The size dependence of flexural properties of cement mortar and concrete beams is investigated. Bazant’s size effect law and modified size effect law by Kim and Eo give a very good fit to the flexural strength of both cement mortar and concrete. As observed in the test results, a strong size effect in flexural strength is found in cement mortar than in concrete. A modification has been suggested to Li’s equation for describing the stress-strain curve of cement mortar and concrete by incorporating two different correction factors, the factors contained in the modified equation being established empirically as a function of specimen size. A comparison of the predictions of this equation with test data generated in this study shows good agreement.
It is widely believed that the true fracture properties of concrete structures can be unequivocally determined only by means of uniaxial tensile stress [
It is important to consider the effect of size while estimating the ultimate strength of a concrete member under various loading conditions. Presently, most design codes for concrete structures do not consider the effect of size. Since quasibrittle materials fail by formation of cracks, size effect has to be implemented. The influence of specimen diameter [
The size effects on the strength of concrete materials have been generally explained as a macroscopic phenomenon resulting from internal microscopic defects such as microcracks and inclusions in the material. Mazars et al. [
Knowledge of the stress-strain relationship of concrete is vital to achieve a rational design of structures and structure components involving concrete. The nonlinear stress-strain behavior, size-dependent strength, fracture mode transition, and other phenomena are not adequately explained by the classical concepts of the mechanical behavior of materials and mechanics of fracture [
In this paper, the authors discuss the size effects on cement mortar and concrete flexural behavior quantitatively. The equations to estimate stress-strain curves were also proposed. Results of the investigation on the validity of the existing models are reported as well. This paper brings together the results of past studies and the results of new experiments and interpretations.
Ordinary Portland cement (OPC) was used in the production of cement mortar specimens. The cement was the most widely used cement in general concrete construction works in China. The fine aggregate was river sand consisting mainly of quartz, with 10 percent feldspar. The gradation test showed that the particle size of the sand was continuously distributed within the range of 0.4–2.5 mm with 80% of sand. The water-cement ratio (
Test specimens of three different sizes were cast. Their dimensions were (width × depth × length):
Concrete specimens with beam depths of 40, 100, and 150 mm.
As aforementioned, in order to study the influence of specimen size on flexural properties of cement mortar and concrete, four-point bending tests on beams were conducted. The testing was conducted using a dynamic loading system that applied a programmable controlled loading. All these tests were performed with a closed loop servo-controlled stiff testing machine. The load was applied at an approximately constant deformation rate of 0.25 kN/s. The test setup is shown in Figure
Picture of test setup for four-point loading.
Typical load time curves for concrete specimen.
Failure mode of concrete specimens after flexural tests.
The discussion of the results is organized as follows. First we present and describe the strength results obtained from the tests. Then we proceed to discuss the size effect in stress-strain curves. In addition, the test results were compared with existing scaling laws.
The flexural strength of cement mortar and concrete measured from different sizes of beams are shown in Figures
Influence of beam size on flexural strength of cement mortar.
Influence of beam size on flexural strength of concrete.
Glucklich and Cohen [
The other explanation of the size effect on flexural strength is to be found in the theory of quasibrittle fracture, describing materials of heterogeneous microstructure in which the formation of distinct fractures is preceded by distributed cracking. The failure of a beam begins by distributed cracking that develops in a boundary layer. The thickness of this layer for different beam sizes is about the same, provided the same concrete is considered. Hillerborg et al. [
Some relationships involving strength and size of engineering materials have been reported in the literature. Historically, several general types of model have been developed for cement-based materials. As stated above, the aim of the investigation is to judge the range of applicability of the various size effect formulae available in the literature.
On the basis of the experimental results, an empirical expression was proposed by Xu and He [
Picture of test setup for direct tension test.
Based on a similar strain-gradient approach, a size dependent empirical relationship for the flexural strength has been specified in the CEB-FIP Model Code 1990 [
In the analysis of the flexural strength of concrete, Rokugo et al. [
According to Bazant [
In fact, (
By considering a constant maximum flaw size, as is likely to occur in real materials, Kim and Eo [
Chen et al. [
Karihaloo et al. [
By plotting these relationships against the experimental results in this study (Figure
Comparison of test data with existing scaling laws: (a) cement mortar; (b) concrete.
The stress-strain curves of cement mortar and concrete with different sizes are shown in Figures
Influence of beam size on stress-strain curves of cement mortar: (a) beam depth 40 mm; (b) beam depth 100 mm; (c) beam depth 150 mm.
Influence of beam size on stress-strain curves of concrete: (a) beam depth 40 mm; (b) beam depth 100 mm; (c) beam depth 150 mm.
Damage mechanics has become a powerful tool for the modeling of the nonlinear behavior of materials subjected to a progressive microcracking process. The theory itself was initially developed in 1958 [
Among many types of concrete damage models at hand, one established model [
According to the equivalent strain principle in damage mechanics, the damage constitutive relationship of concrete was established by Li et al. [
The effect of specimen size was taken into account by assuming that the threshold strain ( for cement mortar,
for concrete,
The predictions of the analytical model thus computed are compared with the experimental data for different sizes of cement mortar and concrete in Figures
Comparison of predicted results with test data for cement mortar: (a) beam depth 40 mm; (b) beam depth 100 mm; (c) beam depth 150 mm.
Comparison of predicted results with test data for concrete: (a) beam depth 40 mm; (b) beam depth 100 mm; (c) beam depth 150 mm.
In this study, the flexural behavior of cement mortar and concrete with different sizes were investigated. From the test results, the following conclusions can be drawn. Test results of flexural strength show size effect. Large specimens resist less in terms of stress than the smaller ones. A strong size effect in flexural strength is found in cement mortar than in concrete. The size effect in flexural strength is compared with existing scaling laws; it is found that Bazant’s size effect law and the modified size effect law proposed by Kim and Eo are in agreement with the test results. A modification has been suggested to Li’s equation for describing the stress-strain curve of cement mortar and concrete by incorporating two different correction factors, the factors contained in the modified equation being established empirically as a function of specimen size. A comparison of the predictions of this equation with test data generated in this study shows good agreement.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the National Natural Science Foundation of China (Grant no. 51178162) for the financial support.