In this work the compression behavior of masonry was investigated. After a detailed review of code approaches and different research works, a new formula was suggested to describe the compression strength of masonry, based on the mechanical and geometrical properties of its components, when deformation properties of units are larger than the ones of mortar. Later on, a new model, Extended Drucker-Prager Cap Yielding Function, is suggested to describe the three axial compression stress state of mortar in masonry in case deformation properties of mortar are larger than the ones of mortar, and to describe the three axial compression stress state of brick in the other case. This includes defining its parameters based on test diagrams of the mortar material, implementing the model in the numerical software ANSYS, and the numerical results are evaluated for simple cube example. The controlling equations of creep based on the visco-elastic creep theory are presented in the general case of three axial creep under three axial loading conditions. The special case of three axial creep under axial loading is also presented. The “transversal creep” relevant for the compression strength of masonry was discussed and numerical examples have been added to show the effect of changed time-dependent Poisson’s ratio. In another chapter, many examples are presented showing the application of the suggested material models and discontinuous numerical method named eXtended finite element method. Conclusions and recommendations are given in the last chapter.

In this work the compression behavior of masonry was investigated. After a detailed review of code approaches and different research works, a new formula was suggested to describe the compression strength of masonry, based on the mechanical and geometrical properties of its components, when deformation properties of units are larger than the ones of mortar. Later on, a new model, Extended Drucker-Prager Cap Yielding Function, is suggested to describe the three axial compression stress state of mortar in masonry in case deformation properties of mortar are larger than the ones of mortar, and to describe the three axial compression stress state of brick in the other case. This includes defining its parameters based on test diagrams of the mortar material, implementing the model in the numerical software ANSYS, and the numerical results are evaluated for simple cube example. The controlling equations of creep based on the visco-elastic creep theory are presented in the general case of three axial creep under three axial loading conditions. The special case of three axial creep under axial loading is also presented. The “transversal creep” relevant for the compression strength of masonry was discussed and numerical examples have been added to show the effect of changed time-dependent Poisson’s ratio. In another chapter, many examples are presented showing the application of the suggested material models and discontinuous numerical method named eXtended finite element method. Conclusions and recommendations are given in the last chapter.

The main goal of CIC-BREL project (Cracked and Inelastic Calculation of BRacing Elements) is to develop an analytical method to distribute horizontal forces on bracing elements, in this case reinforced masonry shear walls, of a building considering the cracked and inelastic state of material. The moment curvature curve of the wall section is created first depending on the section geometry and material properties of both the masonry units and steel reinforcement. This curve will start with an elastic material behavior, then continue in inelastic material behavior where the masonry crushes and the steel start to yield, until the maximum bending moment M_p is reached. Due to reinforced masonry wall ductility, post maximum capacity is also considered assuming a maximum curvature of 0.1%. From the moment curvature curve, the force displacement curve could be extracted depending on the wall height and wall boundary conditions. Matrix formulation has been developed for both elastic and damaged stiffness matrix, considering different boundary conditions. Fixed-fixed boundary condition which usually exists at the middle stories or last story with strong top diaphragm, fixed-pinned which is the case of the last story that has a relatively soft top diaphragm, and pinned-fixed in the first story case. Other boundary conditions could be considered depending on the degree of fixation on the wall both ends at the top and the bottom. The matrix formulation combined with the force-displacement curve which considers different material stages (elastic, inelastic, ductile post peak force) is used to define forces in each bracing element even after elastic behavior. After elastic phase of each wall the stiffness of the element will degrade leading to a less portion of the total lateral force; other elastic walls, i.e., stronger walls, will receive more portion of the total force leading to a redistribution of the total force. This process will be iterated until the total force is distributed on each bracing element depending on the wall section state: elastic, inelastic and ductile post-peak capacity. Flowcharts clearly will show this process. Finally, a Fortran code is developed to show examples using this method. The developed analytical method will be verified by the results of shake table tests held at the University of California in San Diego, USA. Last test performed in the year 2018 uses T-section reinforced masonry walls, subjected to shakings with increased intensity. The total applied force for each shaking could be defined depending on the structural weight and shaking intensity (acceleration). The damage and displacement at each intensity has been recorded and evaluated. Depending on these test results, the results of the analytically developed method will be compared and evaluated. Total system displacement at different lateral load values has been compared for analytical calculations and shake table tests; furthermore, each wall state at increased load has been compared, good agreement could be noticed.:Acknowledgement 5 1. Introduction 7 1.1. State of the Art 9 1.2. Elastic Formulae 9 1.3. Example, Elastic Calculation 12 1.3.1. Stiffnesses of the System 13 1.3.2. Torsion due to Eccentric Lateral Loading 14 1.3.3. Distribution of the Lateral Load on Wall “j” and Floor “i” 15 2. Force Displacement Curve of RM Shear Wall 19 2.1. Introduction 19 2.2. Cantilever Wall 19 2.2.1. Cantilever Elastic Wall 19 2.2.2. Cantilever Inelastic Wall 21 2.2.3. Cantilever Post-Peak Wall 22 2.3. Fixed-Fixed Wall 23 2.3.1. Fixed-Fixed Elastic Wall 23 2.3.2. Fixed-Fixed Inelastic Wall 24 2.3.3. Fixed-Fixed Post-Peak Wall 26 2.4. Moment – Curvature Analysis 26 2.5. Example, Rectangle Cross Section, Cantilever 29 a) Moment Curvature Curve 29 b) Force Displacement Curve 32 2.6. Example, Rectangle Cross Section, Fixed-Fixed 33 a) Moment Curvature Curve 33 b) Force Displacement Curve 33 2.7. Example, T Cross Section, Cantilever 35 a) Moment Curvature Curve 35 b) Force Displacement Curve 41 2.8. Example, T Cross Section, Fixed-Fixed 43 a) Moment Curvature Curve 43 b) Force Displacement Curve 43 3. Matrix Formulation 47 3.1. Procedure 47 3.2. Structure Discretization 47 3.3. Element, i.e.; Wall, Local Stiffness Matrix 48 3.4. Stiffness Matrix of Fixed-Pinned Beam 52 3.4.1. Elastic 52 3.4.2. Pre-Peak Inelastic 54 3.4.3. Post-Peak Inelastic 55 3.4.4. Normal Force Part in the Stiffness Matrix 56 3.5. Stiffness Matrix of Pinned-Fixed Beam 57 3.5.1. Elastic 57 3.5.2. Post-Peak Inelastic 57 3.6. Stiffness Matrix of Fixed-Fixed Beam 58 3.6.1. Elastic 58 3.6.2. Post-Peak Inelastic 60 3.7. Summary of Stiffness Matrices 61 3.7.1. Fixed-Fixed 61 3.7.2. Fixed-Pinned 62 3.7.3. Pinned-Fixed 63 3.8. Transformation Matrix 63 3.9. Assemble the Structure Stiffness Matrix 65 3.10. Assemble the Structure Nodal Vector 66 3.11. Solve, Get Nodal Displacements and Forces 66 4. Matrix Formulation and Deformation Based Method 69 4.1. Elastic Method in Distributing Lateral Force 69 4.2. Elastic and Inelastic Method in Distributing Lateral Force 70 5. Shake Table Tests 73 5.1. Introduction 73 5.2. Design of Test Structure 73 5.3. Material Properties 75 5.4. Tests and Observations 75 5.4.1. Tests up to Mul-90% 76 5.4.2. Tests with Mul-120% 76 5.4.3. Tests with Mul-133% 76 5.5. Deformations 77 6. Verification 81 6.1. T Cross Section, Dimensions, Reinforcement and Materials 81 6.2. Moment Curvature Curve 82 6.3. Force Displacement Curve 85 6.4. Force Displacement Curve of the Structure 88 7. Conclusions and Suggestions 91 8. References 93 Appendix 1, Timoshenko Beam 95 • Fixed-Fixed 95 • Fixed-Pinned 95 • Pinned-Fixed 96 Appendix 2, Bernoulli Beam 97 • Fixed-Fixed 97 • Fixed-Pinned 97 • Pinned-Fixed 98