Research Article | Open Access
A Stress-Strain Model for Brick Prism under Uniaxial Compression
This study proposes a simple and rational stress-strain relationship model applicable to brick masonry under compression. The brick prism compression tests were conducted with different mortar strengths and with constant brick strength. From the observation of the test results, shape of the stress-strain curve is assumed to be parabola. In developing the stress-strain model, the modulus of elasticity, the strain at peak stress, and the strain at 50% of the peak stress on the descending branch were formulated from regression analysis using test data. Numerical and statistical analyses were then performed to derive equations for the key parameter to determine the slopes at the ascending and descending branches of the stress-strain curve shape. The reliability of the proposed model was examined by comparisons with actual stress-strain curves obtained from the tests and the existing model. The proposed model in this study turned out to be more accurate and easier to handle than previous models so that it is expected to contribute towards the mathematical simplicity of analytical modeling.
Masonry is a material built from units and mortar that induce an anisotropic behavior for the composite. The lack of knowledge on the properties of the composite material imposes low assessments of the strength capacity of the masonry wall. Atkinson et al.  state that the prediction of compressive and deformation of full-scale masonry based on compressive test of stack-bond masonry prism and the interpretation of the results prism tests have a significant influence on the allowable stress and stiffness used in the masonry design. When structural masonry is subjected to vertical and horizontal loading, one of the most important parameters for design is the stress-strain relationship. Especially, the stress-strain relationship of concrete brick prism in compression is essential for the analysis of masonry structures. The relationship is generally known to depend on several interrelated test parameters including compressive strength of bricks and mortar. Many mathematical models have been proposed for accurate finite element models and structural analysis of concretes in compression. Existing stress-strain models for concretes [2–5] used the basic expression established by Popovics  or Sargin et al. , and the constants in the basic expression were determined empirically. In some models, the ascending and descending branches were dealt with separately with nonlinear equations; in this case, the test data were essential to establish the empirical constants. Hence, some limitations such as applicable ranges of concrete strength and concrete density exist. Knutson  evaluated the stress-strain diagrams for various materials and showed that they can be cast into a mathematical form. However, Mohamad et al.  mentioned a complete understanding of the mechanisms involved in the deformation and failure which are not fully explained. It is believed that the development of a theoretical model of universal application is a rather hard task, although there have been very nice efforts to propose simplified mathematical models for the stress-stain relation [10, 11]. When modeling masonry structure in common FEM software such as Abaqus  and LS-DYNA , it is not possible to correctly model and predict the behavior of masonry structures primarily due to the lack of the references that fully define the plastic behavior of masonry. In design process, just as in analysis, the accurate design code considering elastic and plastic properties of the masonry is not given, either.
Although the brittle materials such as concrete have similar issues, a model proposed by Yang et al.  explained the stress-strain behavior of it in compression quite successfully. The study calibrated the mathematical equation for the stress-strain curve using material test results. As mentioned above, the pure theoretical development for the behavior of the concrete brick prism is rather a difficult task. However, the theoretical approach with aid of the material test can make a satisfactory result. The present study aims to propose a simple and rational model for nonlinear stress-strain curves of concrete brick masonry in compression with various mortar strengths (). For this model, a key parameter that determines the slopes of the ascending and descending branches is formulated using a parametric numerical analysis, where different mortar strengths are considered, including the modulus of elasticity and secant modulus joining the origin and the 0.5 point after the peak stress, where is a strength of prism. For the material properties used to define the stress-strain relationship, a regression analysis is performed on an extensive amount of test data collected from a wide variety of concrete specimens. The reliability of the developed model is examined using a normalized root-mean-square error obtained from a comparison of model estimates with the experimental data. Finally, the existing empirical models are reviewed and compared with the developed model.
To evaluate the compressive strength of the concrete brick prism, the specimens (Figure 1) were prepared with three different mortar strengths: (1) twice of the minimum concrete brick strength (8 MPa) required by KS F404 ; named specimen Cp-2.0, (2) two and half times of the minimum concrete brick strength, named specimen Cp-2.5, and (3) three times of the minimum concrete brick strength, named specimen Cp-3.0. Here, the specimen notations include two parts as follows: the first part, “Cp,” refers that the specimen is in compression and the second part refers to the mortar identification.
Before evaluating the strength of concrete brick prism, material test of each component, brick and mortar, was performed. The test protocol of compressive strength of standard concrete brick (190 × 90 × 57 mm) followed KS F404 . The test result shows that the average compressive strength of 11 specimens was 8.23 MPa and the standard deviation of them was 0.198 (Table 1). The volumetric mixture ratio of cement and sand of the joint mortar was 1 : 2.7. The water-cement ratio was decided through the premixing procedure (Table 2). The cylindrical specimen (ф100 × 200 mm) test result showed that the strength of the mortar was more than 10.8 MPa which is the minimum required mortar strength for masonry by KS L5220 . The resulting compressive stress of the mortar was 2-3% more than the planned strength and also that of the concrete brick was 3% more than originally planned strength. As a result, the exact ratios of the mortar to concrete brick strength of Cp-2.0, Cp-2.5, and Cp-3.0 were 2.0, 2.4, and 2.8, respectively. As the compressive strength of mortar increased, the strain at the maximum strength decreased by 8.16% in 2.5 and 10.20% in 3.0, when they were compared to 2.0 (Table 3), where is the strength of brick. The stress-strain relationship of mortars and brick is shown in Figure 2. The brick was the most ductile material among them showing the lowest strength but the highest strain.
Note. = compressive strength of brick.
Note. = compressive strength of mortar and = required compressive strength of concrete brick.
2.3. Loading and Measurement
The test specimens were prepared with caution to align the loading point with the center of the specimen to avoid eccentricity. Two linear variable differential transformers (LVDT) which can measure displacement up to 25 mm were installed at both sides of the prism (Figure 3). The data from each LVDT were compared to exam if there occurred any eccentricity. The load was applied by 500 kN capacity universal testing machine (UTM). The loading rate was 0.1 mm per minute.
3. Test Results and Discussion
3.1. Failure Mode
Typical failure mode by compression is shown in Figure 4. The first crack started at the brick near the steel attachment at 40%∼50% of the peak stress. The number of cracks increased mainly around it. The cracks developed sharply along the loading direction at 85% of the peak stress, which was accompanied by a rapid increase in the strain. The fracture process zone developed to the middle as reaching the peak strength. Most of the cracks were observed in the concrete bricks. At last, the cracks from one surface of the specimen developed to reach the other surface to conclude its fracture. These tendencies were equally observed regardless of the mortar strengths.
3.2. Prism Strength
The strengths of the 16 prism specimens are listed in Table 3. Gumaste et al.  noted that the brick masonry strength increases with increase in brick and/or mortar strength. In this study, only one parameter, i.e., the strength of the mortar (fm), was introduced. Because all other conditions were fixed other than that, the strength of the prism (fpm) would be expressed as a function of the strength of a mortar and a brick as
Using the test data, with a constant brick strength, a regression analysis  was performed as shown in Figure 5, and the relationship between the prism strength and the mortar strength was found to be
3.3. Stress-Strain Relationship
The compressive stress-strain curve of prism obtained for the concrete mixes is plotted in Figure 6. The shape of the curve was a second-degree parabola with its vertex at the peak stress point. The slopes of the ascending and descending branches of the curve mostly depend on . The curve was almost linear up to approximately one-half of the peak stress point, showing that their initial slope increased as increased. The strength of Cp-2.5 and Cp-3.0 were 7.3% and 11.5% higher than that of Cp-2.0, respectively. The strain at the peak stress also increased in ascending branch (the strain of Cp-2.5 and Cp-3.0 was 7.4% and 11.1% more than that of Cp-2.0, respectively) but the strain at 0.5 fpm in descending branch reduced as the compressive strength increased (the strain at 0.5 fpm of Cp-2.5 and Cp-3.0 was 2.1% and 4.3% less than that of Cp-2.0, respectively). It shall be noted that the strength of the prism was lower than that of the brick or mortar, against expectation. The innate nature of each materials as well as the way of assemblage of them may cause inevitable uneven contact condition and develop local cracks.
3.4. Modulus of Elasticity,
From above inference, it is thought that the slope of the curve in the earlier stage, i.e., the modulus of elasticity, is directly related with the prism strength (). Considering a parabolic trend of the stress and strain relationship, the equation for modulus of elasticity can be expressed as
Similar studies have been conducted on determining the factors and by, for example, Yang et al.  and Noguchi et al. . In this study, based on the test results, a regression analysis was conducted to find a best-fit value of and in equation (3), as shown in Figure 7, finding = 1513 and = 0.33. The test results for and the analysis results from equation (5) are compared in Table 4. The averages of differences between them for Cp-2.0, Cp-2.5, and Cp-3.0 are 0.2%, 1.0%, and 0.9%, respectively. It can be concluded that the analysis equation for elastic modulus derived above matched with the test results with accuracy.
Equation (3) was compared with existing equations found in internationally accepted documents such as FEMA306 , which proposes Epm 550fpm. International Building Code  and the MSJC document  recommend Epm as 700 times fpm, while Eurocode6  suggest conservatively higher values of Epm (1,000 times fpm). The Canadian masonry code S304.1  recommends Epm as 850 times fpm with an upper limit of 20,000 MPa. The proposed Epm in this study was compared with some selective existing models as shown in Figure 8. All featured models showed higher Epm than the proposed model in most of the ranges of fpm; in other words, the proposed model estimates the Epm rather conservatively.
3.5. Strain at Peak Stress and at 50% of Peak Stress of Descending Branch
MacGregor and Wight  established that the strain at peak stress () of concrete increases with increase in concrete strength. The same trend is observed in brick prism made with concrete. As it was revealed from the test results shown in Table 3, for the ascending branch of the stress-strain curve, the strain at the peak stress was proportional to . On the other hand, for the descending branch, decreased as increased, i.e., they are in reverse proportion to each other. Their relationships can be expressed aswhere = 10 MPa is a reference value for prism strength.
The test results for and and the analysis results from equations (5) and (6) are compared in Table 5. The averages of differences of between them for Cp-2.0, Cp-2.5, and Cp-3.0 are 3.1%, 2.0%, and 0.7%, respectively. Those of between them are 7.1%, 4.9%, 4.7%, respectively. It can be concluded that the analysis equation for strains derived above represents the test results with high fidelity.
The differences and the difference averages are the ones compared with the test results listed in Table 2.
4. Mathematical Equation for Stress-Strain Relationship
4.1. Generalized Equation
The shape of a compressive stress-strain curve of concrete is generally characterized as a parabola with its vertex at the peak stress . This physically means that the tangential modulus of elasticity has maximum value at the origin, gradually decreases to zero at the peak stress, and becomes negative in the descending branch of the curve (Figure 11).
In this study, the same assumption and the following nonlinear equation (7) were applied in generating a complete curve of concrete brick prism:where is the normalized stress, is the normalized strain, and is the prism stress corresponding to strain .
The physical meaning of the equation gives the following boundary conditions: (1) , for ; (2) for ; and (3) , for . From the first and second conditions, it can be said that is equal to . From the tangential modulus at a point, , and the third boundary condition, it can be inferred that is equal to . Therefore, the stress-strain curve of concrete can be expressed in the following basic form with the key parameter :
Note that the slopes of the ascending and descending branches of the curve depend on the value of ; however, the value of differs for each branch. To determine the slope of the ascending branch, the elastic modulus of prism, , can be regarded as a more adequate reference parameter than the initial tangent modulus because of the lack of available test data for . Following ASTM C1314 , was decided as the slope of the line joining the 5% and the 33% of the peak strength. This statement is thought to be reasonable because the stress-strain curve of prism in compression would remain linear up to 0.33fpm . Substituting the defined in equation (8) gives the following equation for the key parameter of the ascending branchwhere .
In contrast to the ascending branch slope, there is no consensus on the reference point to determine the slope of the descending branch. For mathematical simplicity, Tasnimi  used an inflection point as a reference, but it is difficult to identify the location of the inflection point. Van Gysel and Taerwe  employed the secant modulus joining the origin and 50% of the peak stress to derive the descending branch slope. Furthermore, CEB-FIP  describes the descending branch only up to 0.5 point. Following these researchers, the present study selected the secant modulus at 0.5 as a reference point for evaluating the descending branch slope and formulated an equation for the key parameter defining the descending branch as follows:where and is the strain corresponding to 0.5 after the peak stress.
4.2. Key Parameter
The equations for , , and derived in the preceding subsections were substituted in equations (9) and (10). These two nonlinear equations, which incorporate , were then solved for using the Newton–Raphson method. Based on the analytically obtained results, a statistical optimization was carried out as shown in Figure 12 to derive the following best-fit equations for : equation (5) for the ascending branch and equation (6) for the descending branch.
4.3. Comparisons with Existing Models
In this section, the test and analysis results provided above are compared with another notable model. Knutson  assessed the masonry stress-strain diagram for different combinations of mortar and brick and concluded that the stress-strain relationship could be approximated aswhere σ is the normal stress, ε is the normal strain, is the masonry compressive strength, and is the elastic modulus.
The stress-stain relationship of the test specimens and the analytical results from the proposed model and Knutson’s model are compared in Figure 13. The normalized root-mean-square errors obtained from each stress-strain curve are listed in Table 6.
Note. Normalized root-mean-square error (NRMSE) = , where is the mean stress in the measured stress-strain curve, and are experimental and predicted stress, respectively, and n is number of points in experimental stress-strain curve.
The stress-strain graphs generated by the analysis model match with the test results quite well from the beginning, through ascending branch, to the 50% of peak stress in the descending branch. In the final stage after the in the descending branch, the analysis results showed rather decreased ductility than the test results. It is thought that the confined condition of the specimens in test affected the ductility after the crush had happened. Further experimental study with mode specimens under same or different test set-ups is required to find out the cause of difference.
The Knutson model would deal with the stress-strain relationship of the brick prism from loading commencement, only up to the peak stress. The curve matched with the test results well until 60% of the peak stress. After that, the decreased stiffness moved the peak point far away from that of the tests. For example, the strain at the peak point from the Knutson model was 23% more than that from the test of specimen Cp-2.5. The model did not provide the descending branch of a stress-strain curve.
In summary, the above comparison reveals some limitations of Knutson’s model: (1) only ascending branch can be modeled in Knutson’s model, as is often the case with; (2) compared with the earlier stage of the stress-strain relationship, the final stage of it is not well explained. On the other hand, the predictions from the model proposed in this study are in better agreement regardless of compressive strength. The calculated normalized root-mean-square error (NRMSE) by the proposed model ranged between 0.239 and 0.257, while in Knutson model, it was between 0.357 and 0.358 (Table 6).
In this study, concrete brick prisms with three different mortar strengths and with the same brick strength were tested under compressive load. An analytical model was proposed to provide a stress-strain relationship of them. Based on the research summarized in this paper, the following conclusions were drawn:(1)The compressive strength of the prism differed according to the mortar strength when the brick unit strength was constant. However, the increase rate of the prism strength was not exactly proportional to the increase rate of the mortar.(2)The strength of a brick prism was not a summation of both brick strength and mortar strength. Rather, it was lower than the individual strength of a brick unit or a mortar. The contact condition of both nonhomogeneous materials is thought to cause local cracks under compressive condition.(3)The proposed stress-strain model for brick prism in compression predicted the relationship accurately, regardless of mortar strength, although some discrepancies were observed after in the descending branch.(4)The key parameter , which is an exponential function of , defines the stress-strain curve. Two equations for were provided for ascending and descending branches, separately.(5)The proposed stress-strain relationship model contributes towards the mathematical simplicity of analytical modeling.(6)The authors considered that the comparison between Ewing and Kowalski , Kaushik et al.’s  modeling based on the “modified” Kent–Park model proposed by Priestley and Elder , and their own model should be given on a future assignment.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (no. 2015R1A5A1037548) and by Kyonggi University’s Graduate Research Assistantship 2018.
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