(ProQuest Information and Learning: ... denotes formulae omitted.)

INTRODUCTION

In recent years, considerable interest has developed in the fatigue behavior of concrete because of its increasing use in such structures as dams, bridges, crane beams, airport/highway pavements, and pressure

vessels. Most fatigue studies carried out in the past, however, dealt with cases for concrete under uniaxial cyclic loading (ACI Committee 215 1974; Tepfers and Kutti 1979; Hsu 1981; Slowik et al. 1996; Comite Euro-International du Beton Bulletin d'Information 1993), only a few investigations concerning fatigue under triaxial loads have been conducted (Buyukozturk and Tseng 1984; Su and Hsu 1988; Taliercio and Gobbi 1996; Yu et al. 1998). Limited information is available on the behavior of concrete subjected to nonproportional multiaxial fatigue loadings, which is usual in a practical constructal structure (for example, the lateral loads subjected to a concrete dam are constant when the water level keeps constant, whereas the vertical loads will be changed when a vertical dynamic vibration takes place). Therefore, some reliable information on fatigue strength, deformation, failure mode, life, and constitutive relationship is required for nonproportional multiaxial fatigue loadings.

Su and Hsu (1988) reported the biaxial compression fatigue tests for concrete plates of 15.2 ? 15.2 ? 3.8 cm subjected to four principal compression stress ratios (the ratio of the stress s^sub 1^ in the vertical direction to the stress s3 in the horizontal direction), that is, s^sub 1^/s^sub 3^ = 0, 0.2, 0.5, and 1.0. It was concluded that the fatigue strength of concrete under biaxial compression is greater than that under uniaxial compression for any given number of load cycles, the fatigue failure envelopes have shapes similar to the envelope for static strength, and the size of the envelope reduces as the fatigue life increases. Nelson et al. (1988) conducted an experimental investigation on the behavior of high strength concrete subjected to proportional biaxial-cyclic compression. They found that the biaxial state of stress does not enhance the fatigue strength of high-strength concrete below approximately 75% of the peak stress. Yin and Hsu (1995) performed tests on steel fiber-reinforced concrete specimens subjected to biaxial compressive fatigue loading. Their experiments were focused on the comparison of fatigue strength, failure modes, and deformation characteristics between plain and fiber reinforced concretes. It was found that the fatigue strength of steel fiber-reinforced concrete in biaxial compression is higher than that in uniaxial compression for any given number of load cycles. Taliercio and Gobbi (1996, 1997, 1998) performed a series of conventional triaxial cyclic tests on cylindrical samples under different mean stress, amplitude, and the ratio of the mean lateral stress to the mean axial stress. They found that the fatigue life of concrete is usually shorter when axial stress and confining pressure vary in phase opposition rather than in phase coincidence.

Subramaniam et al. (1999, 2002) conducted a series of fatigue tests for concrete subjected to compression-tension and tensile compression-tension, respectively. Subramaniam (1999) and Subramaniam et al. (1999) found that the damage in concrete subjected to biaxial fatigue loading in the compression-tension region via a torsion test is localized to a single crack, and the crack growth governs the observed load-deformation responses.

A review of the available literature indicates that most of the experimental results pertain to proportional fatigue loading. In practical structural applications, however, concrete structures may be subjected to nonproportional fatigue loading. But no information could be found in the literature on the response of concrete subjected to nonproportional fatigue loading where the load along the one axis are fixed with the fatigue loading imposed in the orthogonal direction.

Early attempts of damage constitutive modeling of concrete under cyclic loading have been limitedly studied. Fardis et al. (1983) and Yang et al. (1985) applied damage mechanics to concrete under repeated loading. A simple model developed by Fardis et al. (1983) captures the nonlinear characteristics of the monotonic and cyclic behavior of concrete well. Since then, Suaris et al. (1990) developed a damage model for monotonic and cyclic behavior of concrete in which the elastic potential was introduced in terms of principal stresses and a damage dependent compliance tensor. Khan et al. (1998) developed an appropriate damage-effect tensor for concrete in constructing the constitutive equations, in which essential features of concrete, such as degradation of elastic properties, strain softening, gain in strength under confinement and different behavior in tension and compression, have been taken into account effectively. Al-Gadhib et al. (2000) developed an anisotropic damage model capable of predicting the fatigue life of concrete under compression through the adaptation of the constitutive model developed by Khan et al. (1998). Few constitutive laws, however, have been proposed to model the damage accumulation in concrete subjected to nonproportional fatigue loading where the load along the one axis is held fixed while fatigue loading is imposed in the orthogonal direction. Accordingly, the main objective of this paper is to propose a damage constitutive model for concrete subjected to nonproportional biaxial fatigue stresses with one constant through the adaptation of the model developed by Al-Gadhib et al. (2000), and the data obtained from the experiments are used to calibrate the model.

RESEARCH SIGNIFICANCE

The fatigue behavior of concrete has drawn considerable attention because of the increasing use of concrete in bridge and dam engineering. Most fatigue studies carried out in the past, however, dealt with the behavior of concrete under uniaxial cyclic loading. Only a few investigations concerning fatigue under biaxial loads have been conducted, and limited information is available on the behavior of concrete subjected to nonproportional biaxial fatigue stresses with one constant. Also, few constitutive laws have been proposed to model the damage accumulation in concrete under cyclic biaxial loading with one stress constant. The investigation reported herein provides experimental data and presents a damage model for predicting the behavior of concrete nonproportional biaxial fatigue stresses with one constant.

EXPERIMENTAL INVESTIGATION

Test facilities

Specimens were tested in a specially designed, truly triaxial apparatus, as shown in Fig. 1. The testing system consists of separate loading frames, hydraulic actuators, servo valves, and load cells in three directions, as shown in Fig. 2. Each loading frame with leading wheels, consisting of four lead screws and two load-carrying steel plates, can be moved along the fixed guide rails to keep the deformations of the two sides of the specimen equal. The bottom of the vertical loading frame is connected with the main hydraulic compressor and the top of which is laterally restrained through the vertical leading levers fixed to the stiffened steel frame. The loading frames can move freely and independently in any of the three axial directions. Specimens can be loaded via three pairs of pressure levers; spherical hinges are installed between the levers and the heads so as to keep the load exerted exactly in an axial direction. The actuators are designed to act independently or jointly. The nominal capacity of the system is 2000 kN in compression and 500 kN in tension. To prevent lateral restraint of the loaded specimen, all the loaded surfaces are polished and equipped with three layers of plastic sheet with grease of MoS^sub 2^ to reduce the surface friction to a minimum.

A 32-channel data acquisition processor was employed to collect loads, displacements, and the number of cycles to failure for the test. The loads were measured through a load cell with a capacity of 2000 kN. Displacement of a specimen at any direction was measured by two linear variable differential transducers (LVDT) attached on the opposite sides of the specimen.

Test program

Plain concrete cubes with a size of 100 ? 100 ? 100 mm were subjected to both monotonic or cyclic compression loads. The static tests were used to determine the average uniaxial and biaxial compressive strengths. The loads were applied to the specimen with a constant stress rate of 20 MPa per minute. In the uniaxial compression tests, the load was applied in a vertical direction (s^sub 3^), whereas in the nonproportional biaxial compression test, the lateral load (s^sub 1^) was first applied in one of the horizontal directions to a specified value. Then, the lateral stress was kept constant and the vertical load was applied successively.

In the uniaxial cyclic tests, the vertical load was applied sinusoidally with time. In the biaxial cyclic tests, the loading procedure of the lateral stress was the same as that of the biaxial static tests, whereas the vertical load was applied sinusoidally. The maximum stress level f^sub max^/f^sub cc^ (S^sub max^) is defined as the ratio of the major principal stress s^sub 3^ to the static strength f^sub cc^ (f^sub cc^ is defined as the biaxial static compression strength of concrete with a constant lateral stress). Four maximum stress levels 0.75, 0.8, 0.85, and 0.9 were chosen to apply loads in this paper as indicated in Table 1. The reason for choosing the range of the maximum stress levels is to satisfy the needs of seismic analysis for concrete structures, which is still very important for the dynamic analysis of practical structural engineering such as dams and highway pavements. A constant minimum stress of 20% of the maximum applied stress R' = 0.2) was maintained during cyclic testing. For the 85 and 90% maximum stress levels, a frequency of 1 Hz was used. For all other maximum stress levels, a frequency of 5 Hz was used. At least three specimens were tested at each stress level. The adopted combinations of maximum stress levels (denoted as S^sub max^), stress ratio R' ratio of minimum stress to maximum stress), loading frequencies f, and the number of specimens are listed in Table 1.

Test specimens

The content of the fresh concrete of cement, water, sand, and coarse aggregate was 383, 200, 663, and 1154 kg per cubic meter, respectively. Type I cement, general river sand, and crushed rock were employed. The water-cement ratio (w/c) was 0.52, and the maximum size of the aggregate is 20 mm. Steel molds were used to cast the cubic specimens. The specimens were cured in a water tank for 28 days and then stored in normal laboratory air until it was tested.

The 28-day compressive strength of concrete obtained by testing standard cube specimens (150 ? 150 ? 150 mm) is 28 MPa. Also, the cube specimens were tested at an age over 120 days.

TEST RESULTS FOR MONOTONIC LOAD

Typical static stress-strain response of the concrete subjected to various levels of lateral stress is given in Fig. 3. It should be noted that these results represent the relationships of the vertical stress with the accumulated total strains in the vertical and lateral directions, respectively, and, therefore, the strains do not begin from zero but from the values when the lateral load reaches the specified lateral stress (tension strains are designated as negative). The compressive strengths of the concrete are 20.47, 25.8, and 27.4 MPa at lateral stress levels 0, 0.25 and 0.5, respectively.

TEST RESULTS FOR CYCLIC LOAD

S-N diagrams

The results of the compression fatigue tests are statistically analyzed to obtain S-N curves. The S-N curves for uniaxial and biaxial fatigue tests of concrete are given in Fig. 4 through 6, where each experimental data point represents the number of cycles that the specimen sustained until failure at a given maximum stress level. The maximum stress level S^sub max^ (S^sub max^ = f^sub max^/f^sub cc^) has been nondimensionalized with respect to the average static strength f^sub cc^ (f^sub cc^ = f^sub c^ for uniaxial loading) of the compressive cube specimens tested under different lateral stress. S^sub max^ is plotted on a linear scale, while the fatigue life N is plotted on a logarithmic scale.

The covariance (Cov) (S^sub max^, logN) between the maximum stress level and fatigue life for different lateral stress was calculated. These values are 0.045, 0.041, and 0.042 for a = 0, 0.25, and 0.5, respectively. It is obvious that fatigue response of concrete exhibits significant statistical scatter.

The test points in Fig. 4 through 6 show a good correlation between the fatigue strength and the logarithm of the fatigue life. Examination of data indicates that nonlinear analyses would result in a better correlation. The best-fit curves resulting from the nonlinear regression analysis of data are also shown in Fig. 4 through 6, and their mathematical relationships are given in the following.

For a = 0

... (1)

For a = 0.25

... (2)

For a = 0.5

... (3)

Curves denoted by Eq. (1) through (3) are plotted in Fig. 4 through 6. The three S-N curves expressed by Eq. (1) through (3) are plotted together in Fig. 7. They are close in the 1 = N = 10^sup 4^ regions. It should be mentioned that values f^sub max^/f^sub cc^ > 1 calculated by Eq. (1) through (3) for logN = 1 are induced just by the regressing conic method.

To achieve a unified S-N relationship, a new fatigue equation based on the test results is proposed that contains the lateral stress effect as follows

... (4)

The maximum stress levels f^sub max^/f^sub c^ for uniaxial and biaxial tests are nondimensionalized with respect to the average static strength f^sub c^ of the companion cube specimens tested under uniaxial loading in the monotonic tests. The best-fit curves resulting from the nonlinear regression analysis of the data are also shown in Fig. 8.

In uniaxial fatigue, Eq. (1) predicts fatigue strength levels of 1.2 and 0.75 for one and 10^sup 4^ cycles, respectively. The fatigue strength of concrete under biaxial compression is higher than that under uniaxial compression for any given number of load cycles, as shown in Fig. 8. The predicted fatigue strength levels f^sub max^/f^sub c^ are summarized in Table 2 for different lateral stress ratios and for fatigue lives of one and 10^sup 4^ cycles, respectively. It can be seen that the maximum strength under a lateral stress ratio 0.5 is 30% higher than that under uniaxial conditions corresponding to one cyclic loading, and that it is 35% higher for repeated loading of 10^sup 4^ cycles.

Cyclic deformations

Typical stress-strain curves for the concrete specimens in the vertical and horizontal directions with the lateral stress ratios of 0 and 0.25 are shown in Fig. 9 and 10, respectively. It is evident that the vertical strains have the same characteristics at different lateral stress ratios. However, the horizontal strains at uniaxial and biaxial states of stresses are different. They are distinctively different in three principal aspects: for uniaxial compression, loading in the vertical direction of the specimen induces a tensile strain in the horizontal direction due to the Poisson effect, and the total tensile strains increase with the load cycles, and for biaxial compression, the total strain in the horizontal direction stays compressive during cyclic loading and increases with the number of load cycles. For the biaxial compression state, the compressive strain in the horizontal direction decreases with the increase of the vertical load for any load cycle, and the total compressive strain reaches its maximum value within a cycle when unloading the vertical load to the minimum stress level. Accordingly, the maximum total strains (corresponding to the maximum vertical stress level) of the specimens in the horizontal direction increase with the increase of load cycles. The stress-strain curves in vertical and horizontal directions show hysteresis loops for any lateral stress ratios.

The maximum total strain accumulations in the horizontal and vertical directions for the lateral stress ratio of 0.5 are plotted with respect to normalized load cycles n/N, where n is the number of cycles and N is the number of cycle at failure, as shown in Fig. 11. It is shown that the three have behaviors of the strain evolution in vertical and horizontal directions observed in uniaxial compressive fatigue is also obvious in biaxial compressive fatigue. In the first phase (0 < n/N < 0.1), deformations increase rapidly; in the second phase (0.1 < n/N < 0.85), deformations increase slowly and steadily; in the third phase (0.85 < n/N < 1), deformations increase rapidly.

Failure modes

Two main typical failure modes of the concrete cube specimens are shown in Fig. 12. For specimens tested in uniaxial compression, most of the fractures were due to the formation of cracks in the direction of loading (Fig. 12(a)) and the specimen was split into several short columns in the load direction. For specimens tested in biaxial compression, failure occurred only by splitting in planes parallel to the free surfaces of the specimen, and the specimen was split into several slices (Fig. 12(b)). The failure surfaces for cases under quasi-static and fatigue loading are the same with most of the failure are in the interfaces of coarse aggregate and mortar.

DAMAGE FOR CYCLIC LOADING

The damage theory for concrete using the bounding surface concept is a well-established theory that predicts the behavior of concrete under monotonic tension, compression, biaxial loading, and uniaxial cyclic compression adequately (Suaris et al. 1990; Al-Gadhib et al. 2000). In this paper, it is used to model the damage accumulation in concrete owing to biaxial cyclic loading.

Damage evolution

The damage variable is defined as the area of microcracks in a given cross section. A general description for damage evolution was made by Suaris et al. (1990), only the equations derived in this paper.

The compliance matrix for concrete in compression is proposed by Suaris et al. (1990) as

... (5)

where C is the effective compliance of the material with damage; E^sub 0^ and v are the initial elastic modulus and Poisson's ratio of the material, respectively; ?^sub i^ (i = 1, 2, 3) are the principal damage components; and ß is a parameter introduced as the calibration parameter to match the experimental peak strengths for various stress paths.

The complementary energy per unit volume ?? for damaged states may be written as

... (6)

where ? is the mass density of the material, ?? is the complementary energy per unit volume, and s is the stress tensor.

For biaxial compression, the Cauchy stress tensor in the principal coordinate system degenerates to a vector given by

[-s^sub 1^, 0, -s^sub 3^] (7)

where s^sub 3^ and s^sub 1^ are the stresses in vertical and horizontal direction, respectively; also, s^sub 1^ is constant in this paper.

Substitution of Eq. (5) and (7) into Eq. (6) and consideration of ?^sub 3^ = 0 because there is no stress in this direction yields

... (8)

Differentiating Eq. (8) with respect to ?^sub i^, the energy release rate components are given by

... (9a)

... (9b)

and the equation for the loading surface becomes

... (10)

with the mapping parameter b ranging from an initial value of ? to a limit value of 1 on growth of the loading surface to an eventual coalescence with the bounding surface. R^sub c^ is the critical energy release rate of the concrete and can be calibrated by a standard compression test. The value f is the loading surface whose gradient can be expressed as

... (11)

The incremental form of the principal damage components may be written as (Al-Gadhib et al. 2000)

... (12)

The damage modulus H is expressed as a function of the distance between the loading and the bounding surface, given by

... (13)

where D equals a constant; and <> are Macaulay brackets that set the quantity within to zero if the value is negative. The normalized distance d between the loading and bounding surface is given by

... (14)

The normalization of d in the form shown in Eq. (14) results in a constant value of d along a fixed loading surface. The d = d^sub in^ corresponds to R^sub 0^ when the state point first crosses the initial damage surface during any cycle.

Differentiating R^sub i^ in Eq. (9) with respect to ?^sub 1^ and s^sub 1^, and substituting the results along with Eq. (11) into Eq. (12), the following equation can be achieved

... (15a)

... (15b)

where

... (16a)

... (16b)

... (16c)

... (16d)

... (16e)

Constitutive relationships

The incremental form of the elastic damage constitutive equations may be expressed as (Al-Gadhib et al. 2000)

... (17)

Differentiating C^sub ij^ of Eq. (5) with respect to ?^sub k^ and substituting the results along with Eq. (11) through (15b) into Eq. (17), one obtains

... (18a)

... (18b)

... (18c)

where de^sub i^ are the increments of strains.

VERIFICATION OF MODEL

Calibration of model parameters

The material parameters of the model are obtained from the statistical average values of the monotonic and fatigue loading tests. The Young modulus was evaluated E^sub 0^ = 23.4 GPa; Poisson's ratio v = 0.17; and the parameter R^sub c^ is 2.19 ? 10^sup -3^ MPa, 2.81 ? 10^sup -3^ MPa, 3.83 ? 10^sup -3^ MPa corresponding to a = 0, 0.25, and 0.5, respectively, which denotes the magnitudes of energy release rates when the loading surface f = 0 reaches the corresponding bounding surface. Parameters ß = 0.1 (controlling the damage growth rate and influencing the prepeak behavior) and D = 2.65 (controlling the softening phase of concrete response in stress-strain space) are given by Suaris et al. (1990).

For monotonic loading, the threshold of damage is identified by the initial damage surface f^sub 0^ = 0 with size R^sub 0^. For cyclic loading, however, R^sub 0^ is variable and supposed to increase with each successive cycle and denoted as R^sub 0^ = R^sub 0^(?), where ? = (?^sub i^?^sub i^)^sup 2^ is the magnitude of the damage vector ?^sub i^. The function of R^sub 0^(?) is found to have an elliptical form for cyclic loading (Al-Gadhib et al. 2000). The form of the surface in R^sub 0^-? space may be expressed as

... (19)

where R^sub 0^^sup i^ and ?^sub i^ correspond to the initial size of the limit fracture surface and the associated damage, respectively, and R^sub 0^^sup b^ and ?^sub b^ correspond to the bound surface and associated damage, respectively. The limit fracture surface may reach its bounding surface while the loading surface f = 0 may still be remote from its own conjugate bounding surface F = 0. Consequently, further damage is deemed to occur at a fixed size of limit fracture surface (R^sub 0^ ^sup b^) until damage reaches its limiting value ?^sub m^ and the loading surface f = 0 reaches the bounding surface F = 0, defining incipient failure.

In general, crack initiation in compression occurs at approximately 40% of the peak stress for normal strength concrete, and the inherent initial damage ?^sub 1^ and ?^sub 2^ are evaluated as 0.05 resulting in ?^sub i^ = 0.07. Parameters R^sub 0^ ^sup b^ and ?^sub b^ in Eq. (19) are functions of the maximum stress level. For concrete with f^sub c^ = 20.4 MPa presented by Al-Gadhib and Baluch (1995), it was noted that the bounding damage ?^sub 1^ and ?^sub 2^ are set to 0.5, and ?^sub b^ = 0.71.

Substitution of ?^sub i^ and ?^sub b^ into Eq. (20) yields the evolution relationship between R^sub 0^ and ?.

Comparison with experimental data

The theoretical model in this paper is coded into a computer program to simulate the response under biaxial fatigue compression. With the experimental parameters, the theoretical stress-strain relationships of concrete under different stress levels are obtained. The experimental and theoretical cyclic stress-strain curves for biaxial compression (a = 0.5, s^sub max^/f^sub cc^ = 0.8) are given in Fig. 13, where the calculated fatigue life corresponds to that of the statistical average value of the test. In the vertical direction, it can be seen that most of the experimental stress-strain curves, except the first several cycles, are concave to the stress axis, whereas the theoretical ones are convex to the stress axis as the effect of damage softening is considered in the constitutive relationships. In the horizontal direction, the experimental stress-strain curves are convex to the stress axis, whereas the theoretical ones are convex to the stress axis. With the same stress level and cycles, strains predicted by the theoretical model are smaller than those observed in the test. The reason may be that the plastic strain is generated during the fatigue loading and only elastic damage is considered in the constitutive model.

CONCLUSIONS

Based on the study in this paper, the following conclusions are drawn:

1. The fatigue strength of concrete under biaxial compression is higher than that under uniaxial compression for any given number of load cycles. Approximately 28 and 35% strength increases were achieved at the lateral stress ratios of 0.25 and 0.5, respectively, for a fatigue life of 10^sup 4^ cycles;

2. The stress-strain curves in vertical and horizontal directions show hysteretic loops for any lateral stress levels. For biaxial compression fatigue tests, the total strains in the horizontal loading direction stay compressive during cyclic loading, and the maximum total strains in horizontal direction increase with the number of load cycles;

3. The maximum total strain accumulations in the vertical and horizontal directions exhibit three distinctive stages both for uniaxial and for biaxial cyclic loading;

4. For biaxial tests, cube concrete specimens under cyclic loading failed due to tensile splitting with the fracture surfaces parallel to the free planes. The same failure modes for static and fatigue tests occurred; and

5. The analytical trend of the damage evolution law for concrete for biaxial cyclic loading derived in this paper is in good agreement with the experimental results.

ACKNOWLEDGMENTS

This work was performed under grants from the National Natural Science Foundation of China (Grants No. 50078010, 50225927, and 90210010) and China Postdoctor Foundation (Grant No. 2003033161).

NOTATION

b = mapping parameter

D = parameter defining damage modulus

f^sub c^ = uniaxial compressive strength of concrete

f^sub cc^ = static compressive strength of concrete under biaxial loading

f^sub max^ = maximum stress

H = damage modulus

N = fatigue life

n = number of cycles

R = ratio of minimum to maximum fatigue stress

R^sub c^ = critical energy release rate

R^sub i^ = thermodynamic force conjugates of damage components ?^sub i^

r = correlation coefficient

S^sub max^ = maximum stress ratio

a = ratio of stress in horizontal direction to uniaxial compressive

strength of concrete s1/fc

ß = parameter accounting for cross effect of damage in compression

d = normalized distance between loading surface and bounding surface

e^sub i^ = principal strain

? = complement free energy

? = Poisson's ratio

? = density of the material (concrete)

s^sub 1^ = minimum principal stress [arrow down][arrow up]

s^sub 3^ = maximum principal stress [arrow right][arrow left]

?^sub i^ = components of damage