Effect of size on biaxial flexural strength for cement-based materials by using a triangular plate method

Abstract

The effect of size on the biaxial flexural strength (BFS) of Portland cement mortar was investigated by using the recently proposed triangular plate method (TPM). An experimental program was conceived to study the size effect by keeping a constant water-cement ratio of 0.485, cement-sand ratio of 1:2.75, and using unreinforced triangular mortar plates of five different thicknesses and seven different side lengths. The BFS of the produced specimens was tested, and variations of BFS depending on specimen thickness and side length were determined. The results indicated that increases in triangular plate specimen side length and specimen thickness led to a decrease in the BFS of Portland cement mortar. The effect of specimen length increase on BFS was more significant than on the effect of the specimen thickness. The variations in specimens’ thickness indicated a deterministic Type I size effect, while the variations in specimens’ length showed an energetic-statistical Type I size effect.

Graphical abstract

Keywords

testing / apparatus & methods / plain concrete / tensile properties / biaxial flexural strength / triangular plate method

1 Introduction

Tensile strength is an essential property in the analysis and design of structural elements that are produced with cement-based materials [1]. The tensile strength is generally determined by using uniaxial direct and indirect tension test setups such as splitting or flexural (three or four-point loading) tests. Although the uniaxial direct tension test provides more accurate and reliable results in measuring the tensile strength of cement-based materials, such tests are difficult to perform, requiring a tension machine, and so indirect tests (splitting and flexural tensile tests) are widely preferred [2–4]. Moreover, the direct test is quite sensitive to variations in apparatus and loading procedures [5]. The splitting test is believed to provide similar (5% to 12% higher) tensile strength values to those measured using the direct tensile strength test [6,7]. The splitting test is not a single tensile test but an integration of tension and compression tests [5]. Specimens in flexural tests are exposed to three-point or four-point bending load up to the failure of the beam by flexure. The maximum theoretical tensile stress is known as the rupture modulus and is calculated by assuming a linear distribution of flexural stress across the section at failure. The nonlinear stress-strain characteristics of the concrete in the tension zone may cause an overestimation of flexural tensile strength [8]. The maximum uniaxial direct tensile stress has been reported as 0.735 times the rupture modulus [9]. The stress state at failure in the direct tension, flexural, and splitting tests is uniaxial [10]. However, thin plate structures such as pavements, slabs, and roofs are exposed to multi-axial stress states due to the complex loading configurations and the geometry of plate structures. Identifying the correct multi-axial properties of the materials is important for estimating the safety of plate structures in the non-uniaxial stress state [11]. Despite the importance, there are not many reports on the multi-axial behaviour of concrete due to the requirement of a multi-actuator control system [11,12], which is costly and difficult to control [11]. Therefore, the triangular plate method (TPM) [13] has been proposed recently to determine the BFS of quasi-brittle materials such as concrete and mortar. The BFS measured by using the TPM was about 22% lower than that by the flexural test [13]. The uniaxial or biaxial flexural strength of quasi-brittle and defect-sensitive materials such as concrete depends to a significant extent on the dimensions of the specimens tested [10,14–17]. This phenomenon is called size effect and indicates that the probability of finding a crack of critical size and orientation increases with an increase in the number of cracks, i.e., with the volume of the specimen [18,19]. The strengths of quasi-brittle materials are significantly influenced by the imperfections, and the ultimate load carrying capacity of the specimen is related to specimen size [20]. Increasing the size of a specimen is expected to cause a decrease in load carrying capacity due to the higher probability of imperfections. Ince et al. [21] reported that ultimate tensile strength is significantly different when the same tensile strength tests are applied on two geometrically identical specimens with different sizes. However, Bažant [22] and Carloni et al. [23] described the fracture-type size effect at the beginning of 1990 and called it the size effect law (SEL). The SEL has been widely accepted and validated by data in the literature. The SEL was developed based on the energy balance at the onset of crack propagation and the dimensional analysis applied to the geometrically similar specimens [24]. Bažant et al. [25] have described two types of size effect (Type I and Type II) for quasi-brittle structures.

Type I size effect involves the highest peak load when a macro crack starts from one representative volume element (RVE). Type I size effect is mainly observed in a structure with a clear boundary. The complete Type I size effect can be calculated by combination of the energetic size effect and the statistical size effect [26]. Type I size effect can also be detected from a finite weakest link model in which the structure is statistically represented by a finite chain of RVEs. The probability distribution of RVE strength is derived from fracture mechanics of nano-cracks [27,28]. The maximum load of a structure in the Type II size effect can be obtained right after the initiation of a single crack. Type II size effect applies to quasi-brittle structures containing a large notch or a large stress-free crack formed before the maximum load. The biaxial tensile strength of circular plates in different sizes have been investigated by Zi et al. [10]. The size effect on the equi-biaxial tensile strength of concrete was reported as stronger than that on the uniaxial tensile strength.

In summary, previous studies [10,14,15,17,19,29–33] have shown that tensile strength depends on the size of the specimen. For the standard TPM test, it is necessary to come up with a dimension of Triangular plate specimens. This study shows the effect of specimen sizes on tensile strength and helps in choosing the correct specimen sizes. In this study, an experimental program was designed to investigate the size effect on BFS of cement-based materials by using a recently proposed TPM. Unreinforced triangular mortar plates of five different specimen thicknesses and seven different side lengths were produced to determine the BFS. Variations in BFS depending on the specimen thickness and side lengths were examined.

2 Triangular plate method

2.1 General introduction and analytical solution

Testing methods to determine the BFS of concretes, rocks, ceramics and glasses such as piston-on-three-balls test, ball-on-three-balls test, ring-on-ring test, ball-on-ring test, and biaxial flexural tests have been introduced and used in several studies [6,11,29]. However, the aforementioned methods have some application difficulties due to issues such as surface smoothness, contact surface, and eccentricity problems. The load should be uniformly applied to the specimen to maintain the state of equi-biaxial tensile stress. The surface of the specimen should be as smooth as possible during the test. Kim et al. [11] placed an annular soft rubber layer and four Teflon sheets between the rings and top and bottom of the specimen to smooth its surface and to eliminate the eccentricity. Moreover, Kim et al. [11] treated the specimen-ring contact surface with high-strength gypsum, in a similar manner that is used for the capping of concrete cylinders for compression testing. The modified biaxial flexural test (or modified ring-on-ring test) proposed by Kim et al. [11] revealed that too much effort is needed to construct the test design and to prepare specimens such as by capping of disc specimens from the bottom and topside, and installation of teflon and rubber layers. Despite the tedious test arrangement and specimen preparation procedure, the eccentricity problem might not be eliminated due to the use of loading and support rings. Therefore, using ball-on-three-ball test in defining the biaxial tensile strengths of cement-based materials is more reasonable than using a modified biaxial flexural test of Kim et al. [11]. The ball-on-three-ball test does not require any complicated testing arrangement or specimen preparation such as by capping or by use of rubber/teflon sheets. However, support balls in ball-on-three-ball test must be placed far from the side of the specimen to eliminate side cracks. Turker [13] recently proposed a new method namely, triangular plate method (TPM), which uses triangular plate specimens (Fig.1). The load in triangular plate specimens is applied from the center of mass of specimens with the aid of spherical silver ball. The supports are constructed to the one-third points of the triangular medians. Thus, supports are moved away from the sides to prevent the formation of side cracks. Support and loading balls are gripped to the base plate and load-cell by using hexagonal nuts (Fig.1).

A complete picture and testing phase of the suggested design is shown in Fig.2. The main advantages of the TPM are the simplicity of the procedure, high tolerance for an uneven flatness of the specimen surface, and easiness in specimen preparation. The fracture load is measured in the test. The tensile strength is defined as the maximum principal tensile stress in the triangular plate specimen and occurs on the specimen surface across the central loading ball [29]. Turker [13] has derived the Eq. (1) by using the yield line method based on plasticity theory for the maximum BFS on the soffit. In the yield line method, several possible solutions can be accomplished due to each possible failure mechanism. Therefore, understanding the exact mechanism of failure is quite important.

The failure mechanism shown in Fig.3 was identified as the exact failure mechanism for the triangular plate specimens. Based on the work method for failure mechanism (Fig.3), internal and external energy are equilibrated to obtain biaxial flexural tensile strength of the triangular plate specimens. The equation for the maximum stress on the soffit was derived as follow:

(1){\sigma }_{\mathrm{t}}=\frac{2P}{t^2\sqrt{3}},

where P is the peak load and t is the triangular plate thickness. The stress σ is a nonlinear function of the specimen thickness and linear function of the ultimate load. Details of the TPM, numerical determination of the stress field and verification of the TPM might be found in Ref. [13].

Yield line theory has been developed for reinforced concrete, and assumes that the bending moment along a line or lines reaches a yield value, and stays constant until other parts of the line reach that yield value. Thus, when a failure occurs, a pattern of yield lines develops with a constant moment along each line. The yield line method is normally used for ductile materials since it assumes the existence of plastic hinges which are not found in brittle materials. All the assumptions considered in yield line theory have been adopted to derive the Eq. (1). The plastic hinges was replaced with the energy needed to cause crack formation. The failure mechanism was related to the presence and the stable growth of a relatively large fracture-zone placed along the yield lines. The fracture zone grew along the crack tip prior to the failure of the structure.

The failure in experimental specimens took place along the patterns of yielding predicted by yield line theory. Also, strength values predicted by yield line theory for the unreinforced plate had a reasonable agreement with the experiment and Finite Element analysis for many spans and load conditions. The crack patterns causing the failure form at once at the failure load. Since the load versus displacement curves reach the failure load linearly and then the cracking patterns causing the failure are formed immediately, load redistribution is not needed. The linearity of the load-displacement relationship up to the fracture moment indicates that no load transmission or the breaking mechanism occurs at the same time.

2.2 Numerical determination of the stress field

The Eq. (1), derived to find the maximum tensile stress developed in the plate, is based on the assumption that the tensile stresses will be constant along the yield lines at the time of fracture. The analytically obtained equation indicates that the tensile stress depends only on the load and the plate thickness. It means that modulus of elasticity and plate size have no effect on the tensile stress. Therefore, to check the accuracy of this equation, linear and non-linear finite element analyses were performed to determine the tensile stress field in the loaded triangular plates which have different side lengths, thicknesses and elastic modules. The triangular plates were analyzed using both static analysis available in ABAQUS/Standard and quasi-static analysis available in ABAQUS/Explicit. In the static analysis, a constant force was applied through a rigid ball in contact with the upper surface of the triangular plate above the center of gravity. In the quasi-static analysis, an incremental displacement with a low velocity was applied to the center of the plate. The triangular plates were modelled using three-dimensional 8-noded hexahedral solid elements (C3D8R), and the contacts between the balls and the plate were modelled by surface to-surface contact elements. The supporting balls were modeled using rigid elements.

The dependence of the tensile stress field was examined for various combinations of triangular plate side lengths, plate thicknesses, and elastic modules. The Fig.4 shows the stress distribution obtained by linear-elastic analysis. The elastic analysis shows that the stresses are more intense in the direction of the yield lines, but they are not uniform along the lines.

The tensile stress variation in the direction of the yield lines is shown in the Fig.5. The graph demonstrates that if the thicknesses are the same, the stresses are the same even if the plate side lengths and the modulus of elasticity are different. The stresses are different only in the region close to the center of the plate where the point load acts. From the Fig.5, it can be concluded that the tensile stress depends on the load and the plate thickness. In addition, the stresses vary along the yield lines. The equation used to calculate the maximum tensile stress is an oversimplification of the real situation, which can be used for a first estimation of tensile strength. However, considering factors such as the ratio between the size of the ball used for loading and the plate size and the non-uniformity of the stress distribution along yield line, Eq. (1) is re-written as follows:

(2){\sigma }_{\mathrm{t}}=\frac{\beta P}{t^2},

where β is a dimensionless coefficient that depends on the factors mentioned above. Extensive experimental and numerical studies are required to determine the value of β and these are not within the scope of this article. However, preliminary studies indicated that β is likely to vary between 1.2 and 2.

A normalized plot of the 1st principal stresses along the yield line is shown in Fig.6 for two different elastic modules, two different plate side lengths and two different plate thicknesses. Combinations of elastic modules, length and thickness of plate within the given parameter range lead to the same tensile stress within a deviation smaller than 2%.

Nonlinear finite element analysis of the triangular plates under a static load were conducted to investigate the plates’ failure modes in terms of cracking pattern and ultimate load capacity. The maximum principal strain at the bottom surface of the triangular plate is shown in Fig.7. The maximum principal strain develops first at the point on the bottom surface of the plate below the center of gravity. Thus, the cracks are expected to initiate in this region. Next, three cracks initiated and traveled towards the three edges of the plates breaking the specimens into three parts. According to Fig.7, the numerically obtained crack patterns using the concrete damage plasticity model closely predict the experimentally observed cracks in the specimens.

3 Experimental investigations

3.1 Materials and specimen preparation

Triangular plate and prism specimens were produced to test the biaxial and axial flexural strengths of mortar specimens, respectively. The BFS was determined by using TPM while axial flexural strength was measured by three-point bending tests. Mortars with water-cement ratio (w/c) of 0.485 and cement-sand ratio of 1:2.75 by weight were produced. Type I (CEM I/42.5 R) Portland cement and a mixture of river sand and crushed limestone sand with a maximum grain size of 4 mm were used. For the three-point bending test, 4 cm × 4 cm × 16 cm prism specimens were produced. In addition, six 10 cm × 10 cm × 10 cm cubic specimens were produced to determine the compressive strength of the mortar specimens. All specimens were cast from the same fresh mortar batch and cured for a day at room temperature. The specimens were then demolded and stored in a water bath for 28 d. The average compressive strengths of six specimens at the 28th day were measured as 37.5 MPa.

3.2 Specimens for size effect

The dimensions of the specimens used for the ETPM are presented in Tab.1. Size effect was studied with two groups of specimens. In the first group, unreinforced triangular mortar plates were produced with specimen thickness ranging from 2 to 10 cm with a 2 cm increment. The side length of the triangular was kept constant at 40 cm. The minimum thickness was identified to be approximately five times greater than the maximum size of the fine aggregate (4 mm) used in mix proportions. In the second group, however, specimen thickness was kept constant at 4 cm while the triangular plate side length was changed from 20 to 80 cm with a 10 cm increment. Five specimens were produced for each configuration to obtain reliable test results and specimens were tested by using the TPM. Fifteen and twenty specimens were produced for the first and second groups, respectively. In order to prevent failure in the demolding and carrying stages and difficulty of handling, triangular plate specimen thicknesses were set to 2 cm as the lowest, 10 cm as the highest, with maximum side length of 80 cm. The TPM test was performed by applying concentric loads of which the magnitudes were monitored and recorded. The failure of biaxial stress was determined by using Eq. (1).

4 Experimental results and discussion

4.1 Fracture mechanism

Fracture mechanism or fracture pattern of the prism and triangular plate specimens are presented in Fig.4 and Fig.5. The three-point bending test was applied to determine the axial flexural strength of the mortars. The prism specimens failed as expected due to initiation of single cracks at the bottom surfaces and extension of those cracks up to the top of the prism specimen (Fig.8). However, in all fractured triangular plate specimens, the fracture initiated on the tensile surface plane underneath the loading ball (Fig.9). The cracks in almost all tested specimens occurred as symmetric triples regardless of the specimen thickness and side length (Fig.9). Only one specimen had a single straight crack as shown in Fig.9(c). The failure mechanism of the triangular plate specimens involved symmetric triple cracks; whose locations were determined by the three supports (Fig.9). Zi et al. [10] also observed similar fracture patterns in which the biaxial tensile strength of the cement-based disc specimens was tested by the biaxial flexure test. The symmetric triple cracks type failure mechanism was attributed to the minimization of the Helmholtz free energy of a homogeneous isotropic material subjected to an equi-biaxial tension field.

4.2 Flexural strength measurement with TPM and three-point bending test

The details of the TPM have been given in Turker [13]. The triangular plate specimens with a side length of 30 cm and thickness of 4 cm and prisms with a section of 4 cm × 4 cm and length of 16 cm were tested to investigate the relationship between the TPM and three-point bending method. Both types of specimens were produced from the same materials, all cast from the same fresh mortar batch and cured for a day at room temperature. The specimens were then demolded and stored in a water bath for 28 d. The mean compressive strength of six 10 cm × 10 cm × 10 cm cubic specimens was previously determined as 37.5 MPa (see above). Test results showed that the mean biaxial tensile strength values obtained from the testing of triangular plate specimens were considerably lower than those obtained from the prism specimens (Tab.2). The overall tensile strength ranged from 5.25 to 6.66 MPa and 7.48 to 8.70 MPa for biaxial and uniaxial tests, respectively. Mean biaxial tensile strengths of triangular plate and prism specimens tested were 6.10 and 7.91 MPa, respectively. The ratio of mean strength for a three-point bending test and that of the biaxial tensile strength (biaxial/uniaxial) was 0.77. The maximum uniaxial direct tensile stress value was 0.735 times the conventional modulus of rupture value. Presented test results indicated that the TPM is a convenient and dependable way of determining the biaxial tensile strength of cement-based materials.

4.3 Size effect on biaxial flexural strength

The results of TPM depending on the specimen thickness and side length are presented in Tab.3 and Tab.4. Column 6th of Tab.3 and Tab.4 represents the mean values of the five companion triangular plate specimens. The standard deviation of the BFS ranged from 0.16 to 0.62 and from 0.18 to 0.37 for the first and second group specimens (constant specimen thickness and varied side lengths) (Tab.3 and Tab.4). The narrow range of standard deviation values is an indication of the consistent repeatability of the TPM results.

The size effect was observed as expected in both specimen groups. The variation of mean nominal BFS of the mortar specimens depending on the specimen thickness is shown in Fig.10. The mean BFS varied between 5.25 and 6.79 MPa. The BFS decreased from 6.79 to 5.25 MPa as the specimen thickness increased from 2 to 10 cm. Percent reduction in BFS at 10 cm specimen thickness resulted in an asymptote which indicated that size effect due to the specimen thickness increment might be ambiguous when the specimen thickness is higher than 10 cm. The results obtained from the specimens which have no notch indicated a trend consistent with the Type I size effect.

Several researchers investigated the size effect by using a statistical approach. Smaller specimens have a fewer number of micro-cracks and defects compared to larger specimens [16,30]. While others used the Weibull theory of statistical size effect to explain the effect of specimen thickness on the BFS [31,32], the chain model of Bažant et al. [33] where a specimen is hypothetically formed by many elements and fails as the weakest element is broken helps to understand Weibull theory. Nguyen et al. [17] indicated that thick specimens contain more elements in the chain and have higher probabilities of failure; therefore, the thick specimens become weaker than the thin specimens. The thinner specimens are easily cured, dehydrated and dried compared to those of the thicker specimens, which may lead to a considerable increase in the BFS of the thinner specimens at the early ages such as 28 d. However, the aforementioned effect except for the effect of number of micro-cracks and defects might not be observed in the long term.

Since notches were not observed on the specimens, the maximum load is assumed to be reached before micro cracking localized into a distinct macro-crack. In other words, the size effect is assumed to be of Type I effect that can be expressed as:

(3){\sigma }_{\mathrm{N}}=f_{\mathrm{t}}\left(1+\frac{d_{\mathrm{o}}}{D}\right).

The comparison of the Type I SEL and the experimental data on BFS is shown in Fig.11. The data optimally fitted to f_t = 4.87, d_o = 7.94.

The effects of specimen side length on the BFS at constant specimen thickness (4 cm) are illustrated in Fig.12. The mean BFS varied between 5.37 and 9.39 MPa. The increase in the side length caused a gradual decrease in BFS of the triangular plate specimens. Distinctively from the size effect of specimen thickness, an asymptotical decreasing trend was obtained in percent reduction of BFS. A four-fold increase in side length resulted in a 42% decrease in BFS. An asymptotical effect on the size effect of BFS was observed in the effect of side length increase. Furthermore, size effect that resulted from the increase in specimen side length was more evident than that resulting from the increase in specimen thickness. The distance among the supports may have contributed to the remarkable decrease in BFS caused by the side length increase. The specimen thickness in the second group specimens kept constant (4 cm) and side length changed from 20 to 80 cm with a 10 cm increment rate. That means the side length/specimen thickness ratio was not constant and altered for each case. Therefore, the distance among the supports was also increased for each case, which significantly affected the BFS. The highest decrease in BFS caused by the increase in specimen thickness was around 20%. As seen in Eq. (1), the BFS is not dependent on the side length in the TPM. Normally, side length change may not affect the BFS of the cement-based materials. However, the effect of increase in side length led to an evident decrease in the BFS compared the increase in specimen thickness. The presence of micro-cracks due to the plastic shrinkage may also cause such behavior. With constant specimen thickness at 4 cm, increases in the side length from 20 to 80 cm caused an increase in the surface area of the specimens from 173.2 to 2771.3 cm². The plastic shrinkage cracking likely occurs when a high level of evaporation causes the specimen surface to dry out. Therefore, increasing the specimen surface area indicated an increase in the probability of the presence of plastic shrinkage cracking.

The data obtained could be explained in terms of the Type I size effect. Therefore, the data obtained from the experiments fitted to Eq. (1).

The comparison of the Type I SEL and the experimental data on BFS given in Tab.4 is presented in Fig.13. The data are optimally fitted to f_t = 4.43, d_o = 5.9. The data from the experiment showed that nominal strength reduction for the case with length increase was larger than for the case with depth increase. The f_t value obtained for length increase was not consistent with the value of f_t for depth variations.

5 Conclusions

Size effect on the BFS of Portland cement mortar was investigated by using the recently proposed TPM method. Size effect was studied depending on the triangular specimen thickness and side length parameters. Overall, the test results demonstrated that both parameters studied are highly influential on the BFS and showed the size effect. The observations and conclusions drawn from the study are as follows.

1) The BFS values obtained by using the TPM which produced consistent results with low standard deviation values are more relevant to direct tensile strength than those obtained with the three-point bending test.

2) Testing the triangular plate specimens by using the TPM indicated a completely different failure mechanism than that shown in prism specimens. Symmetrically triple cracks were monitored in almost all triangular plate specimens, irrespective of the specimen thicknesses and side lengths studied.

3) The results of measurements on specimens of varying thickness indicated a deterministic Type I size effect.

4) The size effect on the BFS of a specimen length increase was found to be stronger than on the BFS of the specimen’s thickness increase.

5) The increase in specimen thickness caused a decrease in BFS of the triangular plate specimens, and the maximum rate of reduction was 22.7%.

6) Increasing the side length of triangular plate specimens led to a decrease in BFS of Portland cement mortars. The larger the specimen side length, the lower was the observation of biaxial flexural strength. The effect of side length on the BFS was more prominent than the effect of specimen thickness. The decrease in BFS was around 42% due to the four times increase of side length.

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