# A Mesoscale Approach for Concrete Block Masonry

Authors: Xu Yang; Mingming Jia; Bin Chi; Mingzhi Wang; Jianfeng Zheng
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

A comprehensive understanding of the concrete block masonry’s mesostructure on its macroscopic mechanical behavior is meaningful for modern masonry. The mesoscale numerical method provides an extremely powerful analyzing tool for this problem. However, it is still difficult to analyze the mesomodel efficiently due to a large number of operating processes caused by the components’ nonlinear mechanical properties. An efficient finite-element-method-based mesoscale approach for concrete block masonry was introduced in this paper. The benchmark test results obtained show that this approach is able to simulate the concrete block masonry and analyze its failure. According to the simulation studies, a positive effect of mortar and concrete block strength on masonry compressive strength was obtained. Besides, an obvious nonlinear relationship between the masonry compressive strength and the mortar was also observed.


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## Body

## 1. Introduction

Concrete block masonry is widely applied in various modern buildings. Its mechanical behavior has received increased attention in recent years.Currently, there is little doubt that the behavior of concrete block masonry is strongly influenced by the mechanical properties of its components at the mesolevel [1]. Therefore, analyzing this structure at the mesoscale should be extremely powerful for understanding and predicting the observed macroscopic mechanical behavior. Especially, the concrete block masonry is usually treated as quasibrittle material [2]; it is difficult to devise the experimentations. The numerical method is not limited by experimental equipment or sites, and thus, precise results can be conveniently obtained. Researchers only need to establish a reasonable mesomodel of the concrete block masonry, considering the geometrical parameters and material properties of each component. The numerical mesomodel can then be utilized to ascertain the influence of components on the mechanical behavior of concrete block masonry or to simulate loading conditions that are difficult to achieve in the laboratory [3–9].Concrete block masonry is generally considered a three-phase composite on the mesoscale, consisting of concrete block, mortar, and the interfacial transition zone (ITZ) between these two [1, 10, 11]. The ITZ simulation methods of concrete block masonry under compression could then be divided into three categories according to the assumption: tie mode, friction mode, and cohesion mode [1, 12].First, in the framework of the tie mode, the concrete masonry unit and mortar were assumed to be well bonded and there would be none of the deformation slips during the process of concrete block masonry prism under compression [13–16]. Barbosa et al. proposed a combined experimental program and numerical model to study the performance of ungrouted concrete masonry block prisms [17]. They demonstrated the predictions of peak load and failure mode of concrete masonry prism under compression are in good agreement with experimental results [18]. Since this type of simulation method aims to establish a simplified model which could obtain the maximum compressive strength of concrete masonry prism quickly, the deformation of concrete masonry prism could not be precisely simulated.Second, the friction mode proposes an ITZ model by using the discrete cracking model with the friction assumption [19–21]. As introduced in previous studies, the concrete masonry and mortar are modeled separately, while the mechanical behavior between them is employed using the “surface-to-surface” interaction property [22]. The interaction property involved two directional parameters: normal and tangential mechanical behavior. The normal behavior keeps the unit and mortar stay in contact and would not sink into each other when under a compression load. Meanwhile, the separation is allowed between the concrete block and mortar when the ITZ bears a tension load. The coulomb frictional theory was adopted to describe the tangential behavior. This numerical method considered the longitudinal relative displacement between block and mortar when the displacement force exceeds the critical shear stress. The above research studies demonstrate the numerical model could successfully predict the compressive strength, global stiffness, and stress-strain curves of concrete masonry prism under compression. Nevertheless, the neglect of the influence of interfacial adhesion in this simulation method caused its downside in analyzing the concrete masonry prism under shear loading.Recently, some researchers introduced the cohesive method into the contact property between concrete masonry unit and mortar, which is named “cohesion mode” [23–25]. Several researchers expanded the units up to half of the mortar thickness in horizontal and vertical directions as continuum elements, while the interface between the continuum elements was used adhesives-cohesive behavior [26, 27]. The adhesives-cohesive behavior can provide the traction-separation response and fracture modes of the interface, and the relative parameters are determined by the material strength of the mortar. A good agreement could be found between the numerical and experimental results of concrete masonry prism under compression, shear, and diagonal tension. Nevertheless, many parameters in this method are needed to be treated and assumed due to lacking efficient information and the disadvantages of relatively complex modeling, larger calculation time also limits its application scope.In conclusion, because of the large number of the operating process caused by the components’ nonlinear mechanical properties, it is still difficult to analyze the mesomodel efficiently. In the present study, an efficient finite-element-method-based mesoscale approach for concrete block masonry was introduced. The benchmark test results demonstrated that this approach is suitable for simulating the concrete block masonry and analyzing its failure. Furthermore, based on the numerical case study, the effect of the components on the concrete block’s macroscopic mechanical behavior was studied and discussed.


## 2. Method

### 2.1. Mesoscale Geometrical Model of Concrete Block Masonry

In general, as represented in Figure1, the concrete block masonry could be considered as a three-phase composite on the mesoscale, consisting of concrete block, mortar, and the interface transition zone between them [1, 11].Figure 1
Schematic diagram of concrete block masonry’s mesomodel.In particular, concrete block, as a concrete product, could usually be treated as elastic. The constitutive behavior of the concrete block was formulated based on linear elastic damage mechanics in this paper, using the maximum tensile strain criterion as the threshold.The mortar thickness is commonly between 8 and 12 mm. In the numerical model proposed in this study, the mortar was formulated based on linear elastic damage mechanics with the maximum tensile strain criterion as the threshold, as well. As the dimension of mortar was much smaller than that of concrete block, it should be noticed that mesh size should be set carefully.ITZ locates between the concrete block and the mortar. The ITZ’s mechanical properties influence the failure pattern and load-carrying capacity of the masonry. It should be noticed that the ITZ thickness is difficult to be measured [10]. Thus, the ITZ is defined as a thin layer with a finite thickness (larger than zero) in several studies [28], or a zero-thickness boundary layer in other studies [29, 30]. Therefore, the thickness of ITZ was considered as 1 mm to simplify the model, i.e., 10% of the mortar thickness, in this study. The constitutive law for the ITZ employed in this approach conforms to the failure criterion proposed in [31]. Denoting the shear and tensile strengths of ITZ as fsITZ and ftITZ, respectively, the failure surface SfailureITZ is defined as the following three-parameter hyperbola:(1)SfailureITZ=τ2−σ−fsITZtanφ2+ftITZ−fsITZtanφ2,where τ is the shear stress, σ is the normal stress, and φ is the asymptotic friction angle, which is defined as 27° according to [31].


### 2.2. Numerical Method

The geometrical model generated based on the method introduced in Section2.1 could be solved by the finite element method considering failure process, which was named as kill-FEM in this study. The basic iterative algorithm of kill-FEM represented in Figure 2 for finite element analysis could be described as follows.Figure 2
Schematic diagram of solving the numerical model based on kill-FEM.


#### 2.2.1. Based on the Geometrical Dimension, Generate an Initial Finite Element Model

First, as shown in Figure3, the geometrical model generated could be divided into standard four-noded tetrahedral or eight-noded hexahedral finite elements. This meshing step was carried out on ABAQUS (version 2020), which provides an adaptive mesh module with general applicability. After reading and converting the mesh results to the format could be processed in python by implementing meshio tool [32], the global nodal forces F0 and the global nodal displacements d0 in this initial model could be related through the global stiffness matrix K0, which could be given by the classic equation F0=K0d0. Denoting the element stiffness matrix as k of an element, the global stiffness matrix could be obtained by assembling element stiffness matrices simply. According to the definition of the tetrahedral or hexahedral element, k would be influenced by the shape of the element, elastic modulus, Poisson’s ratio, and the constitutive model.Figure 3
Finite element mesh of the concrete block masonry. (a) Four-noded tetrahedral element. (b) Eight-noded hexahedral element.
(a)(b)


#### 2.2.2. Analyze This Finite Element Model, Find the Failure Elements, and Then Kill These Elements

After solving the initial model, all stress and strain values of all the elements could be obtained. Each element would be then validated based on the damage criterion mentioned above. Specifically, the concrete block and the mortar elements would be checked whether their principal strain meets the maximum tensile strain criterion; the ITZ elements would be checked whether they meet the criterion given by equation (1). Once the damage criterion was met, the element would be “killed” by multiplying the element stiffness k by a very small number α, for example, α=10−14 was employed in this study. Therefore, the element stiffness of the killed element k′=αk could be obtained. As well, the global stiffness matrix of the first step K1 could be obtained by assembling k′ of all elements. Obviously, these killed elements could remain in the model but contribute a near-zero stiffness to the whole model.


#### 2.2.3. Analyze the Finite Element Model with Killed Elements

Then, input the global nodal forcesF1 and global nodal displacements d1 in the first step model and relate them through K1. The matrix equation of this model F1=K1d1 could be obtained. The model in the first step could be solved. We could repeat steps (b) and (c) until the finish.This method can be used for efficient simulation of quasibrittle materials subjected to forces.


## 2.1. Mesoscale Geometrical Model of Concrete Block Masonry

In general, as represented in Figure1, the concrete block masonry could be considered as a three-phase composite on the mesoscale, consisting of concrete block, mortar, and the interface transition zone between them [1, 11].Figure 1
Schematic diagram of concrete block masonry’s mesomodel.In particular, concrete block, as a concrete product, could usually be treated as elastic. The constitutive behavior of the concrete block was formulated based on linear elastic damage mechanics in this paper, using the maximum tensile strain criterion as the threshold.The mortar thickness is commonly between 8 and 12 mm. In the numerical model proposed in this study, the mortar was formulated based on linear elastic damage mechanics with the maximum tensile strain criterion as the threshold, as well. As the dimension of mortar was much smaller than that of concrete block, it should be noticed that mesh size should be set carefully.ITZ locates between the concrete block and the mortar. The ITZ’s mechanical properties influence the failure pattern and load-carrying capacity of the masonry. It should be noticed that the ITZ thickness is difficult to be measured [10]. Thus, the ITZ is defined as a thin layer with a finite thickness (larger than zero) in several studies [28], or a zero-thickness boundary layer in other studies [29, 30]. Therefore, the thickness of ITZ was considered as 1 mm to simplify the model, i.e., 10% of the mortar thickness, in this study. The constitutive law for the ITZ employed in this approach conforms to the failure criterion proposed in [31]. Denoting the shear and tensile strengths of ITZ as fsITZ and ftITZ, respectively, the failure surface SfailureITZ is defined as the following three-parameter hyperbola:(1)SfailureITZ=τ2−σ−fsITZtanφ2+ftITZ−fsITZtanφ2,where τ is the shear stress, σ is the normal stress, and φ is the asymptotic friction angle, which is defined as 27° according to [31].


## 2.2. Numerical Method

The geometrical model generated based on the method introduced in Section2.1 could be solved by the finite element method considering failure process, which was named as kill-FEM in this study. The basic iterative algorithm of kill-FEM represented in Figure 2 for finite element analysis could be described as follows.Figure 2
Schematic diagram of solving the numerical model based on kill-FEM.


### 2.2.1. Based on the Geometrical Dimension, Generate an Initial Finite Element Model

First, as shown in Figure3, the geometrical model generated could be divided into standard four-noded tetrahedral or eight-noded hexahedral finite elements. This meshing step was carried out on ABAQUS (version 2020), which provides an adaptive mesh module with general applicability. After reading and converting the mesh results to the format could be processed in python by implementing meshio tool [32], the global nodal forces F0 and the global nodal displacements d0 in this initial model could be related through the global stiffness matrix K0, which could be given by the classic equation F0=K0d0. Denoting the element stiffness matrix as k of an element, the global stiffness matrix could be obtained by assembling element stiffness matrices simply. According to the definition of the tetrahedral or hexahedral element, k would be influenced by the shape of the element, elastic modulus, Poisson’s ratio, and the constitutive model.Figure 3
Finite element mesh of the concrete block masonry. (a) Four-noded tetrahedral element. (b) Eight-noded hexahedral element.
(a)(b)


### 2.2.2. Analyze This Finite Element Model, Find the Failure Elements, and Then Kill These Elements

After solving the initial model, all stress and strain values of all the elements could be obtained. Each element would be then validated based on the damage criterion mentioned above. Specifically, the concrete block and the mortar elements would be checked whether their principal strain meets the maximum tensile strain criterion; the ITZ elements would be checked whether they meet the criterion given by equation (1). Once the damage criterion was met, the element would be “killed” by multiplying the element stiffness k by a very small number α, for example, α=10−14 was employed in this study. Therefore, the element stiffness of the killed element k′=αk could be obtained. As well, the global stiffness matrix of the first step K1 could be obtained by assembling k′ of all elements. Obviously, these killed elements could remain in the model but contribute a near-zero stiffness to the whole model.


### 2.2.3. Analyze the Finite Element Model with Killed Elements

Then, input the global nodal forcesF1 and global nodal displacements d1 in the first step model and relate them through K1. The matrix equation of this model F1=K1d1 could be obtained. The model in the first step could be solved. We could repeat steps (b) and (c) until the finish.This method can be used for efficient simulation of quasibrittle materials subjected to forces.


## 2.2.1. Based on the Geometrical Dimension, Generate an Initial Finite Element Model

First, as shown in Figure3, the geometrical model generated could be divided into standard four-noded tetrahedral or eight-noded hexahedral finite elements. This meshing step was carried out on ABAQUS (version 2020), which provides an adaptive mesh module with general applicability. After reading and converting the mesh results to the format could be processed in python by implementing meshio tool [32], the global nodal forces F0 and the global nodal displacements d0 in this initial model could be related through the global stiffness matrix K0, which could be given by the classic equation F0=K0d0. Denoting the element stiffness matrix as k of an element, the global stiffness matrix could be obtained by assembling element stiffness matrices simply. According to the definition of the tetrahedral or hexahedral element, k would be influenced by the shape of the element, elastic modulus, Poisson’s ratio, and the constitutive model.Figure 3
Finite element mesh of the concrete block masonry. (a) Four-noded tetrahedral element. (b) Eight-noded hexahedral element.
(a)(b)


## 2.2.2. Analyze This Finite Element Model, Find the Failure Elements, and Then Kill These Elements

After solving the initial model, all stress and strain values of all the elements could be obtained. Each element would be then validated based on the damage criterion mentioned above. Specifically, the concrete block and the mortar elements would be checked whether their principal strain meets the maximum tensile strain criterion; the ITZ elements would be checked whether they meet the criterion given by equation (1). Once the damage criterion was met, the element would be “killed” by multiplying the element stiffness k by a very small number α, for example, α=10−14 was employed in this study. Therefore, the element stiffness of the killed element k′=αk could be obtained. As well, the global stiffness matrix of the first step K1 could be obtained by assembling k′ of all elements. Obviously, these killed elements could remain in the model but contribute a near-zero stiffness to the whole model.


## 2.2.3. Analyze the Finite Element Model with Killed Elements

Then, input the global nodal forcesF1 and global nodal displacements d1 in the first step model and relate them through K1. The matrix equation of this model F1=K1d1 could be obtained. The model in the first step could be solved. We could repeat steps (b) and (c) until the finish.This method can be used for efficient simulation of quasibrittle materials subjected to forces.


## 3. Numerical Case Study

### 3.1. Materials

The masonry prism for simulating consisted of concrete block, mortar, and the ITZ between them.The dimension of concrete block was represented in Figure4. Actually, the diagonal dimension of the hole should be controlled by a small round corner, which had been simplified in this study.Figure 4
Dimensions of concrete blocks (all dimensions are in mm). (a) Concrete type A. (b) Concrete type B.
(a)(b)The mechanical properties of concrete were applied for simulating the block, i.e., the elastic modulusEb, Poisson’s ratio μb, and maximum tensile strain ϵmax,b, which would be measured in the standard concrete material testing. It should be noticed that several researchers employed the mechanical properties measured by concrete block directly for simulating the block, we could barely agree on which. A concrete block, for example, which is represented in Figure 4, could already be considered as a structure or a production made by concrete and the mechanical properties of a block must not be equal to those of concrete obviously. Since the block would be divided into a set of elements by employing mesh technology naturally, each element then contributed to the whole masonry prism’s mechanical behaviors. This element type was the concrete, which should be assigned the properties of the concrete, i.e., the base material of the block. Employing wrong mechanical parameters of materials would cause implausible simulation results.Additionally, the following mechanical properties were applied for simulating the block, i.e., the elastic modulusEm, Poisson’s ratio μm, and maximum tensile strain ϵmax,m, which would be measured in the standard construction mortar material testing.Since the elastic modulusEITZ and Poisson’s ratio μITZ were difficult to be measured in the experiments, the value of EITZ was approximately set as 0.8 times Em according to the proposal in previous studies and μITZ was set as 0.15 [33].


### 3.2. Concrete Block Masonry under Compression

In this section, a group of benchmark tests was simulated for validation of the numerical method first. As shown in Figure5, the dimensions of the masonry wallets tested were 590 mm length, 990 mm height, and 190 mm thickness. These corresponded to five courses of concrete block vertically, on and a half blocks horizontally, and on block transversely. The mechanical properties of concrete block and mortar, as well as the shear and tension strength of ITZ, were measured in our previous study. More details about the experimentation could be found in [34]. Besides, the elastic modulus and Poisson’s ratio of ITZ were obtained based on the mortar parameters, according to the manners introduced in Section 3.1. All material parameters mentioned above of these three components are represented in Table 1, based on which the benchmark models for validation were simulated.Figure 5
Dimensions of masonry wallets (all dimensions are in mm).Table 1
Various components’ material parameters used for modeling.
No.EbMPaμbϵb10−6EmMPaμmϵm10−6EITZMPaμITZfsITZMPaftITZMPa1120060.2011759400.1510647520.152.720.292116090.2011256760.1510345410.152.600.293124370.1911661610.1510149290.152.730.304181150.2014259400.1510647520.152.620.295172540.2013560640.1510948510.152.660.286189180.2013960420.1510148340.152.630.297181150.20142152340.15102121870.152.790.308174030.19142154520.15104123610.152.850.309189070.19147146060.14107116850.152.850.2910294350.20106152340.15102121870.152.620.2911294360.19108150860.15106120690.152.610.3012291380.20106145870.14102116690.152.610.28Then, the effect of concrete block and mortar mechanical properties on the masonry was discussed based on different components’ material parameters, i.e., the concrete block and the mortar material properties are assigned as in Tables2 and 3, respectively. The varied combinations were simulated.Table 2
Concrete block’s material parameters.
TypeEbMPaμbϵb10−6MU7.575000.2100MU10100000.2100MU15150000.2100MU20200000.2100MU25250000.2100Table 3
Mortar and ITZ’s material parameters.
TypeEmMPaμmϵm10−6EITZMPaμITZfsITZMPaftITZMPaMb576240.156660990.22.720.29Mb7.598430.157678750.22.770.28Mb10117990.158594390.22.740.28Mb15152330.1598121870.22.790.29Mb201826010.15110146080.22.730.30


## 3.1. Materials

The masonry prism for simulating consisted of concrete block, mortar, and the ITZ between them.The dimension of concrete block was represented in Figure4. Actually, the diagonal dimension of the hole should be controlled by a small round corner, which had been simplified in this study.Figure 4
Dimensions of concrete blocks (all dimensions are in mm). (a) Concrete type A. (b) Concrete type B.
(a)(b)The mechanical properties of concrete were applied for simulating the block, i.e., the elastic modulusEb, Poisson’s ratio μb, and maximum tensile strain ϵmax,b, which would be measured in the standard concrete material testing. It should be noticed that several researchers employed the mechanical properties measured by concrete block directly for simulating the block, we could barely agree on which. A concrete block, for example, which is represented in Figure 4, could already be considered as a structure or a production made by concrete and the mechanical properties of a block must not be equal to those of concrete obviously. Since the block would be divided into a set of elements by employing mesh technology naturally, each element then contributed to the whole masonry prism’s mechanical behaviors. This element type was the concrete, which should be assigned the properties of the concrete, i.e., the base material of the block. Employing wrong mechanical parameters of materials would cause implausible simulation results.Additionally, the following mechanical properties were applied for simulating the block, i.e., the elastic modulusEm, Poisson’s ratio μm, and maximum tensile strain ϵmax,m, which would be measured in the standard construction mortar material testing.Since the elastic modulusEITZ and Poisson’s ratio μITZ were difficult to be measured in the experiments, the value of EITZ was approximately set as 0.8 times Em according to the proposal in previous studies and μITZ was set as 0.15 [33].


## 3.2. Concrete Block Masonry under Compression

In this section, a group of benchmark tests was simulated for validation of the numerical method first. As shown in Figure5, the dimensions of the masonry wallets tested were 590 mm length, 990 mm height, and 190 mm thickness. These corresponded to five courses of concrete block vertically, on and a half blocks horizontally, and on block transversely. The mechanical properties of concrete block and mortar, as well as the shear and tension strength of ITZ, were measured in our previous study. More details about the experimentation could be found in [34]. Besides, the elastic modulus and Poisson’s ratio of ITZ were obtained based on the mortar parameters, according to the manners introduced in Section 3.1. All material parameters mentioned above of these three components are represented in Table 1, based on which the benchmark models for validation were simulated.Figure 5
Dimensions of masonry wallets (all dimensions are in mm).Table 1
Various components’ material parameters used for modeling.
No.EbMPaμbϵb10−6EmMPaμmϵm10−6EITZMPaμITZfsITZMPaftITZMPa1120060.2011759400.1510647520.152.720.292116090.2011256760.1510345410.152.600.293124370.1911661610.1510149290.152.730.304181150.2014259400.1510647520.152.620.295172540.2013560640.1510948510.152.660.286189180.2013960420.1510148340.152.630.297181150.20142152340.15102121870.152.790.308174030.19142154520.15104123610.152.850.309189070.19147146060.14107116850.152.850.2910294350.20106152340.15102121870.152.620.2911294360.19108150860.15106120690.152.610.3012291380.20106145870.14102116690.152.610.28Then, the effect of concrete block and mortar mechanical properties on the masonry was discussed based on different components’ material parameters, i.e., the concrete block and the mortar material properties are assigned as in Tables2 and 3, respectively. The varied combinations were simulated.Table 2
Concrete block’s material parameters.
TypeEbMPaμbϵb10−6MU7.575000.2100MU10100000.2100MU15150000.2100MU20200000.2100MU25250000.2100Table 3
Mortar and ITZ’s material parameters.
TypeEmMPaμmϵm10−6EITZMPaμITZfsITZMPaftITZMPaMb576240.156660990.22.720.29Mb7.598430.157678750.22.770.28Mb10117990.158594390.22.740.28Mb15152330.1598121870.22.790.29Mb201826010.15110146080.22.730.30


## 4. Results and Discussions

### 4.1. Benchmark Test Results

Before all, the numerical modeling method should be validated by comparing the simulation results with the experimental ones. The concrete block masonry’s compressive strength was usually considered as one of the most important indexes in engineering and research, based on which the simulation would be discussed.Denoting the concrete block masonry’s compressive strength measured in the tests as thex-coordinate and the strength obtained from simulation as the y-coordinate, respectively, the comparison point could then be plotted in a scatter plot as shown in Figure 6.Figure 6
Comparison of simulated and experimental compressive strengths.Additionally, a straight reference line ofy=x was added in the scatter. This line could indicate the location that simulation results are equal to the testing results, which made the plot more intuitive. The points plotted in Figure 6 could be observed that located beside the reference line, which demonstrated a good agreement with the testing results. According to this numerical benchmark test results which were consistent with the fact, the numerical method proposed could be basically considered practical.


### 4.2. Failure Pattern

In addition to the compressive strength, the failure pattern could also represent the mechanical characteristics of masonry. The simulation failure pattern would therefore be compared with the experimental results and discussed in this section.By applying the modeling method introduced, the cloud charts of principle tensile strain, displacement magnitude, and1st principle stress could be drawn, which are represented in Figure 7.Figure 7
Simulation results. Cloud chart of (a) principle tensile strain, (b) displacement magnitude, and (c) 1st principle stress.
(a)(b)(c)As a quasibrittle material, the concrete block masonry could be recognized as that meeting the maximum tensile strain criterion. That is to say, the distribution of the principal tensile strain could reflect its failure pattern. Thus, according to the principle tensile strain cloud chart shown in Figure7(a), cracks might be easy to form and distribute on the center part, the ITZ as well as the mortar, and the top as well as the bottom part. Compared with the failure mode shown in Figure 8, the simulation results were demonstrated to be similar to the test. The obvious cracks and damage could be observed in the masonry sample’s center of the front face, the upper corner, and the center of the side face. These damaged parts matched the high-value area in the cloud chart of principle tensile strain. Meanwhile, these experimental and simulation results met the results and discussion reported in our previous studies [34, 35]. Thus, it could be noted that this simulation method would be suitable for analyses of concrete block masonry.Figure 8
Experimental failure pattern. (a) Front view. (b) Side view.
(a)(b)


### 4.3. Effect of Concrete Block and Mortar Types

Based on the extended simulation analysis, the effect of component material properties on the masonry’s compressive strengths would be discussed in this part. The relationship between the strength of masonry and components, i.e., the mortar and concrete block, is drawn in Figure9.Figure 9
Effect of mortar and concrete block strength on masonry compressive strength.The effect of mortar and concrete block strength on masonry compressive strength could then be observed: the increase of these two components’ strength would cause the masonry’s compressive strength to increase. Besides, the curves plotted in Figure9 represented an obvious nonlinear relationship between the masonry compressive strength and the mortar. This simulation study produced results that corroborate the findings of a great deal of the previous work reported [17, 36].Besides, the core procedure in this simulation method introduced was solving algebraic equations. Many efficient and widely used methods for solving this sort of equation have been developed. Compared with other nonlinear solving methods based on iteration, the manner proposed in this study costs less computational resources.


## 4.1. Benchmark Test Results

Before all, the numerical modeling method should be validated by comparing the simulation results with the experimental ones. The concrete block masonry’s compressive strength was usually considered as one of the most important indexes in engineering and research, based on which the simulation would be discussed.Denoting the concrete block masonry’s compressive strength measured in the tests as thex-coordinate and the strength obtained from simulation as the y-coordinate, respectively, the comparison point could then be plotted in a scatter plot as shown in Figure 6.Figure 6
Comparison of simulated and experimental compressive strengths.Additionally, a straight reference line ofy=x was added in the scatter. This line could indicate the location that simulation results are equal to the testing results, which made the plot more intuitive. The points plotted in Figure 6 could be observed that located beside the reference line, which demonstrated a good agreement with the testing results. According to this numerical benchmark test results which were consistent with the fact, the numerical method proposed could be basically considered practical.


## 4.2. Failure Pattern

In addition to the compressive strength, the failure pattern could also represent the mechanical characteristics of masonry. The simulation failure pattern would therefore be compared with the experimental results and discussed in this section.By applying the modeling method introduced, the cloud charts of principle tensile strain, displacement magnitude, and1st principle stress could be drawn, which are represented in Figure 7.Figure 7
Simulation results. Cloud chart of (a) principle tensile strain, (b) displacement magnitude, and (c) 1st principle stress.
(a)(b)(c)As a quasibrittle material, the concrete block masonry could be recognized as that meeting the maximum tensile strain criterion. That is to say, the distribution of the principal tensile strain could reflect its failure pattern. Thus, according to the principle tensile strain cloud chart shown in Figure7(a), cracks might be easy to form and distribute on the center part, the ITZ as well as the mortar, and the top as well as the bottom part. Compared with the failure mode shown in Figure 8, the simulation results were demonstrated to be similar to the test. The obvious cracks and damage could be observed in the masonry sample’s center of the front face, the upper corner, and the center of the side face. These damaged parts matched the high-value area in the cloud chart of principle tensile strain. Meanwhile, these experimental and simulation results met the results and discussion reported in our previous studies [34, 35]. Thus, it could be noted that this simulation method would be suitable for analyses of concrete block masonry.Figure 8
Experimental failure pattern. (a) Front view. (b) Side view.
(a)(b)


## 4.3. Effect of Concrete Block and Mortar Types

Based on the extended simulation analysis, the effect of component material properties on the masonry’s compressive strengths would be discussed in this part. The relationship between the strength of masonry and components, i.e., the mortar and concrete block, is drawn in Figure9.Figure 9
Effect of mortar and concrete block strength on masonry compressive strength.The effect of mortar and concrete block strength on masonry compressive strength could then be observed: the increase of these two components’ strength would cause the masonry’s compressive strength to increase. Besides, the curves plotted in Figure9 represented an obvious nonlinear relationship between the masonry compressive strength and the mortar. This simulation study produced results that corroborate the findings of a great deal of the previous work reported [17, 36].Besides, the core procedure in this simulation method introduced was solving algebraic equations. Many efficient and widely used methods for solving this sort of equation have been developed. Compared with other nonlinear solving methods based on iteration, the manner proposed in this study costs less computational resources.


## 5. Conclusions

This paper presented an alternative mesoscale approach for modeling concrete block masonry in which a simplistic representation of the mesomodel geometrical model consisting of the concrete block, the mortar, and the ITZ was provided. Additionally, the kill-FEM was successfully employed to solve the numerical model. The kill-FEM contributed significantly to reducing the computational cost since only a group of linear equations needs to be solved with a modified element stiffness matrix. According to the benchmark tests, the following conclusions arose:(1)
The mesoscale numerical method proposed was able to predict the main aspect of the failure behavior of concrete block masonry, i.e., the compressive strength as well as the failure pattern of the masonry.(2)
As a quasibrittle material, the concrete block masonry could be recognized as that meeting the maximum tensile strain criterion. According to the principle tensile strain’s cloud chart, cracks might be easy to form and distribute on the center part, the ITZ as well as the mortar, and the top as well as the bottom part.(3)
The positive effect of mortar and concrete block strength on masonry compressive strength was obtained according to the simulation studies. Besides, an obvious nonlinear relationship between the masonry compressive strength and the mortar was also observed.The numerical method introduced in this paper provided a powerful tool for predicting and studying the concrete block masonry’s observed macroscopic behavior. This study contributes to the characterization of the concrete block masonry mesostructure and the understanding of its influence on the macrobehavior. Furthermore, a proper understanding of the relationship between mesostructure and macroscopic behavior can guide the study of other systems containing advanced concrete production.



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