Experimental Study of Plain Concrete under Triaxial Stress

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 Batch 2 0.55

185 185

462 336

830 830

870 976

2398 2404

3.68 5.38

0.61 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

ε′cc f cc ′ -------= β 1 ------- – β 2 where β 1 = 5, β 2 = 0.8 ε′c f 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

Saturated specimens 0

σlat, MPa

f ′cc, MPa

ε ′cc, ε3 , percent percent

f ′cc, MPa

ε ′cc , ε3 0, percent percent

0 3.2

73.35 96.13

–0.325 –0.495

–0.31 –0.445

64.69 80.85

–0.297 –0.277 –0.455 –0.51

6.4 12.8

108.7 125.6

–0.65 –1.045

–0.66 –1.1

96.8 113.51

–0.61 –1.125

–0.8 –1.335

25.6

168.6

–2.025

–2.465

153.87

–2.235

–2.585

38.4

203.95

–3.105

–4.525

190.59

–3.495

–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 4.3

57.65 67.3

–0.43 –0.69

–0.395 –0.585

45.95 53.51

–0.43 –0.65

–0.405 –0.63

8.6 17.2

83.59 118.07

–1.46 –2.53

–1.305 –2.47

72.98 107

–1.66 –2.81

–1.86 –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 1.05

28.62 33.6

–0.26 –0.47

–0.24 –0.315

21.17 25.85

–0.22 –0.36

–0.26 –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 14.7

65.15 92.26

–2.375 –3.425

–2.34 –3.57

55.23 79.4

–2.96 –3.96

–2.32 –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

1 E ------ = --------------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 εcr is the cracking strain of the ε3>ε3lim= ----------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 0 lim ⎝ 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

REFERENCES Akroyd, T. N. W. (1961), “Concrete under Triaxial Stress,” Magazine of Concrete Research, V. 13, No. 39, pp. 111-118. ASTM C 642-92 (1992), “Standard Test Method for Specific Gravity, Absorption, and Voids in Hardened Concrete,” Annual Book of ASTM Standards, 04.02, American Society for Testing and Materials, pp. 310-311. Chen, W. F., and Saleeb, A. (1994), Constitutive Equations for Engineering Materials, V. 1 and 2, Elsevier Science B.V., 1129 pp. Chen, W. F., and Han, D. J. (1988), Plasticity for Structural Engineers, Springer-Verlag Publishing, 606 pp.

ACI Materials Journal/November-December 1996

Chen, W. F. (1988), “Evaluation of Plasticity-Based Constitutive Models for Concrete Material,” Solid Mechanical Architecture, V. 13, No. 1, pp. 1-63. Hoek, E., and Franklin, J. A., “Simple Triaxial Cell for Field or Laboratory Testing of Rock,” Transactions of the Institute of Mining and Metallurgy, 77, pp. A22-A26. Imran, I. (1994), “Applications of Nonassociated Plasticity in Modeling the Mechanical Response of Concrete,” PhD thesis, Department of Civil Engineering, University of Toronto. Karsan, P., and Jirsa, J. O. (1969), “Behavior of Concrete under Compressive Loading,” Journal of the Structural Division, American Society of Civil Engineers, V. 95, No. 12, pp. 2543-2563. Kosaka, Y.; Tanigawa, Y.; and Hatanaka, S. (1984), “Inelastic Deformational Behavior of Axially Loaded Concrete under Low Lateral Confining Stresses,” Transactions of the Japan Concrete Institute, 6(III-5-B), pp. 263-270. Mills, R. H. (1966), “Effects of Sorbed Water on Dimensions, Compressive Strength, and Swelling Pressure of Hardened Cement Paste,” Special Report No. 90, Highway Research Board, pp. 84-111. Pantazopoulou, S. J., and Mills, R. H. (1995), “Microstructural Aspects of the Mechanical Response of Plain Concrete,” ACI Materials Journal, V. 92, No. 6, Nov.-Dec., pp. 605-616. Pantazopoulou, S. J. (1995), “Role of Expansion on Mechanical Behavior of Concrete,” Journal of Structural Engineering, American Society of Civil Engineers, V. 121, No. 12, pp. 1795-1805. Powers, T. C. (1958), “Physical Structure and Engineering Properties of

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Concrete,” PCA Research Department Bulletin No. 90, Chicago, pp. 1-28. Powers, T. C. (1968), Properties of Fresh Concrete, John Wiley and Sons, New York. Rehbinder, P. A.; Schreiner, L. A.; and Zhigach, K. F. (1948), “Hardness Reducers in Drilling,” Colloids-Electrochemical Institute, Academy of Science, USSR, pp. 1-39. Richart, F. E.; Brandtzaeg, A.; and Brown, R. L. (1928), “A Study of the Failure of Concrete under Combined Compressive Stresses,” Engineering Experiment Station Bulletin No. 185, University of Illinois, Urbana. Scavuzzo, R.; Stankowski, T.; Gerstle, K. H.; and Ko, H. Y. (1983), “StressStrain Curves for Concrete under Multiaxial Load Histories,” Structural Research Series, C.E.A.E. Department, University of Colorado, Boulder. Schickert, G., and Winkler, H. (1977), “Results of Test Concerning Strength and Strain of Concrete Subjected to Multiaxial Compressive Stress,” Heft 277, Deutscher Ausschuss für Stahlbeton, Berlin. Smith, S. S. (1987), “On Fundamental Aspects of Concrete Behavior,” Structural Research Series No. 87-12, C.E.A.E. Department, University of Colorado, Boulder. Smith, S. S.; Willam, K. J.; Gerstle, K. H.; and Sture, S. (1989), “Concrete over the Top, or: Is There Life after Peak?” ACI Materials Journal, V. 86, No. 5, Sept.-Oct., pp. 491-497. Van Mier, J. G. M. (1986), “Fracture of Concrete under Complex Stress,” Heron, V. 31, No. 3, pp. 1-90.

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