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INTRODUCTION

The dynamic tensile strength of concrete is of substantial importance in the theory and design of concrete structures. Dynamic behavior of concrete can be attributed to the effects of strain magnitude, strain rate, loading pattern, and so on. The loading pattern plays an important role for the deformation properties of concrete, and three distinct types of it can be identified: 1) monotonically rapid loading that acts in a very short duration, such as explosion and collision; 2) fatigue loading due to a large number of cycles of loading of relatively low stress levels; and 3) relatively small number of loading cycles of rather high stress with varying amplitude, typical of earthquake excitation. The investigation of concrete behavior subjected to cyclically varying loading has great significance for earthquake-resistant structures.

A survey of the technical literature indicates that, though extensive experimental work was conducted on the cyclic behavior of concrete, early research was aimed toward obtaining a fatigue stress level for the material.1,2 Later, considerable effort was expended to derive an empirical formula to simulate general stress-strain behavior of normal concrete under cyclic loading for engineering purpose; more experimental work was placed on the compressive loading3,4 and less was placed on the tensile loading.5 Based on these studies, it was found that within the natural scatter of test results, similar envelopes of stress-strain relationship were obtained for various loading histories. Many researchers have come to the conclusion that a unique envelope curve may well represent the envelopes at all loading histories, and this envelope coincides with the monotonic curve up to failure. This has lead to the material models in modern seismic codes.6 The monotonic loading stress-strain curve is assumed to form the skeleton curve of cyclic loading response. In addition, the test results for confined concrete4,7 show that this assumption is also reasonable for reinforced concrete members.

It is worth mentioning that test results of concrete under random cyclic loading are not yet available up to date.3 Moreover, in the aforementioned research, attention has been focused on the post-peak softening behavior and the general shape of the unloading and reloading curves-the loadings were generally either at very high speeds2 (for fatigue test) or at very slow speed (for stress-strain curve test);2,3 the strain rate effects on the behavior were not taken into consideration. As a consequence, existing codes make the assumption that the dynamic strength of concrete in an earthquake equals the monotonic strength, that is, the cyclic effect is ignored.

RESEARCH SIGNIFICANCE

Earthquake loading is characterized by its cyclic behavior and rate dependency. In this regard, little is known in the current literature. This research aims to increase the nearly nonexistent database of concrete behavior subjected to varying amplitude cyclic tension, and this will be significant for designing structures in seismic active area.

EXPERIMENTAL INVESTIGATION

To simulate the environment of earthquake action as closely as possible, an elevated-amplitude cyclic loading scheme was designed. The influence of the main factors, such as the initial static load intensity, the applied load frequency, and the amplitude increment per load cycle on the dynamic strength of the material were taken into consideration.

The applied loading pattern is shown in Fig. 1. At the first stage, the specimen was subjected to a preliminary static load F0, and then an elevated-amplitude cyclic load was steadily applied until the specimen ended in failure. The cyclic history is expressed as

F(t) F^sub 0^ + A(t) 2(πft) (1)

... (2)

where F(t) is the current load intensity; A(t) is the amplitude of the current cycle, which is varying with time; f is the loading frequency; A^sub 0^ is the amplitude of F at the time instant of t = 1/4f; and ΔA is the amplitude increment per cycle.

Three different loading schemes were employed as given in the following:

1. The frequency of the applied load cycles f was assigned as 0.5 Hz, 2 Hz, 10 Hz, 20 Hz, and 30 Hz, respectively, while the initial static load F0 and the amplitude increment per load cycle ΔA = A^sub i+1^ - A^sub i^(I = 1, 2,) were kept unchanged;

2. Limited by the test system in which simultaneously applying tension and compression in one testing process was not allowed, only two cases of F0 were tested, which correspond to 72.4% and 90.5%, respectively, of the monotonic static failure load. During the test, f was kept as 2 Hz and ΔA was maintained constantly; and

3. Four cases of ΔA were studied.

Test specimens

Dumbbell-shaped specimens were used in the tensile tests, the dimensions of which are shown in Fig. 2. Both ends of the specimens were flared to confine failure to the midheight of the specimen. Three-dimensional finite element analysis shows that, when the tensile load is applied through the end plates, stresses are uniformly distributed throughout the midheight of the specimen. The accompanying specimens used for compressive tests were 3.39 x 3.39 x 3.39 in. (100 x 100 x 100 mm) cubes. The concrete had a characteristic compressive strength of 2900 psi (20 MPa) at 28 days.

Nearly 45 specimens were tested and the fractured location was randomly distributed along the midheight of the specimens. Typical modes of the fractured specimens are shown in Fig. 3. From this, it may be inferred that the preparation for the specimens was adequate.

Ordinary portland cement 32.5R (28-day early strength at 4713.7 psi (32.5 MPa), general mixed sand, crushed gravel with maximum grain size of 0.394 in. (10 mm), and tap water was used. The proportion of the mixture was: cement/water/sand/gravel = 1.00/0.69/2.63/3.39. All specimens were cast in steel molds and compacted by vibration. Having been demolded on the next day, specimens were put in a water tank for 2 days, then cured in a fog room for 28 days. Afterward, they were naturally cured in the laboratory.

The specimens were tested at the age of 300 ± 5 days. It may be reasonably assumed that the properties of the specimens were kept unchanged during the period of testing. The performed cube compression test showed that the compressive strength at the time of experiment was 3959 psi (27.3 MPa). Among the 37 batches of strength test data obtained during the curing period, not one of them was observed with its datum exceeding the average value of the batch by 15%. This ensures the reliability of the test results.

Loading arrangement

Dynamic tests were carried out at the Key Laboratory of Structural Analysis for Industrial Equipment of Dalian University of Technology, China. The loading machine is a closed loop, servo-controlled, advanced material test system with a capacity of 22.48 kips (100 kN). It is programmable displacement control with loading rate up to 5.9 in./second (1.50 mm/second).

The test setup is shown in Fig. 4 and 5. Axial load was applied by a hydraulic actuator. The load was transmitted to the specimen through load platens with ball joints. The specimen was glued to the steel end plates and the end plates were connected to the load platens by bolts. This arrangement produced uniform axial tensile strain in a specimen. After completing the test, the specimen was removed from the steel end plates by putting it into an oven for several hours until the epoxy softened and then it was chipped off the plates.

Instrumentation

The measurement system consists of a strain amplifier, a tape recorder, and an intelligent signal processor. A 10^sup 4^ Hz sampling frequency can be achieved. Four pairs of 1.97 in. (50 mm) long foil strain gauges were used to monitor the longitudinal and transverse strain histories. They were crossly glued onto the four side faces at the midheight of the specimen. Because the specimens were subjected to strains considerably in excess of the normal operating range of the strain gauges after appearance of the cracks, strains were also measured by two pairs of linear variable differential transducers (LVDT) fixed on the side faces of the specimens (refer to Fig. 4 and 5). Two computers were used-one served for control purpose, and the other for data acquisition and processing.

Preparation for test

First, the specimen was fixed in position and the screws of the connecting bolts were adjusted between the steel end plates and the load platens to make the specimen in perfect alignment with the applied load. Then, the specimen was subjected to an initial pressure of the order of 72.5 psi (0.5 MPa) to check whether the readings of deformation of the four longitudinal strain gauges were close to each other. If not, the screws were unloaded and readjusted until the satisfactory state was reached. Finally, the displacement transducers were fixed, the initial voltage of the load amplifier was adjusted, and the initial readings were collected and the computer acquisition system was zeroed. In such a way, the preparations were regarded as adequate. It took 5 to 20 minutes for each tested tensile specimen.

TEST RESULTS AND DISCUSSION

Test results

As described in the preceding section, during the test, specimens were loaded to an initial specified strain and then cycled to produce a given incremental strain per cycle until failure. The recorded loading history is shown in Fig. 6. The typical strain history measured by the strain gauges and the converted stress-strain curves are illustrated in Fig. 7 and 8, respectively. From these readings, the dynamic strength at failure and the strain rate of the loading cycle corresponding to the time instant of failure can be determined. The final results are listed in the first six columns of Table 1, where the tensile strengths are the maximum axial stress at the time instant of the specimen failure. It should be noted that the concrete specimens were always tested in groups of four to five; figures given in the table represent the average value of the test specimens.

Referring to these figures, it can be observed that, although the applied cyclic load stayed symmetrical with respect to the central axis along the time (Fig. 6), the central axis of the cyclic strains gradually shifted upward with an increasing number of cycles (Fig. 7), and the centering position of the cyclic stress-strain curve also shifted with time (Fig. 8). This phenomenon suggests that, under higher initial strain level (more than approximately 70% of the failure strain in this investigation), unrecoverable plastic deformation grows with an increasing number of cycles due to the development of microcracks inside the specimens. This should be taken into consideration in designing concrete structures subjected to earthquake excitation.

The influence of initial static load on the dynamic strength of concrete subjected to varying cyclic loads has been studied. Limited by the test system, only two cases have been tested, where specimens were subjected to an initial static loading of intensity s0 corresponding to 72.4 and 90%, respectively, of the static strength under monotonic loading. The results are given in Table 2. It is apparent that with the rising of initial static loading intensity, the dynamic strength of concrete decreases accordingly.

For comparison, an experiment on the companion specimens subjected to monotonic axial tension with varying strain rates under similar conditions (room temperature was 20 °C [68 °F] and normal water content was 0.3% by weight) was carried out. The strength enhancement with strain rates is typically represented as a dynamic increase factor d^sub f^, that is, the ratio of dynamic to quasi-static strength (refer to Table 3). These results are very similar to those presented in the literature. Details may be referenced to a paper by the authors.8

The relationship between d^sub f^ and the strain rate on the semi-log scale may be approximated by a straight line as shown in Fig. 9. It is expressed as follows

d^sub f^ = 0.134log(ε^sub t^/ε^sub s^) (3)

where d^sub f^ = f^sub t^/f^sub s^; f^sub t^ is the dynamic tensile strength at strain rate ε^sub t^; f^sub s^ is the quasi-static tensile strength at strain rate ε^sub t^;ε^sub t^; is the current strain rate in the range of 10^sup -5^/s to 10^sup -0.3^/s; and is the quasi-static strain rate, 10^sup -5^/s.

Modeling of strain-rate effect

The response of concrete subjected to elevated-amplitude cyclic loading is, in fact, an issue of strain-rate sensitivity. It differs from that of monotonic loading, however. This situation is analyzed in the following.

The stress-strain relation may be expressed as

σ(t) = E(t)ε(t) (4)

where σ, ε, and E represent stress, strain, and elastic modulus, respectively; they are all functions of time t.

Differentiating Eq. (4)

... (5)

where σ = F/S, where S is the cross-sectional area of the specimens. From Fig. 8 it is observed that the slope of the stress-strain curve or the value of E appears nearly unchanged with time. So it may be assumed that dE/dt = 0.

Substituting Eq. (1) into Eq. (5) yields

... (6)

where

... (7)

... (8)

Equation (6) reveals that the effects of strain rate comprise two parts: ε^sub 1^ stands for maximum strain rate of the current cycle and ε^sub 2^ represents the maximum strain rate due to elevated-amplitude increment with time. The phase difference between ε^sub 1^ and ε^sub 2^ is 90 degrees, that is, they do not reach their maximum synchronously.

Because, in general, 2πA [much greater than] ΔA, it can be concluded that the first part ε^sub 1^ dominates. In other words, the strain-rate effect is controlled by the maximum loading rate of the current cycle ε^sub 1^. The rate effect of elevated-amplitude increment ε^sub 2^ is insignificant, and can be neglected. The main physical mechanism of strain-rate effects in the range of ε smaller than approximately 1 s^sup -1^ is due to the viciousness of free water in the nanopores of the hydrates of concrete, or the Stéfan effect.9 In other words, when concrete is subjected to variable-amplitude cyclic loading, once a higher strain-rate in the cycle is reached, the Stéfan effect will lead to the enhancement of the strength. From these findings, it is concluded that the strength enhancement of concrete subjected to elevated-amplitude cyclic loading may be predicted by that of concrete subjected to monotonic loading with the same strain rate.

It is explained in an example as follows. Let concrete be subjected to an elevated-amplitude cyclic loading with initial static stress F^sub 0^/S = 232 psi (1.6 MPa), cyclic frequency f = 9.78 Hz, and amplitude-increment per cycle ΔA/S = 11.6 psi (0.08 MPa).

The tested dynamic strength was f^sub t^ = 417.6 psi (2.88 MPa). This value can be predicted based on the experimental data under monotonic loading conditions. From the recorded amplitude of the cycle at failure A/S = 185.6 psi (1.3 MPa), and the elastic modulus E = 4.175 × 10^sup 6^ psi (29 × 10^sup 3^ MPa), it is found that maximum (ε^sub 1^) = 2.73 × 10^sup -3^/s and maximum (ε^sub 2^) = 2.71 × 10^sup -5^/s. Then, according to Table 3 or Eq. (3), the mean value of predicted dynamic strength σ = 407.5 psi (2.81 MPa) is arrived at and the standard deviation Δσ = 20.4 psi (0.141 MPa). Finally, the predicted upper bound and lower bound of the dynamic strength are determined as σ + Δσ = 427.9 psi (2.9 MPa) and σ - Δσ = 387 psi (2.6 MPa), respectively. The actual strength of 417.6 psi (2.88 MPa) lies within these bounding values. Keeping this in mind, the predicted and the tested results are compared in Table 1. They are also plotted in Fig. 10 and 11. The agreement is fine. Experimental data approach the upper bound of the predicted results. This may be partly explained by the research by Ballatore and Bocca.10 During cyclic loading, all specimens have absorbed energy, which produces strain hardening of the material and partly increases the failure load in consequence of changes to the concrete matrix.

These findings may be extended to predict the dynamic strength of concrete subjected to random cyclic loading, such as earthquake excitation. The prediction may be proceeded in the same manner as before, that is, at each critical loading cycle: first, the maximum strain rate is determined and then, based on Eq. (3), the dynamic strength is estimated. This simplifies the safety assessment of concrete structures to withstand strong earthquake shocks.

CONCLUSIONS

The experiment on the response of concrete subjected to elevated-amplitude cyclic tension has been carried out. The effect of initial static loading intensity, the frequency of loading cycle, and the amplitude increment per cycle on the dynamic strength of concrete are examined. The main conclusions are summarized as follows:

1. The strength enhancement is dominated by the maximum strain rate of the current loading cycle. The rate effect of elevated-amplitude increment is fairly slight and may be neglected;

2. From the limited data in this investigation, it was found that the initial static loading intensity plays an important role in the enhancement of concrete strength. The higher the initial static loading intensity is, the lower the strength enhancement takes place. In addition, for concrete subjected to cyclic loading under higher initial static strain level, unrecoverable plastic deformation grows with an increasing number of cycles; and

3. The strength enhancement of concrete subjected to elevated-amplitude cyclic loading may be predicted by that of concrete subjected to monotonic loading with the same strain rate. These findings may be extended to predict the dynamic strength of concrete subjected to random cyclic loading, such as earthquake excitation.

ACKNOWLEDGMENTS

This research program is supported by the National Natural Science Foundation of China under Grants No. 50139010 and No. 90510018 at Dalian University of Technology.

NOTATION

A(t) = amplitude of cyclic loading

d^sub f^ = dynamic increase factor

E = elastic modulus

F^sub 0^ = initial static load

F(t) = current load

f = frequency

f^sub s^ = quasi-static tensile strength at strain rate ε^sub s^

f^sub t^ = dynamic tensile strength at strain rate ε^sub t^

S = cross-sectional area of specimens

t = variable of time

ΔA = amplitude increment per cycle.

Δσ = standard deviation of s

Δσ^sub e^ = standard deviation of se

ε^sub 1^ = maximum strain rate of current cycle

ε^sub 2^ = maximum strain rate due to elevated-amplitude increment

ε^sub s^ = quasi-static strain rate, 10-5/s

ε^sub t^ = current strain rate within range of 10-5/s to 10-0.3/s

σ = predicted dynamic strength of concrete subjected to elevatedamplitude cyclic loading based on experimental data obtained under monotonic loading conditions

σ^sub 0^ = initial static loading intensity

σ^sub e^ = mean value of test results for specimens subjected to monotonic loading at given strain rate

σ, ε = stress and strain, respectively