CHAPTER 5 FINITE ELEMENT MODELING

Transcription

1 CHAPTER 5 FINITE ELEMENT MODELING 5.1 INTRODUCTION Masonry is a composite material with the building brick units and the mortar as the joining material, which are bonded together. Guinea [2000] 51 reported that the basic mechanical properties of the masonry are strongly influenced by the mechanical properties of its constituents namely, brick and mortar. Utilizing the material properties obtained from the experiments and using actual geometric details of both components and joints, the behavior of the brick masonry was numerically analyzed using ANSYS. There is a need for developing a comprehensive finite element model, as a numerical analysis method becomes more popular in solving numerous engineering problems. The finite element model was developed to understand the behaviour of the brick masonry walls. A three dimensional linear finite element model was developed to determine the strength, lateral displacement and the stress distribution throughout the masonry wall. Masonry itself is a composite material that consists of two materials depending upon the properties of the masonry unit (brick) and the mortar. Paulo Lorenco [2006] 118 has discussed that, there are three approaches towards its numerical representation depending upon the level of accuracy and simplicity desired. They are (i) micro level modeling (ii) meso level modeling and (iii) macro level modeling. The five brick stack bonded clay brick masonry prism and fly ash brick masonry prism is considered to determine the masonry strength. The clay brick masonry prism of size 220mm x 110mm x 400mm is considered. The size of the masonry unit is 220 x 110 x 70mm. The size of the mortar joint is 220 x 110 x 10mm as shown in Fig The finite element model was used to understand the results of the shear compression diagonal compression / tests on masonry wall panel. t b t ba t m (a) (c) Fig. 5.1 (a) Micro level modeling Meso level modeling (c) Macro level modeling 150

2 Zuchini [2009] 145 used these different simulations depend upon the methods offered by different degrees of accuracy and therefore they should be used according to the requirements of individual situations. The first approach offers the detailed interaction between the masonry units (brick) and the mortar as it is most suitable for the current study. The five brick stack bonded prism provides the most detailed accuracy during simulation. The second approach offers a better accuracy of the behaviour of a masonry structure and is suitable for simulation of five brick stack bonded prism to study the concentration of stress. The last approach studies a general behaviour simulation of the structure and is better suited for studying large size structures for the global inplane shear behaviour of the masonry wall. 5.2 FORMULATION OF THE MODEL Masonry strength is dependent upon the characteristics of the masonry unit, the mortar and the bond between them. Empirical formulae as well as analytical and finite element models have been developed to predict the structural behaviour of the masonry. When the masonry is under compression, the masonry unit and the mortar will be under multi-axial state of stress. Hence, the present investigation is an attempt to develop a finite element model to predict the masonry prism compressive strength subjected to concentric compressive loading while using some failure theories developed for brittle materials under multi-axial state of stress. The finite element model is validated by comparing the predicted values with those obtained from the controlled experimental results. In the present study, models with two different material assumptions are presented: in one, masonry as a composite material consisting of brick unit and mortar joint ; the other, treats both phases of the material are replaced with an equivalent material property, assuming it to be a homogenized material, Luisa Berto [2008] 85. Equivalent elastic modulus for brick masonry had been studied assuming that no slippage occurs between the mortar layers and brick unit with the head joints considered to be continuous as considered by Jahangir [2004] 72. In this method, the behavior of the masonry is roughly approximated by linear elasticity and perfect interface bonding hypothesis. Brick unit was modeled using solid 185, eight-node iso-parametric brick element type with three degrees of freedom: translations in the nodal x, y, and z directions as shown in Fig 5.2 (a). The element has plasticity, hyper-elasticity, stress stiffening, creep, large deflection and large strain capabilities. It also has mixed formulation capability for simulating the deformations of nearly incompressible elasto-plastic materials and fully incompressible hyper-elastic materials. Mortar joint was modeled using SOLID45, the 3-D modeling of the 151

3 element with eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions as shown in Fig 5.2. The solid 45 element has plasticity, creep, swelling, stress stiffening, large deflection and large strain capabilities. The unit properties of the brick and the mortar joint required for the analysis were obtained by conducting experiments described in chapter -3 of this thesis. (a) Fig. 5.2 (a) Solid 185, 3D solid element type used for the brick unit Solid 45, 3D solid element type used for the mortar joint The clay brick masonry reinforced with woven wire mesh in the alternate bed course of the brick masonry is modeled using shell 63 element as it has both bending and membrane capabilities in the ANSYS and the element detail is shown in Fig.5.3. Fig 5.3 Shell 63 for the woven wire mesh 152

4 SHELL63 element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities are also included. 5.3 MICRO LEVEL MODELING OF THE BRICK MASONRY The micro level models consider the units and the mortar joints separately, characterized by different constitutive laws; thus, the structural analysis is performed considering each constituent of the masonry material. The mechanical properties that characterize the models adopted for the brick units and the mortar joints are obtained through experimental tests conducted on the single material components. Compressive strength of the brick masonry: The compression testing was performed according to Indian masonry code IS: 1905[1987] 65. Four stack bonded prisms of five bricks each, were constructed and tested under axial loading. The main purpose was to examine the effects of brick unit and the mortar properties on the strength and deformation characteristics of the masonry prisms as discussed by Olivera [2000] 111. The stack bonded brick masonry prism constructed with five clay bricks having dimensions of 220 mm x 110 mm x 70 mm in 10mm thick cement mortar having the ratio of 1:6 with partial replacement of fine aggregate with fly ash. The discretization is such that, the bricks and the mortar joints had been represented by separate layers of elements. Each type of element was represented with its own properties in terms of its respective uni-axial compressive strength and initial modulus of elasticity. The applied loading was on the top surface of the model in the negative y direction representing the compressive loading with the pinned boundary condition as all the nodes at the base of the models are supported in the test platform. Brick and the mortar joint were modeled using the micro-modeling approach representing joints as continuum elements and assuming a perfect bond between the brick unit and the mortar joint. Each model was assumed to be subjected to axial compressive load, as shown in Fig 5.4 (a). 153

5 (a) Fig 5.4 (a) Micro level modeling of brick masonry prism Stress distribution on unreinforced clay brick masonry prism (UCBP) This approach leads to structural analyses characterized by great computational effort. Nevertheless, this approach can be successfully adopted for reproducing experimental tests. The simulation was carried out in sub-steps. The failure of masonry under concentric compression is related to the interaction between the masonry unit and the mortar as a result of their differing deformation characteristics. Shear strength of the brick masonry: In order to determine the shear behaviour parameter, a double shear test method using triplet specimens has been suggested by Gabor [2005] 43. The triplet shear model is assumed to be made from two different materials, namely brick units and mortar. The non-linearity considered in the analysis of the model was only material non-linearity. The 3-D finite element analysis on the unreinforced clay brick masonry with the ratio of 1:6 cement mortar, UCBP was modeled and analyzed. The results are obtained for the maximum deflection, the shear stress and the vertical shear strains in the triplet prism. When the maximum stress level was reached, the behavior of the brick masonry specimens was characterized by a (quasi fragile) softening behavior and by a sliding movement between the adjacent bricks as reported by Fazia Fouchal [2009] 42. The triplet prism modeled indicates that the shear behaviour of masonry is weak along the mortar joint are shown in Fig. 5.5 (a, b & c). 154

6 (a) Fig. 5.5 (a) Deformation of the unreinforced clay brick masonry under triplet shear test Element stress distribution in the clay brick masonry under triplet shear test (c) Fig. 5.5 (c) Nodal stress distribution in the clay brick masonry under triplet shear test The displacement pattern predicted by the finite element model shows vertical sliding between the brick and the mortar of the brick masonry prism. 5.4 MESO LEVEL MODELING OF THE BRICK MASONRY In meso-level modeling expanded units are represented by continuum elements. The behaviour of the mortar joints and the unit/mortar interface are lumped into discontinuum elements as shown in Fig 5.6. In this approach, the units are expanded to retain the initial geometry of the masonry assemblage. Due to the assumption of the zero thickness of mortar joints, the elastic properties of the expanded brick units are adjusted to yield the elastic modulus of the represented masonry. In the stack bonded brick masonry prism, the bricks are assumed as a series of chain 155

7 connection of the components results in the uniform stress distributions both in the brick unit and the mortar. The adjusted average elastic stiffness of the expanded brick units are simply derived from Krit [2007] 79 by considering the elastic properties of the masonry components (brick and mortar) and the expanded thickness as, Adjusted Elastic modulus, E ba = E E ( t t ) t b b E m m b t m E m b ; Adjusted brick thickness, t ba = t b + t m The hexagonal woven wire mesh (reinforcement material) used in the brick masonry was modeled as shell 63 and modeled in between the expanded bricks. The stress behaviour of the clay brick masonry (unreinforced and reinforced) is shown in Fig 5.7 (a & b). Fig. 5.6 Meso level modeling of the brick masonry prism (a) Fig 5.7 (a) Stress distribution on unreinforced clay brick masonry prism (UCBP) Stress distribution on clay brick masonry prism reinforced with wire mesh (RCBP) 156

8 From the finite element modeling of unreinforced clay brick prism, it was understood that the stress is found to be uniform throughout the unreinforced masonry brick prism specimen. Whereas, in clay brick masonry prism reinforced with woven wire mesh the stress is concentrated at the reinforcement (mesh) placed in between the expanded brick units at the bed joint of the brick masonry prism. 5.5 MACRO LEVEL MODELING OF THE BRICK MASONRY In this analysis, the brick masonry which is made from two different materials of the clay bricks and the mortar had been replaced by an equivalent homogenous material. The macro-models are based on the use of constitutive laws for the masonry material; i.e. the stress-strain relationships adopted for the structural analysis are derived by performing tests on masonry, without distinguishing the bricks and the mortar behavior. Equivalent homogeneous material or macrolevel simulation approach of the brick masonry is shown in Fig 5.8. Homogenization involves using the analytical micro models for small masonry assemblages to determine the combined response. The meso level and the macro level modeling technique are unable to model the local failure modes (unlike the micro-modeling technique). (a) Fig. 5.8 (a) Macro level modeling of brick masonry prism Stress distribution on the unreinforced clay brick masonry prism (UCBP) Masonry sustains damage in the form of cracks in the early stage of the loading as the mortar break at the low level of the load compared to the brick units. The micro-modeling has been compared with the meso-level and the macro-level modeling along with the experimental results. Stress - strain behaviour of the unreinforced clay brick masonry prism is predicted using finite element modeling and verified with the experimental data. The comparison of the predicted 157

9 values of masonry compressive strength obtained by different numerical modeling techniques such as (ANSYS) and with the experimental results are shown in Fig. 5.9 Stress, N/mm Fig 5.9 Stress strain curve of the unreinforced clay brick masonry prism (UCBP) The stress-strain curve obtained from the finite element analyses shows that the maximum compressive strength of the brick masonry was slightly higher for the case of homogenized material than that of the composite material. The actual compressive strength of the masonry determined by experimental method was much higher than the strength obtained by numerical method. However, in reality, the brickwork was constructed from two layered materials, namely brick and mortar. Therefore, the idealization of composite material for the analysis should be adopted. The comparison of experimental and numerical results indicates that the compressive strength of masonry obtained by experimental method was 47% higher than those obtained by the finite element method which are in accordance with the theories proposed by Hendry [1990] 56. Therefore, for the design purposes, the strength obtained for the brick masonry from finite element analysis should be magnified with a factor of 1.52 in order to get the actual strength of the brick masonry. Stress strain curve of unreinfoced clay brick masonry prism, UCBP Experimental Micro model Meso model Homogenized model Strain 5.6 IN-PLANE SHEAR STRENGTH OF THE BRICK MASONRY WALL The structural shear walls of a masonry building subjected to horizontal loading commonly present two types of failure. The first one is out-of-plane failure, where cracks appear along the horizontal mortar joints. The second one is in-plane failure, generally characterized by a diagonal tensile crack. Zuchini [2009] 145 stated that if the out-of-plane failure is avoided, then the structural resistance is mainly influenced by the in-plane behaviour of the shear wall. Use of 158

10 macro level modeling requires less number of elements and hence produces quick numerical solutions. The macro-modeling technique is suitable for modeling large sections of masonry, where only a simplified representation of composite behaviour is required and local failure modes are not so important. As macro modeling of masonry is advantageous for the global behaviour of the structure, the macro modeling has been adopted in preference to the micro modeling to study the behaviour of the masonry wall panel as reported by Gabor [2006] 43. In this macro level modeling, the brick masonry made from two different materials of clay bricks and mortar had been replaced by an equivalent homogenous material and analyzed. The finite element model was used to understand the in-plane behaviour of the masonry and is verified with the shear-compression tests and diagonal compression tests. Shear-compression test: Unreinforced masonry shear walls are often used as the main structural component of masonry buildings responsible for carrying the lateral loading such as wind and earthquake loads. The lateral and vertical loads lead to tension and shear combined with compression within the masonry wall. Krit Chaimoon [2007] 79 stated that the fracture and failure of the masonry walls under shear-compression is intricate because of the complex interaction of the shear failure along the mortar joints and compression failure often at the toe of the wall. The modeling of the shearcompression test on brick masonry wall panel predicted the stress behaviour of the clay brick masonry wall and is shown in Fig 5.10 (a & b) and in Fig 5.11(a & b). (a) Fig 5.10 (a) Macro level modeling of brick masonry wall under shear compression test Stress behaviour on unreinforced clay brick masonry wall specimen (CBP) 159

11 Homogenized model behaves as one material and the dispersion of the load in the model will be at about 45 o. However, in the case of models assumed to be made from composite materials, similar dispersion of vertical loads cannot be expected as reported by Luisa Berto [2008] 85. The detailed modeling of the geometrical structure of the masonry requires important computational resources and renders the modeling quite laborious. Thus, if the goal of the modeling is to obtain an approximation of the average behaviour of the masonry in terms of loads and deformations, it is conceivable to build an equivalent material model without considering the internal geometry of the masonry. (a) Fig 5.11 (a) Stress behaviour on unreinforced clay brick masonry wall specimen (CBP10) Stress behaviour on unreinforced clay brick masonry wall specimen (CBP20) From Fig (a & b), the stress distribution of the specimen CBP20 was widened along the diagonal of the wall panel than the specimen CBP10. Finally, the finite element modeling results are compared with the obtained experimental results and discussed the reliability of the finite element models. Diagonal compression test: The diagonal compression test mechanism was composed of a set of metallic elements fixed at the two corners of a diagonal end of the panel. In this test, a compressive force was applied gradually along a diagonal end of the specimen to study the in-plane shear behaviour of the masonry wall panel. The applied compressive force may cause a diagonal tension in the specimen which in turn led to the failure of the specimen with splitting cracks parallel to the direction of the load. The finite element model used to simulate the behaviour of the masonry under diagonal compression tests is described in Fig 5.12 (a). The material properties required 160

12 for the masonry model were determined from the experimental characterization tests in chapter 3 of this thesis. The failure pattern and the load deformation behaviour of the specimen are studied. (a) Fig 5.12 (a) Macro level modeling of brick masonry wall panel under diagonal compression test Stress behaviour on unreinforced clay brick masonry wall specimen (CBP) As the diagonal load was applied on the wall panel specimen, the deformation occured and the stress was formed as the diagonal band along the direction of the application of the load. The stress propagation on the unreinforced clay brick masonry wall panel (with the ratio of 1:6 cement mortar with partial replacement of fine aggregate with fly ash as 0%, 10% and 20%) was in the form of the diagonal band as shown in Fig and Fig 5.13 (a & b). The behaviour of the clay brick masonry reinforced with woven wire mesh is depicted in Fig (a) Fig 5.13 (a) Stress behaviour on unreinforced clay brick masonry wall specimen (CBP10) Stress behaviour on unreinforced clay brick masonry wall specimen (CBP20) 161

13 Load (N) Fig 5.14 Stress behaviour of reinforced clay brick masonry wall specimen (CBPR) The ANSYS modeling was compared with the test results and the behaviour had been observed. The numerical load versus displacement curve is plotted in Fig 5.15 and Fig 5.16 and compared with the experimental results. A reasonably good agreement has been found Load -deformation curve of unreinforced clay brick masonry wall panel (CBP) Experiment Ansys Horizontal deformation (mm) Fig 5.15 Load-deformation curve of unreinforced clay brick masonry wall panel (CBP) 162

14 load (N) Load - deformation curve of brick masonry CBP10 Experiment Ansys Horizontal deformation (mm) Fig 5.16 Load-deformation curve of unreinforced clay brick masonry wall panel with 10% replacement of fine aggregate with fly ash in the mortar (CBP10) The comparison of the shear stress obtained from the ANSYS and the experimental stress value are reported in Table 5.1 Table 5.1 Comparison of shear stress obtained through ANSYS and experiment Type of wall Maximum shear stress through the experiment (MPa) Maximum shear stress through ANSYS (MPa) Shear stress obtained through ANSYS / Shear stress obtained through Experiment CBP CBP CBP CBP0R CBP10R CBP20R FBP FBP FBP FBP0R FBP10R FBP20R Review of the numerical data leads to recognize that, if macro-modeling strategy was applied, the overall response of the masonry panels can be well predicted in terms of the collapse load and the deformation values as well as sufficiently accurate failure mechanisms can be predicted. 163

15 The compressive strength of the brick masonry obtained from ANSYS analysis are magnified by the factor 1.52 in order to obtain the experimental results which are in accordance with Jahanghir et al [2004] 72. Similarly the maximum shear stress obtained from ANSYS is to be magnified by the factor 1.95 to obtain the experimental shear stress. 5.7 CONCLUSIONS (i) The finite element model traces the progressive crack growth and the stress distribution patterns in the masonry units and the mortar. (ii) Monitoring of crack patterns as the applied stress intensity increases shows the progress of crack development and the crack propagation in the masonry prism. The crack pattern predicted by the finite element model leads to vertical splitting cracks in the prism. Such vertical splitting cracks across the prism height were seen in the masonry prisms tested for compressive strength. (iii) The compressive strength of the brick masonry prism (both unreinforced and reinforced clay brick masonry and fly ash brick masonry) predicted by the finite element models is 47% higher than the experimental values which are in accordance with the theories proposed by Hendry [1990] 56. (iv) The comparison of experimental and numerical results indicates that the compressive strength of the masonry derived by the finite element method has to be magnified by 1.52 times to arrive the experimental data. Also to get the actual in-plane shear stress of brick masonry, the finite element analysis results should be enhanced by a factor