Non-linear seismic assessment & retrofitting of unreinforced masonry buildings Master Thesis Eleni Sionti January 2016 Preface This thesis is written under the framework of the Master Degree of Building Engineering in the Civil Engineering Department of Delft University of Technology. The theme concerns the assessment and retrofitting of an existing unreinforced masonry building situated in Loppersum. The research is carried out under the guidance of Delft University of Technology and BAM A&E. TNO supported with the license of the DIANA software and Technosoft with the license of the Tremuri software. Signals are provided by the Nederlandse Aardolie Maatschappij (NAM). Material properties are given by TU Delft. I would like to thank my graduation committee and my colleagues in BAM A&E for their guidance throughout the process. Also I would like to thank my family and friends for their support. 3 Faculty of Civil Engineering and Geosciences Delft University of Technology Personal information Eleni Sionti eleni.sionti@gmail.com Graduation committee Prof. Dr. Ir. J.G. Rots , Department of Structural Engineering Dr.Ir. M.A.N. Hendriks, Department of Structural Engineering Dr. V. Mariani, Department of Structural Engineering Ir. S. Pasterkamp, Department of Building Engineering Ir. M. Spanenburg, BAM A&E 4 J.G.Rots@tudelft.nl M.A.N.Hendriks@tudelft.nl V.Mariani@tudelft.nl S.Pasterkamp@tudelft.nl mark.spanenburg@bam.nl Summary Increasing seismicity is observed the last years in the area of Groningen due to extraction of gas. This has an impact on the building stock of the area which is primarily made by unreinforced masonry. These buildings are not constructed following seismic guidelines and their assessment becomes now a necessity. The present analysis is based on a specific Case Study corresponding to the category of Terraced Houses with the presence of timber diaphragms. The main research objectives are the assessment of the building under seismic loading with two modelling approaches (detailed finite element model and equivalent frame analysis model) and two analysis procedures (pushover analysis and nonlinear time history analysis). The influence of different strengthening methods on the models is also researched. The research questions associated to the primary objectives are further analysed in the report. The methodology developed focuses on a global response approach with the objective to assess the global capacity of the structure. The focus is considered primarily in capacities and secondary in displacements. The need to develop a number of analysis and different analysis procedures resulted in the development of a model with fixed parameters that can produce results in relatively low computational time. This approach is considered suitable for the purpose of this analysis. Specifically, the modelling strategy followed considers 2D elements, conventional pushover analysis with uniform application of loading, fixed supports, a Total Strain Rotating Crack Model and fixed material parameters. The load increment procedure followed is force control and the iterative solution method Regular Newton-Rapson. A displacement convergence norm is set for the pushover analysis and an energy norm for the Time History analysis. Experimental results are not yet available to support this analysis and the applicability of the pushover analysis in buildings with timber diaphragms is considered unexplored. For the model parameters no sensitivity analysis is carried out. The main parameter considered a variable in the analysis is the quality of the connections as it is evaluated to play a key role in the global behaviour. As regards the quality of the results the convergence characteristics of each analysis are reported in terms of forces and displacements. The acceptability of the analysis results is related to the acceptability of the convergence details. For the assessment of the building three types of analysis are performed; a modal analysis, a pushover analysis and a non-linear time history analysis. The analysis is mainly focused on the Pushover analysis, while the modal analysis is used to understand the behaviour of the structure under a free vibration and the Time history analysis as a check tool. The pushover analysis is developed with two modelling approaches, namely a finite element approach (FE) with the use of curved shell elements and an equivalent frame analysis (EF) where each component is modelled as one dimensional beam element. The time history analysis is developed with the FE model and is used for the final assessment of the existing and the improved structure. The FE model is built in the DIANA software and the EF model in Tremuri. In the modal analysis the main parameters observed are modal shapes and eigenfrequences, while in the pushover analysis and the NLTHA the principal strains, failure mechanisms, crack widths and drift limits are the main parameters of interest. The current analysis is based on the National Draft Code (NPR 9998) released on February 2015. (Ontw. NPR 9998, February 2015) The main stages identified in the models developed refer to: (1) Gravity loading; (2) Linear phase; (3) Extensive cracking; (4) Crack propagation; and (5) Collapse. To assess the structure attention was given to the existing connections. The connection that was doubted refers to the connection between wooden 5 beams of the diaphragms and the masonry walls. To capture this uncertainly two analysis are performed referring to: (1) hinged connections and (2) sliding connections. With these analysis the capacity envelope of the structure is assessed. The two extremes give different failure modes, referring to out of plane failure and in-plane shear failure. To get a better understanding of the behaviour of the structure interfaces are inserted in the connections where stiffness is assigned and the behaviour is observed. The effect of the decreased elastic modulus of the diaphragm of 40% is also investigated and no significant influence is noted in the model. Following the EF model is developed and the pushover curve is defined. The base shear showed correlation with the FE model. The failure mechanisms assessed by this approach are based on the drift limits of the elements. For structures where a limited number of elements is influencing the global behaviour the model is found sensitive to assess the actual failure mechanisms. Also capacity is calculated following an analytical approach and the result is compared to the results of the EF and FE model. Finally, the target displacement is calculated following the approach of EC-8 and this is compared to the result by the EF model. According to EC-8 the check of displacements is the main check that needs to be performed for non-linear analysis. After the assessment phase is completed, the reinforcement of the structure is investigated. In the following models a reduced modulus of elasticity is used, to incorporate the reduced in plane stiffness of the diaphragm and full connectivity at the ends of the wooden beams is considered. This is considered as the base model. A weak point of the structure is pointed at the absence of connection longitudinally to the beams and the facades. This is the first point modified considering hinged connections. The addition of connections resulted to an increase of 50% in the direction parallel to the facades. In the direction perpendicular to the facades this measure resulted in protection from out of plane failure and an increase of 120% in the capacity. Following the influence of the addition of wooden planks on the diaphragms is searched. An increase in the total capacity of 35% is found for an addition of 80mm wooden plank. The last measure investigated is the use of steel frames. Specifically three configurations are shown. The main interest lies on assessing the behaviour of the new system and the influence of the presence of the steel frames in the behaviour of the masonry. For the new system three main phases are identified: (1) Masonry contribution; (2) Steel and masonry contribution; (3) Plateau. For the three configurations the corresponding behaviour factors are defined and unity checks are performed in terms of displacements and capacities. The checks are performed for both 67% of NPR requirement and 100% to underline the importance of risk acceptability throughout the assessment process. The final step was to perform the time history analysis. This analysis is performed for the lower boundary (Case 1) and one reinforced solution with steel frames (Configuration 1). Case 1 is not considered adequate to perform seismically under a signal of 67% NPR and the hysteretic loop of this analysis is found in correlation to the pushover analysis results. The analysis is stopped with the presence of divergence. For Configuration 1 with 67% NPR divergence occurred and the failure is related to numerical instability. It is recommended that further research focuses on the assessment of the ductility factors of the buildings under consideration with the use of the FE model. A more refined modelling strategy is also suggested with the application of a cyclic pushover analysis and the adaptation of displacement control with arc-length control in the analysis. The research on different iteration procedures and convergence criteria in the NLTHA is also recommended. 6 Table of contents 1. Introduction ..........................................................................................................................................17 1.1. Research objectives ..........................................................................................................................19 1.2. Research method .............................................................................................................................20 1.3. Case study ........................................................................................................................................21 1.4. Structure of the report .....................................................................................................................23 2. Literature study ....................................................................................................................................25 2.1. Masonry behaviour .........................................................................................................................25 2.1.1. Failure behaviour .....................................................................................................................25 2.1.2. Numeric representation...........................................................................................................26 2.1.3. Possible failure mechanisms ....................................................................................................26 2.1.4. Flange effect.............................................................................................................................28 2.2. Buildings in Groningen .....................................................................................................................29 2.2.1. Timber diaphragms ..................................................................................................................29 2.2.2. Cavity walls...............................................................................................................................30 2.3. Computational modelling of masonry structures ............................................................................31 2.4. Analysis of seismic behaviour...........................................................................................................34 2.4.1. Pushover analysis .....................................................................................................................35 2.4.2. Nonlinear time history analysis................................................................................................36 2.5. Modelling approaches ......................................................................................................................38 2.5.1. Approaches overview...............................................................................................................38 2.5.2. Comparison of approaches ......................................................................................................39 2.6. Seismic assessment ..........................................................................................................................45 2.6.1. Ductility factor .........................................................................................................................45 2.6.2. Force reduction factors ............................................................................................................45 2.6.3. Drift limits ................................................................................................................................46 2.6.4. Target displacement ................................................................................................................46 2.6.5. Analytical approaches ..............................................................................................................50 2.7. Seismic rehabilitation .......................................................................................................................51 2.7.1. Framework ...............................................................................................................................51 2.7.2. Retrofitting methods................................................................................................................52 3. FE modelling .........................................................................................................................................57 3.1. FE model parameters .......................................................................................................................58 3.2. Eigenvalue analysis ...........................................................................................................................71 3.3. Pushover analysis .............................................................................................................................72 3.3.1. Capacity envelope of building ..................................................................................................73 3.3.2. Analysis of capacity curves.......................................................................................................74 3.3.3. Case 1: Non-connected (x) .......................................................................................................76 3.3.4. Case 3: Fully connected (x) ......................................................................................................79 7 3.3.5. Case 3: Fully connected (y) ......................................................................................................80 3.3.6. Case 2: Semi-connected ...........................................................................................................81 3.3.7. Reduced in-plane stiffness of diaphragms ...............................................................................84 3.4. Nonlinear time history analysis ........................................................................................................85 3.4.1. Accelerogram ...........................................................................................................................85 3.4.2. Case 1: Non-connected ............................................................................................................86 4. EF modelling .........................................................................................................................................89 4.1. EF model parameters .......................................................................................................................89 4.2. EF model results ...............................................................................................................................92 5. Assessment ...........................................................................................................................................97 5.1. Building capacity ..............................................................................................................................97 5.1.1. Comparison of models .............................................................................................................97 5.1.2. Capacity from codified equations ............................................................................................99 5.1.3. Comparison of capacities .......................................................................................................100 5.2. Target displacement .......................................................................................................................101 5.3. Ductility and behaviour factor........................................................................................................101 5.4. Base shear check ............................................................................................................................102 6. Retrofitting .........................................................................................................................................103 6.1. Seismic demand .............................................................................................................................103 6.2. Improvement of existing connections ............................................................................................104 6.3. Addition of connections .................................................................................................................105 6.4. Improved in plane stiffness of floors ..............................................................................................107 6.5. Strengthening of walls with steel frames .......................................................................................108 6.5.1. Pushover analysis ...................................................................................................................108 6.5.2. Nonlinear time history analysis..............................................................................................118 7. Conclusions .........................................................................................................................................119 Acronyms ....................................................................................................................................................125 Definitions ...................................................................................................................................................126 Appendix A: Dead loads calculation............................................................................................................128 Appendix B: Capacity hand calculations .....................................................................................................129 Appendix C: Target displacement calculation.............................................................................................132 Appendix D: Convergence quality ...............................................................................................................138 Appendix E: Case study drawings ...............................................................................................................143 References ..................................................................................................................................................145 8 List of figures Figure 1: Number of events per year, magnitude per year and gas production. (KNGMG & NWO-ALW, 2014) ............................................. 17 Figure 2: Epicentres and gas fields of the North east provinces. (Royal Netherlands Meteorological Institute, 2012) ................................... 17 Figure 3: Typical Dutch Terraced House. ......................................................................................................................................................... 18 Figure 4: Contour plot of the peak ground acceleration in [ for a return period of 475 years. (Ontw. NPR 9998, February 2015) ................................................................................................................................................................................................ 18 Figure 5: Building plan. .................................................................................................................................................................................... 21 Figure 6: Building views................................................................................................................................................................................... 22 Figure 7: Building section. ............................................................................................................................................................................... 22 Figure 8: Overview of models developed. ....................................................................................................................................................... 23 Figure 9: Yield criterion and a typical stress-strain model for brick unit. (Lawrence Livermore National Laboratory, 2009). ......................... 25 Figure 10: Modelling strategies for masonry structures: (a) detailed micro-modelling; (b) simplified micro-modelling; (c) macro-modelling. (Lourenço , 2013) ............................................................................................................................................................................................ 26 Figure 11: In-plane failure mechanisms. (Elgawady, Badoux, & Lestuzzi, 2006; Magenes & Penna, 2009) ..................................................... 27 Figure 12: Out of plane failure mechanisms. (Calvi, Pinho, Magenes, Bommer, Restrepo-Vélez, & Crowley, 2006)....................................... 27 Figure 13: Failure of URM related to diaphragms. (Oliver, 2010) .................................................................................................................... 27 Figure 14: Walls separation and failure of gamble. (NZSEE, 2015) .................................................................................................................. 28 Figure 15: Traditional layout of timber floors: (1) One way; and (2) Two way. (Brignola, Podesta, & Pampanin, 2008) ................................. 29 Figure 16: Contributions to the flexibility of diaphragm. (Brignola, Podesta, & Pampanin, 2008) .................................................................. 30 Figure 17: Angular deformation of masonry unit and expulsion of building corners. (Brignola, Podesta, & Pampanin, 2008) ....................... 30 Figure 18: Typical cavity wall and related out of plane failure modes. (The University of Auckland, 2015) .................................................... 30 Figure 19: Force control (left) versus displacement control (right). (Palacio, 2013) ........................................................................................ 32 Figure 20: Arc-length control (left) and load increment methods characteristics (right). (Palacio, 2013) ....................................................... 32 Figure 21: Iteration process. (TNO DIANA BV., 2014) ...................................................................................................................................... 33 Figure 22: Regular Newton-Raphson method. (Palacio, 2013) ........................................................................................................................ 33 Figure 23: Two degrees of freedom system. (adapted from Chopra A., 2012) ................................................................................................ 34 Figure 24: Load – displacement response of wall. (Facconi, Plizzari, & Vecchio, 2013) ................................................................................... 35 Figure 25: Force distribution in a Monotonic pushover analysis. (University of Buffalo, 2009) ...................................................................... 35 Figure 26: Hysteritic loop of Cyclic Pushover analysis. (University of Buffalo, 2009)....................................................................................... 36 Figure 27: Variation of modal damping ratios with natural frequency: (a) mass-proportional damping and stiffness-proportional damping; (b) Rayleigh damping. (Chopra, 2012) ............................................................................................................................................................. 37 Figure 28: Stress-strain relation for compression and tension. (TNO DIANA BV., 2014) ................................................................................. 39 Figure 29: CQ40S curved shell element, CQ24TM translation mass element and CL18B beam element. (TNO DIANA BV., 2014) ................. 40 Figure 30: Topology and displacements in linear interface element. (TNO DIANA BV., 2014) ........................................................................ 40 Figure 31: Displacements, relative displacements and tractions in the definition of interface. (TNO DIANA BV., 2014) ................................ 40 Figure 32: Example of equivalent frame idealization. (Lagomarsino, Penna, Galasco , & Cattari, 2013)......................................................... 41 Figure 33: 3D assembly of masonry walls. (Lagomarsino, Penna, Galasco , & Cattari, 2013) .......................................................................... 41 9 Figure 34: Sketch of idealization of masonry panels response according to the multilinear constitutive laws implemented in Tremuri. (D26, 2012) ............................................................................................................................................................................................................... 42 Figure 35: Nonlinear beam degradation. (S.T.A.DATA) ................................................................................................................................... 42 Figure 36: 4-node membrane element as average of 3-node. (Lagomarsino, Penna, Galasco , & Cattari, 2013)............................................ 43 Figure 37: Capacity spectrum method. (Chopra & Goel, 1999) ....................................................................................................................... 47 Figure 38: Bilinear approximation of force displacement curve. (Allen, Masia, Derakhshan, Griffith, Dizhur, & Ingham, 2013) .................... 48 Figure 39: Face to face connector of wall with two layers. (Meireles & Bento, 2013) .................................................................................... 56 Figure 40: Schematization of the FE model. .................................................................................................................................................... 59 Figure 41: Overview of the FE model. ............................................................................................................................................................. 60 Figure 42: Meshed elements of the FE model. ................................................................................................................................................ 60 Figure 43: Correction of generated mesh........................................................................................................................................................ 60 Figure 44: Definition of layers in the curved elements and local axis. ............................................................................................................. 61 Figure 45: As built configuration of cavity wall. .............................................................................................................................................. 61 Figure 46: Modelling of cavity wall. ................................................................................................................................................................. 62 Figure 47: Fixed base with the use of links. ..................................................................................................................................................... 62 Figure 48: As-built connection of floors to walls and modelling considerations ............................................................................................. 63 Figure 49: As built connection of wooden beams and modelling cases developed......................................................................................... 64 Figure 50: Connections modelling with the use of links. ................................................................................................................................. 64 Figure 51: As built configuration and modelling set up of interface................................................................................................................ 64 Figure 52: Modelling set up of connection to intermediate wall. ................................................................................................................... 65 Figure 53: As built floor longitudinal connection and modelling set up. ......................................................................................................... 65 Figure 54: As built roof connection and modelling choices. ............................................................................................................................ 66 Figure 55: Modelling set up of roof connection to wall................................................................................................................................... 66 Figure 56: Modelled wooden floor in the FE model. ....................................................................................................................................... 66 Figure 57: Application of load and position of plotted displacements. ........................................................................................................... 67 Figure 58: Variable loads. ................................................................................................................................................................................ 67 Figure 59: Walls numbering. ........................................................................................................................................................................... 68 Figure 60: Steel frame configuration 1. ........................................................................................................................................................... 69 Figure 61: Steel frame configuration 3. ........................................................................................................................................................... 69 Figure 62: Mode shapes of Case 1. .................................................................................................................................................................. 71 Figure 63: First mode shape for Case 3 : . .............................................................................................................................. 71 Figure 64: Tied wooden beams to masonry walls (left) and non-tied (right)................................................................................................... 72 Figure 65: Capacity curve per connection type till first drift limit reached. (x)................................................................................................ 73 Figure 66: Capacity curve until out of plane failure occurs. – Case 3 (y) ......................................................................................................... 73 Figure 67: Stress strain relationship assigned. ................................................................................................................................................ 74 Figure 68: Pier dimensions. ............................................................................................................................................................................. 75 10 Figure 69: Stress-strain relationship of steel elements and definition of yield strain. ..................................................................................... 75 Figure 70: Displacements and principal tensile strains at collapse stage. - Case 1 (x) ..................................................................................... 76 Figure 71: Failure modes identified. ................................................................................................................................................................ 76 Figure 72: Capacity curve analysis. – Case 1 (x) ............................................................................................................................................... 77 Figure 73: Displacements and principal tensile strains at first step. - Case 1 (x) ............................................................................................. 77 Figure 74: Displacements and principal tensile strains at linear stage. - Case 1 (x) ......................................................................................... 77 Figure 75: Displacements and principal tensile strains at extensive cracking phase. - Case 1 (x) .................................................................... 78 Figure 76: Displacements and principal tensile strains at crack propagation stage. - Case 1 (x) ..................................................................... 78 Figure 77: Drifts per storey and load step.- Case 1 (x)..................................................................................................................................... 78 Figure 78: Displacements and principal tensile strains at collapse stage. - Case 3 (x) ..................................................................................... 79 Figure 79: Behaviour of building for fully connected timber floor. (Piazza, Baldessari, & Tomasi, 2008)........................................................ 79 Figure 80: Capacity curve analysis. - Case 3 (x)................................................................................................................................................ 79 Figure 81: Displacements and principal tensile strains at collapse stage. - Case 3 (y) ..................................................................................... 80 Figure 82: Capacity curve of Case 3-y until out of plane failure occurs. ......................................................................................................... 80 Figure 83: Capacity curves per shear stiffness of connection. ......................................................................................................................... 81 Figure 84: As built configuration and modelling set up of interface................................................................................................................ 81 Figure 85: Building behaviour for flexible diaphragm. (Piazza, Baldessari, & Tomasi, 2008) ........................................................................... 82 Figure 86: Displacements and principal tensile strains at collapse stage. - Normal stiffness 0.01 N/mm3 ...................................................... 82 Figure 87: Displacements of left wall for unconnected, semi-connected and fully connected beams. ........................................................... 82 Figure 88: Capacity curve for assigned stiffness at both ends. ........................................................................................................................ 82 Figure 89: Interface stresses Stx of ridge beam............................................................................................................................................... 83 Figure 90: Interface stresses Stz of ridge beam. ............................................................................................................................................. 83 Figure 91: Capacity curve for reduced modulus of elasticity. - Case 3 (x) ....................................................................................................... 84 Figure 92: Set 1 of signals provided by NAM. (67%) ........................................................................................................................................ 85 Figure 93: Interstory drifts versus time in the x (left) and y (right) direction. - Case 1 .................................................................................... 86 Figure 94: Base shears versus time in the x (left) and y direction (right). - Case 1 .......................................................................................... 86 Figure 95: Maximum crack widths per 50 steps. - Case 1................................................................................................................................ 87 Figure 96: Maximum tensile strains (left) and maximum compressive strains(right) per 50 steps. - Case 1 ................................................... 87 Figure 97: Displacements and principal strains at last step of time history. ................................................................................................... 87 Figure 98: Comparison between Pushover and NLTHA. – Case 1 .................................................................................................................... 88 Figure 99: Geometry definition of unit in EF model. ....................................................................................................................................... 90 Figure 100: Wooden floors definition in EF model. ......................................................................................................................................... 91 Figure 101: Discretization in EF model. ........................................................................................................................................................... 91 Figure 102: Capacity curve of EF model in the x direction............................................................................................................................... 92 Figure 103: Progression of failure in front and back facade. ........................................................................................................................... 92 Figure 104: Internal forces of pier 19. ............................................................................................................................................................. 93 11 Figure 105: Capacity curve of EF model in the y direction. ............................................................................................................................. 95 Figure 106: Comparison of capacity curves between FE and EF model. .......................................................................................................... 98 Figure 107: Relation of failure modes of FE and EF model at back façade. (x) ................................................................................................ 98 Figure 108: Relation of failure modes of FE and EF model. (y) ........................................................................................................................ 99 Figure 109: Definition of the seismic demand. .............................................................................................................................................. 103 Figure 110: Tensile strains before and after connectivity is assured. ............................................................................................................ 104 Figure 111: Connectivity of wooden beams. (ARUP, 2013) ........................................................................................................................... 104 Figure 112: As built connectivity longitudinally to the wooden beams and modelling with links. ................................................................ 105 Figure 113: Connection of roof and floor before and after reinforcement method. ..................................................................................... 105 Figure 114: Capacity curves of Case 3 (x) and connectivity along beams. ..................................................................................................... 105 Figure 115: Displacements and tensile strains at collapse stage. – Connection longitudinally (x) ................................................................ 106 Figure 116: Capacity curves for Case 3(y) and addition of connection. ......................................................................................................... 106 Figure 117: Displacements and tensile strains at collapse stage. Connection longitudinally (y) ................................................................... 106 Figure 118: In plane stiffness of floors. (Brignola, Podesta, & Pampanin, 2008) ........................................................................................... 107 Figure 119: Capacity curves for improved in plane stiffness. ........................................................................................................................ 107 Figure 120: Steel configurations examined. .................................................................................................................................................. 108 Figure 121: Capacity curves for strengthening with steel frames. ................................................................................................................ 108 Figure 122: Displacements and tensile strains at collapse stage. – Configuration 1 ..................................................................................... 109 Figure 123: Stress-strains diagram for steel elements. – Configuration 1 ..................................................................................................... 109 Figure 124: Developed moments in steel frame at collapse stage of masonry. ............................................................................................ 109 Figure 125: Critical steps of the masonry behaviour. - Configuration 1 ........................................................................................................ 110 Figure 126: Capacity curves for steel and masonry. – Configuration 1.......................................................................................................... 111 Figure 127: Capacity curve of masonry with and without steel. – Configuration 1 ....................................................................................... 111 Figure 128: Displacements and tensile strains at collapse stage. – Configuration 2 ..................................................................................... 113 Figure 129: Stress-strains diagram for steel elements. – Configuration 2 ..................................................................................................... 113 Figure 130: Differences in the behaviour of Configuration 1 and 2. .............................................................................................................. 114 Figure 131: Displacements and tensile strains at collapse stage for Configuration 2. ................................................................................... 116 Figure 132: Crack widths versus drift limits. .................................................................................................................................................. 117 Figure 133: Interstory drifts versus time in the x (left) and y (right) direction. – Configuration 1 ................................................................. 118 Figure 134: Comparison between Pushover and NLTH. – Configuration 1 .................................................................................................... 118 Figure 135: Crack widths and principal tensile strains of masonry at last steps. – Configuration 1 .............................................................. 118 Figure 136: Modelling approaches used in the retrofitting phase. ............................................................................................................... 121 Figure 137: Single degree of freedom system. (Chopra, 2012) ..................................................................................................................... 127 Figure 138: Fundamental mode of a multi-mass system (left) and equivalent single mass system (right). (ATC-40, 1996) .......................... 127 Figure 139: Pier dimensions. ......................................................................................................................................................................... 130 Figure 140: Selected PGA in analysis. (Ontw. NPR 9998, February 2015)...................................................................................................... 133 12 Figure 141: Horizontal elastic response spectrum. ....................................................................................................................................... 134 Figure 142: Capacity curves and bilinear representation of SDOF until drift limit of 0.5 %. .......................................................................... 136 Figure 143: Capacity curve of SDOF and spectrum ........................................................................................................................................ 137 Figure 144: Convergence characteristics. – Case 1 (x) ................................................................................................................................... 138 Figure 145: Convergence characteristics. – Case 3 (x) ................................................................................................................................... 138 Figure 146: Convergence characteristics. – Case 1 (y)................................................................................................................................... 139 Figure 147: Convergence characteristics. – Case 2 (Stiffness 0.01 N/mm3)................................................................................................... 139 Figure 148: Convergence characteristics. – Case 2 (Stiffness 0.1 N/mm3)..................................................................................................... 139 Figure 149: Convergence characteristics. – Case 2 (Stiffness 0.1 N/mm3 at both ends) ................................................................................ 139 Figure 150: Case 3 – Reduced stiffness. ........................................................................................................................................................ 140 Figure 151: Convergence characteristics. – Connection longitudinally (x) .................................................................................................... 140 Figure 152: Convergence characteristics. – Connection longitudinally (y) .................................................................................................... 140 Figure 153: Convergence characteristics. – Plank 40mm .............................................................................................................................. 140 Figure 154: Convergence characteristics. – Plank 80mm .............................................................................................................................. 141 Figure 155: Convergence characteristics for Steel frames. - Configuration 1 ................................................................................................ 141 Figure 156: Convergence characteristics for Steel frames. - Configuration 2 ................................................................................................ 141 Figure 157: Convergence characteristics for Steel frames. - Configuration 3 ................................................................................................ 141 Figure 158: Energy variation at last steps of time history. - Case 1 ............................................................................................................... 142 Figure 159: Energy variation at last steps of time history. - Configuration 1 ................................................................................................. 142 Figure 160: Connections of timber beams to cavity walls at roof level. ........................................................................................................ 143 Figure 161: Longitudinal connection of timber beams. ................................................................................................................................. 143 Figure 162: Building plans ............................................................................................................................................................................. 144 13 14 List of tables Table 1: Research method overview. .............................................................................................................................................................. 20 Table 2: Components and materials of unit under consideration. .................................................................................................................. 21 Table 3: Element drift limits according to Eurocode. (EN 1998-3 , 2005) ........................................................................................................ 46 Table 4: Drift limits for in-plane walls and wall piers according to ASCE 41-06. (ASCE/SEI41-06, 2007) ......................................................... 46 Table 5: Target displacement definition formulas. (EN 1998-1, 2004) ............................................................................................................ 49 Table 6: Pier failure mechanisms. (NZSEE, 2015) ............................................................................................................................................ 50 Table 7: Limit states definition. (EN 1998-3 , 2005) ........................................................................................................................................ 51 Table 8: Strengthening of floor to wall connections. (Brignola, Podesta, & Pampanin, 2008) ........................................................................ 52 Table 9: Strengthening of timber floors. (Brignola, Podesta, & Pampanin, 2008) ........................................................................................... 53 Table 10: Methods of in-plane strengthening of masonry walls. .................................................................................................................... 54 Table 11: Strengthening of URM with modification of openings. ................................................................................................................... 55 Table 12: Mortar strengthening. ..................................................................................................................................................................... 55 Table 13: Analysis choices. .............................................................................................................................................................................. 57 Table 14: Material properties of masonry. ...................................................................................................................................................... 58 Table 15: Material properties of wooden elements. ....................................................................................................................................... 58 Table 16: Material properties of steel elements. ............................................................................................................................................ 59 Table 17: Mass and dynamic mass in DIANA. .................................................................................................................................................. 62 Table 18: Connections between wooden beams and walls. ............................................................................................................................ 63 Table 19: Variable loads at masonry walls. ..................................................................................................................................................... 67 Table 20: Fictitious densities calculation. ........................................................................................................................................................ 68 Table 21: Steel profiles for configuration 1. .................................................................................................................................................... 69 Table 22: Critical values of tensile and compressive strains. ........................................................................................................................... 74 Table 23: Drift limits per element. .................................................................................................................................................................. 75 Table 24: Material properties in the EF model. ............................................................................................................................................... 89 Table 25: Applied loads in EF model................................................................................................................................................................ 89 Table 26: Horizontal elastic response spectrum.............................................................................................................................................. 90 Table 27: Computational parameters in EF model. ......................................................................................................................................... 92 Table 28: Exceedance of bending drift for pier 19........................................................................................................................................... 93 Table 29: Capacities of Pier 19 according to EF model formulas. .................................................................................................................... 94 Table 30: Exceedance of shear drift for pier 11. .............................................................................................................................................. 94 Table 31: Failure mechanisms of EF model in y direction................................................................................................................................ 95 Table 32: Mass and dynamic mass of models. ................................................................................................................................................ 97 Table 33: Periods and mass participation of models. ...................................................................................................................................... 97 Table 34: Maximum base shear and critical failure mode in x direction. ...................................................................................................... 100 15 Table 35: Maximum base shear and critical failure mode in y direction. ...................................................................................................... 100 Table 36: Ultimate & target displacement in the x direction. (100% NPR) .................................................................................................... 101 Table 37: Calculated ductility and behaviour factors. ................................................................................................................................... 101 Table 38: Unity check of Base Shears. – Case 2 (stiffness at both ends) ....................................................................................................... 102 Table 39: Design elastic and plastic moments calculation. – Configuration 1 ............................................................................................... 110 Table 40: Critical values at collapse stage. – Configuration 1 ........................................................................................................................ 111 Table 41: Target displacement before and after reinforcement. – Configuration 1 (100% NPR) .................................................................. 112 Table 42: Ductility and behaviour factors before and after reinforcement. – Configuration 1 ..................................................................... 112 Table 43: Unity check of Base Shears. – Configuration 1............................................................................................................................... 112 Table 44: Unity check for steel profiles at last step. – Configuration 2 (100% NPR) ...................................................................................... 113 Table 45: Target displacement before and after reinforcement. – Configuration 2 ...................................................................................... 114 Table 46: Ductility and behaviour factor. - Configuration 2 .......................................................................................................................... 114 Table 47: Unity check of Base Shears. – Configuration 2 (100% NPR) ........................................................................................................... 115 Table 48: Critical values at collapse stage. – Configuration 2 ........................................................................................................................ 115 Table 49: Critical values at collapse stage. .................................................................................................................................................... 116 Table 50: Target displacement before and after reinforcement. (100% NPR)............................................................................................... 116 Table 51: Ductility and behaviour factors before and after reinforcement. .................................................................................................. 117 Table 52: Unity check of Base Shears. ........................................................................................................................................................... 117 Table 53: Outcomes of assessment phase. ................................................................................................................................................... 120 Table 54: Outcomes of retrofitting phase. .................................................................................................................................................... 122 Table 55: Calculation of floor weight. ........................................................................................................................................................... 128 Table 56: Calculation of roof weight. ............................................................................................................................................................ 128 Table 57: Material properties in NZSEE calculation. ...................................................................................................................................... 129 Table 58: Calculation of failure mechanisms of pier 1. (x direction) ............................................................................................................. 129 Table 59: Calculation of failure mechanisms. (x direction) ............................................................................................................................ 131 Table 60: Importance factors per consequence classes. ........................................................................................................................... 132 Table 61: Consequence classes parameters. ................................................................................................................................................. 133 Table 62: Parameters of horizontal response spectrum................................................................................................................................ 134 Table 63: Spectrum in ADRS format. ............................................................................................................................................................. 135 Table 64: Equivalent SDOF capacity curve..................................................................................................................................................... 136 Table 65: Idealized curve. .............................................................................................................................................................................. 136 16 Introduction 1. Introduction The Netherlands is a country with no significant natural seismicity. The exploitation though of gas reservoirs which started in 1960s has caused number of small magnitude seismic events the last decades. The Groningen area which holds the largest gas field in the region is related to these seismic events. These induced events have caused damage to the existing building stock and are subject of investigation. A relation between the number of seismic events and the gas extraction is presented in the following figure. As can be noted the seismic activity is proportional to the gas extraction. The number of seismic events for the years 1995 to 2013 show a maximum of 120 and the magnitude is reported at 3.6 for 2013. In later research the estimated magnitude for the next years is 5. (KNMI, 2013) Figure 1: Number of events per year, magnitude per year and gas production. (KNGMG & NWO-ALW, 2014) The epicentres of the seismic activities are found in the North east provinces of the Netherlands. In the following map the epicentres are shown (orange) with relation to the gas fields (green). Figure 2: Epicentres and gas fields of the North east provinces. (Royal Netherlands Meteorological Institute, 2012) The type of buildings present in the area are primarily unreinforced masonry. The residential buildings are classified in different categories, including terraced houses, semi-detached, detached, cottages, mansions and villas. Terraced houses are predominant and are two-story units of buildings in series developing a building block. Their diaphragms are usually constructed by concrete or wood. These buildings are found vulnerable to seismic action as they do not follow any seismic regulation. Peculiarities 17 Introduction of these structures are related to the presence of cavity walls, the quality of the material which influences the capacity and the layout which has supporting walls only in one direction. Figure 3: Typical Dutch Terraced House. The peak ground acceleration is considered a representative measure to express the seismic intensity in a region. In the present analysis the draft regulation released in February 2015 is taken into account. (Ontw. NPR 9998, February 2015). The contour of accelerations presented in this document is illustrated in the following figure. As can be seen Loppersum is the area with the most conservative peak ground acceleration set at . Figure 4: Contour plot of the peak ground acceleration in [ NPR 9998, February 2015) for a return period of 475 years. (Ontw. Under these circumstances the assessment of the buildings of the area and the investigation of ways to be reinforced become a necessity. 18 Introduction 1.1. Research objectives The main research objectives of this thesis are the assessment of the seismic performance and the retrofitting of an existing unreinforced masonry building subjected to seismic loading. The area under consideration is Groningen with epicentrum in Loppersum. For the assessment two modelling approaches are evaluated; a detailed finite element model and an equivalent frame analysis model. Also the analysis procedures investigated are a Pushover analysis and a Non-linear time history analysis, with main focus on Pushover analysis. To satisfy the above mentioned general objectives the following questions are considered important to be answered by this analysis: What is the capacity of the Case Study under seismic loading with the use of a Pushover analysis? Which are the main expected failure modes? What is the ductility of the building? What is the influence of the connections in the seismic performance of the building? Is the building adequate to perform seismically? How can the results of a Pushover analysis be compared to a Nonlinear Time history analysis? Is an equivalent frame model suitable for the analysis of the building? Can the results be compared to an analytical approach? What is the capacity when existing connections are improved? What is the capacity when connections are added? What is the capacity when the in plane stiffness of floors is improved? What is the seismic performance when steel frames are added? Is the building adequate to perform seismically after the addition of steel frames? 19 Introduction 1.2. Research method To answer the above mentioned research questions a research method needed to be developed. An overview of this method is shown in the following table. The choices made throughout the process are elaborated in more detail in the report. The presented results are considered valid for the specific case study and the specific methodology. Table 1: Research method overview. FE Model - DIANA Analysis aspects Choices Geometry of elements 2-D curved shell elements Integration scheme 3 integration points Modelling approach Macro -modelling Load application Uniform Supports Fixed Connectivity of elements Variable Constitutive law Total strain rotating crack model Material parameters Fixed parameters Material properties Non-linear Type of analysis Force control Numerical method Implicit Iterative solution method Regular Newton Raphson - 1-D beam elements Diaphragm Flexible Comparison to experimental results No comparison Maximum displacement of capacity curves Yes - Case 1 (x) till Collapse Case 3 (x) and Steel configurations till NC Other cases till 0.5% drift limit Limit States Eurocode Drift limits Relevant to limit state NC Spectrum NPR February 2015 - Unity checks - 20 Displacement for Pushover Energy for NLTHA Geometry Use of analytical methods Assessment Primarily conventional pushover NLTHA as check tool Load increment procedure for Pushover Convergence criteria EF Model - Tremuri - Displacement with calculation of target displacement Capacity for comparison Introduction 1.3. Case study The case study under consideration consists of a block of eight identical URM terraced residential buildings (units) situated in the area of Loppersum. These houses are classified as terraced houses and are built in 1966. The present analysis is focused on one unit and is illustrated in Figure 5. Each unit consists of two floors and an attic. The structure has timber diaphragms consisting of timber beams and a timber plank (both floors and attic) and a concrete foundation. The face walls of each unit are cavity walls of and the separating walls are uniform walls of . The walls at left and right end of the whole building are also cavity walls. An intermediate supporting wall of is also present in each unit. The outer leaf of the cavity wall consists of clay brickwork and the inner leaf of calcium-silicate brickwork. The geometric characteristics are summarized in the following table: Table 2: Components and materials of unit under consideration. Components Material Dimensions External leaf of cavity wall Clay brickwork 100 mm Internal cavity wall Calcium silicate 100 mm Cavity left wall Calcium silicate 100 mm Separating right wall Calcium silicate 200 mm Intermediate wall Calcium silicate 100 mm Diaphragms Timber Beams 71 x 196 mm Plank 22 mm Roof Timber Beams 71 x 196 mm Ridge beam 71 x 246 mm Plank 22 mm Ceramics The general dimensions of one unit are: Width: Depth: First floor height: Second floor height: Total height: An overview of the block is presented in the following figure: Figure 5: Building plan. 21 Introduction Also the views and sections of each building are presented below: Figure 6: Building views. Figure 7: Building section. Details concerning the connections and building plans are presented in the relevant Appendix. 22 Introduction 1.4. Structure of the report The document is organized in the following way. Firstly the framework is presented in Chapter 1, including the problem statement, the case study and the research objectives. Following in Chapter 2 a literature study is shown where the main themes analysed include the behaviour of masonry, the special characteristics of the buildings in Groningen, the ways the seismic behaviour of a structure can be analysed and the main differences of the modelling approaches followed. Finally, the main seismic assessment parameters and the rehabilitation process are introduced. In the following chapters the modelling approaches are developed. In Chapter 2 the FE model is shown. The pushover analysis is performed for different systems taking into account the uncertainty of the quality of the connections. The scope here is to show a range of capacity curves and failure modes that the structure might experience. Initially the two extremes are modelled considering unconnected to fully connected wooden beams to masonry walls. Following a new model is developed where interfaces are introduced and the stresses developed at the interface are shown. Also the elastic modulus of the wooden elements is reduced as a correction. For the system with the lower capacity a Time history analysis is performed and the behaviour of the building is discussed. In Chapter 4 the modelling in Tremuri is presented and a comparison is shown between the two modelling approaches. To understand the failure mechanism a single element is analysed and the behaviour is compared to the theoretical diagram assigned by the program. Also the exceedance of drift limits is verified. The following Chapter 5 focuses on the assessment of the structure. Here the capacity curves developed and the failure modes are summarized. Parameters such as target displacements, ductility and behaviour factors are also presented. Chapter 6 concerns the retrofitting of the structure, where four main directions are investigated. These include assuring connectivity at ends of wooden beams, adding connections longitudinally, improving in plane stiffness of the diaphragms and introducing steel frames to improve the in plane capacity of the walls. Finally the conclusions of this analysis are summarized in Chapter 7. The document is supplemented by a definition list and Appendixes where the supporting calculations are presented. Retrofitting NLTHA Improvement of existing connections FE Models Case 1: Non connected Assessment - Capacity - Ductility - Failure modes - Drifts Case 2: Semi connected Increase of in plane stiffness - One end - Two ends Case 3: Connected Addition of connections Reduced timber modulus Steel frames - Configuration 1 - Configuration 2 - Configuration 3 EF Model Analytical approach NLTHA Figure 8: Overview of models developed. 23 Introduction 24 Literature 2. Literature study The scope of this literature study is to give an understanding on the seismic behaviour of unreinforced masonry buildings and the way they need to be assessed. As a first step it is considered important to understand the characteristics of the material and analyse the failure modes. Following the specific characteristics of the building under consideration are discussed. The main methods to analyse the seismic behaviour are presented and emphasis is given to nonlinear methods, including: (1) Pushover analysis and (2) Nonlinear time history analysis. As a next step two modelling approaches are presented: (1) Finite element analysis with the use of 2-D curved shell elements and (2) an Equivalent frame approach with the use of 1-D beam elements. Following information on the assessment of URM buildings and the main assessment parameters are discussed. Finally an overview of strengthening methods is shown. 2.1. Masonry behaviour Masonry is a composite material of brick units and mortar. Brick units can be made out of clay, compressed earth, stone or concrete. Mortar can be lime or a mixture of cement, lime, sand and water. As a result masonry properties can vary depending on the type of brick units and mortar used. Other factors influencing the behaviour of masonry are the dimensions of the units, the mortar width and the arrangement of units. (Mosalam, Glascoe, & Bernier, 2009) Masonry can be classified in three main categories depending on the construction method followed. These include: Unreinforced masonry (URM) which refers to stand-alone masonry units and is used traditionally for the construction of masonry structures; Reinforced masonry where steel bars are usually used for the reinforcement of the units. Confined masonry which consists of masonry walls and horizontal and vertical RC members built on all sides. In unreinforced masonry the interaction between mortar and units defines the behaviour of the material. 2.1.1. Failure behaviour Masonry units are characterized by a quasi-brittle behaviour. This refers to the way the force is transferred through the material. Specifically, after the peak load is reached the force gradually decreases to zero. This way of softening is related to localized deformations that cause the quick growth of microcracks to macro-cracks and finally open cracks. (Bakeer, 2009) The stress-strain relation of unreinforced brick masonry and the yield criterion are presented in the following figure. Figure 9: Yield criterion and a typical stress-strain model for brick unit. (Lawrence Livermore National Laboratory, 2009). 25 Literature When the tensile behaviour is observed this is related to two main phases: (1) Pre-peak stage where micro-cracks are developed; and (2) Post-peak stage where softening is observed at the fracture zones. At this stage the micro cracks begin to bridge forming macro-cracks. When the behaviour under compression is observed, this shows again a (1) Pre-peak stage and (2) Postpeak stage with the presence of softening. The pre-peak stage can be further discretized to: (a) Closure of existing micro-cracks; (b) Linear elastic phase; (c) Crack initiation and stable crack growth; (d) Crack damage and unstable crack growth. In the last phase a quick increase of strains is observed till the reach of the peak load. 2.1.2. Numeric representation Masonry is a composite material showing an anisotropic behaviour. This is related to the specific arrangement of units and mortar joints. Numeric representation of masonry can be achieved by modelling masonry sub-elements separately following a micro-modelling approach, or by applying a macro-modelling approach where the whole structure is modelled as a continuum. (Nicolini, 2012) In the later approach the whole material is considered as orthotropic and the model is characterized as smeared. (Pela, Cervera, & Roca, 2011) This approach is considered suitable for the analysis of large structures but excludes the representation of local elastic and inelastic mechanisms of the mortar. Figure 10: Modelling strategies for masonry structures: (a) detailed micro-modelling; (b) simplified micro-modelling; (c) macro-modelling. (Lourenço , 2013) Also other models have been proposed for modelling the cracking behaviour of URM. The distributed stress field model (DSFM) gives the possibility to simulate the global average behaviour but also take into account the local nonlinear shear slip response. (Facconi, Plizzari, & Vecchio, 2013) 2.1.3. Possible failure mechanisms The general modes of failure associated with URM buildings include: (Boussabah & Bruneau, 1992) Lack of anchorage Anchor failure In-plane failure Out-of plane failure Combined in-plane and out-of plane failure Diaphragm-related failures In-plane failure mechanisms of URM can have three main forms: (1) Shear failure, (2) Sliding failure, and (3) Flexural (rocking) failure. These are also defined as global response mechanisms. 26 Literature Shear Sliding Flexural Global response Figure 11: In-plane failure mechanisms. (Elgawady, Badoux, & Lestuzzi, 2006; Magenes & Penna, 2009) Vulnerability of existing URM structures to seismic loading is associated to some extend to local failure modes, mainly out-of-plane response of walls. Buildings can be governed by this type of failure mechanism due to poor connections between walls, or walls and floors. Examples of out of plane failure modes are shown below: Figure 12: Out of plane failure mechanisms. (Calvi, Pinho, Magenes, Bommer, Restrepo-Vélez, & Crowley, 2006) When the failure is associated to the connections of diaphragms to the masonry walls three of the above mentioned failure modes are identified: (1) parapet failure; (2) wall-diaphragm tension-tie failure; (3) wall-diaphragm shear failure. Figure 13: Failure of URM related to diaphragms. (Oliver, 2010) Roof and floor diaphragms can be considered as: (1) Flexible, (2) Semirigid, (3) Rigid. Diaphragms are considered flexible when the maximum lateral displacement along its length is greater than twice the 27 Literature average inter-storey drift of the vertical lateral load resisting elements. Unreinforced masonry bearing wall buildings with timber floors and roofs can be considered flexible. (FEMA 356 , 2000) The connections between masonry walls are a weak point and are expected to separate during cyclic loading. This is related to the incompatibility of stiffness between the two elements. As a result the flange effect is lost resulting to softening of the building and change of the dynamic characteristics. This separation causes damage but is not necessarily related to structural damage. The structural capacity is related to whether the elements have enough out of plane capacity. Another failure mode related to URM buildings is the failure of the gamble, which is the part of the wall supporting the pitched roof. Inadequate connection of the gamble to the roof causes rocking of the element as a free cantilever and can result to collapse. Figure 14: Walls separation and failure of gamble. (NZSEE, 2015) 2.1.4. Flange effect Flange effect refers to the influence of perpendicular walls to the failure mode of in-plane loaded walls. Testing has been conducted to investigate the effects of the boundary conditions and specifically of the flange effect to the in plane behaviour of masonry walls. What is found is that simplified predictive techniques like the New Zealand Guidelines cannot accurately take into account this effect and can result to incorrect prediction of failure. It is observed that failure mode can change from rocking for unrestrained walls to shear cracking for flanged walls. (Russell & Ingham, 2008) Other testing efforts also reported that these equations can be conservative when the flange effect is neglected. Specifically it is reported that walls with flanges can support higher lateral force than unsupported. All the walls tested also failed in shear failure and confirmed that a drift limit of 0.4 % is suitable for walls failing in shear. The length of the flange can be determined according to the Masonry Standard Joint Committee (MSJC) as , where is the thickness of the wall. (Russell & Ingham, 2010) 28 Literature 2.2. Buildings in Groningen The buildings in Groningen have been classified in categories. The case study under consideration belongs to terraced houses with timber diaphragms. These building show two main peculiarities related to the diaphragms and the cavity walls. 2.2.1. Timber diaphragms Timber floors in URM buildings typically consist of: (1) beams; and (2) cross boards nailed to the main elements. These can typically be one-way or two-way as illustrated in the following figure: Figure 15: Traditional layout of timber floors: (1) One way; and (2) Two way. (Brignola, Podesta, & Pampanin, 2008) The observation of past earthquakes on similar typologies of buildings has shown the key role of diaphragms flexibility on the overall response. Two main features are considered critical: (1) in-plane stiffness; and (2) the connections contribution. The flexibility for a single straight sheathing nailed in a single layer to the cross beams, can be evaluated by considering three contributions namely: Flexural deformation of the single board; Shear deformation of the single board; Rigid rotation of the board caused by nails slip. These three contributions can be expressed by the following equation (Brignola, Podesta, & Pampanin, 2008). ( ) Where: nail slip resulting from shear force ( ); nail deformability that can be determined with experimental tests; shear factor; shear modulus of timber planks; flexural modulus parallel to the grain of the planks; area of plank section; moment of inertia of plank section; wheelbase between beams; and nails spacing. 29 Literature The three contributions are also schematized as presented in the following figure: Figure 16: Contributions to the flexibility of diaphragm. (Brignola, Podesta, & Pampanin, 2008) In FEMA 356 and the NZSEE Guidelines an equivalent shear modulus is defined to account for these three contributions. Generally it is recognized that a highly flexible diaphragm with inadequate connections between wall and floors can lead to excessive displacement at floor level, possibly causing over turning of the perimeter wall. Stiffening the diaphragm can limit the out of plane failure mode but still generate undesirable failure mechanisms. Specifically expulsion of the corners can be caused, activating a torsion mechanism. Figure 17: Angular deformation of masonry unit and expulsion of building corners. (Brignola, Podesta, & Pampanin, 2008) 2.2.2. Cavity walls Most of the building stock in Groningen consists of cavity walls. These comprise of an inner load bearing leaf and an outer non load bearing leaf. Connectivity of these leaves can be considered a variable as is dependent on the extend of corrosion of the ties used and the construction process followed. A detailed review of 125 cavity walls showed damage due to weak mortar and lack of wall restrains. (The University of Auckland, 2015) The main failure modes observed are related to out of plane failure. Specifically three types of failure modes are shown: (1) One way bending type failure in long walls and/or walls without side supports; (2) Two way bending type failure in walls restrained in all boundaries; (3) Cantilever type failure where the top section of the wall collapses. Figure 18: Typical cavity wall and related out of plane failure modes. (The University of Auckland, 2015) In plane failure was less widely observed in this study and includes mainly: (1) Shear failure; and (2) Shear sliding failure on mortar joints or between storeys. In general buildings with cavity walls are considered to be more vulnerable to seismic loading than solid walls and need a close evaluation. (ARUP, 2013) 30 Literature 2.3. Computational modelling of masonry structures The development of a numerical model to represent masonry structures behaviour subjected to seismic loading involves a number of choices and considerations. (Bull, 2001) Some of these considerations are presented in this section. These points are used as a basis for the development of the appropriate modelling strategy. Geometry definition: Here the development of a two or three dimensional model is decided. The selection of two or three dimensional elements and the integration scheme is also defined at this stage. Modelling of masonry: The selection of a micro, macro or simplified-micro approach is chosen at this phase as introduced in Section 2.1.2. Loading application: The seismic load can be applied in a number of ways depending on the type of analysis and the modelling approach followed to represent the seismic behaviour. These are further analysed in Section 2.4. Boundary conditions: The definition of the foundation is a critical point especially when settlements take place. Connectivity: The way the elements are connected play a significant role in the analysis results. The use of interfaces and the assignment of relevant stiffness in the connections can be an advance in the developed model. Constitutive law: Material models that can be used for masonry are: (1) Total strain crack models; (2) Rankine –Hill material model; (3) Coulomb friction model. In a Total Strain Cracking Model two types of cracking behaviour can be defined; the fixed and the rotating. The fixed model considers that the rotation of the crack is fixed after the first crack occurs. When this model is used the shear retention factor needs to be defined to account for the stiffness that remains after the first cracking. In a rotating model the direction of the crack changes every time the stress-strain relationship is defined. The Rankine-Hill model incorporates also the anisotropic behaviour, while the Coulomb friction model takes into account the properties of the bond between bricks and mortar. The selection of the constitutive model is related to the modelling strategy followed and the level of detail of the analysis. Model parameters: After the definition of the constitutive law the model needs to be fed with material parameters. Factors influencing the material properties can be related to the thickness of bricks and mortar layers and the inhomogeneity of the masonry in the thickness of the structural element. Especially when old masonry needs to be assessed the permanent damage needs to be incorporated in the parameters. Sensitivity analysis is often carried out to define these parameters. Experimental results: The development of a sophisticated numerical model for masonry requires advanced testing in order to obtain the mechanical behaviour of the material. For the unreinforced masonry structures present in The Netherlands no experimental results are yet available therefore the evaluation of the models is restrained by this lack. Analysis procedures: These can include the choice of: (1) load increment procedure, including a force control, displacement control or arc-length control; (2) Iterative solution methods, including Newton-Raphson method, Modified Newton-Raphson method, Secant or Linear Stiffness method; and (3) Convergence criteria; where a force norm, displacement norm or energy norm can be defined. 31 Literature In a force control analysis loads are incrementally applied. For models experiencing softening the method cannot lead to a solution when the load applied is higher than the capacity. In a displacement control analysis the displacement of a reference point is incrementally applied. When a snap-through behaviour is expected this analysis is more adequate. The way the two procedures work are captured in the following figure. Figure 19: Force control (left) versus displacement control (right). (Palacio, 2013) When the load displacement curve is almost horizontal, the predictions of the displacements increment can become very large. When the load increment is fixed this will result to large predictions of the displacements. This problem can be overcome with an arc-length control, where the increment is adjusted. This method is also capable of tracing snap-back behaviour. The way this method works is illustrated in the following figure. Also an overview of the methods characteristics are presented. Figure 20: Arc-length control (left) and load increment methods characteristics (right). (Palacio, 2013) The iterative process as defined in DIANA is presented in the following figure. In all processes the total displacement increment is adapted iteratively by the increment till equilibrium is achieved. The total incremental displacement at iteration is therefore defined as: Where: Total displacement increment at iteration Total displacement increment at iteration ; Iterative increment. 32 ; Literature Figure 22: Regular Newton-Raphson method. (Palacio, 2013) Figure 21: Iteration process. (TNO DIANA BV., 2014) The basic difference between the iterative methods is the way the iterative increment is calculated. A reference is made here to the Newton-Raphson method as this is the method used in this analysis. The reader is referred to the relevant literature for insight in the other methods. The increment is calculated as: Where: Iterative increment at iteration ; Stiffness matrix at iteration , representing the tangential stiffness Out-of balance force at iteration . ; The relative displacement variation reported in each iteration refers to . The displacement control procedure offers advantages over the force control method as it can pass points where the force control fails. Also it is reported that the method can have better conditioned tangent stiffness matrixes. However the method fails when snap-back behaviour needs to be captured. In this case arc-length control is suitable. Recommendations on the use of the software for models with cracking focus on arc-length control analysis with indirect displacement control. 33 Literature 2.4. Analysis of seismic behaviour Earthquake is a sudden slip on a fault which results to ground shaking and radiated seismic energy. The seismic action can be caused by any sudden stress state in the earth. In case of induced earthquakes these are related to human activity. (USGS, 2005) The impact of the seismic action to a structure can be captured by different methods. The main categories include: Lateral force analysis: static analysis where the seismic action is applied as a concentrated force at the centre of mass for each floor; Response spectrum analysis: linear dynamic analysis where the seismic action is given as a spectrum; Pushover analysis: load is applied statically but non linearity of the material is taken into account; Non-linear time history analysis: load is applied as an accelerogram and the nonlinearity of the material is considered. A two storey structure under seismic loading can be modelled as a two degree of freedom system. Figure 23: Two degrees of freedom system. (adapted from Chopra A., 2012) The equation of motion which describes this problem is presented in the following equation: ̈ ̈ ̇ ̇ ̂ ̂ ̈ Where: ̂ ̈ mass matrix; damping matrix; non-linear stiffness matrix; relative displacements matrix between nodes; effective earthquake force; and earthquake acceleration. For the different analysis procedures other parts of this equation are neglected. In the time history analysis the full equation is taken into account. 34 Literature 2.4.1. Pushover analysis In a pushover analysis the lateral forces are distributed to the height with the load increased to push the structure untill an ultimate displacement is reached. This analysis provides information about the peak response in terms of storey drift, floor displacements and other deformation quantities. (Chopra, 2012) A characteristic curve to be defined by a pushover analysis is the Capacity Curve, where the displacements are plotted versus the developed base shear. An example of capacity curve where the difference between experimental and numerical results is emphasized is illustrated in the following figure. Figure 24: Load – displacement response of wall. (Facconi, Plizzari, & Vecchio, 2013) The application of the load can be performed in different ways defining a different type of pushover analysis. A monotonic pushover analysis considers a monotonic lateral load pattern which pushes the structure till the lateral capacity is reached. The capacity of the structure is dependent on the loading pattern. Types of loading patterns are presented in the following figure: Figure 25: Force distribution in a Monotonic pushover analysis. (University of Buffalo, 2009) In an adaptive monotonic pushover analysis, the loading pattern reflects the deformation pattern of the structure at the end of each load step. The structural capacity is therefore independent of the initial loading. A cyclic pushover analysis is performed by a number of chained pushover analysis. Here each pushover analysis pushes the structure in the opposite direction to the previous one. Also each pushover load case uses the stiffness at the end of the previous load case. From this analysis the equivalent viscous damping can be defined, as the characteristic hysteretic loop shown in the following figure. 35 Literature Figure 26: Hysteritic loop of Cyclic Pushover analysis. (University of Buffalo, 2009) Regulations are mainly focused on monotonic pushover curves. In Eurocode the pushover analysis is defined as a non-linear static analysis with constant gravity loads and monotonically increasing horizontal loads. (EN 1998-1, 2004). For masonry buildings capacity is defined in terms of roof displacement. The ultimate displacement capacity is taken at roof displacement where total lateral resistance (base shear) has dropped below 80% of peak resistance, due to progressive damage and failure of lateral load resisting elements. (EN 1998-3 , 2005) The above mentioned methods are initially developed considering rigid diaphragms. The applicability of the pushover analysis in unreinforced masonry buildings with flexible diaphragms is considered unexplored. Relevant studies show that the method becomes less reliable when flexibility is considered. In this case the reduced in plane stiffness of the diaphragm can influence the response of the building which is then dominated by higher modes. This comes in opposition with the consideration of a single degree of freedom system considered in the conventional method. Studies have investigated the type of pushover analysis that seems more relevant to these structures. The approach that seems to be more suitable for unreinforced masonry structures with flexible diaphragms is the modal pushover analysis. The reader is referred to these studies for further insight. (Nakamura, Magenes, & Griffith, 2014) 2.4.2. Nonlinear time history analysis Time history analysis considers the seismic action as a time history, a function between acceleration and time applied as a base excitation. Theoretically time histories have complete information about the seismic event in a certain location and record three traces: (1) Two horizontal ones; and (2) One vertical one. (Chen & Lui, 2005) During a seismic event energy dissipation takes place in the structure and sub-structure. The damping in the inelastic range is a combination of: (1) Viscous damping; and (2) Hysteritic damping. Hysteritic damping accounts for the area inside the loops that are formed when the earthquake force is plotted against displacement and can also be expressed as equivalent viscous damping using equations available in literature. (ATC-40, 1996) Hysteritic damping is tackled by the nonlinear dynamic analysis of the finite element model. The structural components dissipate a large amount of energy through hysteretic behaviour due to inelasticity, but also cracking or internal friction between constitutive materials. The damping-models available to represent the remained un-modelled energy dissipation (in the form of equivalent viscous damping) can be categorized as: (1) Mass-proportional, (2) Initial stiffnessproportional, (3) Tangent proportional and (4) Rayleigh damping. (Correia, Almeida, & Pinho, 2013) A type of damping often used in Time history analysis is the Rayleigh damping and can be expressed by the following equation: (Chopra, 2012) 36 Literature Where: Where: mass matrix; stiffness matrix; mass proportional coefficients; stiffness proportional coefficient; damping ratios; and natural frequencies. Figure 27: Variation of modal damping ratios with natural frequency: (a) mass-proportional damping and stiffnessproportional damping; (b) Rayleigh damping. (Chopra, 2012) This hysteretic energy is absorbed by the system which undergoes quasi-static or dynamic loading and can be a useful measure of the seismic performance of a structure. 37 Literature 2.5. Modelling approaches 2.5.1. Approaches overview URM buildings show presence of cracking at low levels of earthquake demand which highlights the need for nonlinear assessment methods. This type of construction has often insufficient strength to resist lateral earthquake load and specifically lacks the ability to dissipate energy and exploit ductility. (Cattari, et al., 2015) Different approaches are proposed to model masonry structures. The main differences lie on the scale of analysis and the way masonry is described. The main modelling approaches can be categorized as follows: (D26, 2012) Continuum constitutive laws model (CCLM); where masonry is considered as homogeneous. This constitutive law can be defined either by experimental results following a phenomenological approach or though homogenization procedures following a micromechanical approach. Discrete interface models (DIM); where masonry is considered heterogeneous. Here each part of the material (brick units and mortar) is modelled separately and finally assembled by interface elements. Structural elements models (SEM); where the definition of elements (spandrels and piers) is required. Here the equilibrium of the elements is defined in terms of internal forces instead of continuum stresses. The elements cracking and rotations are described with the use of nonlinear constitutive laws. Macro-blocks model (MBM); where a number of elements are considered connected through interfaces. Here the non-linear behaviour is defined at interfaces which are considered not to resist tensile forces and in some cases can deliver friction forces. The modelling approaches analysed in the present document refer to: (1) A continuum constitutive law model with the use of DIANA software and referred to as Finite Element model (FE); and a (2) Structural elements model with the use of Tremuri software and referred to as Equivalent Frame model (EF). The main characteristics of these approaches are presented in the following paragraph. These are summarized per category to clarify the main differences. 38 Literature 2.5.2. Comparison of approaches The modelling approaches used in this analysis are further analysed and the main differences are highlighted. The information is organized per category. Material properties The FE strategy followed focused on a macro-model approach and the cracking mechanism is expressed by smeared cracking as usually adopted for concrete. The non-linear behaviour of masonry is modeled with the use of a constitutive model based on total strain, the Total Strain Rotating Crack model. In this model after the exceedance of the tension criterion assigned the element is considered cracked and the orientation of the crack is continuously changed. This model is suited for analysis of materials where cracking and crushing are governing. (TNO DIANA BV., 2014) For tension a linear softening response is chosen and defined through the definition of the tensile strength of masonry and the fracture energy . For compression a parabolic softening curve is used, where the compressive strength and compressive fracture energy are defined. The stress strain relationship is presented in the following figure: Figure 28: Stress-strain relation for compression and tension. (TNO DIANA BV., 2014) Timber elements are modelled as isotropic for simplification. In the EF model the material properties assigned refer to compressive and shear strength, while tensile strength is not taken into account. Discretization The FE strategy followed considers a mesh of 200 mm. Masonry elements are modelled as curve-shell elements. The curved shell element chosen is type CQ40S which is an eight-node quadrilateral isoparametric element. This element is based on quadratic interpolation and Gauss integration. The integration scheme over the area is by default 2 x 2 . The default scheme in the direction perpendicular to the element is a 3-point Simpson which is adopted in the present analysis. For non-linear analysis also higher integration schemes are recommended. (TNO DIANA BV., 2014) The external leaf of the cavity wall is assigned as a translational mass. For this the CQ24TM element is used, which is an eight-node quadrilateral acting as a surface boundary. This is decided since the external leaf does not participate in the load bearing capacity of the building but participates as a mass. Therefore the external leaf will not be part of the mass of the building but will be part of the dynamic mass which is important in the time history analysis. The timber beams are assigned as CL18B which is a three-node, three-dimensional class-III beam element. The elements are illustrated in the following figure: (TNO DIANA BV., 2014) 39 Literature Figure 29: CQ40S curved shell element, CQ24TM translation mass element and CL18B beam element. (TNO DIANA BV., 2014) The structural interface elements used follow a linear interpolation function and are defined by only two nodes. The element used is an N6IF which is defined for a 3D configuration. Figure 30: Topology and displacements in linear interface element. (TNO DIANA BV., 2014) When looking at tractions, the normal traction and are tangential to the interface. is perpendicular to the interface and the shear tractions Figure 31: Displacements, relative displacements and tractions in the definition of interface. (TNO DIANA BV., 2014) The variables of the interfaces to the curved elements are located in the local axes. In comparison to the 2D interface element this element has an additional rotational degree of freedom to account for the compatibility with the curved element. The relevant matrixes are shown in the following equations. { } { } { } Where: Nodal displacements; Relative displacements; and Tractions. In the EF model the discretization follows a different approach and is larger. Specifically the following elements are defined: 40 Piers, referring to vertical elements; Spandrels, horizontal elements which couple the piers; and Rigid nodes; which connect spandrels and piers. Literature An example of equivalent frame idealization is illustrated in the following figure: Figure 32: Example of equivalent frame idealization. (Lagomarsino, Penna, Galasco , & Cattari, 2013) Degrees of freedom In the EF approach the walls are modelled as plane frames and the nodes are considered 2D with 3 degrees of freedom (d.o.f.) each. When looking at the corner nodes these are characterized by 5 d.o.f. The rotational degree of freedom around the z axis is neglected because of the membrane behaviour adopted between walls and floors. Figure 33: 3D assembly of masonry walls. (Lagomarsino, Penna, Galasco , & Cattari, 2013) In the FE model the curved elements used have 5 degrees of freedom per node with a total of 8 nodes per elements, resulting to 40 degrees of freedom per element. It is therefore clear that for this model a large number of dofs results and therefore the computational time required is high. The CL18B is a 3node element with 6 degrees of freedom per node, resulting to 18 degrees of freedom per element. Nonlinear response In the EF approach the progression of the nonlinear response is defined by a multi-linear constitutive law. This law describes the response of masonry until severe damaged is caused, associated to a certain strength decay and corresponding drift limit. The damage is described in 5 levels, ranging from 1 (nonstructural damage) to 5 (total collapse). The reach of a damage level is defined in terms of drift limits ( ), which is associated to a certain strength decay ( ). These parameters are defined separately for piers and spandrels related to the dominating failure mode. When damage level 5 is reached the element contributes to the overall strength only in terms of capacity to bear vertical loads. (Bento, Simoes, Lagomarsino, & Cattari, 2012) The response is captured in the following figure: 41 Literature Figure 34: Sketch of idealization of masonry panels response according to the multilinear constitutive laws implemented in Tremuri. (D26, 2012) The element expiration is considered at ultimate drift without interruption of the global analysis. The nonlinear beam degradation is illustrated in the following figure: Figure 35: Nonlinear beam degradation. (S.T.A.DATA) The nonlinear behaviour starts when one of the nodal forces reaches its maximum value, estimated according to the minimum of the following strength criteria: (1) Flexural rocking, (2) Shear- sliding, (3) Diagonal-cracking shear. The ultimate strength of the panel is defined through the definition of the maximum drift which is associated to the governing failure mechanism present in the panel. (Lagomarsino, Penna, Galasco , & Cattari, 2013) The drift usually ranges between 0.4% and 0.8%. For limit state NC this is increased by a factor of 4/3 taking into account the relevant norm. (Ontw. NPR 9998, February 2015) When the element collapses it is considered a strut. Specifically, no residual shear and bending strength is considered and only checks on the ability of the element to resist the vertical loads are considered. The element failure can be selected to interrupt or not interrupt the global response. The Ultimate Limit State (ULS) value can be defined at an 80% decay of the base shear or at the point where the first element fails. In the FE model the non-linear response is associated to the reduction of stiffness due to the degradation of the material and the formation of cracks. Initially cracks form and in the crack propagation stage existing cracks open and new cracks are created. Finally cracks reduce the stability of the structure and collapse is observed. The macro modelling strategy followed cannot describe all the failure modes that can occur. Specifically the failure modes related to the failure of mortar (sliding) need a micro modelling approach. Loading The FE strategy followed for the Pushover analysis is load control with uniform application of forces. This is decided to have a more satisfying load transfer. The application of a force at end nodes of the building led to local failure at the corners. In the EF approach a displacement is applied at a control node and subsequently forces are calculated at all nodes. The outcome presented in the pushover analysis is assigned to be the average displacements of all nodes of each level of the control node. It can be noted 42 Literature that the capacity curves do not present the displacement at roof level as prescribed by Eurode. A crucial parameter which needs attention is the selection of the control node. This node needs to be determined at the weaker walls where maximum deformation is expected. (Galasco, Lagomarsino, & Penna, 2006) Diaphragms In the FE approach timber elements are modelled as isotropic. Timber beams and timber planks are modelled separately and merged at each node. A reduced modulus of elasticity is also assigned to account for the reduced in plane stiffness. In the EF approach diaphragms are modelled as orthotropic membrane plane stress elements, with two degrees of freedom at each node. The orthotropic matrix assigned by the program for three nodes membranes is presented in the following formula: ̂ [ ] Where: Young modulus along the floor spanning direction; Young modulus along the perpendicular direction; Poisson ratio; Shear modulus; ratio . The actual orientation of the diaphragms is defined by the following matrix: [ ] Based on these two matrixes the final stiffness matrix of the diaphragm is calculated as: [ ] The program can calculate for a given typology of floor the stiffness of the diaphragm taking into account the modulus of elasticity of the material. For this case study a single way timber floor is assigned and only the modulus of elasticity is given. The software assigns a reduced modulus of elasticity based on the given typology. The shear modulus is not taken into account in the developed approach. (Lagomarsino, Penna, Galasco , & Cattari, 2013) Also in the EF model nodes are always considered connected. Figure 36: 4-node membrane element as average of 3-node. (Lagomarsino, Penna, Galasco , & Cattari, 2013) 43 Literature Cavity walls The EF model has no specific module to account for cavity walls. Here only the internal bearing leaf is modelled. Since only pushover analysis is performed in this software, where there is no contribution of the mass in the calculation, this is considered adequate. In the FE model the external leaf is assigned as a translation mass to take into account its contribution in the time history analysis. In the pushover analysis the mass of the building is not influenced by the assignment of the external leaf. Foundation In both approaches foundation is considered fixed. Both models give the possibility to assume a certain stiffness but this is not considered in this analysis. Assessment The EF model presents an assessment of the structure following the methodology proposed by Eurocode. The safety check followed compares the structure ultimate displacement capacity ( ) and the target displacement . The ultimate displacement is taken at roof level at which the base shear drops below 80% of the peak resistance as defined by Eurocode or at the point where the ultimate drift is reached. The seismic action is defined by the user. The FE model is an analysis tool and the assessment is performed by the user by close evaluation of in plane and out of plane failure, strains developed, crack widths and exceedance of maximum drifts. The target displacement is also calculated by the user. 44 Literature 2.6. Seismic assessment In this section three main parameters related to the seismic performance of structures are introduced. These include the ductility factor, force reduction factor (expressed as behaviour factor in Eurocode) and the target displacement. 2.6.1. Ductility factor Ductility of the structure refers to its ability to undergo large deformations beyond the elastic range and maintain its strength without degradation and sudden failure. (ATC-40, 1996) The ductility factor can be calculated based on the following formula: (EN 1998-3 , 2005) Where: displacement at the formation of the plastic mechanism; and yield displacement of the idealised SDOF system. Ductility factors can show a great variation depending on the configurations of clay and calcium silicate of unreinforced masonry buildings. According to shaking tests to various configurations can range between 3.2-10. (Allen, Masia, Derakhshan, Griffith, Dizhur, & Ingham, 2013) 2.6.2. Force reduction factors Force reduction factors are considered one of the most important aspects of seismic design. The general formula used by most codes is the following: (Allen, Masia, Derakhshan, Griffith, Dizhur, & Ingham, 2013) Where: component of reduction factor associated to the inherent energy; overstrength factor. There are two principles followed to define the force reduction factor: (1) Equal energy or (2) Equal displacements principle. According to these principles the behaviour factor can be defined according to the following formulas: Equal energy: Equal displacements: √ In Eurocode the force reduction factor is expressed in terms of behaviour factor . Behaviour factors are introduced in seismic design to reduce the forces from the linear analysis in order to take into consideration the non-linear response of the structure. (EN 1998-1, 2004) According to NPR 9998 the behaviour factor for unreinforced masonry buildings is considered 1.5, where a multiplication factor of 1.33 is used. (Ontw. NPR 9998, February 2015) Other codes such as the Italian seismic code OPCM 3274 give a range of q values between 2.1 – 5. Most masonry structures fall into the accelerations region (short period where ) and equation (1) is applicable. When structures show a longer period then equation (2) can be considered. In the NPR only the equal energy formula is mentioned considering that structures fall into the accelerations region. 45 Literature 2.6.3. Drift limits Interstory drift limits are considered a principal design consideration in performance-based design. The system performance level is actually evaluated through this parameter. Control of the interstory drifts can give information about the distribution of the ductility in the different floors. For masonry structures Eurocode refers to element storey drifts. The formula is associated to a specific limit state and the type of failure. These are presented in the following table. The definition of limit states is presented in Section 2.7.1. Table 3: Element drift limits according to Eurocode. (EN 1998-3 , 2005) Limit state Shear ⁄ Significant damage (SD) Near Collapse (NC) Bending ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ Where: In plane horizontal dimension of the wall (depth); Distance between the section where the flexural capacity is attained and the contra flexure point. Drift values are also presented in ASCE 41-06. For unreinforced masonry walls these are defined as follows: Table 4: Drift limits for in-plane walls and wall piers according to ASCE 41-06. (ASCE/SEI41-06, 2007) Limit state Life safety (LS) Collapse prevention (CP) Rocking Rocking Primary components (%) Secondary components (%) ( ⁄ ) ⁄ ( ⁄ ) ⁄ Where: Wall effective height; and Length of wall or wall pier. 2.6.4. Target displacement Pushover curves are considered a key element in the overall assessment process of the seismic performance of buildings. According to performance based design, seismic demand needs to be calculated. There are various methods available to assess the seismic demand. These methods refer to structures with rigid diaphragms at each floor level. The seismic demand is expressed in terms of target displacement. Methods presented at different standards include: (1) Coefficient method (ASCE/SEI41-13, 2014); (2) Capacity spectrum method (ATC-40, 1996) and (3) N2 method followed by Eurocode. (Parisi, 2010) The reader is referred to the relevant codes for insight into the different methods. Here the scope is to show the main characteristics of each approach. 46 Literature The capacity spectrum method is a procedure that involves a graphic representation of the expected seismic performance of the structure. The displacement demand is expressed as the intersection between the structures capacity spectrum and the response spectrum. This is defined as the performance point and the displacement coordinate is the displacements demand for a level of seismic hazard. The method involves four main steps: (a) Development of the Pushover Curve; (b) Conversion of the pushover to the capacity diagram; (c) Conversion of the elastic response from standard to A-D format; and (d) Definition of displacement demand. This process is illustrated in the following figure: Figure 37: Capacity spectrum method. (Chopra & Goel, 1999) 47 Literature In the coefficient method the target displacement is expressed by the following formula: Where: Modification factor to relate the spectral displacement of an equivalent single degree of freedom system (SDOF) to the roof displacement of the multi degree of freedom system (MDOF); Modification factor to relate expected maximum inelastic displacements for those calculated for linear elastic response; Modification factor to capture the effect of pinched hysteresis shape, cyclic stiffness degradation and strength deterioration on maximum displacement response; Response spectrum acceleration; Effective fundamental period of the building; and Acceleration of gravity. The N2 method as presented in Eurocode involves the following steps: 1. 2. 3. 4. 5. Transformation to an equivalent Single Degree of Freedom (SDOF) system; Determination of the idealized elasto-perfectly plastic force-displacement relationship; Determination of the period of the idealized equivalent SDOF system; Determination of the target displacement for the equivalent SDOF system; and Determination of the target displacement for the MDOF system. The definition of the bilinear relationship of the capacity curve can be based on a simplified approach presented by several authors. (Allen, Masia, Derakhshan, Griffith, Dizhur, & Ingham, 2013) This approach considers an effective yield force as an acceptable approximation to the equal energy method for unreinforced masonry structures and . The values are illustrated in the following figure. The capacity curve can be reproduced by experimental results or by the development of a finite element model. The approach presented in Eurocode is based on the actual deformation energy up to the formation of the plastic mechanism. Figure 38: Bilinear approximation of force displacement curve. (Allen, Masia, Derakhshan, Griffith, Dizhur, & Ingham, 2013) The formulas used by Eurocode are summarized in the following table, where the relevant step is indicated. In the present analysis this approach is implemented. The same approach is followed by the EF model and the results are compared. 48 Literature Table 5: Target displacement definition formulas. (EN 1998-1, 2004) Step Value Formula Mass of equivalent SDOF 1 Parameters Mass in the i-th storey ∑ Transformation factor Normalized displacements ∑ Force of SDOF system Base shear force Control node displacement of MDOF system Displacement of SDOF system 3 Yield force Period of idealized equivalent SDOF system √ Yield displacement [ For (medium and long period range) [ 4 Target displacement of SDOF For Target displacement of the structure with period T* ] ] Elastic acceleration response spectrum at period T* (short period range) If If ( 5 ( )) Target displacement of MDOF 49 Literature 2.6.5. Analytical approaches In parallel to the development of numerical models to assess URM structures, analytical procedures mechanically based are presented in standards. These processes focus on both in-plane and out of plane failure while different formulas are applied for piers and spandrels. International standards presenting analytical approaches include NZSEE 2006, Eurocode 8, ASCE 41-13 2014, Italian Building Code NTC. The reader is referred to the relevant documents for an insight on the different approaches followed. (Cattari, Lagomarsino, Bazzurro, Porta, & Pampanin, 2015) For the purpose of this analysis a simplified pier-only method is applied and the results are compared to the modelling approaches developed. The main interest lies on observing the applicability of analytical approaches for URM buildings with unloaded facades and flexible diaphragms. This approach considers that spandrels are indefinitely stiff and therefore piers govern the behaviour. The relevant formulas applied for piers are presented in the following table. Table 6: Pier failure mechanisms. (NZSEE, 2015) Failure mode Formula Parameters Factor to correct nonlinear stress distribution Diagonal tensile cracking Area of net mortared/grouted section of the wall web √ Masonry diagonal tension strength Axial compression stress due to gravity loads calculated at the base of the wall /pier Masonry bed-joint cohesion Masonry coefficient of friction Strength of wall or wall pier based on rocking Factor equal to 0.5 for fixed-free cantilever wall, or equal to 1.0 for fixed-fixed wall pier Rocking capacity Superimposed and dead load at the top of the wall/pier under consideration Self-weight of the wall/pier Length of the wall/pier Height of the resultant of seismic force Factor equal to 0.5 for fixed-free cantilever wall, or equal to 1.0 for fixed-fixed wall pier Superimposed and dead load at top of the wall/pier Toe crushing capacity Self-weight of wall/pier Length of wall/pier Height to resultant of seismic force Axial compression stress due to gravity loads at mid height of wall/pier Masonry compression strength Masonry coefficient of friction Bed-joint sliding shear capacity 50 Superimposed and dead load at top of the wall/pier Self-weight of wall/pier above the sliding plane being considered Literature 2.7. Seismic rehabilitation 2.7.1. Framework Seismic rehabilitation of buildings is based on a performance-based design approach which differs from seismic design procedures. The rehabilitation process can be based on the following steps: (ASCE/SEI4106, 2007) Review of Initial considerations; including structural characteristics, economical, historic, results from previous studies etc. Selection of Rehabilitation objectives; including target Building Performance level and Seismic Hazard; Obtaining As-Built information; Selection of rehabilitation method; Performance of Rehabilitation design; and Verification of Rehabilitation design. In this process the definition of the rehabilitation objective can be considered crucial and is expressed by: (1) a target Building Performance Level; and (2) an Earthquake Hazard Level. The target Building Performance levels are expressed in EC-8 as fundamental requirements and refer to the state of damage of the structure. These requirements are defined in terms of limit states (LS) and are summarized in the following table: Table 7: Limit states definition. (EN 1998-3 , 2005) Near Collapse (NC) Significant Damage (SD) Damage Limitation (DL) Structure is heavily damaged. Structure is significantly damaged. Structure is lightly damaged. Structural components Low residual lateral strength and stiffness, although vertical elements can sustain vertical loads. Some residual lateral strength and stiffness is present and vertical elements can sustain vertical loads. Structural elements are prevented from yielding and retaining their strength and stiffness properties. Non-structural components Most non-structural components have collapsed. Non-structural components are damaged although partitions and in-fills have not failed out-of plane Non-structural elements like partitions and in-fills may show distributed cracking but the damage will be economic to repair. Large permanent drifts present. Moderate permanent drifts are present. Permanent drifts are negligible. Structure would probably not survive another earthquake even of moderate intensity. The structure can sustain after-shocks of moderate intensity. The structure is likely to be uneconomic to repair. The structure does not need any repair measures. Overall damage Drifts General Earthquake hazards refer to hazards that can exist at the building site which could damage the building and are not directly related to the seismic shaking. These hazards include fault rupture, liquefaction, soil failures, landslides and inundation from off-site effects like dam failure or tsunami. (ASCE/SEI41-13, 2014) The seismic hazard in Groningen has been defined by a Probabilistic Seismic Hazard Analysis. The definition of the peak ground acceleration is based on a 2% probability of exceedance in the next ten years. This can be considered equivalent to a 10 % probability of exceedance in 50 years hazard level and a return period of 475 years. (ARUP, 2013) 51 Literature 2.7.2. Retrofitting methods The retrofitting methods reviewed are related to unreinforced masonry buildings with timber diaphragms and cavity walls. A brief overview of these methods is given in the following paragraphs. The reader is referred to the relevant literature for further insight on each method. The methods can be categorized in the following main directions: (1) Strengthening of floor to wall connections; (2) Increase of in-plane stiffness of diaphragms; (3) In plane strengthening of masonry walls; (4) Connection of inner and outer leaf of cavity wall; and (5) Base Isolation. 2.7.2.1. Strengthening of floor to wall connections The strengthening of the floor to wall connection can be related to the lateral connection or the connections at the ends of the wooden beams. A description of relevant methods is presented in the following table: Table 8: Strengthening of floor to wall connections. (Brignola, Podesta, & Pampanin, 2008) Method Description 1 Steel ties at ends Connections at ends of wooden beams to masonry walls are improved with the use of steel ties. 2 Steel ties for lateral protection Steel ties are placed perpendicular to the wooden beams. L-shape steel element Elements are connected to the floor with screws. The ends of the profiles are connected to the lateral masonry unit with threaded steel bars of 20-30 mm chemically or mechanically connected. 3 52 Photo Literature 2.7.2.2. Improvement of in-plane stiffness of diaphragms Different retrofitting methods are available for the improvement of the in-plane stiffness of the diaphragms. (OPCM 3274, 2005; Brignola, Podesta, & Pampanin, 2008). Some options are shown in the following table: Table 9: Strengthening of timber floors. (Brignola, Podesta, & Pampanin, 2008) Method Description 1 Cross laminated plywood sheet Superposition of a new layer of wood planks or plywood on the existing sheeting. Usually the new boards are crossly positioned to the existing ones and screwed. 2 Fibre reinforced Polymers (FRP) or steel plates Application of diagonal bracing. The sheets of the FRP can be glued to the wooden planks with the use of epoxybased resin. The light steel plates can be nailed to the planks. Concrete topping Lightweight concrete topping of usually 40-50 mm thick with or without the use of steel connectors. Reinforcement is given with the use of a wire-mesh of 56 mm diameter. 3 Photo 53 Literature 2.7.2.3. In-plane strengthening of masonry walls There are different ways to influence the in-plane behaviour of the masonry walls. Some options are presented in the following table. Table 10: Methods of in-plane strengthening of masonry walls. Method Description Photo Internal reinforcement Steel bars inserted in holes drilled in plane of the unreinforced masonry walls. In this way both in plane and out of plane flexural capacity are improved. - 1 Steel bracing system Addition of steel bracing to influence stiffness and improve the ductility factor. - 2 3 FRP covering or X strips This method concerns the covering of the full surface with composites or diagonal “X” retrofitting configuration. (Elgwady, Lestuzzi, & Badou, 2005) 4 5 6 54 Shotcrete overlay The overlay is sprayed on the surface of a masonry wall over a mesh of reinforcing bars. The thickness of the layer can be adapted to the seismic demand. The overlay thickness is recommended to be at least 60 mm. (Elgawady, Badoux, & Lestuzzi, 2006) Centre Core Method that can be applied as follows: (1) Vertical holes with certain tolerances are perforated on the walls to the footing; (2) Reinforcing steel bars are embedded in the holes; and (3) Cement grout is injected to create a bond strength between wall and bars. (Amiraslanzadeh, Ikemoto, Miyajima, & Fallahi, 2012) RC Jackets Technique based on the application of single-sided or double-sided RC walls or coatings. When reinforcing steel is used the following process is followed: (1) Removal of plaster and cleaning of mortar joints; (2) Grouting of cracks if present and build of anchor ties; (3) Cleaning of surface, moistened and spattered with cement milk; (4) Application of two layers of concrete with reinforcing mesh in between; (5) Connection of mesh on both sides with the steel anchors by welding or tying the wire. (Churilov & Dumova-Jovanoska, 2012) Literature 7 Post – Tensioning with Rubber Tyres The method involves the application of a compressive force to masonry walls. This force counteracts the tensile stresses produced by lateral loads. The method is used to enhance the tensile and flexural capacity of URM walls and includes: (1) Core drilling from the top of the masonry walls; and (2) Vertically post-tensioning the walls to the foundation. (Smith & Redman, 2009) Modification of openings: The modification of openings can also have a positive effect on the in-plane behaviour of masonry walls. Some options are presented in the following table: Table 11: Strengthening of URM with modification of openings. Method Description Photo 1 Infill openings Infill of unnecessary windows and door openings. The stress concentrations at the corners which are a cause of cracks are avoided. - 2 Enlarge openings Used to increase the aspect ratio of a pier in order to change the failure from shear to flexure. The mode of failure is therefore changed from brittle to ductile. - 3 FRP reinforced openings Placing FRP strips around windows and doors with the addition of intermediate strips along the walls. This method is proven to improve outof plane stiffness and concentrations of stresses at the corners. (Bouchard, 2007) Mortar strengthening: The enhancement of the mortar can give a positive effect on the masonry behaviour as the properties of the material are improved. A method to strengthen the mortar is shown in the following table: Table 12: Mortar strengthening. 1 Method Description FRP Structural Repointing Applied to enhance the mortar, usually when aesthetics needs to be preserved. The technique is applied as follows: (a) Grinding of masonry joints, (b) Masking to avoid staining, (c) Application of epoxy based paste to masonry joint, (d) Installation of GFRP Rods. (Tumialan , Huang, Nanni, & Silva, 2001) Photo 55 Literature Repair of cracks: This technique involves the filling of the voids and cracks with grout or epoxy. The result is dependent on the injection technique adopted. Epoxy resin is used for cracks less than 2mm wide. Cement paste grout is appropriate for filling of larger cracks, voids and empty collar joints. Walls retrofitted with epoxy tend to be 10-20 % stiffer than unreinforced. The method is advised only when the consequences of the increase in strength of certain cracks to adjustment portions is studied. (Elgawady, Lestuzzi, & Badoux, 2004) 2.7.2.4. Connection of inner and outer leaf of cavity walls The two leaves of the cavity wall can be better connected with the use of transversal anchorage. This aims at avoiding the separation of the inner and outer leaf. Figure 39: Face to face connector of wall with two layers. (Meireles & Bento, 2013) 2.7.2.5. Base isolation This technique aims at reducing the acceleration transferred to the masonry walls from the ground and therefore prevents the relative displacements in the walls and improves the energy dissipation in the building. Various methods have been proposed to reach base isolation in unreinforced masonry buildings. (Yekrangnia, Mahdizadeh, Seyri, & Raessi, 2012) The reader is referred to the relevant literature for insight in these methods. 56 FE modelling 3. FE modelling The development of the Finite Element model involved a number of choices which determined the modelling strategy followed. An overview of these choices is presented in the following table: Table 13: Analysis choices. Analysis aspects Choices Geometry of elements 2-D curved shell elements Integration scheme 3 integration points Modelling approach Macro -modelling Load application Uniform Supports Fixed Connectivity of elements Variable Constitutive law Total strain rotating crack model Material parameters Fixed parameters Material properties Non-linear Type of analysis - Primarily conventional pushover NLTHA as check tool Load increment procedure for Pushover Force control Numerical method Implicit Iterative solution method Regular Newton Raphson Convergence criteria - Displacement for Pushover Energy for NLTHA The analysis is primarily based on the assessment of forces. To that end a force control analysis is selected. Also a number of analysis needed to be developed considering the existing and the reinforced structure and this led to the development of a generalized modelling approach. A more detailed approach would involve displacement control analysis with the assignment of an arc-length control. The uniform application of loading is selected due to the presence of the flexible diaphragms. In a pushover analysis the load is theoretically applied at points and the diaphragms are rigid. Application of a point load in this model led to local damage of the masonry and not satisfactory load transfer and was abandoned early in the process. Choices considering the adaptation of 2-D elements, fixed supports and fixed material properties are selected for simplification. A more detailed approach would consider 3-D elements, stiffness at the supports and sensitivity analysis for the material properties. Also the integration scheme followed is composed of 3-layers as this is the default scheme of the DIANA software. As found later in the process for non-linear analysis higher integration schemes are recommended. This is not followed in the current analysis but is recommended in future analysis. Choices considering the iterative solution method and the convergence criteria are not investigated in detail due to time constrains. A more refined approach would involve the comparative analysis of different iterative methods and convergence criteria and adaptation of the most suitable for each analysis. Finally in the present analysis no comparison is made to experimental results. 57 FE Modelling 3.1. FE model parameters In this paragraph the models properties and the modelling choices made throughout the process are discussed in detail. As shown in the previous paragraph most of the parameters are fixed. The influence of the connections quality to the global response is considered of interest and is further investigated. Materials A macro modelling approach is followed in the model and a rotating total strain crack model is adopted. In this approach a homogenization of the material is followed where the properties of the bricks and mortar are smeared out over the element. For linear two-dimensional elements the bandwidth is considered by the FE model. This is calculated as: . The material √ √ properties taken into account are presented in the following table: Table 14: Material properties of masonry. Property Unit Value Young modulus 4000 Poisson ratio - 0.2 Crack orientation - Rotating Tensile curve - Linear crack energy Tensile stress 0.15 Fracture energy tension 0.015 Compression curve - Parabolic Compression strength 6 Fracture energy compression 2.5 The focus of this analysis is on the masonry elements. The nonlinear behaviour of the timber floors is neglected. The wooden elements are considered isotropic for simplification, although timber shows an orthotropic behaviour. The weight of the timber floors is transferred to the masonry walls where fictitious densities are assigned and the timber density is assigned as 0. The material properties taken into account for the wooden elements are presented in the following table: Table 15: Material properties of wooden elements. Property Unit Young modulus Value 10000 Poisson ratio - 0.3 Density - 0 Steel elements are used for the improvement of the in-plane behaviour of the walls. The material properties used are shown in the following table: 58 FE modelling Table 16: Material properties of steel elements. Property Unit Value Class - S235 Modulus of elasticity 220000 Density 7850 Poisson ratio - 0.3 Material model - Von Mises and Tresca plasticity Hardening Hypothesis - No hardening Yield stress 235 Schematization The developed model is based on the following schematization. Specifically the centre of the inner leaf is considered for the cavity walls and the centre of the wall for the separating wall. The levels are considered at the top of each floor and beams are inserted with the right eccentricity. The supports are considered at the level of the floor of the basement and the foundation is excluded from this analysis. Figure 40: Schematization of the FE model. Mesh The main interest of the analysis lies on the behaviour of the masonry elements of the modelled system. To that end the mesh of the masonry elements is defined at 200 mm taking into account the real dimensions of the bricks. The meshes of the wooden planks and beams are also assigned at 200 mm. An overview of the developed model is presented in the following figure and the meshed elements are analysed. 59 FE Modelling Figure 41: Overview of the FE model. Roof wooden planks 22mm Roof beams: 71 x 196 mm2 Ridge beam: 71 x 246 mm2 Cavity walls 100 x 50 x 100 mm Intermediate wall 100 mm Separating wall 200 mm Floor wooden planks 22 mm Floor beams 71 x 196 mm2 Figure 42: Meshed elements of the FE model. The generated mesh initially showed some irregularities at the position of the windows. The mesh quality is checked and these irregularities are corrected by subdividing the mesh area. In addition the mesh is adapted to concern the right position of the wooden beams as the generated mesh was resulting to eccentricities for certain beams. The difference in the quality of the initial and the improved mesh is illustrated in the following figure. Figure 43: Correction of generated mesh. 60 FE modelling Layers The integration scheme followed over the thickness of the element is a 3-point. This implies that the element is defined with three layers and therefore the results are generated separately. A right interpretation of the results requires the clarification of the layers definition. In the following schematic representation the layers as defined are shown aligned with the definition of the local axis. All results in this analysis are shown for layer 3. Figure 44: Definition of layers in the curved elements and local axis. Cavity walls Cavity walls comprise of an inner load bearing leaf and an outer non load bearing. The as built configuration is presented in the following figure. To take into account the dynamic effect of the outer leaf due to the mass participation in the time history analysis, the mass of the outer leaf is transferred to the inner leaf. Therefore there is a difference between mass in the static pushover analysis and the dynamic mass in the time history analysis. Figure 45: As built configuration of cavity wall. 61 FE Modelling External leaf Translation mass Element CQ24TM Internal leaf Curved shell element Element CQ40S Figure 46: Modelling of cavity wall. Table 17: Mass and dynamic mass in DIANA. Property Unit Value Mass 49.94 Dynamic mass 66.40 Connection of base The base of the building is considered fixed, without taking into account stiffness parameters. To reduce the number of elements no plank is defined. One node is fixed and the rest are tied to the fixed node. This configuration is illustrated in the following figure: Figure 47: Fixed base with the use of links. 62 FE modelling Overview of connections of timber beams The as built configuration of the connections between timber beams and masonry walls is presented in the following figure. In the modelling the connections to the right wall and to the intermediate wall are considered hinged, while the connections to the left end are considered variable to take into account the connectivity quality. Hinged Variable Hinged c c c Figure 48: As-built connection of floors to walls and modelling considerations Variable connections of timber beams For the connections between the timber beams and the walls links are created and different situations are considered. The translation in the x direction at one end is considered variable. Rotations are considered free. The three cases developed are presented in the following table: Table 18: Connections between wooden beams and walls. Floor beams x y z Case 1 Free Tied Tied Case 2 Stiffness Tied Stiffness Case 3 Tied Tied Tied ux uy uz Free The two extreme situations consider a timber beam rolling on the top of the cavity wall and a beam working together with the cavity wall in terms of translations. The modelling of these situations is done by modelling a physical gap at these points and introducing links. The schematization of the three cases is presented in the following figure: 63 FE Modelling As- built configuration Case 1: Non connected Case 2: Semi-connected Case 3: Fully connected Figure 49: As built connection of wooden beams and modelling cases developed. The relevant modelling set-up is described in the following figure. Figure 50: Connections modelling with the use of links. The wooden beams are modelled following the as built geometry. In reality shear will be developed between the beams and the masonry elements. To capture this situation the model is redefined, where now interfaces are included. For the interface a normal direction is defined and values are given for normal stiffness and shear stiffness. This schematization is illustrated in the following figure: Normal direction Figure 51: As built configuration and modelling set up of interface. ⁄ The shear stiffness is not considered critical in this configuration and a high value of is assigned. The normal stiffness is considered critical as it defines the stiffness of the spring. The value of the as built configuration is not known and for this reason values are chosen to reproduce the expected reduced capacity. The interface surface is defined considering that the beam will be placed at half of the area of the depth of the wall and is defined as . Physically the problem is related only to the development of shear. The spring definition is used as part of the modelling strategy to represent the problem. The academic interest here is to check the influence on the model results when the normal stiffness of the interface is reduced. 64 FE modelling Connections at middle wall The connections at the intermediate wall are considered hinged to take into account the presence of two beams connected at the same node. The modelling set up is presented in the following figures. Figure 52: Modelling set up of connection to intermediate wall. Longitudinal connections of timber beams The end beams of the floors are unconnected longitudinally to the masonry facades. The as built configuration and the modelling set-up are presented in the following figure. A physical gap is created at these points considering the centre to centre distance of the two elements. The longitudinal connection will be evaluated as a retrofitting method where links will be introduced. As-built configuration Modelling set-up Figure 53: As built floor longitudinal connection and modelling set up. Connection of roof The connections of the roof beams follow the same approach as for the floor beams. The roof end beams are modelled merged to the masonry walls. In reality friction will also be developed between the beam and the masonry wall, but this is excluded from the present analysis. The roof planks are linked to the front masonry walls with links, considering that they will be tied only in the z direction. This is decided to take into account the worst case where no support is given by the existing nails. In reality the nails will restrain the plank also in the x and y direction. The connectivity of these elements will be considered as a reinforced method and will also be evaluated. The as-built configuration and the relevant modelling choices are shown below. 65 FE Modelling As- built configuration Free in y Free in x Restrained in z Figure 54: As built roof connection and modelling choices. Merged end beam to wall Roof plank to end beam connection with links Figure 55: Modelling set up of roof connection to wall. Timber floor The wooden plank and floor beams are considered merged and only the wooden beams are connected at the two ends at the masonry elements. Only the top plank of 22mm is modelled considering that the bottom plank will not play a structural role. The floor configuration is illustrated in the following figure: Figure 56: Modelled wooden floor in the FE model. 66 FE modelling Loads In a static pushover analysis the load can be applied in different ways. In this analysis a uniform application of loading is considered and the load is applied as a horizontal acceleration. The application of the seismic load as point loads led to local damage of the masonry and was abandoned from an early stage. The application of load and the position of the plotted displacements are illustrated in the following figure: Figure 57: Application of load and position of plotted displacements. For the NLTHA the load is applied as a base excitation with three components in the x,y,z direction. The dead loads as assigned as gravity loads. The dead loads are calculated and are assigned as: Timber floors dead load: Roof dead load: The calculation is summarized in the relevant Appendix. Variable loads are assigned as line loads on the relevant masonry walls. The assigned loads are presented in the following table: Table 19: Variable loads at masonry walls. Variable load 1.75 Left wall Load length 1.835 Load 3.21 Intermediate wall Load length 2.825 Load 4.94 Separating wall Load length 0.99 Load 1.73 Figure 58: Variable loads. For the pushover analysis the first step corresponds to the application of gravity load and variable load in a load combination of . In the next steps the external uniform force is implemented in steps. 67 FE Modelling Fictitious densities The main focus of this analysis is on the masonry elements. As presented before the density of the timber elements is neglected. Instead the loads are assigned to the walls by modifying the assigned density, creating fictitious densities. The densities are calculated as presented in the following table: Table 20: Fictitious densities calculation. Units Values Wall 1&4 2 5 3&6 7 8 Wall type cavity uniform cavity uniform cavity uniform Diaphragm dead load Diaphragm load 0.36 0.36 0.36 0.36 0.78 0.78 Load width 1.835 2.825 2.825 0.99 2.83 2.83 Diaphragm load 0.6606 1.017 1.017 0.3564 2.20 2.20 Length 6.92 6.92 6.92 6.92 8.15 8.15 4571 7038 7038 2466 17954 17954 Load Masonry self-weight Density 1920 1920 1920 1920 1920 1920 Acceleration g 9.81 9.81 9.81 9.81 9.81 9.81 Specific weight γ 18835 18835 18835 18835 18835 18835 Wall thickness 0.1 0.1 0.1 0.2 0.10 0.20 Wall length 6.92 6.92 6.92 6.92 8.15 8.15 Wall height 2.7 2.7 2.7 2.7 2.15 2.15 Openings area 0 1.86 3.72 0 0 0 Volume 1.8684 1.6824 1.4964 3.7368 0.88 1.75 Weight 35192 31688 28185 70383 32996 65992 Total weight 39763 38726 35223 72850 50950 83946 Relevant specific weight 21282 23018 23538 19495 58168 47919 Relevant density 2169 2346 2399 1987 5929 4885 The walls definition is illustrated in the following figure: Figure 59: Walls numbering. 68 FE modelling Steel elements Steel elements are used to support the retrofitting methods developed. Specifically three configurations of steel frames are checked. Configuration 1: The first concept concerns steel moment frames, covering the full length of one unit. Here the main focus is to reduce the interstory drifts. The steel elements used are , for the steel frames and for the connection to the masonry wall. The configuration checked is presented in the following figure. The connection of the RHS profiles to the masonry walls are considered hinged leaving rotations free. This is defined with the use of links. Hinged connection Figure 60: Steel frame configuration 1. Table 21: Steel profiles for configuration 1. Top beams IPE300 Beams and columns IPE400 Connections to masonry RHS 150 X 100 X 6.3 Configuration 2: The same configuration as 1 but now the steel profiles used are for all beams. Configuration 3: In this approach the focus is on limiting the elements drifts. Profiles used are , for the diagonals, for the connection to the base. Connections to the masonry are defined as . In this configuration the stiffness is mainly determined by the stiffness of the foundation. The foundation is assigned as fixed in the analysis and therefore the results are expected advantageous. In reality the stiffness of both the foundation and this configuration will be reduced. ` Figure 61: Steel frame configuration 3. 69 FE Modelling Analysis For the pushover analysis the convergence norm is assigned as displacement norm and the iteration method followed is Regular Newton Raphson. Iterations are assigned at 30. The analysis is assigned at continuing when convergence is not reached. For the NLTHA an energy norm is assigned, the maximum iterations are set to 20 and the analysis is also assigned to continue when not converging. The convergence quality is checked at each analysis and reported. The checks refer to: (1) the applied force versus the resultant base shear, to indicate the quality in terms of forces; (2) the displacement variation of the non-converged steps, to determine the quality of the resultant displacements. The focus of this analysis is on determining the capacity of the structure, therefore non converged steps are accepted and the variation is reported. 70 Eigenvalue analysis 3.2. Eigenvalue analysis The results of an eigenvalue analysis can help understand the behaviour of the modelled structure and the interaction between the curved elements. This analysis is used at an early stage to better control the developed models and point out deficiencies throughout the process. In a modal analysis usually the first mode shape has the highest participation of mass for rigid diaphragms. In the eigenvalue analysis of Case 1 the first mode is related to the out of plane failure of the gamble, while higher modes show an out of plane failure of the front shear wall. This analysis is also a starting point to better understand the results from the time history analysis. In this analysis participation of 60% of the mass is observed for the x direction at mode 6 and for y direction at mode 36. This is related to the poor connectivity between the elements which results to local deformations. Mode 1: Mode 4: Mode 2: Mode 3: Mode 5: Mode 6: Figure 62: Mode shapes of Case 1. The observed mode shapes can be described as: Mode 1: Longitudinal bending of left wall; Mode 2: Longitudinal bending of left wall and front façade; Mode 3: Tranversal bending of left wall and longitudinal bending of front façade; Mode 4: Longitudinal bending of left façade; Mode 5: Tranversal bending of left wall and longitudinal bending of front façade; Mode 6: 60% participation of mass achieved in the x direction. For Case 2 where interfaces are inserted participation of 60 % in the x direction is again observed at mode 6. When analysing Case 3 a participation of 60% is observed at mode 1. This can be explained due to the full connectivity assigned at this model which results to the suppression of the localized first mode shapes. Figure 63: First mode shape for Case 3 : . 71 FE model – Pushover analysis 3.3. Pushover analysis In this paragraph the results of the pushover analysis are presented. An important point in the modelling of the case study under consideration is the definition of the connections between timber beams and masonry walls. To account for this uncertainty different situations are analyzed and different capacity curves are presented. This is decided in order to understand the influence of the connections quality in the global response of the building. To that end three situations are developed where the x translation in the left end of the wooden beam is considered crucial. The situations analyzed are the following: 1. Case 1: Non connected beams to masonry walls at the left end, considering that the beams can slide; 2. Case 2: Semi-connected beams to masonry walls at left end, where there is stress developed between beams and masonry walls (modelled with the introduction of interfaces); and 3. Case 3: Fully connected beams to masonry walls at left end, considering that masonry walls and beams have the same translations in all directions. The scope here is to capture the overall behaviour of the structure. Therefore the focus is on the maximum base shear that each system can take and the failure mechanisms that occur. Case 1: Non connected Case 2: Semi connected Case 3: Fully connected Figure 64: Tied wooden beams to masonry walls (left) and non-tied (right). Initially only the left connection of the wooden beams is considered a variable and the three cases are compared. At a later stage for Case 2 also the assignment of interfaces at both ends is investigated as this is closer to the real situation. 72 FE model – Pushover analysis 3.3.1. Capacity envelope of building As a first step the two extremes of the expected capacities are captured, corresponding to Case 1 and 3 of the above mentioned cases. The capacity curves obtained till the first drift limit is reached are illustrated in the following figures. This corresponds to an interstory drift of 0.5 % related to shear failure. The capacity is estimated between 37 to 47 KN considering a modulus of elasticity for the planks of 10000 . In practise the modulus of elasticity of the diaphragms will be reduced. This correction will be further analysed in Section 3.3.7. Base Shear (KN) 60 50 40 30 20 Case 1 Case 3 10 0 0 5 10 15 20 Displacement at roof level (mm) Figure 65: Capacity curve per connection type till first drift limit reached. (x) In the y direction all systems showed the same behaviour where out of plane failure of the back façade is governing. Indicatively the capacity curve of Case 3 is shown. As can be seen the system is in the linear phase, showing that the out of plane failure is premature and the maximum capacity of the system is not yet reached. This is therefore a point for intervention, which will be further elaborated in the reinforcement of the building. 250 Base Shear (KN) 200 150 100 50 0 0.0 0.5 1.0 1.5 2.0 Displacement at roof level (mm) Figure 66: Capacity curve until out of plane failure occurs. – Case 3 (y) 73 FE model – Pushover analysis 3.3.2. Analysis of capacity curves The critical points of each analysis are observed to understand the failure process of the structure. The main points observed are the following: Exceedance of ultimate principal tensile strain at element level (E1); Exceedance of ultimate principal compressive strain at element level (E3); Extensive crack widths; Out of plane failure; and Exceedance of drift limits. For the strains the values that are observed refer to the yield and ultimate strains in both tension and compression. The ultimate values are calculated according to the assigned values of fracture energy and ultimate strength. 1 0 Stress (N/mm2) -1 -2 -3 -4 -5 -6 -0.00375 -0.00275 -0.00175 -0.00075 0.00025 Strain Figure 67: Stress strain relationship assigned. Table 22: Critical values of tensile and compressive strains. Property Value Yield tensile strain Ultimate tensile strain Yield compressive strain Ultimate compressive strain The analysis of the capacity curves for each case is discussed in the following paragraphs. The presented results are the displacements and the principal tensile strains to indicate the locations of the cracks. For the case study under consideration the drift limits per element are calculated and are presented in the following table. This calculation can give an indication of the most vulnerable element per floor. 74 FE model – Pushover analysis Table 23: Drift limits per element. Height Width Pier Shear drift Pier Bending drift [mm] [mm] - - Comment Front façade – 1st floor 1 Left pier 2150 680 0.017 0.034 2 Middle pier 1900 795 0.013 0.025 3 Right pier 2450 980 0.013 0.027 Middle pier attains drift limit first nd Front façade – 2 floor 4 Left pier 2070 680 0.016 0.032 5 Middle pier 1020 1050 0.005 0.010 6 Right pier 1650 1130 0.008 0.016 Middle pier attains drift limit first st Back façade – 1 floor 7 Left pier 1910 480 0.021 0.042 8 Middle pier 1900 795 0.013 0.025 9 Right pier 2150 680 0.017 0.034 Middle pier attains drift limit first nd Back façade – 2 floor 10 Left pier 2070 480 0.023 0.046 11 Middle pier 1440 1495 0.005 0.010 12 Right pier 2070 680 0.016 0.032 Middle pier attains drift limit first As can be noted drift limits are exceeded firstly for (1) Middle pier of front and back façade for the second floor; following by (2) Middle pier of front and back façade of first floor. A drift limit of 0.5 % is considered as the lowest boundary. The dimensions taken into account are shown in the following figure: Figure 68: Pier dimensions. For the steel elements the stress-strain relationship assigned corresponds to the following scheme: Property Yield stress Yield strain Value Figure 69: Stress-strain relationship of steel elements and definition of yield strain. 75 FE model – Pushover analysis 3.3.3. Case 1: Non-connected (x) This case indicates poor connectivity of the wooden beams to the masonry walls. The masonry left wall is free to move and it fails out of plane. Subsequently in plane failure of front and back façade is observed. The failure modes observed are diagonal cracking related to shear failure and toe crushing. Also extensive cracking is observed at the connection of the masonry elements. The displacements of the structure and the principal tensile strains are shown in the following figure. Figure 70: Displacements and principal tensile strains at collapse stage. - Case 1 (x) The failure modes identified are illustrated in the following figure: Figure 71: Failure modes identified. 76 FE model – Pushover analysis The critical points of the capacity curve are illustrated in the following graph. Converged steps are found only in the linear phase. The convergence details are reported in the relevant Appendix. Capacity curve - Case 1 (x) 45 40 Base Shear (KN) 35 Step 10: Ultimate compressive strength 30 25 Step 10: Crack width 7.78 mm 20 15 Step 23: Interstory drift limit 0.5% of first floor reached Step 24: Interstory drift limit 0.8 % of first floor reached Step 10: Out of plane failure 10 Step 25: Collapse Step 8: Ultimate tensile strength 5 0 0 10 20 30 40 50 60 70 Displacements at roof level (mm) Figure 72: Capacity curve analysis. – Case 1 (x) From this curve different phases in the behaviour of the structure can be identified. The main phases can be summarized as follows: 1. 2. 3. 4. 5. Gravity loading; Linear phase; Extensive cracking; Crack propagation; and Collapse. Gravity loading: The first step is related to the application of the gravity loads and the result is a negative displacement and a negative base shear. The displacement is a value of -0.2 mm and the base shear of 21 N. The displacements and the developed tensile strains of this step are shown in the following figure: Figure 73: Displacements and principal tensile strains at first step. - Case 1 (x) Linear phase: Here an almost linear behaviour can be identified in the capacity curve. In this part the formation of the first cracks is noted which shows that the behaviour is actually nonlinear. The first cracks are identified at the corners of the left wall which shows a movement of the wall out of plane. Figure 74: Displacements and principal tensile strains at linear stage. - Case 1 (x) 77 FE model – Pushover analysis Extensive cracking: In this phase the first big cracks can be noted and the stiffness of the structure decreases significantly. The extensive cracks are identified at the left wall and this is where out of plane failure is pointed. Figure 75: Displacements and principal tensile strains at extensive cracking phase. - Case 1 (x) Crack propagation: Here existing cracks open and new cracks form. The propagation of cracks is associated with the loss of energy for the structure. This part of the curve defines the total capacity. Crack formation is also identified in the front and back façade starting from the corners of the openings and propagating till the closest corners. The in plane behaviour of these walls is governed by shear failure as the characteristic diagonal cracking is identified. The sequence of failure shows initially failure of the middle and right pier of the first floor. Figure 76: Displacements and principal tensile strains at crack propagation stage. - Case 1 (x) Collapse: This phase is characterized by a sudden crack which reduces the stability of the structure. The results are shown in Figure 68. The sequence of failure shows firstly failure of the middle and right pier of the first floor at the front and back façade. Then failure of the elements of the second floor is identified. The way the drift limits are exceeded per step are illustrated in the following figure. Drifts 0.020 First floor Second floor Roof 0.010 0.000 0 10 20 30 Steps Figure 77: Drifts per storey and load step.- Case 1 (x) 78 FE model – Pushover analysis 3.3.4. Case 3: Fully connected (x) The improvement of the connectivity results to the suppress of the out of plane failure. Now the structure deforms more uniformly and the masonry walls fail only in plane. Shear failure is observed at front and back façade in the middle and right pier. Toe crushing is shown at the left side of the right pier of the front façade. Shear failure is also observed at the second floor starting at the corners of the windows and ending at the closest opening. At the connection of the front façade to the intermediate wall cracking is also noted. Now high tensile strains are developed at the shear walls mainly, while on the left wall cracking is observed along the edges showing the presence of the flange effect. Figure 78: Displacements and principal tensile strains at collapse stage. - Case 3 (x) The behaviour of the structure is associated to the following scheme. Figure 79: Behaviour of building for fully connected timber floor. (Piazza, Baldessari, & Tomasi, 2008) The critical points of this analysis are presented in the following figure. An increase in the capacity of 27% is observed and a delay in the development of extensive cracking and in the exceedance of the compressive strength. Also more converged steps are noted. Base Shear (KN) Capacity curve - Case 3 (x) 50 45 40 35 30 25 20 15 10 5 0 Step 24: Crack width 7.67 mm Step 27: Interstory drift limit 0.5 % of first floor reached Step 24: Ultimate compressive strength Step 28: Interstory drift limit 1 % of first floor reached Step 10: Ultimate tensile strength 0 10 20 30 40 Displacements at roof level (mm) Figure 80: Capacity curve analysis. - Case 3 (x) 79 FE model – Pushover analysis 3.3.5. Case 3: Fully connected (y) Out of plane failure of the back cavity wall is observed at this analysis. At collapse stage the displacement at roof level is noticed at 0.40 mm and a base shear of 227 KN. The failure in this direction is observed at the points where the timber diaphragms are situated on the facade. The principal strains developed show extensive cracking at the position where the façade is connected to the masonry walls and the displacement developed at the middle is 370 mm. The same failure mechanism and the maximum capacity of the structure is observed for all cases of un-connected to fully connected beams, as the wooden beams are always unconnected longitudinally to the masonry wall. As can be observed from the capacity curve the out of plane failure is noted at the linear phase. The structure is not passing to the post peak phase and the model results cannot be trusted after the out of plane failure occurs. The displacements and principal strains are shown in the following figure: y Figure 81: Displacements and principal tensile strains at collapse stage. - Case 3 (y) This failure is characterized by two main features including extensive cracking in the connection of the masonry members and out of plane failure at the middle. The capacity curve for this case is shown in the following figure: 250 Base Shear (KN) 200 150 100 50 0 0.0 0.5 1.0 1.5 2.0 Displacement at roof level (mm) Figure 82: Capacity curve of Case 3-y until out of plane failure occurs. When looking back to the modelling assumptions, the connection between the plank and the end beam of the roof is considered tied only in the z direction. In reality the nails between the wooden roof plank and the end beams will provide some restrain in the y direction resulting to a smoother failure mode and not a complete detachment of the wall at the top. This modelling assumption is chosen to consider the worst case scenario where the connection is not adequate, therefore the capacity of 227 KN observed is a low boundary of the expected capacity. In any case though out of plane failure will occur at the middle of the wall. This analysis also helps to identify the week points of the structure and is used as a basis for the development of the strengthening strategy. 80 FE model – Pushover analysis 3.3.6. Case 2: Semi-connected After underlying the importance of the connections in the structural behaviour of the case study, it is considered interesting to study the effect of the connections stiffness to the global behaviour of the structure. The normal stiffness is altered in each case, while the shear stiffness is considered constant and a value of 1000 is given to account for a stiff connection. The normal stiffness is considered a variable. The values assigned are not correlated to the as-built connections stiffness but are used as indicative values to study the influence. As it can be seen as the stiffness of the connection decreases the overall base shear of the structure drops. Also the deformed shape of the structure is more realistic. The position of the cracks is almost the same to the previous models and out of plane failure is not present. The capacity curves as generated from the different models are summarized in the following figure. 60 Base Shear (KN) 40 20 Normal stiffness 0.01 N/mm3 Normal stiffness 0.1 N/mm3 Case 3 Case 1 0 0 5 10 15 20 25 Displacement at roof level (mm) Figure 83: Capacity curves per shear stiffness of connection. The way the interfaces are defined is shown in the following figure. In the as built configuration the timber beams are supported half way to the masonry wall. In the model developed the beams are designed at a distance from the wall and the normal direction is defined to match the new set up. As mentioned before no loads are assigned at the timber beams therefore the distance of the timber beam from the wall creates no extra bending moment. The normal stiffness assigned is related to the friction of the wooden beam on the supported area. The local axis are defined to match the global system. Normal direction Normal stiffness: 0.1 - 0.01 Shear stiffness: 1000 Figure 84: As built configuration and modelling set up of interface. The introduction of interfaces can allow to capture the behaviour of the diaphragm more realistically than before. Now the left masonry wall deforms according to the following theoretical scheme. 81 FE model – Pushover analysis Figure 85: Building behaviour for flexible diaphragm. (Piazza, Baldessari, & Tomasi, 2008) Specifically the masonry wall on the left side is deformed following a curved shape and the corners show a deformation heading outwards from the building. From the displacements it can be noted that now the out of plane failure is delayed and it is observed after the walls fail in plane. The results are illustrated in the following figure. Figure 86: Displacements and principal tensile strains at collapse stage. - Normal stiffness 0.01 N/mm 3 A closer look at the interface can indicate the result of the stiffness in the connection. Now the displacement of the beam is more regulated in comparison to case 1. The deformation of the gamble for the three cases under consideration is shown in the following figure: Figure 87: Displacements of left wall for unconnected, semi-connected and fully connected beams. In reality relative displacements will be observed at both ends of the wooden beams. The introduction of interfaces at both ends is therefore considered to better describe the actual behaviour. The new system shows a reduced initial stiffness. This reduced stiffness can play a significant role in the assessment process as it influences the definition of the bilinear configuration. As a result the definition of the ductility factor, the behaviour factor and the target displacement will be influenced. 50 Base Shear (KN) 40 30 20 10 Normal stiffness 0.1 N/mm3 Normal stiffness 0.1 N/mm3 - both ends 0 0 5 10 15 20 25 30 Displacement at roof level (mm) Figure 88: Capacity curve for assigned stiffness at both ends. 82 FE model – Pushover analysis The stresses at the ridge beam are observed versus the relative displacements in the normal direction. It can be noted that the stress developed in the normal direction increases linearly when the relative displacement increases. This is expressed by the following formula: Where: Normal stiffness ⁄ ; Relative displacement of interface Developed stress in normal direction ; and ⁄ . Interface stress Stx (N/mm2) For the model where stiffness is assigned at both ends the structure is more flexible and the relative displacements are higher at the interface. This results to an increase at the developed stress in relation to the case where stiffness is assigned at one end. Normal stiffness 0.01 N/mm3 Normal stiffness 0.1 N/mm3 Normal stiffness 0.1 N/mm3 - both ends 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -15 -10 -5 0 Interface relative displacement in normal direction Figure 89: Interface stresses Stx of ridge beam. Interface stress Stz (N/mm2) When the stresses are analysed it is noted that no stress in developed. This comes in accordance to the modelling assumptions considered. Specifically as discussed before no loads are applied to the wooden elements but instead the densities of the masonry elements are adjusted assigning fictitious densities. Normal stiffness 0.01 N/mm3 Normal stiffness 0.1 N/mm3 Normal stiffness 0.1 N/mm3 - both ends -15 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -10 -5 0 Relative displacements variation in normal direction Figure 90: Interface stresses Stz of ridge beam. 83 FE model – Pushover analysis 3.3.7. Reduced in-plane stiffness of diaphragms The flexibility of the timber diaphragm as mentioned in the Literature Study can be influenced by the connectivity of the floor to the masonry wall and the in plane stiffness. After connectivity is assured the flexibility is mainly dependent on the in plane stiffness. In the model developed the timber diaphragm is considered elastic and the modulus of elasticity is assigned at 10000 , therefore only the parameter EI is expressed. In reality this modulus of elasticity will be reduced. For consistency a reduced modulus of elasticity of 6000 is assigned at Case 3 and the difference in the overall capacity is evaluated. The value is chosen taking into account the calculated value of the EF model. As can be noted the reduction of the modulus of elasticity causes a negligible effect on the overall capacity. After connectivity of the diaphragm is achieved, a rigid diaphragm will improve the load transfer from masonry wall, through connections to the diaphragm and again on the next masonry wall. Therefore a rigid diaphragm will have a positive effect in the distribution of forces, overall stability and the suppression of the out of plane failure modes. Nevertheless the capacity will be mainly governed by the failure modes of the masonry walls, which are mainly influenced by the way the diaphragm is connected to them and the number of elements participating in failure. The result is illustrated in the following figure. The differences in the displacements are related only to convergence differences. 60 Base Shear (KN) 50 40 30 20 10 E=10000 N/mm2 E=6000 N/mm2 0 0 5 10 15 20 Displacements at roof level (mm) 25 30 Figure 91: Capacity curve for reduced modulus of elasticity. - Case 3 (x) For consistency Case 3 with a reduced modulus of elasticity will be used as the basis model for further reinforcement of the structure. 84 FE model – NLTHA 3.4. Nonlinear time history analysis Time history analysis involves the application of a real signal to the structure and can give information about the actual behaviour under the specific seismic action. The time history performed at the assessment phase is referring to Case 1 which corresponds to the lowest boundary. The main scope of this analysis is to identify the failure mechanisms and the critical points which determine the failure of the structure. Also the convergence quality is reported in the relevant Appendix. The results will be associated to the pushover analysis to check the correspondence between pushover analysis and the NLTHA. 3.4.1. Accelerogram The NPR suggests to strengthen an existing building with a short term goal of a risk level of and a long term goal of individual risk level of . The assessment must be performed in terms of linear or non-linear analysis considering the 67% of the NPR requirement as well as the 100% of the NPR requirement. (NAM, 2015) For the purpose of this assessment it is decided to use only one set of 67% of the NPR requirement for the lowest boundary. (Case 1) The applied signal is presented in the following figure: Acceleration (m/s2) Accelerograms (67% NPR) 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0 5 10 15 Time (s) S1 - x S1 - y S1 - z Failure Figure 92: Set 1 of signals provided by NAM. (67%) Three signals are used in the x,y and z direction. The signal was given with a time step of 0.005 s. This is the time step adopted in this analysis. 85 FE model – NLTHA 3.4.2. Case 1: Non-connected For the assigned signals the program reproduced 847 steps relevant to the first 4.23 seconds of the analysis. The analysis showed divergence at this point and the duration of the analysis is reported at 48 hours. When observing the signal it can be seen that this is the point where the highest accelerations occur. To evaluate whether the interruption is due to structural or computational instability, the most important measures are plotted and the behaviour of the structure is observed at critical points. Firstly, the interstory drifts are calculated and are presented in the following figure. As can be seen the drift limits in both x and y direction reaches a value of 0.5%. Drifts (x) vs time Drifts (y) vs time 0.005 Drift Drift 0.005 0.000 -0.005 0.000 -0.005 0 1 2 3 4 0 1 Time (s) First floor 2 3 4 Time (s) Second floor Roof level First floor Second floor Roof level Figure 93: Interstory drifts versus time in the x (left) and y (right) direction. - Case 1 Following the values of the base shears are observed. The result from the Pushover analysis gave a capacity of the structure of 37 KN. The results show that the capacity is exceeded in the x direction at 2.5s. In the y direction the pushover analysis showed out of plane failure with a capacity of 227 KN. The base shear developed in this analysis showed a maximum of 200 KN indicating that out of plane failure of the facades is likely to have occurred. This is cross checked with the displacements developed in the structure. Base Shear vs time (y) 60 40 20 0 -20 -40 -60 200 Base Shear (KN) Base Shear (KN) Base Shear vs time (x) 100 0 -100 -200 0 1 2 Time (s) 3 4 0 1 2 3 4 Time (s) Figure 94: Base shears versus time in the x (left) and y direction (right). - Case 1 After 2s extensive cracking is observed as can be verified from the following graph. This graph shows the maximum crack widths observed every 50 steps in the time history. After 3.5s the results show poor convergence and are not trusted. The position of the maximum cracks is spotted at the connection of masonry walls. This is also in agreement with the principal strains graph presented in Figure 97. 86 Crack widths (mm) FE model – NLTHA 80 60 40 20 0 0 1 2 3 4 Time (s) Figure 95: Maximum crack widths per 50 steps. - Case 1 The principal tensile and compressive strains developed in the structure throughout the time history are shown in the following graphs. Maximum compressive strain E3 per 50 steps 0.20 0.15 0.10 0.05 0.00 0 1 2 Time (s) 3 4 Comopressive strain E3 Tensile strain E1 Maximum tensile strain E1 per 50 steps 0.00 -0.02 -0.04 -0.06 -0.08 0 1 2 3 4 Time (s) Figure 96: Maximum tensile strains (left) and maximum compressive strains(right) per 50 steps. - Case 1 Out of plane failure of the left wall is not observed in this analysis. The tensile strains at collapse stage showed shear failure of the right pier of the first floor. Extensive cracking is also observed at the connections of the wall elements and out of plane failure at both front and back facades at the levels of the floors. Cracking is shown at the spandrels connecting the left windows of first and second floor. The type of failure observed in a time history analysis is related to the characteristics of the applied signal and this is verified by the analysis results. Specifically a combination of the failure modes shown in the pushover analysis is observed. In addition the presence of the vertical component of the loading influences the failure modes that occur. The displacements and the tensile principal strains at collapse stage are presented in the following figure. Figure 97: Displacements and principal strains at last step of time history. 87 FE model – NLTHA As discussed in the literature study the hysteretic loop can provide an indicative measure for seismic performance. To compare the result of the time history analysis to the static pushover the hysteretic loop is plotted. As can be observed these show good correlation. Base Shear (KN) Force - displacement curve in x 60 40 20 0 -20 -40 -60 NLTH Pushover (+) Pushover (-) -40 -20 0 20 40 Displacement at roof level (mm) Figure 98: Comparison between Pushover and NLTHA. – Case 1 From the analysis results it can be concluded that structural failure is present leading to numerical failure and final divergence of the model. Further research is proposed for this analysis with the use of different convergence norms and iteration methods. 88 EF modelling 4. EF modelling The EF model is developed to verify the results and to check whether this method can be applicable for URM buildings with cavity walls. The emphasis is given on the definition of the failure mechanisms, the total base shear developed and the target displacements defined. 4.1. EF model parameters The input required by the program is limited and refers to material properties, loads, geometry definition, the elastic spectrum and some control parameters. Materials The material inserted are presented in the following table: Table 24: Material properties in the EF model. Symbol Units Value Masonry properties Modulus of elasticity 4000 Shear modulus 2000 Density 1920 Mean compressive resistance 6 Shear Strength 0.29 Wood properties Modulus of elasticity 10000 Loads The applied loads are presented in the following table. Table 25: Applied loads in EF model. Parameter Symbol Unit Value Floors dead load 0.3 Floors variable load 1.75 In the second floor the roof loads are applied on the corresponding masonry walls. Spectrum The parameters of the horizontal elastic response spectrum are inserted according to NPR and are presented in the following table. (Ontw. NPR 9998, February 2015) The spectrum is used by the program for the definition of the target displacement. 89 EF modelling Table 26: Horizontal elastic response spectrum. Parameter Symbol Units Value Peak ground acceleration 4.20 Soil factor - 1.00 Period 0.10 Period 0.22 Period 0.45 Importance factor - 1.20 Control parameters For the ductility control the drift limits for masonry walls corresponding to limit state Near Collapse are taken equal to: (EN 1998-3 , 2005) Shear: Normal force and bending: ⁄ ⁄ Geometry The structure is defined as presented in the following figure. The structural elements are: (1) Masonry wall 20 cm on the right side, (2) Masonry walls of 10 cm for the cavity walls and the intermediate wall, (3) Timber diaphragms and (4) Concrete base. As can be noted there are some differences with the initial plans of the structure. This is decided since the program could not identify successfully the spandrels and piers and therefore no results could be generated. This was noticed in case of small windows or windows attached to doors Also the roof is excluded in this model as this can only be assigned as rigid. The geometry built-up is illustrated in the following figure: Figure 99: Geometry definition of unit in EF model. 90 EF modelling Diaphragms are defined as one-way timber floor with single wood plank as presented below. Where: Figure 100: Wooden floors definition in EF model. To account for the flexibility of the diaphragm the program assigns a reduced modulus of elasticity. In this analysis the given modulus is and the computed parameter of the modulus of elasticity is with an equivalent thickness of . After the model is defined the discretized model is generated with the definition of the piers and spandrels. Figure 101: Discretization in EF model. 91 EF model – Pushover analysis 4.2. EF model results The analysis in the EF model is based on a frame analysis and the failure is related to the exceedance of capacities and the relevant drift limit. For this model a displacement is applied as load. The load per step is considered important as it defines which element will fail. Another parameter that is consider critical is the selection of the control node. For this reason three trials are made, where the control node and the load per step are altered to check the differences. The load per step is defined by the model as the ratio of the total displacement to the number of substeps. Table 27: Computational parameters in EF model. Units Displacement Trial 1 Trial 2 Trial 3 9 20 20 Substeps - 200 200 200 Iterations - 400 400 400 Control node - N3 N3 N12 The capacity curves as resulted from the analysis are presented in the following figures. In the x direction no significant difference is observed in terms of capacity and this is assessed at 40 KN. The difference observed is in terms of failure mechanism. 50 Base Shear (KN) 40 30 20 Trial 1 Trial 2 Trial 3 10 0 0 10 20 Displacement at second floor level (mm) 30 Figure 102: Capacity curve of EF model in the x direction. In the x direction the failure is associated to the exceedance of the drift limits set. The failure progression of the elements is presented in the following figure. As it can be observed, firstly shear failure of the middle pier of the second floor is assessed, following by the right pier. This comes in agreement with the drift limits calculation presented in Table 23, where also shear drift of the middle pier of the second floor is calculated as the most unfavourable. Failure is finally associated to the bending failure of all piers of the back façade. Figure 103: Progression of failure in front and back facade. 92 EF model – Pushover Analysis A difference is noted for Trial 2 and 3 where the shear failure of the middle pier is followed by the bending failure of the piers of the back facades. These differences are expected where the load per step is increased and can show the sensitivity of the model especially when a small number of elements are present in the structure. Better results are expected for bigger structures where more elements control the failure. Also in these buildings the control node can be selected in the middle, leading to a uniform distribution of load. Shear force (KN) To understand the failure of a single element the forces are shown versus the horizontal displacements. As can be noted these follow the theoretical diagram presented in the Literature Study. Also bending moments are exceeding first the bending moment capacities calculated in Table 29. 16 14 12 10 8 6 4 2 0 0 5 10 15 20 25 20 60 Normal forces (KNm) Bending moments (KNm) Displacements at top node (mm) 15 10 5 50 40 30 20 10 0 0 0 5 10 15 20 25 0 Displacements at top node (mm) 5 10 15 20 25 Displacements at top node (mm) Figure 104: Internal forces of pier 19. The failure of the pier is related to the exceedance of the bending drift limit. The displacements and rotations at the element are presented in the following table and the drifts are calculated. The generated results refer to one step before failure. As can be seen the drift limit for rocking set at 1,07% is almost reached. At step 53 the pier is assigned at rocking failure. Table 28: Exceedance of bending drift for pier 19. Step 52 (mm) 0 (mm) 20.9 (rad) (rad) 0 0 H (mm) Element Drift (%) 2150 As can be noted the formula calculating the drift in the EF model does not involve the width of the element as defined by Eurocode. The calculation of the capacities are shown in the following table: 93 EF model – Pushover analysis Table 29: Capacities of Pier 19 according to EF model formulas. Symbol Calculation Pier characteristics Length Thickness Axial load Compressive strength Shear resistance Friction coefficient 0.75 Cohesion of mortar 0.3 Stress distribution factor 1 Bending capacity Bending capacity Shear failure √ Fracture of brick √ = Fracture of mortar For pier 11 the failure is associated to the exceedance of the shear drift limit. The drift limits at the step before and at failure are shown in the following table together with the calculated drifts. Table 30: Exceedance of shear drift for pier 11. Step 94 (mm) (mm) (rad) (rad) H (mm) Element Drift (%) 48 12.5 21.2 0.0001 0.0002 1700 0.00527 49 12.8 21.6 0.0001 0.0002 1700 0.00533 EF model – Pushover Analysis Analysis in y direction Base Shear (KN) In the direction perpendicular to the facades the failure is associated with the drop of the base shear at a value lower than 80% and is assessed at 280 KN. Also it can be noted that the maximum displacement is assessed at 8mm. When looking at the failure mechanism it can be seen that only the pier of the first floor of the left façade is participating in the failure. Therefore the capacity is related to the capacity of only one element. This issue is related to the way the flexible diaphragms is defined. As discussed in the literature study the diaphragms are considered as 4-noded membrane elements. In reality the flexible diaphragm will show a maximum displacement at the middle. In the model there is no present node in the middle resulting to a maximum displacement at one corner and the failure of the wall of that side. This is also illustrated in the deformation of the building in plan. This assessment is considered underestimating the capacity in the y direction, as in reality all elements will participate resulting to significantly higher capacity. This problem is expected to be overcome when the diaphragm is assigned as rigid. 300 250 200 150 100 50 0 0 2 4 6 8 Displacement at second floor level (mm) Figure 105: Capacity curve of EF model in the y direction. Table 31: Failure mechanisms of EF model in y direction. The special characteristics of the case study and the fact that the software is not yet widely applied in the Netherlands caused different difficulties throughout the modelling process. In case of flexible diaphragms it is recommended that the roof is excluded from the analysis and the loads are applied at a two storey building. This is recommended as the roof can be defined as rigid therefore the results are not considered reliable. Another point that needs attention is the control node. This needs to be defined at the point where maximum deformation is expected. The software seems to work better when a number of elements are present in each direction. In the case study the y direction is defined by only one element therefore the result is considered conservative. In the assessment process the critical parameters defining the target displacement need to be critically checked as in some cases the participation factors are noted unrealistic. Also the periods resulting from the eigenvalue analysis need to be checked. Considering the low computational time needed to run an analysis and considering that the designer has knowledge of the modelling process followed by the software is considered a promising modelling tool for assessment of URM. The tool is under development therefore some of the difficulties pointed out before are expected to be overcome. 95 EF model – Pushover analysis 96 Assessment 5. Assessment 5.1. Building capacity The capacity of the structure is assessed with the two modelling approaches and calculated with analytical formulas. As a first step the models are compared. Following the calculated capacity according to the Pier-only method presented by NZSEE is presented and finally the calculated capacity is compared to the models outcome. 5.1.1. Comparison of models To compare the two models in terms of seismic behaviour it is considered important to initially evaluate basic characteristics. To that end firstly the weight and the results of the eigenvalue analysis are presented for the two approaches. Following the capacities, ultimate displacements and failure mechanisms are compared. Weight No significant difference is observed in terms of weight. In the FE model the dynamic mass is different than the actual mass, as the dynamic mass includes the assignment of the external leaf as a distributed mass. These values are shown in the following table: Table 32: Mass and dynamic mass of models. Value Units DIANA Tremuri Mass kg 49.94 50.38 Dynamic mass kg 66.40 - Eigenvalue analysis Differences are observed in the eigenvalue analysis. In the EF model the first modes show a high participation of mass. In the FE model for Case 1 where no connectivity is assigned the modes are localized and a percentage of 60% mass participation is reached after a number of modes. For Case 3 a high participation is observed from the first modes. The eigenvalue analysis plays no significant role in the pushover analysis but can show how the two models behave under a free vibration. The EF model shows good correlation with Case 3 as both models assume full connectivity. Table 33: Periods and mass participation of models. Model Analysis Mode Tx Mx % Mode Ty My % FE Model Case 1 6 0.149 66.46 36 0.049 60.29 Case 2 6 0.137 62.50 37 0.049 60.38 Case 3 1 0.181 63.51 27 0.049 60.00 - 1 0.189 52.5 5 0.051 62.36 EF Model 97 Assessment Pushover analysis The pushover curves resulted from the two models are shown in this paragraph. To compare the models displacements are plotted at the second floor level also for the FE model. As can be seen the models show good agreement in terms of base shear in the x direction. The EF model can be compared to Case 3 with a reduced modulus of elasticity of the timber floors, as both approaches consider connectivity between the elements. Base Shear (KN) 60 50 40 30 20 Case 1 Case 3 EF model 10 0 0 5 10 15 20 25 Displacement at second floor level (mm) Figure 106: Comparison of capacity curves between FE and EF model. For the EF model a lower linear phase is shown. This is related to the absence of the tensile strength in the model, which plays an important role in the linear phase. The initial stiffness of the EF model is given by the elastic (cracked) properties, defined with the use of a stiffness reduction factor. In the y direction no comparison is shown between the models as in the FE model out of plane failure of the front and back façade is observed, while in the EF model the capacity assessed is related to only one element and is considered underestimated. Failure In the EF approach failure is related to the drift limit set and the capacity of the element. In FE approach failure is captured as a process related to the crack formation, propagation and final collapse of the structure. In the EF model both bending and shear failure are observed in the x direction. Bending of the piers of the back façade are considered critical to govern the failure. In the FE model the failure is related to the piers of the back façade, where now shear failure is predominant. Figure 107: Relation of failure modes of FE and EF model at back façade. (x) 98 Assessment In the y direction the EF model can be compared to the FE model after connectivity is assured along the wooden beams. This is presented as a strengthening method as these connections are not present in the current geometry. The FE model shows a more detailed failure mechanism, where rocking is observed at left and right wall, with the characteristic cracking on the base longitudinally. Also diagonal cracking is shown at the same wall indicating that shear failure can be present at a later stage. In addition shear failure of the right pier of the front façade and out of plane failure of the left side of the first floor is noted. For the EF model, the failure is related to shear failure of the left wall. As discussed before, the assessment in y direction is doubted as it is related to the failure of one element. Figure 108: Relation of failure modes of FE and EF model. (y) 5.1.2. Capacity from codified equations In order to verify the results the total base shear of the unit is calculated based on the NSZEE formulas. To analyse the in-plane loaded URM walls and perforated walls the “pier only” model is used. In the calculation the superimposed load due to flange effect is taken into account. In this calculation rocking capacity is considered the critical failure mechanism and the calculated base shear at x direction is defined at: ∑ When no flange effect is taken into account the rocking capacity is calculated 10449 N. The consideration of the flange effect gives an increase almost 300% to the capacity. This is related to the typology of the building under consideration. Specifically there is no load transfer from the diaphragms to the facades therefore the superimposed load when no flange effect is considered is relatively low. In any case when walls are considered interlocked the flange effect needs to be taken into account when these analytical formulas are applied. The detailed calculation is shown in the relevant Appendix. 99 Assessment 5.1.3. Comparison of capacities In this paragraph the results of the EF models, the FE models and the analytical approach are summarized. As can be observed the analytical formulas give a good estimation of the expected capacity when the flange effect is taken into account. From the EF analysis rocking is the critical failure mechanism and the base shear is defined at 40 KN. As can be concluded the analytical formulas and the EF model results show correlation in terms of failure mechanisms. This is due to the fact that both approaches are based on an equivalent frame analysis although the exact formulas differ. From the FE model a base shear of 47 KN is shown when connectivity is assured. (Case 3) Here the governing failure mechanism is shear failure. As discussed in the Literature study the presence of the flange effect can alter the failure mode from rocking to shear and this is observed in the results. The results are summarized in the following table: Table 34: Maximum base shear and critical failure mode in x direction. Approach Base Shear (KN) Critical failure modes NZSEE Rocking EF model Rocking FE model- Case 1 Out of plane of gamble FE model - Case 3 Shear Table 35: Maximum base shear and critical failure mode in y direction. Model 100 Base Shear (KN) Critical failure modes EF model Shear failure of left wall FE model Out of plane failure of front and back facade Assessment 5.2. Target displacement The definition of the target displacement involves an accurate definition of the capacity curve in terms of displacements. This has an influence on the bilinear configuration determined and the characteristic of the equivalent single degree of freedom system. The modelling strategy followed is based on forces following a Force control approach and therefore these values cannot be estimated with precision. The scope here is to point out the procedure followed by Eurocode and give an estimation of the expected values. Also a comparison between the values presented by the EF model is shown. To define the target displacement of the FE model the model considering a stiffness at both ends is used. To make a comparison of the results between the FE model and the EF model the assessment is performed till the exceedance of the first drift limit. Table 36: Ultimate & target displacement in the x direction. (100% NPR) Model Case Ultimate displacement ( ) u.c. Target displacement ( ) EF model - 0.0247 0.0418 ⁄ FE Model Case 2 - Stiffness at both ends 0.030 0.038 ⁄ The results give an indication that the structure cannot perform seismically and that reinforcement is necessary. The reader is referred to Appendix C for the complete calculation. 5.3. Ductility and behaviour factor The ductility and behaviour factor define the ability of the structure to undergo deformations after the yield point. The definition of the yield point requires the definition of the bilinear configuration of the equivalent single degree of freedom system. (SDOF) The reader is referred to Appendix C for this calculation. The definition of these factors involves a displacements control analysis and is not considered under the framework of the current analysis. Nevertheless the values presented by the EF model and calculated by the FE model can give an indication that larger ductility factors can occur than the proposed value of proposed by the NPR. This is expected as the code gives a low boundary of the expected values. Table 37: Calculated ductility and behaviour factors. Model Case Ductility μ Behaviour factor q EF Model - √ FE Model Case 2 - Stiffness at both ends √ 101 Assessment 5.4. Base shear check The definition of the behaviour factor gives the possibility to perform the unity check also in terms of capacity. The calculation is summarized in the following table. In Eurocode the check is prescribed only in terms of displacements with the calculation of the target displacement. This check is only shown for comparative reasons. Also the difference in the unity checks for different acceptability of risk is highlighted. To comply with the provisions of the current NPR the results are shown for 100% of the PGA prescribed in NPR and for the 67 %. Table 38: Unity check of Base Shears. – Case 2 (stiffness at both ends) Symbol Units 100 % NPR 67% NPR - Behaviour factor Period of structure Elastic spectral acceleration Inelastic spectral acceleration ⁄ Mass Demanded Base Shear Resisted Base Shear Unity check 102 - - ⁄ Retrofitting 6. Retrofitting The assessment of the case study pointed out the main deficiencies of the structure and the failure modes that are likely to occur. This analysis will be used as the basis for the retrofitting method, which will be based on two main directions: (1) Improve the capacity of the existing building by improving the existing elements; (2) Increase the capacity with the use of additional elements. In the literature study an overview is presented of different reinforcement methods applicable to masonry structures. The methods that will be investigated have as a main scope to reduce the flexibility of the floors and to improve the in plane behaviour of the walls. To that end the following methods are checked: Improvement of existing connections, where the connections between wooden beams and walls are assured; Addition of connections, where connections along the wooden beams and the facades are added; Stiffening of floors, with the use of extra planks; Steel frames, to increase the in-plane capacity of the masonry walls. Improvement of the in plane stiffness of the roof will not be part of this analysis. The model used as basis for the investigation of the different strengthening options is Case 3 with a reduced modulus of elasticity set at . 6.1. Seismic demand Ground Acceleration S(T) (m/s2) The evaluation of the different methods involves the definition of the seismic demand of the structure. According to NPR a behaviour factor of 1.5 is proposed multiplied by a factor of 1.3. In the previous section it is shown that the ductility factors in practice can show higher values. Although the analysis is based on a Force controlled strategy and therefore it can be said that displacements are not trusted, it gives an indication that the behaviour factor can be higher. The accelerations that results from the design spectrums for a behaviour factor of 2 and 3 are presented in the following figure. This is an advantage of the nonlinear methods as the behaviour factors can be assessed. 15 Design spectrum for q=2 Structure period Design spectrum for q=3 Elastic spectrum 10 5 0 0 1 2 3 4 Period (s) Figure 109: Definition of the seismic demand. The reader is referred to the relevant Appendix for the definition of the elastic spectrum. The design spectrum is defined by its division with the behaviour factor under consideration. 103 Retrofitting 6.2. Improvement of existing connections The importance of the wooden beams end connections in the overall capacity of the building is already highlighted in Section 3. There an increase in the capacity of almost 27% is shown. Therefore the first retrofitting method proposed is the check and improvement of these connections. In this way out of plane failure will be suppressed resulting to only in plane failure mechanisms which can be easier controlled. Considering that full connectivity will be reached Case 3 with a reduced in-plane stiffness of the floors can be taken into account for the further retrofitting. The difference in the development of the strains is captured in the following graph. Figure 110: Tensile strains before and after connectivity is assured. A typical configuration of this solution is illustrated in the following figure: Figure 111: Connectivity of wooden beams. (ARUP, 2013) 104 Retrofitting 6.3. Addition of connections In the assessment of the structure it is pointed out that there is no connectivity along the wooden beams to the masonry walls. This resulted to out of plane failure of the front and back wall when the seismic load is applied in the direction perpendicular to the facades. Also in all cases participation of both floors is noted in the failure mechanisms of the masonry. In this section the influence of the improvement of this connection in the global capacity is investigated. To model this situation links are created between facades and beams. Now all translations are considered tied and rotations free. Figure 112: As built connectivity longitudinally to the wooden beams and modelling with links. The roof planks in the initial model are connected to the end beams only in terms of vertical translation. Now both roof and floor planks are connected to the masonry walls at the same point, with the use of two links. Figure 113: Connection of roof and floor before and after reinforcement method. The global capacity of the structure shows an increase of 50%. The capacity curves are shown in the following figure: Pushover curves (x) Base Shear [KN] 80 60 40 20 Longitudinally connected Case 3 - E=6000 N/mm2 0 0 5 10 15 20 Displacements [mm] 25 30 Figure 114: Capacity curves of Case 3 (x) and connectivity along beams. The box-type behaviour is now present as can be seen from the deformed shape. The failure is sudden and is related to shear failure of the right pier of the first floor. The element that fails is dependent on the ratio height to width of the piers. The failure is observed at a displacement of 17 mm, while the capacity is increased to 75 KN. As can be noted the addition of connection results to higher capacity for the 105 Retrofitting structure and lower ductility. Now the first floor is mainly participating in the failure mode. The failure starts with the failure of the right pier, then the failure of the middle pier and finally the left pier. This measure results to a more controlled behaviour of the structure. Figure 115: Displacements and tensile strains at collapse stage. – Connection longitudinally (x) This solution can be supported with the use of perimetric L-shape beams as illustrated in Figure 112. Other options are also mentioned in the Literature Study. The analysis in the y direction showed out of plane failure of the back façade. The connectivity of the diaphragm along the wooden beams can protect from the out of plane failure in this direction and result to a significant increase in the global capacity. The capacity curve is illustrated in the following figure. This shows an increase of 120 %. Pushover curve (y) Base Shear [KN] 600 500 400 300 200 Longitudinally connected 100 Case 3 - E=6000 N/mm2 0 0 5 10 15 Displacements at roof level [mm] 20 Figure 116: Capacity curves for Case 3(y) and addition of connection. At collapse stage shear failure of the right pier is observed, with the characteristic diagonal cracking. At the left, intermediate and right wall, longitudinal cracking is observed at the base, indicating bending failure. Also the front façade showed out of plane failure at the position of the windows of the first floor. Extensive cracking is also observed at the position of the connections added at the level of the floors. Figure 117: Displacements and tensile strains at collapse stage. Connection longitudinally (y) 106 Retrofitting 6.4. Improved in plane stiffness of floors The flexibility of the diaphragm can be further improved by increasing the in plane stiffness. The influence of the in plane stiffness in the overall capacity is investigated in this section. The model with a reduced modulus of elasticity for timber is used as basis for the analysis. The only parameter changed is the thickness of the timber planks, considering the use of wooden boards on top of the existing planks as presented in the Literature Study. This solution can be supported as presented in the following figure: Figure 118: In plane stiffness of floors. (Brignola, Podesta, & Pampanin, 2008) In the following figure the results for an extra plank of 40 and 80 mm are shown. For comparative reasons the results from the previous section are also presented. It can be observed that both measures can give an increase in the overall capacity. 80 70 Base Shear [KN] 60 50 40 30 Longitudinally connected Case 3 - Extra plank 80 mm Case 3 - Extra plank 40 mm Case 3 - E=6000 N/mm2 20 10 0 0 5 10 15 20 25 30 Displacements [mm] Figure 119: Capacity curves for improved in plane stiffness. In comparison to Case 3 no difference is observed in the failure mechanism and the way the building deforms. The effect of adding wooden boards on top showed an effect which can be achieved with only connectivity along the beams. More effective ways would involve the use of FRP or steel plates as presented in the Literature Study. 107 Retrofitting 6.5. Strengthening of walls with steel frames 6.5.1. Pushover analysis In plane strengthening of existing walls can be achieved with different ways as discussed in the Literature Study. In this section the influence of the use of steel frames on the behaviour of the structure is checked. For this purpose three possible configurations are analysed and the main differences in the behaviour of the new systems are discussed. The role of the steel frames is related to the increase in the total capacity of the system combined with limitation of the developed drifts. The main interest here is to observe the interaction of the two materials. The configurations of steel frames that are examined are presented in the following figure. Configuration 1 Configuration 2 Configuration 3 Figure 120: Steel configurations examined. The resulting capacity curves are shown in the following figure. The system which is used as base model in these analysis is after longitudinal connection is added presented in Section 6.3. 350 300 Base Shear [KN] 250 200 150 100 Configuration 1 50 Configuration 3 Configuration 2 Longitudinally connected 0 0 10 20 30 40 50 60 Displacements at roof level [mm] Figure 121: Capacity curves for strengthening with steel frames. . 108 Retrofitting Configuration 1 This configuration showed a total capacity of 300 KN. The failure is related to shear failure of the elements of the first floor and a participation of all piers is noted, showing that the capacity of the floor is completely exhausted. Cracking starts from the middle pier, following by the right pier and finally by the left pier. On the second floor cracking is noted at the corners of the openings and diagonal cracking towards the closest corners. Out of plane failure is observed at the left cavity wall at the position of the gamble. This is an indication that also this part will need to be strengthened. One possible solution for this part can be the connection of inner and outer leaf. The displacements and principal tensile strains developed for the masonry are shown in the following figures. Figure 122: Displacements and tensile strains at collapse stage. – Configuration 1 The observation of the principal strains for the steel elements shows that the material is in the elastic branch at failure of the masonry. This indicates that the capacity of the frame is not yet exhausted. The stress-strain relationship of the steel elements compared to the theoretical diagram assigned are shown in the following diagram. Stresses Sxx (N/mm) 250 Developed stress-strain at steel elements 200 150 100 Theoretical diagram 50 0 0.00 0.01 0.02 Strains Exx Figure 123: Stress-strains diagram for steel elements. – Configuration 1 The moments developed in the steel frame at collapse stage are shown in the following figure. Figure 124: Developed moments in steel frame at collapse stage of masonry. To understand the relation of the developed moments in comparison to the capacities of the profiles used, the elastic and plastic moments are calculated for the profile where the maximum moments are 109 Retrofitting developed. The plastic moment can be considered theoretically the maximum moment that the section can resist and is related to the formation of a plastic hinge. Loading beyond this point will result to infinite plastic deformation. In reality the material will have some hardening resulting to even higher moment resistance till it fails. Design according to Eurocode is restricted for cross-section Class 3 to the elastic moment resistance and this can be considered for design. For comparative reasons both moments are shown. Table 39: Design elastic and plastic moments calculation. – Configuration 1 Profile Symbol Units Value - - IPE400 Yield strength Elastic section modulus - Partial factor Design elastic moment Plastic section modulus Design plastic moment Therefore it can be verified that the steel sections are in the elastic range and the unity check is satisfied: What is considered interesting at this point is to observe the difference in the behaviour of the masonry due to the presence of the steel elements. As can be seen the two materials work in parallel and the degradation of the masonry is delayed due to the presence of steel. In the following diagram the behaviour of the structure is shown and the critical points are illustrated. 350 300 Base Shear [KN] 250 Interstory drift limit 1 % of 2nd floor Inerstory drift 0.5 % of 1st floor Cracks 5mm 200 Exceedance of compressive strength 150 100 Configuration 1 50 Longitudinal connected Exceedance of tensile strength 0 0 10 20 30 40 50 60 Displacements at roof level [mm] Figure 125: Critical steps of the masonry behaviour. - Configuration 1 After drift limits are exceeded for both floors, the first plastic hinges are observed in the steel structure and this is where failure of the system is considered. 110 Retrofitting The critical values at the last step of the analysis and corresponding behaviour factor are shown in the following table: Table 40: Critical values at collapse stage. – Configuration 1 Units Drift limit of first floor - Drift limit of second floor - Crack widths Value mm The analysis of the developed base shears in the two materials is reported in the following figure. As can be seen initially masonry gives the highest stiffness to the system. After the capacity of masonry is exhausted the steel structure continues raising the capacity of the system. 300 250 Base Shear [KN] 200 150 100 Configuration 1 Masonry Steel 50 0 0 20 40 Displacements at roof level [mm] 60 Figure 126: Capacity curves for steel and masonry. – Configuration 1 For the new system three main phases are identified: (1) Masonry contribution; (2) Steel and masonry contribution; (3) Plateau. It is considered interesting to observe the capacity curve of the masonry before reinforcement is applied and that after the steel frames are added. Here it can be noted that an extra capacity is added to the masonry walls when the steel frames are introduced. The presence of the steel frames will result to a more controlled deformation of the masonry in the horizontal direction resulting to higher capacity. Also in the vertical direction the deformation of the masonry will be reduced. To give a more complete justification further research needs to be carried out. 120 Base Shear [KN] 100 80 60 40 20 Masonry - No reinforcement Masonry - Configuration 1 0 0 5 10 15 20 Displacements at roof level [mm] 25 Figure 127: Capacity curve of masonry with and without steel. – Configuration 1 111 Retrofitting To assess whether this configuration is adequate to resist the seismic loading, the target displacement is defined for the new system. The results before and after reinforcement are summarized in the following table: Table 41: Target displacement before and after reinforcement. – Configuration 1 (100% NPR) Model Ultimate displacement Target displacement Case 2 - Stiffness at both ends 30 38 Configuration 1 31 24 Unity check According to the calculation of the target displacement the new system is capable of resisting the seismic demand. The behaviour factor and the ductility are also calculated for this system. A decrease is now observed in the ductility of the system in comparison to the case without any intervention. Table 42: Ductility and behaviour factors before and after reinforcement. – Configuration 1 Model Ductility μ Behaviour factor q Case 2 - Stiffness at both ends √ Configuration 1 √ The new system will reach a higher capacity but the ductility will be decreased. This is related to the bilinear configuration. Now the term is higher due to the presence of steel. When the check is performed in terms of capacities it can be seen that for 100% of the NPR requirement (associated to a probability of exceedance of ) the unity check is not satisfied. When a higher probability of exceedance ( ) is accepted the unity check is satisfied. This is the geometry that will be further checked with a time history analysis, to observe the behaviour of the structure under a real seismic loading. Table 43: Unity check of Base Shears. – Configuration 1 Symbol Units 100 % NPR - Behaviour factor Period of structure Elastic spectral acceleration Inelastic spectral acceleration ⁄ Mass Demanded Base Shear Resisted Base Shear Unity check 112 - - 67 % NPR Retrofitting Configuration 2 In this configuration the beams have an equal profile size of IPE300. From this analysis it can be seen that the adaptation of the same profiles can have a positive effect on the failure mechanism of the system as it involves the participation of the elements of both first and second floor. Figure 128: Displacements and tensile strains at collapse stage. – Configuration 2 Here the material strength is exhausted and the system shows a decreased capacity. Stresses Sxx 250 200 Developed stress-strain at steel elements Theoretical diagram 150 100 50 0 0.00 0.01 0.02 Strains Exx Figure 129: Stress-strains diagram for steel elements. – Configuration 2 The relation of the developed moments to the design elastic moments are presented below. As can be seen now the ratio is higher than in configuration 1, indicating that the design is optimized. Table 44: Unity check for steel profiles at last step. – Configuration 2 (100% NPR) Profile Symbol Units Value - - IPE300 Yield strength Elastic section modulus Partial factor - Design elastic moment Maximum developed moment Unity check - - 113 Retrofitting No significant difference is observed in the behaviour of masonry as can be seen in the following graph. What can be noted is that Configuration 2 has lower stiffness and lower ductility. Also drift limits are exceeded for the first floor earlier for this system in comparison to Configuration 1. 350 300 Base Shear [KN] 250 200 150 100 Cracks 5mm 50 Interstorey drift limit 0.5 % of 2nd floor Interstorey drift limit 0.5 % of 1st floor Configuration 1 Configuration 2 0 0 10 20 30 40 50 Displacements at roof level [mm] 60 Figure 130: Differences in the behaviour of Configuration 1 and 2. The calculation of the target displacement shows that this configuration is not adequate to resist the seismic demand. This is summarized in the following figure. Table 45: Target displacement before and after reinforcement. – Configuration 2 Model Ultimate displacement Target displacement Unity check Case 2 -Stiffness at both ends 30 38 ⁄ Configuration 2 19 24 ⁄ The calculation of the behaviour factor is shown below. As can be seen this Configuration has a lower initial stiffness resulting to lower behaviour factor. This in terms of seismic demand means that more load will be demanded by the structure. Table 46: Ductility and behaviour factor. - Configuration 2 Model Ductility μ Behaviour factor q Case 2 - Stiffness at both ends √ Configuration 2 √ The check of the base shear shows that the capacity is not sufficient for both requirements. 114 Retrofitting Table 47: Unity check of Base Shears. – Configuration 2 (100% NPR) Symbol Behaviour factor Units 100 % NPR 67 % NPR - Period of structure Elastic spectral acceleration Inelastic spectral acceleration ⁄ Mass Demanded Base Shear Resisted Base Shear Unity check - - The critical values at collapse stage of the structure are shown in the following table: Table 48: Critical values at collapse stage. – Configuration 2 Units Drift limit of first floor Drift limit of second floor Crack widths Value mm 115 Retrofitting Configuration 3 In this configuration the focus is on limiting the drifts at element level. The analysis results show that the capacity reached is equivalent to Configuration 1 but now crack widths are smaller at collapse stage and shear drift limits are exceeded at the last step. Cracking is observed at the elements of both the first and second floor and the left wall, indicating that this configuration takes advantage of the existing capacity of the structure in a better manner. Out of plane failure of left and intermediate wall is noted at collapse stage. These are the walls with an assigned thickness of 10cm. Strengthening of these two walls is recommended. The strengthening of these walls will also increase the capacity of the system. The resultant displacements and principal tensile strains at collapse stage are shown below. As mentioned before the stiffness of this system is determined by the foundation. The consideration of fixed foundation will result to advantageous results for the system. Figure 131: Displacements and tensile strains at collapse stage for Configuration 2. Drift limits and crack widths at the collapse stage are shown in the following table: Table 49: Critical values at collapse stage. Units Drift limit of first floor - Drift limit of second floor - Value mm Crack widths This system is capable of resisting the seismic demand and complies to both the target displacement and the capacities check as can be noted in the following tables. Table 50: Target displacement before and after reinforcement. (100% NPR) 116 Model Ultimate displacement Target displacement Unity check Case 2 - Stiffness at both ends 30 38 ⁄ Configuration 3 37 24 ⁄ Retrofitting Table 51: Ductility and behaviour factors before and after reinforcement. Model Ductility μ Behaviour factor q Case 2 - Stiffness at both ends √ Configuration 3 √ Table 52: Unity check of Base Shears. Symbol Units Behaviour factor 100 % NPR 67 % NPR - Period of structure Elastic spectral acceleration Inelastic spectral acceleration ⁄ Mass Demanded Base Shear Resisted Base Shear Unity check - - It is considered interesting to observe the relation between developed crack widths and interstory drift limits for the three configuration. From the following figure it can be noted that an interstory drift of 0.5 % is related to crack widths between 12 -25 mm. This limit can therefore be considered for design. Higher values will result to even more extensive cracking and are not recommended. Crack width [mm] 60 Configuration 1 Configuration 2 Configuration 3 50 40 30 20 10 0 0.000 0.005 0.010 Interstory drift limit at first floor level Figure 132: Crack widths versus drift limits. 117 Retrofitting 6.5.2. Nonlinear time history analysis The reinforced structure is checked with 67% of Set 1 and time steps of 0.005s. The analysis stopped showing divergence. To understand whether this is caused by numerical or structural failure the state of the structure is assessed throughout the time history. As it can be noted the base shears and the drift limits cannot justify structural failure. Specifically drift limits are observed lower than 0.2 %, maximum base shear is 113 KN in the x direction and 253 KN in the y direction. The comparison to the pushover analysis shows that maximum capacity is not reached. 0.005 Drift Drift 0.002 0.000 -0.002 0.000 -0.005 0 1 2 3 4 0 1 Time (s) First floor 2 3 4 Time (s) Second floor Roof level First floor Second floor Roof level Base Shear (KN) Figure 133: Interstory drifts versus time in the x (left) and y (right) direction. – Configuration 1 300 200 100 0 -100 -200 -300 -60 -10 40 Displacement at roof level (mm) Figure 134: Comparison between Pushover and NLTH. – Configuration 1 Crack widths (mm) The crack widths observed show a maximum of 20mm up to 3.3 s. The principal tensile strains show shear failure of the right pier at the front façade. Extensive cracking is also noted at the masonry walls connections and at the position of the timber floor. 100 50 0 0.00 1.00 2.00 3.00 4.00 Time (s) Figure 135: Crack widths and principal tensile strains of masonry at last steps. – Configuration 1 In this analysis the divergence is related to numerical instability as the results showed poor convergence after 3.3 s. Further research is recommended with the use of different convergence criteria and iteration procedures. 118 Conclusions 7. Conclusions The analysis developed in this report focused on assessing the seismic performance of an unreinforced masonry building with timber floors and evaluate the impact of certain strengthening methods on the results. An effort is also made to underline the main parameters of the assessment process. For the assessment two modelling approaches are used a finite element model and an equivalent frame model. Also a comparison is shown between the results from a pushover analysis and a nonlinear time history analysis. The need to develop a number of analysis and follow different modelling approaches resulted to the adaptation of a fixed modelling strategy with the use of the conventional pushover analysis where no sensitivity analysis is carried out. This approach is considered suitable for the needs of this analysis. Specifically, the modelling strategy followed considers 2D elements, uniform application of loading, fixed supports at the foundations, a Total Strain Rotating Crack Model and fixed material parameters. The load increment procedure followed is force control and iterative solution method Regular Newton-Rapson. For the pushover analysis a displacement convergence norm is adopted and for the time history an energy norm. The parameter that is considered a variable in this analysis is the connectivity between timber beams and masonry walls and the influence on the global capacity is assessed. As discussed in the literature study the applicability of a pushover analysis in a structure with flexible diaphragms is unexplored. Also there are no experimental results available to compare the results. Considering these limitations the modelling strategy followed and the generated results are considered satisfactory. Assessment In the assessment phase the main conclusions driven are: The connectivity of the wooden beams to masonry walls influences the global capacity. The capacity envelope is assessed 37-47 KN. The FE models developed capture a range of possible behaviours of the structure. These models are used to understand the behaviour of the case study and the impact of the modelling choices on the change of the behaviour. Failure modes differ depending on the quality of the connections. For the lower boundary where poor connectivity is assigned out of plane failure of the gamble is shown. When connectivity is assured shear failure of elements is observed, separation of the connection of masonry elements and cracking at the corners showing the presence of the flange effect. An interstory drift limit of 0.5% corresponding to shear failure is shown to capture the extensive cracking stage of the structure, although actual failure is expected at higher displacements. The assignment of reduced stiffness in the connections shows a more realistic behaviour as stress is developed in the interface. In this analysis interfaces are inserted in specific connections. No influence is observed in the global behaviour due to a 60% reduction of the elastic modulus of the timber elements. Out of plane failure of the facades in the level of the floors is observed. When the load is applied perpendicularly to the facades out of plane failure at a load of 227 KN is shown in the FE model. This proved that connectivity longitudinally to the wooden beams needs to be assured and is incorporated in the strengthening options. The EF model assessed shear failure of the left wall and a total capacity of 280 KN. This result is doubted as the failure is associated to the failure of only one element and is considered underestimating the global capacity. The structure is assessed inadequate to perform seismically. For the assessment of the structure the N2 method as prescribed by the EC is followed and compared to the EF model. Here a model 119 Conclusions with an assigned stiffness at both ends is used. For the FE model the assessment of target displacements and ductility factors is accepted as an indication as the modelling approach followed focuses on forces following a force control approach. The results can be therefore accepted after the displacements variations are accepted. The capacity assessment of the EF is found in agreement with the FE model. This was assessed at 40 KN and was found in the range of capacities assessed by the FE model. Different failure modes are observed in the FE and the EF model. Specifically, shear failure is governing in the FE model while bending failure is assessed governing in the EF model and the analytical approach. The result from the FE model are trusted as this model takes into account the flange effect. This change in behaviour is also reported in literature. The use of analytical approaches requires the incorporation of the flange effect. An increase in the capacity of 300% is found in the results when flange effect is considered from 10 to 40 KN. The results of the NLTHA are found in agreement with the pushover analysis in terms of capacities. The analysis is performed for the lower boundary and verified that the structure is not capable of resisting the seismic loading. The failure mechanisms observed are in accordance to the Pushover in the y direction. Out of plane failure of the left wall is not observed in this analysis although this is observed in the relevant pushover analysis in the x direction. These differences in the results are expected as in the NLTHA three components of loading are applied and the load is cyclic applied at the base. In the pushover analysis the base is considered fixed and the load is uniformly applied at every mass. The characteristics of the applied accelerogram in the three directions can determine which failure modes will be present first. The outcomes of the assessment phase are summarized in the following table. The model considered the most adequate to assess the structure behaviour is Case 2 with stiffness assigned at both ends. Table 53: Outcomes of assessment phase. Analysis Methods Pushover analysis FE Case 2* x 4 FE Case 2* x 5 FE Case 3 x 6 FE Case 3 x 7 EF - x 8 Analytical - x 9 Analytical - x 10 FE Case 3 y 11 FE - Y Maximum crack widths (mm) 3 Maximum drift (%) x Maximum capacity (KN) Case 2* u.c. capacity – 67% NPR FE u.c. capacity – 100% NPR 2 Unconnected 37 Out of plane - - - - - 50 0.5 40 40 Shear - - - - - - - - 44 Shear - - - - - - - - 44 Shear 7.5 3.7 1.27 4.48 3 - - - 47 Shear - - - - - - - - Stiffness at one end Stiffness at one end Stiffness at both ends Connected Reduced E modulus for timber Connected No flange effect considered Flange effect considered Behaviour factor u.c. displacements – 100% NPR x Ductility Direction Case 1 Critical failure mode Model FE Capacity (KN) Approach 1 NLTHA** Details Number Approaches & Models 47 Shear - - - - - - - - 40 Bending 3.61 2.5 1.69 - - - - - 10 Bending - - - - - - - - 40 Bending - - - - - - - - Connected 227 Out of plane - - - - - - - - Connected 280 Shear - - - - - - - - *Case 2: Stress developed between beams and masonry walls (modelled with introduction of interfaces). **Divergence occurred. Acceptability of results related to convergence details. 120 Conclusions Retrofitting After the assessment of the structure is completed the building is strengthened with various methods. The approach followed made use of the conclusions driven by the assessment phase, where the weak points of the structure are indicated. An overview of the methods used are shown in the following figure. 1. Improvement of existing connections Case 1: Non connected Case 3: Connected Reduced timber E modulus 2. Addition of connections 3. Improved in-plane stiffness Longitudinally connected Addition of boards - 40 mm - 80 mm 4. Steel frames Configuration 1 Configuration 2 Configuration 3 Pushover analysis NLTHA Figure 136: Modelling approaches used in the retrofitting phase. The conclusions driven in this phase are: Connectivity of elements resulted to 27% increase in global capacity. In this model out of plane failure of the gamble is suppressed. The addition of connections between facades and floors resulted to 50% increase in capacity in the direction parallel to the facades and a box-type behaviour. In the direction perpendicular to facades this measure prevented out of plane failure of the back façade in the level of the floors and an increase of 150% in global capacity. The improved in-plane stiffness of the floors with the addition of boards showed an increase of 30% in the global capacity for an 80% increase in the height of the board. (80mm board) To reach the demanded capacity by the seismic action, different configurations of steel frames are investigated. A parameter that is considered critical in the dimensioning of the steel frames is the behaviour factor q. A linear approach would follow a behaviour factor of suggested by NPR. From the nonlinear analysis higher behaviour factors resulted. Although the main focus of this analysis is on forces as a force-controlled strategy is followed, still there is an indication that the behaviour factor can be higher than 2. The interaction of the steel frames to the masonry showed three discrete phases in the capacity curves. Initially capacity is given by masonry, following masonry and steel work in parallel and finally a plateau is observed. The combination of the right capacity and ductility factor is found critical in the development of a retrofitting strategy with steel frames. The aim is to achieve high capacity with high ductility. 121 Conclusions When looking at table 51 a difference is observed between Configuration 1 and 3. While both configurations achieve almost the same capacity, configuration 3 is more ductile resulting to the satisfaction of all unity checks. Out of plane failure is shown in all Configurations. Configuration 1 and 2 showed an out of plane failure of the gamble, while configuration 3, out of plane failure of left and intermediate wall. The increase of the capacity of the structure is noted at 300% for Configuration 1 and 3 and 233% for Configuration 2. This increase cannot be supported by the masonry elements that fail out of plane. The use of diagonals and the limitation of the drift limits at element level can have a positive effect on the failure mechanism and the observed crack widths as shown from Configuration 3. The acceptability of risk and the adaptation of the relevant spectrum corresponding to 67% or 100% of NPR can be decisive in design. As can be noted in the following table the unity check of the capacities for Configuration 1 is satisfied for 67% of NPR requirement but not for 100%. A discrepancy is found between the unity checks of displacements and capacities. For nonlinear methods the check of displacements is proposed in EC with the definition of the target displacement. This result is trusted for the assessment of the structure. The check of Configuration 1 with a signal corresponding to 67% of NPR showed divergence. The results proved that there is numerical instability as they do not justify structural failure. Table 54: Outcomes of retrofitting phase. Analysis Methods Pushover Direction Details Capacity (KN) Ductility u.c. capacity – 100% NPR u.c. capacity – 67% NPR Maximum Capacity (KN) Maximum drifts (%) Maximum crack widths (mm) 4 Case 2* x Stiffness both ends 44 7.5 3.7 1.27 4.48 3 - - - 12 Improvement of connections x Between beams & masonry walls 47 - - - - - - - - x Connections between facades & floors 13 14 15 Addition of connections y – 100% NPR Model NLTHA** Number Behaviour factor u.c. displacements Approaches & Models 75 - - - - - - - - 500 - - - - - - - - Improved in-plane stiffness x Extra plank 40mm 53 - - - - - - - - x Extra plank 80mm 60 - - - - - - - - 17 Configuration1 x - 295 3.3 2.4 0.77 1.13 0.77 100 0.15 20 18 Configuration 2 x - 242 2.8 2.1 1.26 1.58 1.06 - - - 19 Configuration 3 x - 300 4.7 2.9 0.64 0.98 0.62 - - - 16 *Case 2: Stress developed between beams and masonry walls (modelled with the introduction of interfaces). **Divergence occurred. Numerical instability assessed. Acceptability of results related to convergence details. It can be concluded that the models developed give an overview of how the structure might behave. The scope was to create an envelope of different expected behaviours and show how these can be influenced with interventions. The connectivity of the elements showed to influence the global response of the building and result to different failure mechanisms. This parametric assessment can help identify the weak points of the structure and intervening where necessary. The reinforcement with steel frames showed a collaboration of the two materials, where the degradation of the masonry is delayed due to the presence of steel. 122 Conclusions Discussion The results presented in this report are considered valid only for the Case Study under consideration. In the presented approach the supports are considered fixed. In the Netherlands where there is a weak soil, the soil-structure interaction will play an important role in the actual behaviour of a Terraced House. Also in case of pile foundations special research is necessary. The material properties are considered fixed in this analysis. A specialized study would consider site specific material properties with the definition of damaged-based properties. Also in this analysis 2-D elements are used and a macro-modelling approach is followed. The actual behaviour of masonry would suggest the definition of 3-D elements at the actual dimensions of the bricks and the representation of the mortar following a micro-modelling approach. The application of load in the pushover analysis is defined uniform and is applied in one perpendicular direction. In a real seismic event the load is cyclic. This would involve the application of a more advanced pushover analysis. In the NLTHA only one accelerogram is used and considered as a check tool. A focus on this analysis would suggest the application of a number of accelerograms with different characteristics. Also the accelerograms used are in accordance with the NPR released in February. The new version of the code suggests lower accelerations and longer plateau. This will result in changes in the applied signals and will affect the unity checks performed. Interfaces are used to some connections and the main focus was to check the differences in the results. The actual behaviour of a Terraced House would suggest definition of interfaces in almost all connections. Also the diaphragm flexibility is not taken into account in a direct manner, as timber beams and planks are considered merged. To assess the behaviour of the timber diaphragm the effects of shear and flexural deformation of the boards and the rotation due to the nails slip need to be incorporated. In the present analysis the load increment procedure followed is a force control with the main focus to assess capacities. The development of a retrofitting strategy though requires an accurate estimation of the behaviour factor to define the seismic demand. To that end the adaptation of a displacement control analysis would be more appropriate. Recommendations Recommendations regarding the modelling strategy followed include: Use of a displacement control analysis with the use of arc length control is recommended. This analysis can help to identify more precisely the behaviour factors developed and further adopt it in the dimensioning of the steel elements. Convergence problems identified in the present analysis can be overcome. Adaptation of more integration points along the thickness of the curved shell elements is proposed. In the current analysis three points are used. For non-linear analysis more than three points are recommended. Interfaces could be added to more connections to make the model more realistic. Also interfaces at the foundation level are recommended as settlements play an important role in The Netherlands. Sensitivity analysis of the material properties. More detailed research on the structural behaviour of the timber floor system. In this analysis timber beams and planks are considered merged. A more refined approach would involve the definition of the presence of smaller timber boards and the presence of nails. 123 Conclusions More refined iteration processes could be explored. The used iteration process is a Regular Newton Raphson. As discussed more iteration processes are available influencing the results. Investigation of different convergence norms. Validation of the results through experiments. In depth analysis with the use of the Nonlinear time history analysis. Recommendations regarding the assessment and retrofitting phase: 124 The use of nonlinear methods can give the advantage of adopting a more realistic behaviour factor and developing a retrofitting strategy that can better suit the seismic demand. On the other hand these analysis are case specific. When a general strengthening strategy needs to be defined a behaviour factor of 2 can be adopted as a low boundary. A drift limit of 0.5% is recommended for design when extensive cracking is accepted. The models developed showed that the structure is at the extensive cracking phase when a drift limit of 0.5% is present. The acceptability of risk needs to be defined at an early stage in the design process as it defines the seismic demand. The use of analytical formulas proposed by the NZSEE following the pier only method can give an estimation of the expected capacity of the structure. The flange effect is recommended to be incorporate in the superimposed load when walls are interlocked and the failure modes to be critically assessed. The equivalent frame model can give information about the behaviour of the structure but needs a careful consideration in the application to similar buildings. In these structures walls are composed by a limited amount of elements and therefore the failure mechanisms assessed can be inaccurate. The current version of the software is recommended to be used with a critical view on the generated parameters and knowledge of the modelling process followed. The program is at a development stage for buildings similar to the case study therefore is expected to overcome some of the pointed out deficiencies. At the moment the application of the EF model is recommended to be accompanied by an FE model for comparison of results. Connectivity of masonry and timber elements needs to be assured when a retrofitting strategy is developed. Out of plane failure of masonry walls needs to be suppressed, with improvement of existing connections and/or addition of connections. The in-plane stiffness of floors with the addition of boards can give a significant increase in the global capacity. The combination with FRP or steel elements in the new boards could give even more significant increase. Steel frames retrofitting method is recommended to be accompanied by the support of vulnerable masonry walls, including thin supporting walls and cavity walls. Investigation on the connectivity of the inner and outer leaf of the masonry is also recommended. Acronyms Acronyms URM Unreinforced Masonry FE Finite element EF Equivalent frame NLTHA Nonlinear time history analysis LS Limit State NC Near Collapse dof Degree of freedom NZSEE New Zealand Society of Earthquake Engineering ATC Applied Technology Council (California Seismic Safety Commission) ASCE American Society of Civil Engineers NAM Nederlandse Aardolie Maatschappij (Dutch Petroleum Company) TNO Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek (Netherlands Organisation for Applied Scientific Research) NPR Nederlandse praktijkrichtlijn (Dutch Code of Practice) FEMA Federal Emergency Management Agency (United States) KNMI Koninklijk Nederlands Meteorologisch Instituut (Royal Netherlands Meteorological Institute) KNGMP Koninklijk Nederlands Geologisch Mijnbouwkundig Genootschap (Royal Netherlands Geological and Mining Society) NWO-ALW Nederlandse Organisatie voor Wetenschappelijk Onderzoek – Aard en Levenswetenschappen (Netherlands Organization for Scientific Research – Earth & Life Science)) OPCM Ordinanza del Presidente del Consiglio dei Ministri (Order of the President of the Council of Ministers) USGS United States Geological Survey EC Eurocode MSJC Masonry Standard Joint Committee 125 Definitions Definitions Ductility (ATC-40, 1996) The ability of a structural component, element, or system to undergo both large deformations and/or several cycles of deformations beyond its yield point or elastic limit and maintain its strength without significant degradation or abrupt failure. These elements only experience a reduction in effective stiffness after yielding and are generally referred to as being deformation controlled or ductile. Behaviour factor (EN 1998-1, 2004) Factor used for design purposes to reduce the forces obtained from linear analysis, in order to account for the non-linear response of the structure, associated with the material, the structural system and the design procedures. Drift (ASCE/SEI41-13, 2014) Horizontal deflection at the top of the storey relative to the bottom of the storey. Importance factor (EN 1998-1, 2004) Factor which relates to the consequences of a structural failure. URM (NZSEE, 2015) A masonry wall containing no steel, timber, cane or other reinforcement. An unreinforced wall resists gravity and lateral loads solely through the strength of the masonry material. Cavity wall (NZSEE, 2015) A cavity wall consists of two skins separated by a hollow space (cavity). The skins are commonly both masonry, such as brick or concrete block, or one could be concrete. The cavity is constructed to provide ventilation and moisture control. Non-structural elements (EN 1998-1, 2004) Architectural, mechanical or electrical element, system and component which, whether due to lack of strength or to the way it is connected to the structure, is not considered in the seismic design as load carrying element. Single Degree of Freedom system The motion of a linear SDF system subjected to ground acceleration ̈ formula: ̈ 126 ̇ ̈ is governed by the following Definitions A representation of the system is illustrated in the following figure: Figure 137: Single degree of freedom system. (Chopra, 2012) Equivalent single degree of freedom system (ATC-40, 1996) The definition of this equivalent single degree of freedom system is illustrated in the following figure: Figure 138: Fundamental mode of a multi-mass system (left) and equivalent single mass system (right). (ATC-40, 1996) Capacity curve (ATC-40, 1996) The plot of the total lateral force (V) on the structure, against the lateral displacement (d) of the roof of the structure. This is often referred as pushover curve. Flexible diaphragm (NZSEE, 2015) A diaphragm which for practical purposes is considered so flexible that it is unable to transfer the earthquake loads to shear walls even if the floors/roof are well connected to the walls. Floors and roofs constructed of timber, steel, or precast concrete without reinforced concrete topping fall in this category. Eigenvalue analysis Eigenvalue analysis refers to a free vibration, a motion of the structure without any dynamic excitation. The free vibration starts by applying some initial displacements. The main parameters defined in this analysis are the frequencies and mode shapes. The equation that describes the matrix eigenvalue problem is the following: (Chopra, 2012) Where: mass matrix; stiffness matrix; eigenvector; natural frequency. 127 Appendixes Appendix A: Dead loads calculation Table 55: Calculation of floor weight. Table 56: Calculation of roof weight. Timber floor weight Timber roof weight beams beams width 0.071 width 0.071 height 0.196 height 0.196 length 6.92 length 6.92 Number Volume - 12 Number 1.156 plank - 8 Volume 0.77 ridge beam width 5.65 width 0.071 height- top plank 0.022 height 0.246 height - bottom plank 0.022 length 6.92 length 6.92 Number Volume 1.72 Volume 0.121 - 1 planks Total Volume Weight q 2.88 width 4.385 density 500 height 0.022 g 9.81 length 6.92 14106 Number 0.36 Volume - 1.34 Total volume 2.23 Density g Weight 128 2 kg/m3 500 9.81 10920 q wood 0.28 Ceramic tiles 0.50 q total 0.78 Appendixes Appendix B: Capacity hand calculations The properties considered in the calculation are the following: Table 57: Material properties in NZSEE calculation. Masonry properties Symbol Units Value Density 1920 Compressive strength bricks 14 Cohesion mortar / lime 0.3 Friction coefficient 0.75 Compressive strength masonry 6 Young's modulus of masonry 4000 Tensile strength 0.15 The calculation of pier 1 is presented in the following table to show the calculation process. For the superimposed load from the flanges the fictitious densities as calculated in Table 20 are used. Table 58: Calculation of failure mechanisms of pier 1. (x direction) Symbol Calculation Pier characteristics Width Total floor height Effective height Thickness Self weight Superimposed load nd from 2 floor Flange thickness Flange width Superimposed load from flange Total superimposed load Diagonal tensile capacity Area of net mortared/grouted section of wall web Factor to correct nonlinear stress distribution Axial compression stress due to gravity calculated at the base of the pier For ⁄ ⁄ ⁄ Masonry diagonal 129 Appendixes tension strength Maximum diagonal tensile strength √ √ Toe crushing capacity Factor for fixed-free wall or fixed-fixed pier (0.5,1) Length of the pier ( Toe crushing capacity ) ( ) ( ) ( ) Rocking capacity Rocking capacity ⁄ Bed-joint sliding shear capacity Bed-joint sliding shear capacity ( ) ( ) The calculation of the piers capacities in the x direction for different failure modes are presented in the following table. The total capacity is calculated as the sum of all capacities of the piers of the first floor. In the case study the facades are not loaded from the floors and this results to a relative low assessment of the overall capacity when the flange effect is not taken into account. The piers are numbered as shown in the following figure: Figure 139: Pier dimensions. 130 Appendixes Table 59: Calculation of failure mechanisms. (x direction) Units Piers 1 2 3 7 8 9 680 795 980 480 800 680 2700 2700 2700 2700 2700 2700 2150 1900 2450 1910 1900 2150 100 100 100 100 100 100 2807 2900 4610 1760 2918 2807 3525 4121 5080 2488 4147 3525 100 100 200 200 100 100 600 600 1200 1200 600 600 8153 7774 33164 33164 7774 8153 11678 11895 38245 35653 11921 11678 Diagonal tensile capacity - 68 79.5 98 48 80 68 0.67 0.67 0.67 0.67 0.67 0.67 0.21 0.19 0.44 0.78 0.19 0.21 0.31 0.29 0.48 0.73 0.29 0.31 18334 19769 43428 33916 19855 18334 Toe crushing capacity - 1 1 1 1 1 1 680 795 980 480 800 680 3928 5337 14531 7477 5385 3928 8263 5071 3724 24591 21885 Rocking capacity 3724 5026 14598 Bed-joint sliding capacity 21885 24463 43079 29722 As can be observed rocking capacity is governing. The total base shear can therefore be calculated as: ∑ When no flange effect is taken into account the rocking capacity is calculated . 131 Appendixes Appendix C: Target displacement calculation Elastic spectrum according to NPR The elastic response spectrum is defined based on the following equations: (Ontw. NPR 9998, February 2015) [ ] [ ] [ ] Where: design spectrum; soil factor; ground acceleration; the lower limit of the period of the constant spectral acceleration branch; the upper limit of the period of the constant spectral acceleration branch; and the value defining the beginning of the constant displacement response range of the Spectrum. The design ground acceleration is calculated based on the following formula: Where: importance factor; peak ground acceleration. The importance factor is obtained from the following table for consequence class CC1B. Table 60: Importance factors 132 per consequence classes. Appendixes Table 61: Consequence classes parameters. Figure 140: Selected PGA in analysis. (Ontw. NPR 9998, February 2015) 133 Appendixes The parameters taken into account for the horizontal elastic spectrum are presented in the following table: Table 62: Parameters of horizontal response spectrum. Factor Value 1 1 2 ( ) 0.1 3 ( ) 0.22 4 ( ) 0.45 5 ( 5.04 ) Se (g) The horizontal elastic spectrum based on the above mentioned formulas and parameters is presented below: 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2 3 Period T (s) Figure 141: Horizontal elastic response spectrum. 134 4 Appendixes Transformation of elastic Spectrum in ADRS format For the transformation of the spectrum the following formula is considered: Table 63: Spectrum in ADRS format. T 0 0.1 0.1 0.22 0.22 0.25 0.30 0.35 0.40 0.45 0.45 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 Sa (m/s2) 5.04 15.12 15.12 15.12 15.12 13.31 11.09 9.50 8.32 7.39 7.39 5.99 4.16 3.05 2.34 1.85 1.50 1.24 1.04 0.89 0.76 0.67 0.58 0.52 0.46 0.41 0.37 0.34 0.31 0.28 0.26 0.24 0.22 0.21 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.12 0.11 0.10 0.10 0.09 Sd (m) 0.000 0.004 0.004 0.019 0.019 0.021 0.025 0.029 0.034 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 0.038 135 Appendixes Transformation of system to an equivalent SDOF system ∑ ( ∑ ( ) ( ⁄ ) ) ( ⁄ ) Table 64: Equivalent SDOF capacity curve. Fb (KN) dn (m) F* (KN) d* (m) 0 5 9 14 18 23 27 32 36 41 45 45 45 47 47 46 45 45 0.000 0.000 0.001 0.001 0.002 0.002 0.003 0.003 0.004 0.005 0.008 0.010 0.011 0.014 0.018 0.022 0.026 0.030 0 4 7 11 15 19 23 26 30 34 37 38 38 39 39 38 37 37 0.0000 0.0003 0.0007 0.0010 0.0014 0.0017 0.0021 0.0025 0.0032 0.0046 0.0068 0.0084 0.0094 0.0119 0.0152 0.0187 0.0220 0.0254 Idealized elasto-perfectly plastic force-displacement Table 65: Idealized curve. 2 F* (KN) d* (m) F*/m* (m/s ) 0 0 0.0022 0.0034 0.0254 0.00 0.63 0.97 0.97 36 36 Force of t SDOF system (F*) The resulting bilinear relation is illustrated in the following figure: 50 40 30 20 10 0 0.000 Capacity curve - SDOF Bilinear approximation Capacity curve - MDOF 0.020 0.040 Dispalcement of equivalent SDOF system (d*) Figure 142: Capacity curves and bilinear representation of SDOF until drift limit of 0.5 %. 136 Appendixes Period of SDOF √ √ SDOF Target Displacement [ and [ ] : Acceleration (Se(T)) For ] 20 15 10 5 0 0.00 0.01 0.02 0.03 0.04 Displacement of equivalent SDOF system (d*) Spectrum Capacity curve of SDOF Figure 143: Capacity curve of SDOF and spectrum MDOF Target Displacement 137 Appendixes Appendix D: Convergence quality Pushover analysis The analysis developed are primarily focused on assessing the base shear of the system. To understand the quality of the results in this section two main graphs are presented. The resultant base shear in comparison to the applied force is shown, versus the developed displacements. The distance between the two curves can show in a direct manner the quality of the convergence in terms of base shear. In a force control analysis the loads are increased continuously and the load increment is determined by the load step definition. The iterations assigned are 30. 50 Force [KN] 40 30 20 Applied force 10 Resultant base shear 0 0 20 40 60 80 Displacements Variation Also the displacements variation versus the resultant displacements are illustrated. For these analysis a Displacement convergence norm is applied. The converged steps refer to a displacements variation of 0.01. As can be observed from the graphs the resultant forces are in agreement with the resultant base shears. In most analysis an overshoot is observed at some point but the curve returns back to a good agreement with the applied force. For displacements convergence is mainly found in the first steps and for the non-converged steps the variation is reported. The acceptance of the presented results is related to the acceptance of the below presented convergence characteristics of the analysis. Displacements are referring to roof level as expressed in every capacity curve. 10.000 1.000 0.100 0.010 0.001 0.000 0 Dispalcement [mm] 20 40 60 80 Displacement [mm] 60 50 40 30 20 10 0 Applied force Resultant base shear 0 10 20 30 Dispalcement [mm] 40 Displacements Variation Force [KN] Figure 144: Convergence characteristics. – Case 1 (x) 10.000 1.000 0.100 0.010 0.001 0.000 0 10 Figure 145: Convergence characteristics. – Case 3 (x) 138 20 30 Displacement [mm] 40 Force [KN] 250 200 150 100 Applied force 50 Resultant base shear 0 0.00 0.10 0.20 0.30 0.40 0.50 Displacements Variation Appendixes 10.000 1.000 0.100 0.010 0.001 0.000 0.00 Dispalcement [mm] 0.20 0.40 Displacement [mm] Figure 146: Convergence characteristics. – Case 1 (y) Displacments variation 50 Force [KN] 40 30 20 Resultant base shear 10 Applied force 0 0 5 10 15 20 10.000 1.000 0.100 0.010 0.001 0.000 0 Resultant displacements [mm] 10 20 Displacements [mm] 3 Figure 147: Convergence characteristics. – Case 2 (Stiffness 0.01 N/mm ) Displacments variation 50 Force [KN] 40 30 20 Resultant base shear 10 Applied force 0 0 10 20 30 10.000 1.000 0.100 0.010 0.001 0.000 0 Resultant displacements [mm] 10 20 30 Displacements [mm] 3 Figure 148: Convergence characteristics. – Case 2 (Stiffness 0.1 N/mm ) Displacments variation 50 Force [KN] 40 30 20 Resultant base shear 10 Applied force 0 0 10 20 30 Resultant displacements [mm] 40 1.000 0.100 0.010 0.001 0.000 0 20 40 Displacements [mm] 3 Figure 149: Convergence characteristics. – Case 2 (Stiffness 0.1 N/mm at both ends) 139 60 Force [KN] 50 40 30 20 Applied force 10 Resultant base shear 0 0 10 20 30 Displacements Variation Appendixes 10.000 1.000 0.100 0.010 0.001 0.000 0 10 Dispalcement [mm] 20 30 Displacement [mm] Force [KN] 80 60 40 Applied force 20 Resultant base shear 0 0 5 10 15 20 Displacements Variation Figure 150: Case 3 – Reduced stiffness. 10.000 1.000 0.100 0.010 0.001 0.000 0 Dispalcement [mm] 5 10 15 20 Displacement [mm] 600 Force [KN] 500 400 300 200 Applied force 100 Resultant base shear 0 0 5 10 15 20 25 Displacements Variation Figure 151: Convergence characteristics. – Connection longitudinally (x) 10.000 1.000 0.100 0.010 0.001 0.000 0 Dispalcement [mm] 5 10 15 20 Displacement [mm] 60 Force [KN] 50 40 30 20 Applied force 10 Resultant base shear 0 0 10 20 Dispalcement [mm] 30 Displacements Variation Figure 152: Convergence characteristics. – Connection longitudinally (y) 10.000 1.000 0.100 0.010 0.001 0.000 0 10 Figure 153: Convergence characteristics. – Plank 40mm 140 20 Displacement [mm] 30 70 60 50 40 30 20 10 0 Displacements Variation Force [KN] Appendixes Applied force Resultant base shear 0 2 4 10.000 1.000 0.100 0.010 0.001 6 0 Dispalcement [mm] 10 20 30 Displacement [mm] 350 300 250 200 150 100 50 0 Applied force Resultant base shear 0 20 40 60 Displacements Variation Force [KN] Figure 154: Convergence characteristics. – Plank 80mm 1.000 0.100 0.010 0.001 0 Dispalcement [mm] 20 40 60 Displacement [mm] 300 Force [KN] 250 200 150 100 Applied force 50 Resultant base shear 0 0 20 40 60 Displacements Variation Figure 155: Convergence characteristics for Steel frames. - Configuration 1 1.000 0.100 0.010 0.001 0.000 0 Dispalcement [mm] 20 40 60 Displacement [mm] 350 300 250 200 150 100 50 0 Applied force Resultant base shear 0 20 40 Dispalcement [mm] 60 Displacements Variation Force [KN] Figure 156: Convergence characteristics for Steel frames. - Configuration 2 1.000 0.100 0.010 0.001 0 20 40 Displacement [mm] Figure 157: Convergence characteristics for Steel frames. - Configuration 3 141 Appendixes Time history analysis For the Time history analysis an energy convergence norm is used. The converged steps are related to a displacement variation of 0.0001. For Case 1 convergence is observed till 2.11 s and energy variation is kept at values of a magnitude of 10-4 till 3.65 s. After that poor convergence is observed till divergence occurs. Energy variation 10.0000 1.0000 0.1000 0.0100 0.0010 0.0001 3.65 3.85 4.05 Time (s) Figure 158: Energy variation at last steps of time history. - Case 1 Energy variation For Configuration 1 convergence is observed till 3,295 s. Energy variation is kept at values of a magnitude of 10-4 till 3.295 s. Following poor convergence is observed and energy variation fluctuates till divergence occurs. 10.0000 1.0000 0.1000 0.0100 0.0010 0.0001 0.0000 3.30 3.50 3.70 3.90 Time (s) Figure 159: Energy variation at last steps of time history. - Configuration 1 142 Appendixes Appendix E: Case study drawings Figure 160: Connections of timber beams to cavity walls at roof level. Figure 161: Longitudinal connection of timber beams. 143 Appendixes Figure 162: Building plans 144 References References Allen, C., Masia, M., Derakhshan, H., Griffith, M., Dizhur, D., & Ingham, J. (2013). What ductility value should be used when assessing unreinforced masonry buildings? NZSEE Conference. Amiraslanzadeh, R., Ikemoto, T., Miyajima, M., & Fallahi, A. (2012). A Comparative Study on Seismic Retrofitting Methods for Unreinforced Masonry Brick Walls. 15 WCEE. Lisboa. ARUP. (2013). Implementation Study. ASCE/SEI41-06. (2007). Seismic Rehabilitation of Existing Buildings. ASCE/SEI41-13. (2014). Seismic Evaluation and Retrofit of Existing Buildings. ATC-40. (1996). Seismic evaluation & retrofit of concrete buildings, Volume 1. Bakeer, T. (2009). Collapse analysis of masonry structures under earthquake actions (8th ed.). Dresden: Technische Universitat Dresden. Bento, R., Simoes, A., Lagomarsino, S., & Cattari, S. (2012). Seismic Pushover Analysis of "Gaioleiro" Buildings in Lisbon. International Conference on Seismic Engineering, SE-EEE. Skopje. Bouchard, K. (2007). A Performance-Based Approach to Retrofitting Unreinforced Masonry Structures for Seismic Loads. Massachusett: MIT. Boussabah, L., & Bruneau, M. (1992). Review of seismic performance of unreinforced masonry walls. Earthquake Engineering, Tenth World conference. Rotterdam: Balkema. Brignola, A., Podesta, S., & Pampanin, S. (2008). In-plane stiffness of wooden floor. NZSEE Conference. Bull, J. (2001). Computational Modelling of Masonry, Brickwork and Blockwork Structures. UK: Saxe-Coburg Publications. Calvi, G., Pinho, R., Magenes, G., Bommer, J., Restrepo-Vélez, L., & Crowley, H. (2006). Development of seismic vulnerability assessment methodologies over the past 30 years. ISET Journal of Earthquake Technology, pp. 75-104. Cattari, S., Giongo, I., Marino, S., Lin, Y., Schiro, G., Ingham, J., et al. (2015). Numerical simulation of the seismic response of an earthquake damaged URM building. NZSEE Conference. Cattari, S., Lagomarsino, S., Bazzurro, A., Porta, F., & Pampanin, F. (2015). Critical review of analytical models for the inplane and out-of-plane assessment of URM buildings. 2015 NZSEE Conference. Chen, W., & Lui, E. M. (2005). Handbook of Structural Engineering, Second edition. Chopra, A. (2012). Dynamics of Structures. Theory and Application of Earthquake Engineering. (Fourth ed.). University of California at Berkeley. Chopra, A., & Goel, R. (1999). Capacity-Demand-Diagram Methods for Estimating Seismic Deformation of Inelastic Structures: SDF Systems. University of California, Berkeley. Berkeley: Pacific Earthquake Engineering Research Center. Churilov, S., & Dumova-Jovanoska, E. (2012). Analysis of masonry walls strengthened with RC jackets. 15 WCEE. Lisboa. Correia, A., Almeida, J., & Pinho, R. (2013). Seismic Energy Dissipation in Inelastic Frames: Understanding State-of-thePractice Damping Models. (I. A. Engineering, Ed.) Structural Engineering International, 23(2), 148-158(11). D26, D. (2012). Modelling strategies for seismic global response of building and local mechanisms. Perpetuate Project. Elgawady, M., Badoux, M., & Lestuzzi, P. (2006). Retrofitting of Masonry Walls Using Shotcrete. NZSEE Conference. Elgawady, M., Lestuzzi, P., & Badoux, M. (2004). A review of conventional seismic retrofitting techniques for URM. 13th Brick/Block Masonry Conference. Amsterdam, Holland. 145 References Elgawady, M., Lestuzzi, P., & Badoux, M. (2005). Seismic performance of URM walls retrofitted using FRP. NZSEE Conference. Elgwady, M., Lestuzzi, P., & Badou, M. (2005). Dynamic In-Plane Behavior of URM Wall Upgraded with Composites. Journal of Composites for Construction, 9(6). EN 1998-1. (2004). Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. EN 1998-3 . (2005). Eurocode 8: Design of structures for earthquake resistance – Part 3: Assessment and retrofitting of buildings. Facconi, L., Plizzari, G., & Vecchio, F. (2013). Disturbed Stress Field Model for Unreinforced Masonry. Americal Society of Civil Engineers (ASCE). FEMA 356 . (2000). Prestandard and commentary for the seismic rehabilitation of buildings. Virginia: ASCE. Galasco, A., Lagomarsino, S., & Penna, A. (2006). On the use of pushover analysis for existing masonry buildings. First European Conference on Earthquake Engineering and Seismology. Switzerland. KNGMG & NWO-ALW. (2014). Geo-brief. KNMI. (2013). Report on the expected PGV and PGA values for induced earthquakes in the Groningen area. Lagomarsino, S., Penna, A., Galasco , A., & Cattari, S. (2013). TREMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings. Engineering Structures, 56, pp. 1787–1799. Lawrence Livermore National Laboratory. (2009). Mechanical Properties of Unreinforced Brick Masonry. Section 1. United States: U.S. Department of Energy. Lourenço , P. (2013). Computational Strategies for Masonry Structures: Multi-Scale modelling, dynamics, engineering applications and other challenges. Congreso de Métodos Numéricos en Ingeniería. Bilbao, España. Magenes, G., & Penna, A. (2009). Existing Masonry Buildings: General Code Issues and methods of analysis and assessment. Eurocode 8 Perspectives from the Italian Standpoint Workshop, (pp. 185-198). Napoli, Italy. Meireles, H., & Bento, R. (2013). Rehabilitation and strengthening of old masonry buildings. Mosalam, K., Glascoe, L., & Bernier, J. (2009). Mechanical Properties of Unreinforced Brick Masonry. Nakamura, Y., Magenes, G., & Griffith, M. (2014). Comparison of pushover methods for simple building systems with flexible diaphragms. Australian Earthquake Engineering Society 2014 Conference. Lorne, Victoria. NAM. (2015). Ontwerpuitgangspunten. Bouwkundig versterken van bestaande gebouwen tegen aardbevingsbelasting in de Groningen regio. Nicolini, L. (2012). Equivalent viscous damping and inelastic displacement for Strengthened and reinforced masonry walls. Trento, Italy.: University of Trento. NZSEE. (2015). Assessment and Improvement of the Structural Performance of Buildings in Earthquakes. Section 10 Revision Seismic Assessment of Unreinforced Masonry Buildings. Oliver, S. (2010). A design methodology for the assessment and retrofit of flexible diaphragms in Unreinforced Masonry buildings. SESOC Journal, 23(1). Ontw. NPR 9998. (February 2015). Nederlandse praktijkrichtlijn. Assessment of buildings in case of erection, reconstruction and disproval - Basic rules for seismic actions; Induced earthquakes. Delft: Nederlands Normalisatie-instituut. OPCM 3274. (2005). Code for the seismic design, assessment and retrofitting of buildings (Appendix 2). Ordinance of the Prime Minister. Palacio, K. (2013). Practical Recommendations for Nonlinear Analysis in DIANA. 146 References Parisi, F. (2010). Non linear seismic analysis of masonry buildings. Naples. Pela, L., Cervera, M., & Roca, P. (2011). Continuum damage model for orthotropic materials: Application to masonry. Computer Methods in Applied Mechanics and Engineering, 200, pp. 917-930. Piazza, M., Baldessari, C., & Tomasi, R. (2008). The role of in-plane floor stiffness in the seismic behaviour of traditional buildings. The 14th World Conference on Earthquake Engineering. Beijing, China. Royal Netherlands Meteorological Institute. (2012). Monitoring induced seismicity in the North of the Netherlands. Russell, A., & Ingham, J. (2008). Flange effects of an unreinforced masonry wall subjected to pseudo-static in-plane seismic forces. The 14th World Conference on Earthquake Engineering. Beijing, China. Russell, A., & Ingham, J. (2010). The influence of flanges on the in-plane seismic performance of URM walls in new Zealand buildings. NZSEE Conference. S.T.A.DATA. 3Muri Manual. Release: 5.0.1. Smith, A., & Redman, T. (2009). A Critical Review of Retrofitting Methods for Unreinforced Masonry Structures. EWB-UK Research Conference. The University of Auckland. (2015). Seismic Improvement of Loadbearing Unreinforced Masonry Cavity Walls. New Zealand: Branz. TNO DIANA BV. (2014). DIANA user's manual. Release 9.6. Tumialan , G., Huang, P.-C., Nanni, A., & Silva, P. (2001). Strengthening of Masonry Walls by FRP Structural Repointing. Non Metallic Reinforcement for Concrete Structures - FRPRCS-5. Cambridge. University of Buffalo. (2009). Retrieved nonlinearanalysisstatic.pdf from http://civil.eng.buffalo.edu/cie619/3-handouts/cie619-lecture12- USGS. (2005). Earthquake glossary. Retrieved from http://earthquake.usgs.gov/learn/glossary/?term=earthquake Yekrangnia, M., Mahdizadeh, A., Seyri, H., & Raessi, M. (2012). Seismic performance improvement of masonry buildings by WSBI Method. 15 WCEE. Lisboa. 147