A Stress-Strain Model for Brick Prism under Uniaxial Compression

-is study proposes a simple and rational stress-strain relationship model applicable to brick masonry under compression. -e 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. -e reliability of the proposedmodel was examined by comparisons with actual stress-strain curves obtained from the tests and the existingmodel.-e 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.


Introduction
Masonry is a material built from units and mortar that induce an anisotropic behavior for the composite.e lack of knowledge on the properties of the composite material imposes low assessments of the strength capacity of the masonry wall.Atkinson et al. [1] 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.e 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][3][4][5] used the basic expression established by Popovics [6] or Sargin et al. [7], 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 [8] evaluated the stress-strain diagrams for various materials and showed that they can be cast into a mathematical form.However, Mohamad et al. [9] 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 [12] and LS-DYNA [13], it is not possible to correctly model and predict the behavior of masonry structures primarily due to the lack of the references that fully de ne 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. [14] explained the stress-strain behavior of it in compression quite successfully.e 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 di cult task.However, the theoretical approach with aid of the material test can make a satisfactory result.e present study aims to propose a simple and rational model for nonlinear stressstrain curves of concrete brick masonry in compression with various mortar strengths (f m ).For this model, a key parameter that determines the slopes of the ascending and descending branches is formulated using a parametric numerical analysis, where di erent mortar strengths are considered, including the modulus of elasticity and secant modulus joining the origin and the 0.5f pm point after the peak stress, where f pm is a strength of prism.For the material properties used to de ne the stress-strain relationship, a regression analysis is performed on an extensive amount of test data collected from a wide variety of concrete specimens.
e 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.

Experiment
2.1.Specimens.To evaluate the compressive strength of the concrete brick prism, the specimens (Figure 1) were prepared with three di erent mortar strengths: (1) twice of the minimum concrete brick strength (8 MPa) required by KS F404 [15]; 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 rst part, "Cp," refers that the specimen is in compression and the second part refers to the mortar identi cation.

Materials.
Before evaluating the strength of concrete brick prism, material test of each component, brick and mortar, was performed.
e test protocol of compressive strength of standard concrete brick (190 × 90 × 57 mm) followed KS F404 [15].e 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).e volumetric mixture ratio of cement and sand of the joint mortar was 1 : 2.7.e water-cement ratio was decided through the premixing procedure (Table 2).
e cylindrical specimen (v100 × 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 [16].e 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 2.5f b and 10.20% in 3.0f b , when they were compared to 2.0f b (Table 3), where f b is the strength of brick.e stressstrain relationship of mortars and brick is shown in Figure 2.
e brick was the most ductile material among them showing the lowest strength but the highest strain.

Loading and Measurement.
e test specimens were prepared with caution to align the loading point with the center of the specimen to avoid eccentricity.Two linear variable di erential transformers (LVDT) which can measure displacement up to 25 mm were installed at both sides of the prism (Figure 3).e data from each LVDT were compared to exam if there occurred any eccentricity.e load was applied by 500 kN capacity universal testing machine (UTM).e loading rate was 0.1 mm per minute.2 Advances in Civil Engineering sharply along the loading direction at 85% of the peak stress, which was accompanied by a rapid increase in the strain.e 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   Advances in Civil Engineering its fracture.ese tendencies were equally observed regardless of the mortar strengths.

Prism Strength.
e strengths of the 16 prism specimens are listed in Table 3. Gumaste et al. [17] 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 (f m ), was introduced.Because all other conditions were xed other than that, the strength of the prism (f pm ) 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 [18] was performed as shown in Figure 5, and the relationship between the prism strength and the mortar strength was found to be f pm 0.09f m + 3.92. (2)

Stress-Strain
Relationship.e compressive stress-strain curve of prism obtained for the concrete mixes is plotted in Figure 6. e shape of the curve was a second-degree parabola with its vertex at the peak stress point.e slopes of the ascending and descending branches of the curve mostly depend on f pm .
e curve was almost linear up to approximately one-half of the peak stress point, showing that their initial slope increased as f pm increased.e strength of Cp-2.5 and Cp-3.0 were 7.3% and 11.5% higher than that of Cp-2.0, respectively.e 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 f pm in descending branch reduced as the compressive strength increased (the strain at 0.5 f pm 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.
e innate nature of each materials as well as the way of assemblage of them may cause inevitable uneven contact condition and develop local cracks.

Modulus of Elasticity, E pm .
From above inference, it is thought that the slope of the curve in the earlier stage, i.e., the

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Advances in Civil Engineering modulus of elasticity, is directly related with the prism strength (f pm ).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 A 1 and α by, for example, Yang et al. [14] and Noguchi et al. [19].In this study, based on the test results, a regression analysis was conducted to nd a best-t value of A 1 and α in equation ( 3), as shown in Figure 7, nding A 1 1513 and α 0.33.
e test results for E pm and the analysis results from equation ( 5) are compared in Table 4.
e averages of di erences 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 [20], which proposes E pm ≈ 550f pm .International Building Code [21] and the MSJC document [22] recommend E pm as 700 times f pm , while Eurocode6 [23] suggest conservatively higher values of E pm (1,000 times f pm ).
e Canadian masonry code S304.1 [24] recommends E pm as 850 times f pm with an upper limit of 20,000 MPa.
e proposed E pm in this study was compared with some selective existing models as shown in Figure 8.All featured models showed higher E pm than the proposed model in most of the ranges of f pm ; in other words, the proposed model estimates the E pm rather conservatively.

Strain at Peak Stress ε 0 and at 50% of Peak Stress of
Descending Branch ε 0.5 .MacGregor and Wight [25] established that the strain at peak stress (ε 0 ) of concrete increases with increase in concrete strength.e 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 ε 0 was proportional to f pm .On the other hand, for the descending branch, ε 0.5 decreased as f pm increased, i.e., they are in reverse proportion to each other.eir relationships can be expressed as where f 10 10 MPa is a reference value for prism strength.
To derive equations for ε 0 and ε 0.5 , nonlinear regression analysis (Figures 9 and 10) were conducted and the following best-t equations was developed as ε 0.5 0.004 exp 0.25 where E pm is given in equation ( 3). e test results for ε 0 and ε 0.5 and the analysis results from equations ( 5) and ( 6) are compared in Table 5. e averages of di erences of ε 0 between them for Cp-2.0,Cp-2.5, and Cp-3.0 are 3.1%, 2.0%, and 0.7%, respectively.ose of ε 0.5 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 delity.

Mathematical Equation for
Stress-Strain Relationship

Generalized Equation.
e shape of a compressive stress-strain curve of concrete is generally characterized as a parabola with its vertex at the peak stress [14]. is physically means that the tangential modulus of elasticity E t 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 y (f pc /f pm ) is the normalized stress, x (ε pc /ε 0 ) is the normalized strain, and f pc is the prism stress corresponding to strain ε pc .e physical meaning of the equation gives the following boundary conditions: (1) y 0, for x 0; (2) y 1 for x 1; and (3) (df pc /dε pc ) 0, for x 1.From the rst and second conditions, it can be said that β 3 is equal to β 1 + 1.From the tangential modulus at a point, d(f pc )/d(ε pc ), and the third boundary condition, it can be inferred that β 2 is equal to β 1 + 1.
erefore, the stress-strain curve of concrete can be expressed in the following basic form with the key parameter β 1 : Advances in Civil Engineering Note that the slopes of the ascending and descending branches of the curve depend on the value of β 1 ; however, the value of β 1 di ers for each branch.To determine the slope of the ascending branch, the elastic modulus of prism, E pm , can be regarded as a more adequate reference parameter than the initial tangent modulus E ti because of the lack of available test data for E ti .Following ASTM C1314 [26], E pm was decided as the slope of the line joining the 5% and the 33% of the peak strength.is statement is thought to be reasonable because the stress-strain curve of prism in compression would remain linear up to 0.33f pm [27].Substituting the de ned E pm in equation (8) gives the following equation for the key parameter β 1 of the ascending branch 0.4 X a where X a (0.4f pm /E pm ε 0 ).

Advances in Civil Engineering
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 [11] used an in ection point as a reference, but it is di cult to identify the location of the in ection point.Van Gysel and Taerwe [28] employed the secant modulus joining the origin and 50% of the peak stress to derive the descending branch slope.Furthermore, CEB-FIP [29] describes the descending branch only up to 0.5f pm point.Following these researchers, the present study selected the secant modulus at 0.5f pm as a reference point for evaluating the descending branch slope and formulated an equation for the key parameter β 1 de ning the descending branch as follows: where X d (ε 0.5 /ε 0 ) and ε 0.5 is the strain corresponding to 0.5f pm after the peak stress.
e value of β 1 in nonlinear equations ( 9) and ( 10) can be calculated via numerical analysis, such as the Newton-Raphson method, using the given values of f pm .

Key Parameter β 1 .
e equations for E c , ε 0 , and ε 0.5 derived in the preceding subsections were substituted in equations ( 9) and (10).ese two nonlinear equations, which incorporate f pm , were then solved for β 1 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-t equations for β 1 : equation ( 5) for the ascending branch and equation (6) for the descending branch.
In summary, a stress-strain relationship model for the prism in compression is proposed as follows: where ε pc is a strain, ε 0 is given by equation ( 5), and β 1 is given by equations (11) or (12).

Comparisons with Existing Models.
In this section, the test and analysis results provided above are compared with another notable model.Knutson [8] assessed the masonry stress-strain diagram for di erent combinations of mortar and brick and concluded that the stress-strain relationship could be approximated as (x a , 0.33) Advances in Civil Engineering where σ is the normal stress, ε is the normal strain, f cmas is the masonry compressive strength, and E 0 is the elastic modulus.e stress-stain relationship of the test specimens and the analytical results from the proposed model and Knutson's model are compared in Figure 13.
e normalized root-mean-square errors obtained from each stress-strain curve are listed in Table 6.
e 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 nal stage after the ε 0.5 in the descending branch, the analysis results showed rather decreased ductility than the test results.It is thought that the con ned condition of the specimens in test a ected the ductility after the crush had happened.Further experimental study with mode specimens under same or di erent test set-ups is required to nd out the cause of di erence.e Knutson model would deal with the stress-strain relationship of the brick prism from loading commencement, only up to the peak stress.e curve matched with the test results well until 60% of the peak stress.After that, the decreased sti ness 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.e 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) Best fit curve for descending branch, y = 0.31e 1.53x   Best fit curve for ascending branch, y = 0.62e 0.91x , where (f pm ) m is the mean stress in the measured stress-strain curve, (f pm ) Exp and (f pm ) Pre are experimental and predicted stress, respectively, and n is number of points in experimental stress-strain curve.8 Advances in Civil Engineering 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.e 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).

Conclusions
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) e 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) e 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.e contact condition of both nonhomogeneous materials is thought to cause local cracks under compressive condition.
(3) e proposed stress-strain model for brick prism in compression predicted the relationship accurately, regardless of mortar strength, although some discrepancies were observed after ε 0.5 in the descending branch.(4) e key parameter β 1 , which is an exponential function of (f pm ) 0.67 , defines the stress-strain curve.Two equations for β 1 were provided for ascending and descending branches, separately.(5) e proposed stress-strain relationship model contributes towards the mathematical simplicity of analytical modeling.(6) e authors considered that the comparison between Ewing and Kowalski [10], Kaushik et al.'s [27] modeling based on the "modified" Kent-Park model proposed by Priestley and Elder [30], and their own model should be given on a future assignment.

Figure 2 :
Figure 2: Compressive stress-strain curves measured in the mortars and the brick element.

Figure 7 :
Figure 7: Regression analysis for E pm .

Figure 8 :Figure 9 :
Figure 8: Comparison of proposed E pm with existing models.
Note. f brick compressive strength of brick.

Table 2 :
Compressive strength of mortars (MPa).Identi cation of mortar W/C (%) f m (MPa) Average of f m (MPa) Standard deviation Strain at f m , ε 0 Note.f m compressive strength of mortar and f b required compressive strength of concrete brick.

Table 3 :
Test parameters and test results.

Table 4 :
Comparison of elastic modulus from test and analysis.

Table 5 :
Comparison of ε 0 and ε 0.5 from test and analysis.edierences and the di erence averages are the ones compared with the test results listed in Table2. * Figure 12: Best-t equation for key parameter β 1 obtained from numerical analysis.

Table 6 :
Comparisons of normalized root-mean-square error obtained from each stress-strain curve.