains εA3E and lower amounts of stiffness degradation
during unloading. Hence, by linking the degree of stiffness
degradation to the expansion strain variable [i.e., area strain
in Eq. (6)], it is possible to describe this process for both
confined and unconfined compressive cyclic stress states. The
success of Eq. (8) in capturing stiffness degradation in uniaxially loaded concrete is evidenced by the observed agreement
with other more restricted empirical models of the same
phenomenon; Fig. 12 compares the behavior of Eq. (8) with the
well-established model of Karsan and Jirsa (1969), which
summarizes the experimental data from a series of uniaxial
cyclic tests conducted by the authors on cylinder specimens.
Results of multistep tests
The objective of this test series was to assess the dependence of mechanical behavior on the history of loading,
ACI Materials Journal/November-December 1996
commonly referred to as loading path. Typical loading histories are depicted in the schematic of Fig. 2(b) and (c); results
obtained for concrete with w/c of 0.4 from the two alternative
load paths are plotted in Fig. 13. In the case of Load Path -b, the
specimens were subjected to an initial confining stress of
6.4 MPa, which was subsequently increased in a stepwise
manner (alternating with a stepwise increase of the axial
load) to 25.6 MPa; the confining stress was kept constant
thereafter up to specimen failure. In Load Path -c, the specimens were subjected to an initial confining stress of 25.6
MPa, which was subsequently reduced in a stepwise manner
(alternating with the stepwise increase of the axial load) to
6.4 MPa. For the sake of comparison, results obtained from
monotonic tests at constant confining stresses of 25.6 and 6.4
MPa are also plotted in Fig. 13. The pattern shown was
typical for all multistep tests; whereas the monotonic
strength was reached in the multistep tests, it was observed
that at the same level of imposed axial strain Load Path -b
caused greater amounts of volumetric expansion than the
corresponding monotonic tests. Note that before it reached
the specified value (i.e., 25.6 MPa), the confining stress
applied by Load Path -b was always lower than that applied
in the corresponding monotonic case, and therefore during
this stage the multistep specimen was weaker and experienced cracking at earlier stages than the monotonic one. The
reverse observation can be made when comparing the results
of Load Path -c with those of the monotonic case; because of
the higher initial confining pressure, the multistep specimen
was the more ductile, and experienced cracking (or lateral
expansion) at a later stage than the monotonic specimen.
These observations suggest that whereas the loading path
does not influence the mechanical strength of concrete in
axial compression (i.e., strength criteria of concrete are pathindependent), it has an effect on deformability and thus
should be considered in determining the strain state in the
material resulting from application of mechanical stress.
Model of volumetric behavior of concrete
The preceding review of the experimental program illustrated that it is possible to organize the experimental trends
for triaxially loaded concrete using as a state variable the
degree of expansion in the material microstructure, which is
an effective index of accumulated damage. Results from the
test series support the concept that the volumetric strain
history of concrete is characterized by the level of applied
confining stress. Initially, the εv-ε3 relationship appears to be
linear; however, development of microcrack-related expansion in the cross section supporting the compressive stress is
manifested by progressive nonlinearity in the εv-ε3 curve.
The following expressions summarize the available experimental evidence (Imran, 1994):
a) Prior to cracking in the lateral direction (i.e,
1 – ν-σ′ - ε cr where ε is the cracking strain of the
ε3>ε3lim= ---------------cr
νE lat ν
material in direct tension, and ε3lim is the axial strain at which
cracking occurs in the lateral direction)
2σ′lat
- + ε 3
ε ν = ( 1 – 2ν )  ------------ E

(9)
b) after cracking (ε3 < ε3lim)
(a)
(b)
Fig. 13—Sample results of multistep tests: (a) Load Path Type -b; (b) Load Path Type -c.
ACI Materials Journal/November-December 1996
599
Fig. 14—Comparison of experimental and calculated
volumetric strain histories.
2σ′lat
0
- + ε3
ε ν = [ 1 – 2ν ] ------------E
lim
2
 ε3

ε3 – ε 3
- 
- – ----------------------- -----0
lim
0
 ε3

ε3 – ε3
(10)
In the previous equation, ε30 represents the axial compressive
strain at the instant of zero volumetric strain [based on Fig. 7,
ε30 is taken equal to ε′cc , which in turn is given by Eq. (2)].
The effectiveness of the model given by Eq. (9) and (10) is
illustrated in Fig. 14, which compares calculated and experimentally obtained volumetric responses of concrete with w/c
of 0.75, and for different levels of confinement.
CONCLUSIONS
An extensive experimental program was undertaken to
characterize the behavior of concrete under multiaxial states
of stress. The effect of pores (or voids) on the mechanical
behavior of concrete was one of the primary variables
explored. Other variables considered were the water content
of concrete at the time of testing, the level of lateral
confining pressure, and the type of loading history. Performance of the test specimens was gaged from recorded
stresses and strains and by the extent and history of damage
in the material microstructure. It was found that the most
effective variable in organizing the strength data was the
area strain developing in the cross section of the principal
compressive strut. Based on the experimental evidence it
was concluded that:
a) Failure of the uniaxial specimens was marked by uncontrolled volumetric expansion. The rate of increase of volumetric strain in the postpeak regime was higher for higher
strength concrete.
b) The strength of concrete specimens tested under triaxial
states of stress was essentially limited by the initiation of
volumetric expansion.
c) Under increasing lateral confinement, concrete experienced enhancement of strength and apparent ductility. This
is due to the restraining action of the confining mechanism,
which impeded and consequently slowed down the development of lateral expansion in the material.
d) Saturated concrete specimens were observed to develop
lower strength than dry specimens of the same batch. This is
due to the development of pore pressure in the wet concrete,
which reduced the confining effectiveness of the applied
lateral stress. The weakening influence of moisture content
on the triaxial strength of concrete increased as the confining
600
stress increased. This weakening effect was more significant
for concrete with higher w/c ratio.
e) Concrete subjected to cyclic load reversals exhibited a
gradual decrease of its elastic modulus near and beyond peak
stress due to the development and accumulation of damage
as the level of induced deformation increased. The gradual
change of elastic modulus was found to be primarily a function
of geometrical change and relatively insensitive to the type
of concrete considered.
f) The volumetric-axial strain history of specimens
subjected to cyclic loading exhibited hysteretic loops, associated in shape and size with those commonly observed in
cyclic stress-strain relationships. The size of these loops
increased with increasing confining stress.
g) By comparing the results of the cyclic and multistep
tests, it was found that the deformation behavior of concrete
was path-dependent. On the other hand, the strength of
concrete was observed to be basically path-independent. It
was also observed that at similar levels of imposed axial
deformation, concrete specimens subjected to cyclic loading
experienced larger volumetric expansion than those
subjected to monotonic loading. This indicates that the
cyclic nature of the load promotes damage buildup at faster
rates than the monotonic loading.
ACKNOWLEDGMENTS
The experimental study presented in this paper was carried out in the
Structural Testing Laboratories of the University of Toronto. The study was
partially funded by the Natural Sciences and Engineering Research Council
of Canada; financial support provided to the first author for his PhD studies
by the government of Indonesia is gratefully acknowledged. The authors are
indebted to Professor Emeritus R. H. Mills for his intellectual contributions
to the development of this research.
NOTATION
Eo
Es
f ′c
f ′cc
I1, J2
=
=
=
=
=
w/c
εA3
=
=
εc′
=
ε′cc
=
εcr
εv
ε3E
=
=
=
ε2 , ε3
ε3
ε3lim
=
=
=
ε3R
ε30
σlat
σ3E
=
=
=
=
elastic modulus of uncracked concrete
secant modulus
uniaxial compressive strength
axial strength of confined cylinders
hydrostatic and deviatoric stress invariants (as defined in
Chen et al. [1988])
water-cement ratio
area strain in cross sections orthogonal to longitudinal (3)
axis
axial strain corresponding to peak strength of unconfined
cylinders
axial strain corresponding to peak strength of confined
cylinders
direct tension cracking strain of concrete
volumetric strain
axial strain corresponding to points at envelope of stressstrain diagram
principal strains in cross section of cylinder
axial strain
axial strain at which cracking first occurs in radial direction of cylinder
residual axial strain after unloading from envelope
axial strain at zero volumetric strain
lateral pressure
axial stress corresponding to points at envelope of stressstrain diagram
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