NUMERICAL ANALISIS OF THE STRAINS AND STRESS STATES REINFORCED CLAY BRICK MASONRY WALLS HORIZONTALLY SHEARED ANALISIS OF THE STRAINS AND STRESS STATES REINFORCED CLAY BRICK MASONRY WALLS

: This paper presents the results: stress and strain of bed joints mortar, masonry units, reinforcement bars and mechanism of cracking of numerical simulations using ANSYS of reinforced brick wall in the horizontal shear. Willam-Warnke (WW-5) failure criterion for mortar and bricks and Huber-Mises-Hencky (HMH) plasticity surface for bed joints steel reinforcement (steel smooth bars and truss type reinforcement) has been used. Coulomb-Mohr (CM) criterion in the contact surfaces of mortar and bricks have been used in the interface elements. Numerical calculations showed satisfactory convergence of research in the patterns of cracking; and the load-strain relationship was similar to the results of research with regard only to the cracking time. Destructive force numerical models Hu,cal correspond to the forces observed at the first cracks in the walls Hu,mv. Reduction of transverse and longitudinal deformation of mortar bed joints have been reported in the immediate vicinity of the bars, and the reduction of the main stress of the bricks is reported in the case of the use reinforcement .


Introduction
The FEM analysis allows tracing of the work of construction of the linear range, by cracking until its destruction. Nonlinear material models based on experimental observations, different constitutive relationships and failure criteria are mainly used. Several different modeling methods in terms of the complexity of real masonry construction can be identified. Micromodel is the most complex method of modeling the masonry structure. The test samples are treated as a heterogeneous material with reinforcement taking into account issues of contact between the mortar and the masonry units. Different non-linear behavior of brick, mortar and the reinforcement used in this case ( [1,2,3,4,5,6,7,8,19]. For macromodel and mezomodel characterized by lower complexity, the wall has the same mechanical properties of the periodic structure [18], these models may be used in the practical calculations of large masonry structures, not just parts of the wall, but also whole buildings [1,9,10,2,3,5,6,11,12,13,14]. Reinforcement of the macromodel walls are modeled as a linear elements, representing the reinforcement bars [8,19,4] or with reinforced bed joints changing the mechanical properties [18,20]. Theoretical analysis of reinforced walls suggest that the local deformations of the mortar reduced due to the presence of reinforcement will increase the intensity of cracking joints and increase the value of the cracking stress. The most positive impact of reinforcement is visible in the walls under compression. A study of 51 clay brick walls [22] also confirmed the significant influence of reinforcement.
To recognize influence of reinforcement it was necessary to build a FEM numerical model of the fragment wall, allowing for a detailed analysis of the interaction of mortar and masonry reinforcement until the cracking state. The main aim of this study was to conduct numerical analysis of the results of research [22] in terms of: • patterns of cracking, • influence of reinforcement on the mortar strains, • values of cracking and destructive forces, • principal stress (σ1) in the bricks.

Modeling Strategy
Analysis of the stress and strain states were carried out on a separate fragment wall of the central area of the research model. Due to the limited technical possibilities, the calculation process is divided into 3 stages.
• Stage I. Numerical model of the wall sheared horizontally (orthotropic shell model) the real size was built. Models were initially compressed to obtain stress σc = 0 and 1.5 N/mm 2 depending on the series of elements and sheared the horizontal force H. ) lying on the edge of the part 600 × 600 mm obtained from stage I was loaded linear-elastic solid numerical micromodels of wall (unreinforced, reinforced with bars and truss type reinforcement) overall dimensions 600 × 600 × 250 (125) mm. Eight nodes finite elements were used in this stage with parameters brick and mortar and linear elements, which were modeled reinforcement of smooth bars or truss type reinforcement.
) obtained from the solution of models of stage II was loaded the micromodels with overall dimensions 278×310×250(125) mm. Methods of discretization of numerical models in stages II and III are shown in Figure 2.

Material Models of Mortar and Brick in Stage Iii
Elastic-brittle material model with the failure surface of the Willam-Warnke (WW-5) [24,8,26,25] was used for mortar and bricks, the shape of the octahedral stress [27] shown in Figure 3. Meridians of the five parameters surface cone are parabolic in shape (Fig. 4). In the deviatoric cross-sectional shape of the failure surface WW-5 consists of three mutually tangent to the ellipse described an equilateral Rankine equilateral triangle and inscribed the outer circle Drucker-Prager. Meridian where is located the point corresponding to the compressive strength fc the uniaxial state of stress is called compression meridian and the meridian which is located the uniaxial tensile strength ft and biaxial compression fcc called tension meridian. In the deviatoric cross-section shape of the surface is periodic -meridians on the surface are alternately rotated through Lode angle =60°. All meridians intersect the hydrostatic axis at one point which corresponds to the triaxial tensile strength fttt [110]. Mortar and bricks uniaxial compressive strength and tensile strength fc, ft were adopted directly from the research cylinders  60  120 mm. However, biaxial compressive strength was fccfrom the triaxial test, where the samples were destroyed with uniform lateral and vertical compressive stress σver = 0. The results of uniaxial and triaxial tests were plotted on the graph and approximated by polynomials of the second degree least squares method. In addition, the shape of the meridian compression adjustment was made and found a common point of intersection with the tension meridian corresponding to the triaxial tension strength fttt. The adjusted compression and tensile meridians views WW-5 surface of the mortar and brick is shown in Figure 5 and Figure 6 and summarized in Table 1, where also given modules of elasticity E and Poisson ratios ν obtained in the uniaxial study.   "Smared band of crack" forming in the direction in accordance with the directions of principal stress was used in the micromodel. After cracking there followed modification of the properties of the finite element. Initially isotropic material (mortar and brick) transformed in orthotropic.
One of the axes of the material changed the direction perpendicular to the plane of the crack. In this direction occurred initially material weakness and decreased stress on the tensile strength ft to the value defined as the ratio of the tensile strength and the coefficient of weakness Tc = 0.1.

Figure 7: Smeared band of cracks σ -ε relationships
Then the material weakness in the crack described as a function was decreasing linear relationship between tensile strain εt ÷ 6εt. After cracking in subsequent iterations secant modulus ET was taken. Relationships tensile stressstrain in the plane of crack adopted in the numerical model is presented in Figure 7. In the case of cracks, which occurred in the tensile stress (open crack) with the ability to transfer shear stresses, determined the value of the coefficient t (shear transfer coefficient) taking values between 0 to 1. ,,Closed cracks" was similarly given to the properties. Transferring shear coefficient stress ratio defined as a c, whose value in the case of open cracks could adopt values between 0 to 1. Both coefficients t and c assumed equal to 0.6.
The cracking due to compression (,,crushing cracks ") were accepted the total degradation of modulus of elasticity of the material, and cracked finite element was losing the ability to transfer any tension.

Interface Elements
To the reflect the slip occurring at the interface between bricks and mortar observed in the studies [22], it was decided to use the interface elements. Were applied as 10 nodes surface interface elements of Target 174-Contac 170 type available in the ANSYS system -figure 8.
Elements were elastic-plastic (with the weakening) characteristics of the load-displacement - Figure 9 under shear load. Displacements increased proportionally with an increase of load in the elastic range. Weakness -slip represented by the degradation of the static friction coefficient of friction value to the residual friction coefficient of friction occurred after reaching the shear strength. Elements have limitations solely due to tension which meant that when the value of the tensile stresses in the joint exceed the limit was followed separation of connected pre-contact surfaces (mortar -brick) in the case of normal stress.  To determine the shear strength (fv) it was necessary to determine the initial shear strength (fvo) and static coefficient of friction (tgα) in the mortar joints. Coulomb-Mohr (C-M) (1) relationships, as a criterion of shear stress, was used.
The shape of the interface elements the shear criteria are shown in Figure 10. Linear relationship (1) was valid in the range of the normal stress σ greater than the tensile strength ft of the contact to the shear stress not more than fvmax = 10 19 N/mm 2 . Decay coefficient dc - Figure 10b was used to define after cracking the contact was required. Degradation of the static friction coefficient tgα to the residual coefficient of friction μres was described in the course of the exponential relationship.
Parameters interface elements necessary to calculate the shear were determined on the basis of the initial shear strength of the wall according to EN-1052-3:2003 Methods of test for masonry -Part 3: Determination of initial shear strength code. Interface elements parameters in vertical planes were determined by dividing the value of the parameter (except for the dc coefficient) and quality factor [28] ("quality factor"), which was adopted equal to 0.5. Parameters of the interface elements shown in Table 2.

Steel Material Model
Reinforcing bars in the model were modeled using discrete links, with elastic-plastic characteristics. Huber-Mises-Hencky yield surface in the octahedral system coordinate stress is infinitely long cylindrical surface - Figure 11 was applied. Bilinear relationship stress -strain, with associated flow law and the Prandtl-Reuss kinematic strengthening law and was applied. The kinematic strengthen law takes into account the change of the position of the plasticity surface stress in compression due to caused by tension, the so-called Baushinger effect - Table 3.

The Results and Analysis of the Researches and Numerical Calculations
Mean values of cracking Hcr,mv and ultimate forces Hu,mv obtained on all series specimens (described in detail in the work [22]) are shown in Table 4. The results of the numerical calculations as a cracking Hcr,cal and ultimate Hu,cal forces numerical models used in stage III are shown as well. Numerical shell models in stage I loaded with the forces of the average values specified at the time of the destruction of the research elements Hu,mv - Table 4. The result of model calculations carried out in the first stage was to determine the components of the displacements of all the nodes on the edge of area of 600600 mm, which linearly elastic mikromodel charged in stage II, these displacements has been loaded with linearly elastic micromodel in stage II. Displacements of nodes in stage II model obtained along the edge of the model (278×310×250 (125) mm) in stage III was used. Comparison of shear load H nondialatational strain angle Θ tests specimens, at the σc = 1.5 N/mm 2 initially compressions and numerical micromodels used in the phase III are shown in Figure 12.
Strains angle research elements grew in almost direct proportion to the shear loads to load values of 80% Hcr,mv. The highest stiffness characterized by elements reinforced with truss type reinforcement and the lowest stiffness was unreinforced specimens. With the increase of shear load H disproportionate increase angle displacements were visible, especially important after cracking. The reinforced numerical models characterized by greater stiffness than the unreinforced walls.   Due to the modeling strategy in stage III used loaded displacements of nodes obtained from the models in stage I and stage II did not occur proportional increase of walls angle deformation after crcking, and the difference of destructive forces Hu,cal and given in the researches Hu,mv was Hu,cal / Hu,mv. However, size of the numerical micromodel (stage III) model, allows for comparison of the cracking forces specimen's models Hcr,mv to the ultimate forces of numerical models Hu,cal (Hu,cal / Hcr,mv = 0,77 -1,06). Selected results of numerical models sheared horizontally and compressed vertically at σc = 1,5 N/ mm 2 reinforced with steel smooth bars and truss type reinforcement are presented later in this chapter.

Cracking and Destruction
In the model reinforced with bars, cracks were first observed at a force of Hcr,cal = 200 kN N/mm 2 , which is considerably less by about 120% compared to the average force Hcr,mv = 441 kN strength obtained in research. The first cracks - fig. 13a,b formed in the bed joint above of corner the bricks placed in head layers. Where there was the reinforced bar present cracking intensity was significantly lower than in areas without reinforcement mortar. Destruction of the model was observed at a force of Hu,cal = 585 kN of 14% less than the obtained in experimental studies Hu,mv = 668 kN. The increase in load resulted in the progressive development of the cracking areas ( fig. 13c,d) including internal joint and bricks. At the time of the near destruction of large cracking covered almost all the internal joint head, the supporting brick and joint. In the areas where reinforcement is used, the intensity at the time of destruction of cracking was slightly smaller. However, near to the face of the wall there have more damage and secondary cracks (blue and green in the present step load) providing a clear overload these areas.
Reinforcement strains were also analyzed in the following phases of the load (Figure 14).
After an initial compression to the, the σc = 1,5 N/mm 2 the deformation of the reinforced bar located in the bed joint formed to the uniform tension. With the increase of the horizontal load, the deformations of reinforcement were reduced and their distribution on the length of the bar becomes distorted. After cracking, as the rise of the horizontal load on part of the bar situated at the location cracked bed joint tensile deformation occurred the rest of the bar located in the noncracked bed joint compressive deformation occurred.  visible cracks (Figure 15a, b) of the wall reinforced with steel truss type reinforcement was also observed in the area underlying the bed joint corner of brick, and in places a connected bracking and strip of truss. The destruction occurred at the load and Hu,cal = 576 kN and was 29% lower than obtained in the lab model Hcr,mv = 609 kN. In addition, the cracks were visible in bed and vertical joint ( figure 15c, d). With the increase of horizontal load to a growing area of cracking, bed joints and cracks also appear in the bricks. At the time of the destruction cracking covered almost all the head joint. As in the case of walls with steel bars, in places where the truss type reinforcement was located cracking intensity was less than the near the face of wall. Initial deformation of the truss bars, like the smooth bars were tensile and uniform-figure 16a.
With increasing horizontal load there was seen a gradual reduction in strain in bars and change the distribution of the length. On the sections and diagonal bracking and belts lying cracked areas in the bed joints increased tensile strain - fig. 16b after the cracking. At the time of the destruction, strains were significantly changed in relation to the intermediate states. In the place the mortar cracking dominated tensile strains belts and diagonal bracking, while in places where the mortar remains non-cracked compressive strains occurred. Due to the free of the wall deformation in the thickness direction (along the x axis) the greatest strains diagonal bracking was observed.

Bed Joints Deformation
A comparison of longitudinal joint strains was also carried out where reinforcement was introduced and compared with models unreinforced. Figures -fig 17a, 18a and 19a show a map of the longitudinal strains of the bed joint at cracking and in Figures 17b, 18b and 19b show the longitudinal strains visible form the face of the model.
The bed joint areas, which were observed first cracks (masonry unit corners) in the joint without reinforcement can be seen the concentration of strain. However, in the joints, in which the reinforcement introduced strain distribution in these areas was different in terms of distribution and deformation values. Significant changes have occurred in places where the bars crossed an area of concentration, the strain decreased of 20% to 50%. In this case the separation of the mortar from bricks was visible only on the head surfaces. The visible reduction of mortar strains in the bed joints was the reason for reduced damages around bars and increase of cracking load and stiffness with respect to unreinforced masonry.

Principal Stress in the Brick
According to the well-known Mann-Muller model [28,29] for shear and compression masonry units can be cracked if the principal stress exceeds the tensile strength. The following figures ( fig. 20, 21, 22) presented maps of principal stresses σ1 (σ1 > σ2 > σ3) visible on the surfaces of the bricks located above and below the reinforced bed joint. Comparisons were made at a force equivalent to a cracking of unreinforced model HC-15 series (Hcr,cal = 181 kN). The most significant changes occurred in the state of stress bricks placed longitudinally in the wall. When applied reinforcement was visible, a shift (and increase in value) of extreme stress in the direction of the face of the wall surface in relation to the bricks placed in the model without reinforcement, where the extreme values were almost the entire width of the bricks. This phenomenon substantially reduced the loads of bricks cracking inside the wall, but in turn, caused the cracks near the face surface of the wall. The bricks placed transversely reinforcement effect was much less noticeable.  it as a discrete micro model can describe the work of the wall until cracking the real wall. This method of modeling were obtained: • destructive forces of Hu,cal of micro models, determining with reasonable accuracy cracking the real only sheared walls (σc =0 N/mm 2 ) Hcr,mv/Hu,cal = 0,71÷0,77 and the sheared and maximum compressed (σc = 1,5 N/mm 2 ) Hcr,mv / Hu,cal = 0,75÷1,02, • the cracking forces of Hcr,cal considerably smaller than the value obtained in the researches Hcr,mv the only sheared elements (σc = 0 N/mm 2 ) Hcr,mv / Hu,cal = 1,0÷1,66, and the shear and maximum compressed (σc = 1,5 N/mm 2 ) Hcr,mv / Hu,cal = 2,21÷2,71, • relationship load (H)non-dilatational strain angle (Θ) was similar to the results of research only the cracking. 3) A numerical micro model of masonry confirmed the influence (qualitatively) of the applied reinforcement on the mechanism of cracking formation; • cracking each time were created in the same areas of mortar -in the bed joint located at the corners of brick • confirmed the overall variability of the observed deformation of the reinforcement obtained in the researches. However, due to limitation of displacements of external supports of the model was not observed creation the tensile reinforcement strains observed in studies the real wall, • longitudinal strains the bed joint mortar were significantly reduced in places where it was placed reinforcement, • significant reductions in the lateral deformation of mortar in the bed joint observed in the case of truss type reinforcement, • applied reinforcement changed the value of the largest principal stress on the visible surfaces of bricks placed longitudinally in the wall, • load value at which cracking was observed of the numerical models was significantly lower than that observed in studies, • value of the failure load of the horizontal walls of the numerical models was similar to the value of cracking loads obtained in experimental studies.