Calibration of a numerical model for masonry with application to ...

Protection of Historical Buildings, PROHITECH 09 – Mazzolani (ed) © 2009 Taylor & Francis Group, London, ISBN 978-0-415-55803-7

Calibration of a numerical model for masonry with application to experimental results E. Dumova-Jovanoska & S. Churilov University Ss. Cyril and Methodius, Faculty of Civil Engineering, Skopje, Macedonia

ABSTRACT: A calibration of a numerical model for analysis of masonry walls with application to experimental results is presented in this paper. The experimental results used for calibration are derived from the research project PROHITECH. This paper adopts micro-modeling strategy for analysis of masonry specimen by discrete element method and application of different nonlinear material models both for blocks and mortar. In order to determine the values and to calibrate the necessary material data for the used materials that were not obtained experimentally, several numerical investigations and simulations were performed. Numerical analysis of masonry walls was performed by UDEC software. A comparison of numerical and experimental results as well as a comparison of the failure mechanisms is given. With the assumed modeling strategy and numerical method, a satisfactory agreement with the experimental results regarding limit state and developed failure mechanisms is obtained. With the results from this research and literature survey several recommendations regarding material properties necessary for numerical analysis are given in the end of this paper. 1

INTRODUCTION

This paper presents numerical simulation and identification of the dominant failure mechanisms of masonry walls with different geometrical, load and boundary conditions. Further on, a numerical model for calculation of the ultimate bearing capacity by development of capacity curve of masonry walls by application of discrete element method is shown. Calculations presented in this paper are obtained by performing two-dimensional analysis with discrete element method and the UDEC program (UDEC 2005).

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high levels of vertical stress are present, as shown in Figure 1 (Churilov 2007). With the previously described actions, several actions can be distinguished as follows: occurrence of joint failure mechanisms (a, b), block failure mechanisms (c) and combined failure mechanisms that include joint and block failure (d, e). But still, this approach yields one open question and that is how to investigate all those actions in a single model. The assumed approach suggests that the numerical model should concentrate on all joint failure mechanisms when relatively weak joints are present and, if necessary, on potential cracks in the blocks due to tension stress in the blocks in the middle section of each block, Figure 2.

MODELLING STRATEGY

In this research the main emphasis is given to the simulation of the behavior of masonry walls based on numerical models with the use of micro-modeling approach and use of the discrete element method. Micro-modeling of masonry is probably the most important approach to understand the behavior and performance of masonry walls. In this paper a simplified micro-modeling strategy is adopted where expanded units are represented by continuum elements whereas the behavior of the mortar joints and unitmortar interface is lumped in discontinuous elements. An accurate and precise micro-model should include all basic masonry failure mechanisms, i.e. (a) bed joint failure, (b) head and bead joint failure at low levels of the vertical stress, (c) block failure at direct tension, (d) block failure by overturning of the sample and (e) crushing of masonry by opening of gapping joints as a result of mortar dilatancy when

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DISCRETE ELEMENT METHOD

Discrete element method is a numerical method used for simulation of the mechanical behavior of structures composed of particles or blocks and it is well suited for solving problems in which significant part of the deformation happens in the joints or in the contact points. This method treats the structures made from blocks that mutually interpenetrate each other through contacts. This assumption eliminates the main two difficulties naturally connected to finite element method, i.e. creation of compatible finite element mesh between blocks and joints as well as the inability of the method for remeshing in order to update the contact size or to create new contacts if relatively large movements are present. During the analysis with the discrete element method, different contact types can

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Figure 3. Walls geometry and load application: axial compression and shear compression test.

arrangement of free or unattached blocks when it is not possible to compile the stiffness matrix. The main advantages of the discrete element method are: general and universal; nonlinear material behavior and large displacements (change in system interconnections); low data storage requirements; simple programming; same algorithm for static and dynamic analysis; appropriate for parallel data processing.

Figure 1. Masonry failure mechanisms: Shear failure (a, b, c); Compression failure (d); Tension failure (e).

Figure 2. Adopted modeling strategy. Blocks (u) are expanded in both directions by the thickness of the filled joint and modeled by continuum elements. Filled joints (m) and potentional cracks are presented by zero thickness and interface or contact elements.

be expected depending on the initial geometry and displacement history. Discrete element models use explicit solving algorithm, respectively dynamic and quasi-static analyses are conducted by dynamic relaxation. This method uses large viscous damping for solving the differential equations of movement in order to achieve convergence to the static solution or steady failure mechanism. The method was introduced to model the

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CALIBRATION OF A NUMERICAL MODEL WITH EXPERIMENTAL RESULTS

For the requirements of the research (Churilov 2007) validation and confirmation of the adopted strategy for micro-modeling of masonry walls was performed by comparison and calibration of the numerical model with the available experimental results from Jovanovski & Josifovski (2006) and Gramatikov (2006). In this paper special attention is given to experimental results performed by Institute of Earthquake Engineering and Engineering Seismology (IZIIS), University Ss. Cyril and Methodius, Skopje, for the research project PROHITECH (Earthquake Protection of Historical Buildings by Reversible Mixed Technologies). In the numerical simulations, blocks were considered as fully deformable, while the joints were presented as point contacts which means that the blocks are connected by contact points between each other. Several parameters, as well as material models are varied during the simulations. Walls loaded with axial compression load as well as walls loaded by diagonal shear compression load are considered during experimental testing. These experiments are conducted in order to determine the ultimate limit force and modulus of elasticity of masonry as a prerequisite for numerical simulation of the model of the structure Mustapha Pasha Mosque, to be experimentally tested on a shaking table. The wall geometry and load application are presented in Figure 3. Numerical simulations were performed by: – Variation of mechanical properties of constituent components due to limited data from experimental tests on each component;

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Table 1.

Characteristic material models.

Block

Model 1

Model 2

Model 3

Stone

Linearly elastic

Linearly elastic

Brick

Linearly elastic

Mortar

Elastic/Plastic with Coulomb slip failure

Elastic/Plastic Mohr-Coulomb failure Elastic/Plastic Mohr-Coulomb failure Elastic/Plastic with Coulomb slip failure, residual values for friction, cohesion and tension

Table 2.

Elastic/Plastic with Coulomb slip failure, residual values for friction, cohesion and tension

Material properties for Model 1-blocks.

Properties

Unit

Stone

Brick

Concrete

Unit weight Elastic modulus Poisson’s ratio Bulk modulus Shear modulus

t/m3 kN/m2

2.426 7 · 106 0.2 3.89 · 106 2.92 · 106

1.874 2 · 106 0.2 1.11 · 106 8.33 · 106

2.548 3.15 · 107 0.2 1.75 · 107 1.31 · 107

Table 3.

Linearly elastic

kN/m2 kN/m2

Figure 4. Axial compression wall and the failure mechanisms.

Material properties for Model 1-joints.

Properties

Unit

Joint thickness Elastic modulus Poisson’s ratio Normal stiffness Shear stiffness Tensile strength Cohesion Friction angle Dilatancy angle

m kN/m2 kN/m3 kN/m3 kN/m2 kN/m2 deg deg

Horizontal joints

Vertical joints

0.0225 1.3 · 105 0.2 5.78 · 106 2.41 · 106 0.4 · 103 0.22 · 103 6.38 0

0.0225 1.3 · 105 0.2 5.78 · 106 2.41 · 106 0.1 · 103 0.12 · 103 4.91 0

– Variation of numerical models and nonlinear material laws in order to determine the appropriate wall behavior and failure during the experimental test. – Variation of the numerical models regarding the sections of the walls in order to achieve better calibration with the experimental results due to 2-dimensional software used.

4.1 Axial compression test – longitudinal section First, several unreinforced masonry wall specimens were constructed and experimentally tested on axial and shear compression, while in the end one wall was experimentally tested with applied FRP strengthening in two bands. Wall specimens were constructed by limestone and clay brick blocks and lime mortar filled joints. The height of the walls is 88 cm; length 94 cm and thickness 33 cm, while the limestone blocks are 10 × 10 × 15 cm in dimension and clay blocks are 10 × 9 × 3.2 cm.

Figure 5. Real wall layout in longitudinal section.

Mortar joints have thickness of 2.0–2.25 cm. The vertical load is slowly applied until failure. Three different numerical models are simulated; all of them with the same geometric properties, but different material models were used. The following table presents the model properties and material parameters used. The following tables present the calibrated material properties for Model 1 that were taken into account for numerical simulations. Figures 5 and 6 present a comparison between the real wall geometry and the geometry used in the numerical model in UDEC program. One can notice the difference in the joints presented in the numerical model, where the block height has been increased for half of the thickness of the neighboring joints, and

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Figure 6. UDEC wall layout in longitudinal section.

Figure 8. Real vs. UDEC wall layout in transversal section.

Figure 7. Comparison of experimental and numerical force-displacement diagrams for longitudinal section of the wall.

the joint thickness is introduced directly as a material parameter when calculating joint stiffness. During the experimental test as well as in the numerical simulation, vertical displacement of a point in the upper middle section of the wall was monitored. Figure 7 presents comparison between experimental curve and numerical capacity curve. The experimental behavior of the wall is simulated numerically in satisfactory manner and it can be concluded that curves correspond very well. Obviously the best results are obtained by using models 1 and 3 which assume linearly elastic behavior of blocks and nonlinear elastic/plastic relationships for joints. The results from models 1 and 3 are identical. As expected, model 2 yields smaller values for forces at the same displacements as for models 1 and 3, as a result of nonlinear material model used for the blocks and taking into consideration block tensile strength, cohesion, friction and dilatancy angle.

4.2 Axial compression test – transversal section Due to the two-dimensional software used for numerical simulations, a change in the numerical model in regards of the wall cross-section was performed. Namely, a transversal section model of the wall was constructed by using the same material and geometry properties as in the longitudinal section. As in the

Figure 9. Comparison of experimental and numerical force-displacement diagrams for transversal section of the wall.

previous analyses, several material models were used and calibrated. Wall specimens were constructed by limestone and clay brick blocks and lime mortar filled joints. The geometry and block size of the walls is the same as for longitudinal section. The vertical load is slowly applied until failure. The material, geometry and boundary conditions remain the same, without changes and the only difference between them are the values of the joint modulus of elasticity which was varied between 50, 80 and 130 MPa. Figure 8 presents a comparison between real wall layout and the layout of the blocks used in the numerical model. Figure 9 presents a comparison of the forcedisplacement diagrams between numerical simulations and experimental results where the joints were considered with modulus of elasticity of 130 MPa. Again good agreement between experimental and numerical results has been achieved. 4.3 Shear compression test – unstrengthened specimen In addition to experimental testing of masonry walls with vertical load, several specimens were tested in shear compression. It must be noted that these experiments are not fully compatible with the requirements for experimental determination of the

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Figure 12. Numerical wall failure and deformed shape, for an imposed force of 12.5 kN. Figure 10. Wall layout and failure in shear compression.

Figure 11. Wall layout and failure in shear compression (back side).

shear resistance of masonry walls usually performed. A modification of the procedure for testing was made mostly because of lack of suitable testing set-up and equipment. That’s the reason why masonry specimen walls were rotated for 45◦ in relation to the horizontal plane and the load is gradually applied on the trimmed corner edge of the wall until failure. Figures 10 and 11 presents the layout of the specimen and the failure after reaching the ultimate limit state, while Figure 12 shows the magnified deformed block structure in DE simulation, for an imposed force of 12.5 kN. The predicted failure mechanisms by numerical simulation for block and joint failure correspond well with the crack distribution found through the experiment. 4.4

Shear compression test – strengthened specimen

The experimental investigations on a shaking table of the Mustapha Pasha Mosque model structure consisted of acquiring structure behavior with and without

Figure 13. Shear compression test on a strengthened specimen.

strengthening method applied when the structure is subjected on a real earthquake movement and seismic forces. In order to obtain some knowledge about the strengthening method to be used, one wall specimen was experimentally tested in shear compression strengthened with two FRP bars which were embedded on the same position where the cracks appeared in the unstrengthened specimen. The wall was build with the same blocks and mortar properties as in the previous test. The geometry, load and boundary conditions were kept exactly the same as in the unstrengthened model in order to obtain the real contribution of the strengthening method. The wall layout and strengthening is presented in Figure 13. During the experiment it was found that the strengthened model showed almost the same strength properties as the unstrengthened model without significant change in the modulus of elasticity, but the failure mechanism was changed. While the unstrengthened wall showed diagonal crack opening after de-bonding

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Table 4.

Figure 14. Force-displacement diagram of the strengthened wall.

Figure 15. Failure mechanism of the strengthened wall.

between block and mortar layers and the typical head and bed joint failure in shear as in Figure 1b, the strengthened model demonstrated bed joint failure in shear as in Figure 1a in the layers beneath the FRP bars. Moreover, the strengthened wall failed after reaching the ultimate limit force of 10 kN as presented in Figure 14. This is less then the ultimate limit force obtained at the unstrengthened wall of 12.5 kN. The failure mechanism of the strengthened wall is presented in Figure 15. Numerical simulation for the strengthened wall was not performed because of the inability of the software to introduce strengthening elements over blocks or joints.

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CONCLUSIONS

This paper presents calibration of a numerical model and calculation of capacity curves of masonry walls by using discrete element method implemented in UDEC. For the sake of controlling the possibilities of the used software and interpretation of results, a verification of the software in regards to experimental test was

Suggested material properties for clay bricks.

Properties

Unit

Tensile strength

kN/m2

Cohesion Friction angle Dilatancy angle

2

Table 5.

Clay bricks

kN/m deg deg

(0.03–0.1) · compression strength (0.0–2.0) · tensile strength 35–51 0–12

Suggested material properties for mortar.

Properties

Unit

General purpose mortar

Tensile strength

kN/m2

Cohesion Friction angle Dilatancy angle

kN/m2 deg deg

(0.0–0.1) · compression strength (0.1–0.4) · tensile strength 13–40 0

performed.Also, a calibration of some material parameters included in the numerical model is completed. According to the numerical simulation data one can make conclusion that UDEC software as well as the discrete element method is capable to simulate the behavior of masonry walls under monotonous loading conditions. In addition, failure mechanism of masonry specimen walls is achieved very precisely. In this work only monotonous loading has been considered. Due to simplification reasons in the input data monotonous loading in several steps was applied as a vertical load. This method is good enough when loading values are close to the ultimate load which generates structure failure. Numerical analysis involving cyclic loading on a masonry wall specimen should present more accurate results for masonry behavior and should be able to present softening behavior or stiffness degradation in more detail. In order to load the specimen in cycles, one has to create numerical model that is able to include stiffness degradation explicitly. Also, it is recommended to generate a model which will be able to predict unit failure more precisely as well. It has to be emphasized here that in order to determine the behavior of masonry walls more precisely by using discrete element method it is necessary to know all material properties of the components. Recommended values for some material parameters are hard to obtain through literature survey and it is proposed to acquire them experimentally.According to the results from the numerical simulations, several suggestions for masonry clay bricks and general purpose mortar are given in Tables 4 and 5.

ACKNOWLEDGMENTS We wish to express our gratitude to Prof. Dr. Kiril Gramatikov, national coordinator of PROHITECH

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project for the opportunity to participate in the project, and to Prof. Dr. Christoph Butenweg (RWTH Aachen) for making UDEC available and his kind remarks. REFERENCES Bi´cani´c, N. et al. 2002. Discontinuous Modelling of Structural Masonry. Mang H.A. et al (eds),WCCMV, FifthWorld Congress on Computational Mechanics, Vienna, Austria, July 7–12. Butenweg, C. & Mistler, M. 2006. Seismic Resistance of Unreinforced Masonry Buildings. The Fifth International Conference on Engineering Computational Technology, Las Palmas de Gran Canaria, 12–15 September. Churilov S. 2007. Expert System for Seismic Vulnerability Assessment of Masonry Structures. MSc thesis, University “Sts. Cyril and Methodius”, Faculty of Civil Engineering, Skopje, Macedonia. Dumova-Jovanoska E. & Denkovska L. 1995. Classification of Individual Unreinforced Masonry Buildings Considering Their Seismic Vulnerability. Developments in Computer Aided Design and Modelling for Structural Engineering, Civil-Comp Press, Edinburgh, UK, pp. 279–284. Giordano A. et al. 2002. Modelling of Historical Masonry Structures: Comparison of Different Approaches Through a Case Study. Engineering Structures 24, pp. 1057–1069. Gramatikov K. 2006. WP7: Experimental Analysis, Report on Testing of Bricks and Lime Mortar for Mustafa Pasha Mosque Model, PROHITECH Project, Sixth Framework Programme, INCO-CT-2004-509119. Guggisberg, R. & Thürlimann, B. 1987. Experimental determination of masonry strength parameters. Report No. 7502–5, Institute of Structural Engineering, Zürich. Jovanovski M. & Josifovski J. 2006. WP 7: LaboratoryTesting on the Intact Parts of Stone Used in Construction of Large Scale Model For “Mustafa Pasha Mosque” in Skopje, Testing Report, PROHITECH Project, Sixth Framework Programme, INCO-CT-2004-509119. Lemos, J.V. 2004. Modelling Stone Masonry Dynamics with 3DEC. Konietzky (eds): 1st International UDEC/3DEC

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