Numerical and experimental evaluation of masonry prisms by finite element method

Resumo This work developed experimental tests and numerical models able to represent the mechanical behavior of prisms made of ordinary and high strength concrete blocks. Experimental tests of prisms were performed and a detailed micro-modeling strategy was adopted for numerical analysis. In this modeling technique, each material (block and mortar) was represented by its own mechanical properties. The validation of numerical models was based on experimental results. It was found that the obtained numerical values of compressive strength and modulus of elasticity differ by 5% from the experimentally observed values. Moreover, mechanisms responsible for the rupture of the prisms were evaluated and compared to the behaviors observed in the tests and those described in the literature. Through experimental results it is possible to conclude that the numerical models have been able to represent both the mechanical properties and the failure mechanisms.

Numerical and experimental evaluation of masonry prisms by finite element method

Introduction
Masonry structure is a building system with elements that perform both structural and sealing functions, thus allowing a greater rationalization.As this system has been increasingly used in the construction sector, there is a need for innovative materials in the industry.According to (Castro [1]), the use of minerals (silica fume, calcined clay) and plasticizing admixtures has proportioned blocks of greater strength and lower permeability, thus named high strength blocks.Despite the development of new materials and the use of much more slender structures, codes have not undergone many changes to follow such a big development.The main reason is the lack of experimental and mathematical models that explain the complex behavior of units and mortar working together as a composite material.As an experimental program with wall testing is expensive, most of the authors agree that is possible to establish a relation between the load and failure mode of the walls, studying the prisms behavior.It would allow the reduction of testing costs.Still regarding the possibility of cost reduction, an alternative widely used to study specific phenomena is the numeric simulation.Since well calibrated, it is able to supply information about the structural behavior and provides subsidy to determine, with safety, the parameters to be used in project/codes.In order to obtain a reliable and accurate numerical model, a complete description of the material must be made based on experimental results.Once calibrated the model, it is possible to vary the desired parameters and verify the isolated effect of each component.According to (Oliveira [2]), mortar joints represent planes of weakness and are responsible by most of the nonlinear phenomena that occur on the structure, which behavior makes the computer modeling process more complex.Considering this context, the goal of this work is to present the calibration of prisms' numerical models, based on the experimental results obtained by (Oliveira [2]) and also by the authors of this work.With the numerical modeling results, a study will be conducted to compare the mechanisms responsible by failure to the behavior observed during the tests.

Context
In order to base the discussion of the results (item 4), some conclusions about prisms obtained by other researchers will be presented.

Behavior of mortar in the prisms
When the masonry is submitted to a vertical loading, the generated state of stress causes horizontal stresses on the mortar joint due to existing adhesion between the unit and the mortar.So, in a prism or wall, the mortar is subjected to a triaxial state of stress, presenting a behavior different from that observed in uniaxial testing.It is noted that when the mortar is subjected to a triaxial state of stress, changes occur in its mechanical properties, such as: compressive strength, Young's modulus, and Poisson's ratio.According to (Mohamad [3]) and (Khoo [4]), the study of the behavior of confined mortar is essencial to understand the failure mechanisms of prisms and walls, either by reaching the block's tensile strength, or occurring the block's localized crushing, or happening the mortar bed joints crushing.Thus, both authors studied the increase of mortar compressive strength due to the increase of the confining pressure, obtaining the failure envelope.The equations ( 1) and ( 2) correspond to the failure envelope proposed by (Mohamad [3]) and (Khoo [4]) to mortar 1:1:6, respectively.

Shear strength of masonry mortar joint
The shear strength of the mortar joints has a behavior that can be described by Coulomb's law, which is given by equation ( 3), (Riddington [5]). ( where: u t is the shear strength of horizontal or vertical joint; f is the shear strength by initial adherence, also named cohesion; σ is the normal pre-compression stress acting across the joint; φ is the friction angle of the material.

Rupture modes of the prisms
There is a consensus among researchers that the rupture mode of prisms and walls is function of the mechanical properties of their constituent materials, such as Young's modulus and compressive strength.The studies led by (Hamid e Drysdale [6]) suggest that the mechanism of rupture of masonry occurs due to the biaxial tensile-compressive stress state, in consequence of the low stiffness of the mortar.According to the authors, the units prevent the lateral deformation of the mortar, resulting in lateral confining stresses on it, so the mortar strength is considerably increased.Analyzing the experimental results, the authors concluded that when the loads reach approximately 80% of the ultimate load, some microcracking occurs along the entire length of the prisms.As done by other authors, (Mohamad [3]) performed experimental tests on prisms of concrete units with face shell mortar bedding.In the rupture, they observed the development of tensile stresses on the lateral face caused by rotation and crushing of the supports.This rupture mechanism is similar to the beams', in which the mortar's lateral crushing induces horizontal stresses that cause the material to fail in bending, as can be seen in Figure 1.In his tests, (Romagna) [7] evaluated the mechanical behavior of concrete prisms under compression.The author noted the development of intense cracking at the intersection of the blocks' face shells and webs, and also crumbling at the block's surface.The author also reported that there were cracks in spe-cific points of the block's walls due to the loss of mortar loadcarrying capacity, causing the units' overlapping and their consequent crushing.In this context, the research carried out by (Cheema e Klingner [8] apud Juste [9]) can be also mentioned.They developed mathematical expressions able to predict the compressive load related to the occurrence of each kind of rupture of ungrouted prisms (by transverse block splitting, block crushing or mortar crushing), depending on the relationship between the elasticity modulus of mortar and blocks.Based on some expressions it was possible to plot the graph shown in

Materials and methods
This paper performed a numeric simulation of some prisms tested by the authors of this work and others by (Oliveira [2]).In both cases, the prisms were produced using face shell mortar bedding and concrete blocks.The authors of this study used high strength blocks and (Oliveira [2]) used common blocks.For calibration and validation of  Curve of failure of ungrouted prisms, (Juste [9]) Numerical and experimental evaluation of masonry prisms by finite element method numerical models the authors used the experimental results and the description of the rupture modes experimentally certified.

Materials
Table 1 and Table 2 present experimental results used in the construction and validation of the numerical models, obtained by the authors and (Oliveira [2]), respectively.Figure 3ab presents the rupture modes displayed in the experimen-tal tests performed by the authors and (Oliveira [2]), respectively.It should be emphasized that the commercial software ABAQUS was used to perform the numerical modeling, by Finite Element Method.

Methods
A detailed micro-modeling strategy was adopted for numerical modeling of the prisms.In this technique, each component of the prism was represented by its own mechanical properties and dimensions.It was also necessary to supply parameters for complete discrimination of block/mortar interface (item 3.5).

Description of the models
The finite element used to generate the mesh that represents blocks and mortar was C3D8, which is a solid element of eight nodes with linear interpolation and complete integration, able to represent the translations in x, y and z-axis, once the element has three degrees of freedom per node.
In order to facilitate the description of the numerical models, they were identified as PA and PO, which are related to the experimental results used for calibration and validation obtained by the present authors and (Oliveira [2]), respectively.When performing meshing tests, it was decided to use, for PA model, elements with dimensions of 1 cm for the blocks and 5 mm for the mortar.For PO model, elements with 2 cm for the blocks and 5 mm for the mortar proved to be more appropriate.The meshing  difference between the two models occurs due to the lowest number of interfaces in the PA model, allowing further refinement of the block mesh, still having a reasonable computational cost.In Figure 4, the general aspects of the meshes of the models are presented.
In respect to the loading, it was decided to apply a prescribed displacement (1,0 cm) to the entire top surface, thus all nodes of the surface were subjected to a rigid body motion.The analysis method for the models was the Static General, Newton Rapson.

Constitutive model
The constitutive model used to represent the mechanical behavior of blocks and mortars was the Concrete Damaged Plasticity.According to (Kmiecik and Kaminski [10]), this model is an adaptation of the Drucker-Prager model, which is also used to represent the mechanical behavior of brittle materials.
In Concrete Damaged Plasticity (CDP), the equation that defines Numerical and experimental evaluation of masonry prisms by finite element method the surface of Drucker Prager is modified by a K c parameter equal to 2/3, Figure 7.Note that this model has been proposed by (Lubliner et al. [11]).
Still about the constitutive model, it is necessary to define some essential calibration parameters to make it be useful:

Behavior under uniaxial compression in the inelastic region
In the CDP constitutive model, elastic and inelastic deformation ( el ε and inel ε , respectively) are calculated independently, and subsequently summed to obtain the total deformation ( ε ).el ε depends only on the materials' modulus of elasticity and Poisson's ratio and inel ε is obtained from their stress-strain curve.
In many cases, it is difficult to obtain stress-strain curves experi-mentally.Some authors and codes present empirical formulations, in which the researcher only includes parameters that are generally easy to determine.The compression's stress-strain curve used in this paper was proposed by (Guo [12]).It considers the elastic region until 30% of the ultimate load, where the modulus of elasticity is the slope of the stress-strain diagram.The inelastic region is defined by the formulation proposed by (Guo [12]).Figure 8 presents his schematic stress-strain curve.
The program requires the portion of deformation of the stressstrain curve related to the inelastic deformation independently.Then, it is necessary to subtract the elastic deformation of the total deformation, by using the equation ( 10): (10) Applying this equation to calculate a new deformation for each point of the described stress-strain curve, it is possible to build the curve that supplies the software, Figure 10.
The parameters values necessary to build the analytical stressstrain curve of blocks and mortar are shown in Table 3 and Table 4.The values of α d were obtained from the experimental curves obtained by (Oliveira [2]).For the experimental curves obtained in this study, the same α d obtained by (Oliveira [2]) was adopted.

Behavior under uniaxial tensile in the inelastic region
As done for compression, it is necessary to indicate the behavior of the material under tensile.Thus, it is necessary to provide the tensile's stress-strain curve, that also has an elastic and an inelastic portion, (Guo [12]), Figure 11.The elastic region is described by Input parameters for stress-strain curve, Authors Table 3 Input parameters for stress-strain curve, (Oliveira [2]) * Block properties related to the net area; ¹ the strain at peak load was determined based on the stress-strain curve experimentally obtained.
-Elastic region: (11 -Inelastic region: where: (13) 0.312 and: cm E is the initial tangent modulus of elasticity; btm f is the tensile strength of material.
For concrete, the (ABNT NBR 6118:2014 [17]) estimates the tensile strength as . However, there is no specification for tensile strength of blocks and mortar.Then, equation ( 16) was adopted: Input parameters for tensile stress-strain curve, (Oliveira [2])

Authors
Blocks* Mortar  As done for compression, it is necessary to calculate the portion of deformation related to the inelastic one, in order to build the stressstrain curve shown in (Figure 11).Again, the elastic deformation was subtracted of the total deformation, as done to obtain equation (10).
The input parameters used in the modeling are presented in Table 5 and Table 6.

Stiffness degradation of the material
When applied the load and exceeded the elastic region, a stiffness degradation occurs due to the appearance of plastic deformation.This degradation can be determined by unloading the material, calculating the unloading modulus of elasticity, and comparing it to the initial modulus of elasticity.As a simplification, it was assumed that this behavior occurs just in the postpeak stress-strain curve, Figure 13.The decrease of the slope is ruled by two independent variables, c d (damage in uniaxial compression) and t d (damage in uniaxial tensile).It ranges from zero for an undamaged material to one for the total loss of load-bearing capacity.According to (Cardoso [14]), those variables can be defined using the equations ( 17) and ( 18): (17)

Parameters of the constitutive model Concrete Damaged Plasticity
Besides the previously mentioned parameters, it is necessary to provide to ABAQUS some other ones, which are responsible for allowing to apply the equations of the behavior of materials under uniaxial state of stress to materials under multiaxial state of stress (Aguiar [15]): σ σ : Parameter that describes the ratio between the yield strength in biaxial and uniaxial state.This study adopted the value of 1.16 given by ABAQUS, (SIMULIA [13]) as a default value.
n Dilation angle ( ) ψ : According to (Cardoso [13]), this param- eter is related to the slope that the plastic potencial reaches under higher confining stress.(Kmiecik and Kaminski [10]) define the dilation angle as the concrete's angle of internal friction and recommend the adoption of 36 ψ = ° .
n Viscosity parameter ( ) µ : Parameter whose function is facilitate process of numerical models, regulating the constitutive equations through the viscoplasticity study.In this work, a value of 6 10 µ − = was adopted, based on a preliminary study that looked for the lower value of viscoplasticity that facilitated the convergence process, without affecting the results.
n Eccentricity parameter ρ : According to the theory of Drucker Prager, the yield surface in the meridian planes has the form of a straight line ( 0 ρ = ).However, the experimental trials indicate that the yield surface takes the shape of a hyperbole close to the hydrostatic axis ( 0.1 ρ = ).In this study, it was adopted 0 ρ = .

Description of the block/mortar interface
To make block and mortar interact together in the model, it was necessary to define the relationship between them.The Interactions module of ABAQUS was used to do this.The contact between the surfaces was established through the option called surface-to-surface contact, where three contact properties were defined: Hard, Tangential Behavior, Cohesive Behavior and Damage.n The Hard contact has the ability to prevent a surface penetrates other surface and also allows a separation between them, after the contact is established.n The Tangential Behavior contact can reproduce the friction that occurs on the connection between the materials.It is defined by the static friction coefficient ϕ.It obeys the Coulomb law, in which the shear strength increases due to the increase of the compression

Figure 13
Degradation of the material's stiffness, SIMULIA [13] Source: Simulia (2012) A B Numerical and experimental evaluation of masonry prisms by finite element method stress.Two features in this contact must be taken into consideration: the first one is that part of the cohesion is despised, and the second one is the possibility of establishing a critical shear stress máx t . When the shear stress reaches this critical value, the slipping between the surfaces will occur without an increase in shear strength, regardless of the magnitude of the compressive stress, Figure 14.n The Cohesive Behavior works like a zero thickness rubber, for which you can specify the normal stiffness nn K and the tangential stiffness ss K , tt K of the interface.This kind of contact causes a stiffness degradation called Damage, in which it is only necessary to provide the interface's fracture energy.The input parameters to define the interface properties are presented in Table 7 and Table 8.The interfaces locations on the prisms models are shown in Figure 15ab and Figure 16ab.The positions of the interfaces were defined from the experimental results, and correspond to those where the cracks occured.

Compressive strength and modulus of elasticity
The results obtained from the numerical models are presented below and compared to the experimental results presented at the item 3.1.Once the numeric models are validated for each case, the possible failure modes will be discussed.The properties of blocks and mortars used in the numerical modeling of the prisms were obtained from the mechanical characterization experimentally performed.
Remembering the convention used to identify the numerical model: PO is related to the numerical modeling of the prisms tested by (Oliveira [2]) and PA is related to the modeling of the prisms tested by the authors of this paper.
In Table 9 and Table 10, comparisons between experimental and numerical results of the (Oliveira [2]) and this authors' prisms are presented.The data include values for uniaxial compressive stresses and modulus of elasticity, calculated between 0.5 MPa and 30% of the failure load.
It is noted from Table 9 and Table 10 that the numerical models of the prisms represented well the mechanical behavior obtained experimentally for both failure mode and modulus of elasticity.The biggest difference between experimental and numerical results was around 5%.However, it is worthwhile to also make a comparison of the failure modes in order to verify the behavior compatibility between the experimental and numerical model.The principal stresses state at the moment of the rupture will be checked.

Principal stresses in the blocks
Figure 17ab shows the maximum principal stresses state that was acting (only on the blocks) at the time of the rupture, for the models PO and PA. Figure 17a and Figure 17b show that, at the time of the rupture, the indirect tensile stress (related to the net area) experimentally obtained (0.91 MPa for blocks of this authors and 1.47 MPa for [2]) had already been exceeded.This behavior was mainly noted at the blocks' face shells, which were subjected to bending due to the face shell mortar bedding.Such behavior was also demonstrated in the experimental tests, according to the authors and [2].Based on the Figure 17ab, at the time of the rupture, the blocks' face shells supported tensile stresses higher than the indirect tensile stresses of the blocks.This behavior is due to the stress generated by the tendency of expulsion of the mortar's outer edges (due to the vertical load), so that the shear strength between blocks and mortar was not exceeded, due to the friction between them.In the

Figure 14
Tangential Behavior, SIMULIA [13] Source: Simulia (2012) Table 7 Input parameters adopted to define the interface properties, (Oliveira [2]) Table 8 Input parameters adopted to define the interface properties, Authors  outer part of the mortar joint, the confining effect is smaller.Consequently, the mortar tends to present a greater deformation, causing the described effect.In this study, the occurrence of rupture of the block's walls was observed during the tests, thus indicating those tensile effects.Figure 18ab presents the minimum principal stress state that was acting in the blocks at the time of the rupture, for the models PO e PA.
Based on Figure 18a and Figure 18b, at the time of the rupture, the lowest block's minimum principal stress have already presented values close to those obtained in the uniaxial compression test and related to the block's net area (18.64 MPa to PO and 35.37 MPa to PA).In both models, the blocks' face shells presented stresses close to their compressive strength, which indicates that the loads tend to concentrate in there.In the experiments performed

Principal stresses in the mortar joints
In Figure 19ab and Figure 20ab, the principal stresses in the mortar joints are presented, for the models PO and PA.
Analyzing the Figure 19a (model PO), it is possible to observe that mortar is completely compressed in its two directions due to the confining effect.In the model PA (Figure 20a), the mortar joint is not completely subjected to a triaxial state of compression, since its external part is subjected to a 0.9 MPa tensile stress.Then, the mortar is under low confinement in this region of the joint.This effect was not realized in the model PO, due to the lower ratio of mortar and blocks' modulus of elasticity ( arg / 0.375

b E E =
).Based on Figure 19b, all the mortar joint points were subjected to compressive stresses higher than the compressive strength obtained from the uniaxial compression test (PO -3.52 MPa).The same behavior was observed in the Figure 20b, where all the black area corresponds to stresses higher than those obtained in the uniaxial compression test (PA -11.80 MPa).In both cases this situ-   Assuming the equation ( 1 ) proposed by (Mohamad [3]) and the equation ( 2) obtained by (Khoo [4]), the failure envelope will be reproduced, in order to verify if the lower minor principal stress (obtained numerically) exceeds the envelope, at the rupture's load level.The main point is to verify if the confined mortar's crushing did happen.
Using the equations e knowing that the lower minor principal stress is equal to 24.16 MPa and 64.17 MPa, for PO and PA, respectively, the failure envelopes were built in Table 11 and Table 12, Figure 21 and Figure 22 show the failure envelopes for both studied models.
Considering Figure 21 and Figure 22, it is possible to affirm that at the time of the rupture, PO model presented the lower minor principal stress close to the maximum mortar's triaxial compressive strength.For PA model, the lower minor principal stress had already exceeded the two failure envelopes.Note that these envelopes, obtained by (Mohamad [3]) and (Khoo [4]), are only estimates.For better assessment, it is recommended to carry out studies about the mortar triaxial behavior, as those performed by the authors and [ Oliveira 2].However, their failure envelopes can be considered as tools to assist in the determination of possible failure modes.

Avaliation of failure modes
Based on the itens 4.2 and 4.3, it is possible to say that the rupture of prisms PO occurred due to a combination of factors.They presented large flexural cracks in their walls, and also block and mortar crushing.The transverse block splitting was the more evident rupture mode in the experimental test.Comparing the model proposed by (Chema e Klinger [8]) at the item 2.3 to the results obtained numerically, the ratio arg / 0.375 b E E = would indicate a block's tensile failure, which was verified in the numerical analysis.In PA model, at the time of the rupture, some points of the mortar had already exceeded its compressive strength under confining pressure, causing the localized crushing of the mortar joint.This behavior was observed in some points of the prisms during the research.According to (Chema e Klinger [8]), the localized crushing of the mortar can cause some stress concentration, leading to the appearance of block's cracks by compression.Figure 17 indicates that the blocks' lower minor principal stress exceeded the value of their uniaxial compression strength, indicating the localized crushing in the blocks.Thus, the failure mode of PA prism consisted of the simultaneous occurrence of webs rupture, due to the bending caused by the face shell mortar bedding (inducing tensile stresses), and localized crushing of the mortar joint, which induced blocks' localized crushing.Comparing the model proposed by (Chema e Klinger [8]) with the numerical results, for the ratio arg / 0, 46 b E E = (indicated in the item 2.3) the rupture would be given by a block's tensile failure.This behavior was also obtained in the numerical analysis.

Conclusions
Based on the presented results, it is possible to conclude that numerical models are able to reproduce the mechanical behavior and the failure mechanisms of prisms of ordinary and high strength concrete blocks.The difference between the compressive strength and the modulus of elasticity obtained numerically and experimentally was around 3% for the PO model and 5% for the PA model.In addition, the rupture of PO and PA prisms occurred due to a combination of factors.Block's web tensile failure happened in both of the models, due to the face shell mortar bedding, and was the most significant failure mode of both experimental and numerical point of view.Although both models showed the bending effect on the flanges, each one presented some peculiarities when achieved the maximum supported load, due to the mechanical properties of its own components.
In the PO model, the friction developed between block and mortar was enough to keep the mortar completely confined.However, in the imminence of block's crushing resulted of the localized crushing of the mortar, the transverse block splitting occurred.In the PA model, the region close to the inner face of the blocks presented a localized crushing of the mortar followed by the localized crushing of the blocks.In the outer face, there were tensile stresses in both block and mortar, due to the expulsion of the mortar.Such effect was not observed in the PO model, due to the lower ratio of the mortar and block's modulus of elasticity.

Figure 2
, by mortar crushing (region 2 of the Figure2).When arg 0.66 b E E ≤ the rupture occurs by transverse block splitting (region 1 of the Figure2).

Figure 3 Figure 4
Figure 3 Failure mode of concrete block prisms, Authors and (Oliveira [2]) A B For the supports, both models have had the same boundary conditions, shown in Figure5abc.They where: a) z-axis translational restraint at the base of the block, Figure5a; b) x-axis displacement restraint for both block(1) and mortar(2), Figure5b; c) y-axis displacement restraint for both block (1) and mortar (2), Figure5c.Both x and y restraints of blocks and joints were imposed to all of the model's blocks and joints.

Figure 5 Figure 6
Figure 5 Boundary conditions of the prisms B B B A B1 B2

Figure 7 Figure 8 1 cE
Figure 7Influence of the K c parameter on the shape of the yield surface.Source:Aguiar (2014)

Figure 15 Figure 16
Figure 15 Location of the interfaces with Coesive Behavior B B A B

Figure 17
Figure 17 Maximum principal stress numerically obtained, in Pa -only blocks: a) PO model; b) PA model B B A B

Figure 18
Figure 18 Minimum principal stress numerically obtained, in Pa -only blocks: a) PO model; b) PA model B B A B Figure 20 a) Maximum principal stress; b) Minimum principal stress -both numerically obtained, in Pa -only mortar: PA model B B A B

C
. F. R. SANTOS | R. C. S. S. ALVARENGA | J. C. L. RIBEIRO | L. O. CASTRO | R. M. SILVA | A. A. R. SANTOS | G. H. NALON ation is due to the confining effect, since the compressive strength of the material increases significantly when considering its actual triaxial stress state.

Table 1
Experimental results, Authors * Results related to the net area; ** Data related to the gross area

Table 9
Comparison between experimental and numerical results, PO * Results related to the gross area; *** Results related to the net area; ¹ Related to the experimental results.
*** Results related to the gross area; *** Results related to the net area; ¹ Related to the experimental results.

Table 11
Failure envelopes, PO -Mortar [1]he uniaxial compressive strength of mortar was experimentally obtained by Castro[1]; ² The confining strength was obtained by considering the average value of the vertical stresses acting close to the lower minimum principal stress.