non linear seismic assessment & retrofitting of
TRANSCRIPT
Non-linear seismic assessment & retrofitting of unreinforced masonry buildings
Master Thesis
Eleni Sionti
January 2016
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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.
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Faculty of Civil Engineering and Geosciences
Delft University of Technology
Personal information
Eleni Sionti [email protected]
Graduation committee
Prof. Dr. Ir. J.G. Rots , Department of Structural Engineering [email protected]
Dr.Ir. M.A.N. Hendriks, Department of Structural Engineering [email protected]
Dr. V. Mariani, Department of Structural Engineering [email protected]
Ir. S. Pasterkamp, Department of Building Engineering [email protected]
Ir. M. Spanenburg, BAM A&E [email protected]
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
16
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
Introduction
17
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
Introduction
18
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 [ for a return period of 475 years. (Ontw.
NPR 9998, February 2015)
Under these circumstances the assessment of the buildings of the area and the investigation of ways to
be reinforced become a necessity.
Introduction
19
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?
Introduction
20
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.
Analysis aspects Choices
FE Model - DIANA
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
EF Model - Tremuri Geometry 1-D beam elements
Diaphragm Flexible
Assessment
Comparison to experimental results No comparison
Use of analytical methods Yes
Maximum displacement of capacity curves
- 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
- Displacement with calculation of target displacement
- Capacity for comparison
Introduction
21
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.
Introduction
22
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.
Introduction
23
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
FE Models
Case 1:Non connected
Case 2:Semi connected
- One end- Two ends
NLTHA
Reduced timber modulus
Case 3:Connected
- Capacity- Ductility
- Failure modes- Drifts
Assessment
EF Model
Analytical approach
Improvement of existing
connections
Addition of connections
Increase of in plane stiffness
Steel frames
- Configuration 1- Configuration 2- Configuration 3
NLTHA
Figure 8: Overview of models developed.
Introduction
24
Literature
25
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 micro-
cracks 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).
Literature
26
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) Post-
peak 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.
Literature
27
Global response Shear Sliding Flexural
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
Literature
28
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)
Literature
29
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.
Literature
30
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)
Literature
31
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.
Literature
32
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.
Literature
33
Figure 21: Iteration process. (TNO DIANA BV.,
2014)
Figure 22: Regular Newton-Raphson method. (Palacio,
2013)
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.
Literature
34
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.
Literature
35
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.
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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 stiffness-
proportional, (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)
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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 stiffness-proportional 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.
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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 non-
linear 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.
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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)
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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 is perpendicular to the interface and the shear tractions
and are tangential to the interface.
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:
Piers, referring to vertical elements;
Spandrels, horizontal elements which couple the piers; and
Rigid nodes; which connect spandrels and piers.
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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 3-
node 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:
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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
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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)
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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.
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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.
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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 Bending
Significant damage (SD) ⁄ ⁄
Near Collapse (NC) ⁄ ⁄ ⁄
⁄ ⁄ ⁄
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 Rocking
Primary components (%)
Rocking
Secondary components (%)
Life safety (LS) ( ⁄ ) ( ⁄ )
Collapse prevention (CP) ⁄ ⁄
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.
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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)
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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. Transformation to an equivalent Single Degree of Freedom (SDOF) system;
2. Determination of the idealized elasto-perfectly plastic force-displacement relationship;
3. Determination of the period of the idealized equivalent SDOF system;
4. Determination of the target displacement for the equivalent SDOF system; and
5. 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.
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Table 5: Target displacement definition formulas. (EN 1998-1, 2004)
Step Value Formula Parameters
1
Mass of equivalent SDOF
∑
Mass in the i-th storey
Normalized displacements
Transformation factor
∑
Force of SDOF system
Base shear force
Displacement of SDOF system
Control node displacement of MDOF system
3
Period of idealized
equivalent SDOF system
√
Yield force
Yield displacement
4 Target
displacement of SDOF
[
]
Target
displacement of the structure with period T*
For (medium and long period range)
[
]
Elastic acceleration
response spectrum at period T*
For (short period range)
If
If
( (
))
5 Target
displacement of MDOF
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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
Diagonal tensile
cracking √
Factor to correct nonlinear stress distribution
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
Rocking capacity
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
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
Toe crushing capacity
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
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
Bed-joint sliding shear
capacity
Masonry coefficient of friction
Superimposed and dead load at top of the wall/pier
Self-weight of wall/pier above the sliding plane being considered
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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/SEI41-
06, 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)
Overall damage 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.
Drifts Large permanent drifts present.
Moderate permanent drifts are present.
Permanent drifts are negligible.
General 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.
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)
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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 Photo
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.
3 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.
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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 Photo
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 epoxy-based resin. The light steel plates can be nailed to the planks.
3 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 5-6 mm diameter.
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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
1 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.
-
2 Steel bracing
system
Addition of steel bracing to influence stiffness and improve the ductility factor.
-
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 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)
5 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)
6 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
55
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 out-of 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.
Method Description Photo
1 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)
Literature
56
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.
FE modelling
57
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.
FE Modelling
58
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 Value
Young modulus 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:
FE modelling
59
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.
FE Modelling
60
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.
FE modelling
61
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.
FE Modelling
62
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.
FE modelling
63
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
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 ux uy uz
Case 1 Free Tied Tied
Free
Case 2 Stiffness Tied Stiffness
Case 3 Tied Tied Tied
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:
c
c
c
FE Modelling
64
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.
FE modelling
65
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.
FE Modelling
66
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.
FE modelling
67
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.
FE Modelling
68
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
Load 4571 7038 7038 2466 17954 17954
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.
FE modelling
69
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.
FE Modelling
70
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.
Eigenvalue analysis
71
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 2: Mode 3:
Mode 4: 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 : .
FE model – Pushover analysis
72
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.
FE model – Pushover analysis
73
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.
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.
Figure 66: Capacity curve until out of plane failure occurs. – Case 3 (y)
0
10
20
30
40
50
60
0 5 10 15 20
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
Case 1Case 3
0
50
100
150
200
250
0.0 0.5 1.0 1.5 2.0
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
FE model – Pushover analysis
74
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.
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.
-6
-5
-4
-3
-2
-1
0
1
-0.00375 -0.00275 -0.00175 -0.00075 0.00025
Stre
ss (
N/m
m2 )
Strain
FE model – Pushover analysis
75
Table 23: Drift limits per element.
Height Width Pier
Shear drift
Pier Bending
drift Comment
[mm] [mm] - -
Front façade – 1st floor
1 Left pier 2150 680 0.017 0.034 Middle pier attains drift
limit first 2 Middle pier 1900 795 0.013 0.025
3 Right pier 2450 980 0.013 0.027
Front façade – 2nd floor
4 Left pier 2070 680 0.016 0.032 Middle pier attains drift
limit first 5 Middle pier 1020 1050 0.005 0.010
6 Right pier 1650 1130 0.008 0.016
Back façade – 1st floor
7 Left pier 1910 480 0.021 0.042 Middle pier attains drift
limit first 8 Middle pier 1900 795 0.013 0.025
9 Right pier 2150 680 0.017 0.034
Back façade – 2nd floor
10 Left pier 2070 480 0.023 0.046 Middle pier attains drift
limit first 11 Middle pier 1440 1495 0.005 0.010
12 Right pier 2070 680 0.016 0.032
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 Value
Yield stress
Yield strain
Figure 69: Stress-strain relationship of steel elements and definition of yield strain.
FE model – Pushover analysis
76
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.
FE model – Pushover analysis
77
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.
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. Gravity loading;
2. Linear phase;
3. Extensive cracking;
4. Crack propagation; and
5. 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)
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70
Bas
e S
he
ar (
KN
)
Displacements at roof level (mm)
Capacity curve - Case 1 (x)
Step 8: Ultimate tensile strength
Step 10: Crack width 7.78 mm
Step 10: Ultimate compressive strength Step 23:
Interstory drift limit 0.5% of first floor reached
Step 10: Out of plane failure
Step 25: Collapse
Step 24: Interstory drift limit 0.8 % of first floor reached
FE model – Pushover analysis
78
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.
Figure 77: Drifts per storey and load step.- Case 1 (x)
0.000
0.010
0.020
0 10 20 30
Dri
fts
Steps
First floor
Second floor
Roof
FE model – Pushover analysis
79
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.
Figure 80: Capacity curve analysis. - Case 3 (x)
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40
Bas
e S
he
ar (
KN
)
Displacements at roof level (mm)
Capacity curve - Case 3 (x)
Step 10: Ultimate tensile strength
Step 24: Ultimate compressive strength
Step 24: Crack width 7.67 mm
Step 27: Interstory drift limit 0.5 % of first floor reached
Step 28: Interstory drift limit 1 % of first floor reached
FE model – Pushover analysis
80
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:
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.
0
50
100
150
200
250
0.0 0.5 1.0 1.5 2.0
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
FE model – Pushover analysis
81
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.
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.
0
20
40
60
0 5 10 15 20 25
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
Normal stiffness 0.01 N/mm3
Normal stiffness 0.1 N/mm3
Case 3
Case 1
FE model – Pushover analysis
82
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/mm3
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.
Figure 88: Capacity curve for assigned stiffness at both ends.
0
10
20
30
40
50
0 5 10 15 20 25 30
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
Normal stiffness 0.1 N/mm3
Normal stiffness 0.1 N/mm3 - both ends
FE model – Pushover analysis
83
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 ; and
Developed stress in normal direction ⁄ .
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.
Figure 89: Interface stresses Stx of ridge beam.
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.
Figure 90: Interface stresses Stz of ridge beam.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-15 -10 -5 0
Inte
rfac
e s
tre
ss S
tx (
N/m
m2
)
Interface relative displacement in normal direction
Normal stiffness 0.01 N/mm3Normal stiffness 0.1 N/mm3Normal stiffness 0.1 N/mm3 - both ends
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-15 -10 -5 0Inte
rfac
e s
tre
ss S
tz (
N/m
m2
)
Relative displacements variation in normal direction
Normal stiffness 0.01 N/mm3
Normal stiffness 0.1 N/mm3
Normal stiffness 0.1 N/mm3 - both ends
FE model – Pushover analysis
84
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.
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.
0
10
20
30
40
50
60
0 5 10 15 20 25 30
Bas
e S
he
ar (
KN
)
Displacements at roof level (mm)
E=10000 N/mm2
E=6000 N/mm2
FE model – NLTHA
85
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:
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.
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 5 10 15
Acc
ele
rati
on
(m
/s2
)
Time (s)
Accelerograms (67% NPR)
S1 - x S1 - y S1 - z Failure
FE model – NLTHA
86
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%.
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.
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.
-0.005
0.000
0.005
0 1 2 3 4
Dri
ft
Time (s)
Drifts (x) vs time
First floor Second floor Roof level
-0.005
0.000
0.005
0 1 2 3 4
Dri
ft
Time (s)
Drifts (y) vs time
First floor Second floor Roof level
-60
-40
-20
0
20
40
60
0 1 2 3 4
Bas
e S
he
ar (
KN
)
Time (s)
Base Shear vs time (x)
-200
-100
0
100
200
0 1 2 3 4
Bas
e S
he
ar (
KN
)
Time (s)
Base Shear vs time (y)
FE model – NLTHA
87
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.
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.
0
20
40
60
80
0 1 2 3 4
Cra
ck w
idth
s (m
m)
Time (s)
0.00
0.05
0.10
0.15
0.20
0 1 2 3 4
Ten
sile
str
ain
E1
Time (s)
Maximum tensile strain E1 per 50 steps
-0.08
-0.06
-0.04
-0.02
0.00
0 1 2 3 4Co
mo
pre
ssiv
e s
trai
n E
3
Time (s)
Maximum compressive strain E3 per 50 steps
FE model – NLTHA
88
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.
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.
-60-40-20
0204060
-40 -20 0 20 40
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
Force - displacement curve in x
NLTH
Pushover (+)
Pushover (-)
EF modelling
89
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.
EF modelling
90
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.
EF modelling
91
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.
EF model – Pushover analysis
92
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 Trial 1 Trial 2 Trial 3
Displacement 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.
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.
0
10
20
30
40
50
0 10 20 30
Bas
e S
he
ar (
KN
)
Displacement at second floor level (mm)
Trial 1
Trial 2
Trial 3
EF model – Pushover Analysis
93
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.
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.
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 (mm) (mm) (rad) (rad) H (mm) Element Drift (%)
52 0 20.9 0 0 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:
0
2
4
6
8
10
12
14
16
0 5 10 15 20 25
She
ar f
orc
e (
KN
)
Displacements at top node (mm)
0
5
10
15
20
0 5 10 15 20 25
Be
nd
ing
mo
me
nts
(K
Nm
)
Displacements at top node (mm)
0
10
20
30
40
50
60
0 5 10 15 20 25
No
rmal
fo
rce
s (
KN
m)
Displacements at top node (mm)
EF model – Pushover analysis
94
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 (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
95
Analysis in y direction
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.
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.
0
50
100
150
200
250
300
0 2 4 6 8
Bas
e S
he
ar (
KN
)
Displacement at second floor level (mm)
EF model – Pushover analysis
96
Assessment
97
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
EF Model - 1 0.189 52.5 5 0.051 62.36
Assessment
98
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.
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)
0
10
20
30
40
50
60
0 5 10 15 20 25
Bas
e S
he
ar (
KN
)
Displacement at second floor level (mm)
Case 1Case 3EF model
Assessment
99
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.
Assessment
100
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 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
101
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 ( )
Target displacement ( )
u.c.
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
√
Assessment
102
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 - -
Retrofitting
103
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
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.
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.
0
5
10
15
0 1 2 3 4
Gro
un
d A
cce
lera
tio
n S
(T)
(m/s
2)
Period (s)
Design spectrum for q=2
Structure period
Design spectrum for q=3
Elastic spectrum
Retrofitting
104
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)
Retrofitting
105
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:
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
0
20
40
60
80
0 5 10 15 20 25 30
Bas
e S
he
ar [
KN
]
Displacements [mm]
Pushover curves (x)
Longitudinally connected
Case 3 - E=6000 N/mm2
Retrofitting
106
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 %.
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)
0
100
200
300
400
500
600
0 5 10 15 20
Bas
e S
he
ar [
KN
]
Displacements at roof level [mm]
Pushover curve (y)
Longitudinally connected
Case 3 - E=6000 N/mm2
Retrofitting
107
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.
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.
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25 30
Bas
e S
he
ar [
KN
]
Displacements [mm]
Longitudinally connected
Case 3 - Extra plank 80 mm
Case 3 - Extra plank 40 mm
Case 3 - E=6000 N/mm2
Retrofitting
108
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.
Figure 121: Capacity curves for strengthening with steel frames.
.
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60
Bas
e S
he
ar [
KN
]
Displacements at roof level [mm]
Configuration 1
Configuration 2
Configuration 3
Longitudinally connected
Retrofitting
109
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.
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
0
50
100
150
200
250
0.00 0.01 0.02Stre
sse
s Sx
x (N
/mm
)
Strains Exx
Developedstress-strain atsteel elements
Theoreticaldiagram
Retrofitting
110
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
Symbol Units Value
Profile - - 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.
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.
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60
Bas
e S
he
ar [
KN
]
Displacements at roof level [mm]
Configuration 1
Longitudinal connected
Exceedance of compressive strength
Interstory drift limit 1 % of 2nd floor
Exceedance of tensile strength
Cracks 5mm
Inerstory drift 0.5 % of 1st floor
Retrofitting
111
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 Value
Drift limit of first floor -
Drift limit of second floor -
Crack widths 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.
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.
Figure 127: Capacity curve of masonry with and without steel. – Configuration 1
0
50
100
150
200
250
300
0 20 40 60
Bas
e S
he
ar [
KN
]
Displacements at roof level [mm]
Configuration 1
Masonry
Steel
0
20
40
60
80
100
120
0 5 10 15 20 25
Bas
e S
he
ar [
KN
]
Displacements at roof level [mm]
Masonry - No reinforcementMasonry - Configuration 1
Retrofitting
112
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
Unity check
Case 2 - Stiffness at both ends
30 38
Configuration 1 31 24
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 67 % NPR
Behaviour factor -
Period of structure
Elastic spectral acceleration
Inelastic spectral acceleration
⁄
Mass
Demanded Base Shear
Resisted Base Shear
Unity check - -
Retrofitting
113
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.
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)
Symbol Units Value
Profile - - IPE300
Yield strength
Elastic section modulus
Partial factor -
Design elastic moment
Maximum developed moment
Unity check - -
0
50
100
150
200
250
0.00 0.01 0.02
Stre
sse
s Sx
x
Strains Exx
Developedstress-strain atsteel elements
Theoreticaldiagram
Retrofitting
114
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.
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.
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60
Bas
e S
he
ar [
KN
]
Displacements at roof level [mm]
Configuration 1Configuration 2
Cracks 5mm
Interstorey drift limit 0.5 % of 2nd floor Interstorey drift
limit 0.5 % of 1st floor
Retrofitting
115
Table 47: Unity check of Base Shears. – Configuration 2 (100% NPR)
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 - -
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 Value
Drift limit of first floor -
Drift limit of second floor -
Crack widths mm
Retrofitting
116
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 Value
Drift limit of first floor -
Drift limit of second floor -
Crack widths mm
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)
Model Ultimate displacement
Target displacement
Unity check
Case 2 - Stiffness at both ends
30 38 ⁄
Configuration 3 37 24 ⁄
Retrofitting
117
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 100 % NPR 67 % NPR
Behaviour factor -
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.
Figure 132: Crack widths versus drift limits.
0
10
20
30
40
50
60
0.000 0.005 0.010
Cra
ck w
idth
[m
m]
Interstory drift limit at first floor level
Configuration 1
Configuration 2
Configuration 3
Retrofitting
118
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.
Figure 133: Interstory drifts versus time in the x (left) and y (right) direction. – Configuration 1
Figure 134: Comparison between Pushover and NLTH. – Configuration 1
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.
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.
-0.002
0.000
0.002
0 1 2 3 4
Dri
ft
Time (s)
First floor Second floor Roof level
-0.005
0.000
0.005
0 1 2 3 4
Dri
ft
Time (s)
First floor Second floor Roof level
-300
-200
-100
0
100
200
300
-60 -10 40
Bas
e S
he
ar (
KN
)
Displacement at roof level (mm)
0
50
100
0.00 1.00 2.00 3.00 4.00
Cra
ck w
idth
s (m
m)
Time (s)
Conclusions
119
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
Conclusions
120
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
Approaches & Models Pushover analysis NLTHA**
Nu
mb
er
Ap
pro
ach
Mo
del
Dir
ecti
on
Det
ails
Cap
acit
y (K
N)
Cri
tica
l fai
lure
m
od
e
Du
ctili
ty
Beh
avio
ur
fact
or
u.c
.
dis
pla
cem
ents
–
10
0%
NP
R
u.c
. cap
acit
y –
10
0%
NP
R
u.c
. cap
acit
y –
67
% N
PR
Max
imu
m
cap
acit
y (K
N)
Max
imu
m d
rift
(%
)
Max
imu
m c
rack
w
idth
s (m
m)
1 FE Case 1 x Unconnected 37 Out of plane
- - - - - 50 0.5 40
2 FE Case 2* x Stiffness at one end
40 Shear - - - - - - - -
3 FE Case 2* x Stiffness
at one end 44 Shear - - - - - - - -
4 FE Case 2* x Stiffness
at both ends 44 Shear 7.5 3.7 1.27 4.48 3 - - -
5 FE Case 3 x Connected 47 Shear - - - - - - - -
6 FE Case 3 x Reduced E modulus
for timber 47 Shear - - - - - - - -
7 EF - x Connected 40 Bending 3.61 2.5 1.69 - - - - -
8 Analytical - x No flange effect
considered 10 Bending - - - - - - - -
9 Analytical - x Flange effect considered
40 Bending - - - - - - - -
10 FE Case 3 y Connected 227 Out of plane
- - - - - - - -
11 FE - Y 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.
Conclusions
121
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.
Pushover analysis
4. Steel frames
3. Improved in-plane stiffness2. Addition of connections
1. Improvement of existing connections
Case 1:Non connected
Case 3:Connected
Longitudinally connected
Reduced timber E modulus
Addition of boards
- 40 mm- 80 mm
Configuration 1 Configuration 2 Configuration 3
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.
Conclusions
122
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
Approaches & Models Pushover NLTHA**
Nu
mb
er
Mo
del
Dir
ecti
on
Det
ails
Cap
acit
y (K
N)
Du
ctili
ty
Beh
avio
ur
fact
or
u.c
. d
isp
lace
men
ts
– 1
00
% N
PR
u.c
. cap
acit
y –
10
0%
NP
R
u.c
. cap
acit
y –
67
% N
PR
Max
imu
m
Cap
acit
y (K
N)
Max
imu
m
dri
fts
(%)
Max
imu
m
crac
k w
idth
s (m
m)
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 - - - - - - - -
13 Addition of connections
x Connections between facades &
floors
75 - - - - - - - -
14 y 500 - - - - - - - -
15 Improved in-plane stiffness
x Extra plank 40mm 53 - - - - - - - -
16 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 - - -
*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.
Conclusions
123
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.
Conclusions
124
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:
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
125
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
Definitions
126
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 is governed by the following
formula:
Definitions
127
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.
Appendixes
128
Appendix A: Dead loads calculation
Table 55: Calculation of floor weight.
Timber floor weight
beams
width 0.071
height 0.196
length 6.92
Number - 12
Volume 1.156
plank
width 5.65
height- top plank 0.022
height - bottom plank 0.022
length 6.92
Volume 1.72
Total Volume 2.88
density 500
g 9.81
Weight 14106
q 0.36
Table 56: Calculation of roof weight.
Timber roof weight
beams
width 0.071
height 0.196
length 6.92
Number - 8
Volume 0.77
ridge beam
width 0.071
height 0.246
length 6.92
Number - 1
Volume 0.121
planks
width 4.385
height 0.022
length 6.92
Number - 2
Volume 1.34
Total volume 2.23
Density kg/m3 500
g 9.81
Weight 10920
q wood 0.28
Ceramic tiles 0.50
q total 0.78
Appendixes
129
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 from 2
nd 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
For ⁄ ⁄
Axial compression stress due to gravity
calculated at the base of the pier
⁄
Masonry diagonal
Appendixes
130
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.
Appendixes
131
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
Rocking capacity
3724 5026 14598 8263 5071 3724
Bed-joint sliding capacity
21885 24463 43079 29722 24591 21885
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 .
Appendixes
132
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 per consequence classes.
Appendixes
133
Table 61: Consequence classes parameters.
Figure 140: Selected PGA in analysis. (Ontw. NPR 9998, February 2015)
Appendixes
134
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
The horizontal elastic spectrum based on the above mentioned formulas and parameters is presented
below:
Figure 141: Horizontal elastic response spectrum.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 1 2 3 4
Se (
g)
Period T (s)
Appendixes
135
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 Sa (m/s2) Sd (m)
0 5.04 0.000
0.1 15.12 0.004
0.1 15.12 0.004
0.22 15.12 0.019
0.22 15.12 0.019
0.25 13.31 0.021
0.30 11.09 0.025
0.35 9.50 0.029
0.40 8.32 0.034
0.45 7.39 0.038
0.45 7.39 0.038
0.50 5.99 0.038
0.60 4.16 0.038
0.70 3.05 0.038
0.80 2.34 0.038
0.90 1.85 0.038
1.00 1.50 0.038
1.10 1.24 0.038
1.20 1.04 0.038
1.30 0.89 0.038
1.40 0.76 0.038
1.50 0.67 0.038
1.60 0.58 0.038
1.70 0.52 0.038
1.80 0.46 0.038
1.90 0.41 0.038
2.00 0.37 0.038
2.10 0.34 0.038
2.20 0.31 0.038
2.30 0.28 0.038
2.40 0.26 0.038
2.50 0.24 0.038
2.60 0.22 0.038
2.70 0.21 0.038
2.80 0.19 0.038
2.90 0.18 0.038
3.00 0.17 0.038
3.10 0.16 0.038
3.20 0.15 0.038
3.30 0.14 0.038
3.40 0.13 0.038
3.50 0.12 0.038
3.60 0.12 0.038
3.70 0.11 0.038
3.80 0.10 0.038
3.90 0.10 0.038
4.00 0.09 0.038
Appendixes
136
Transformation of system to an equivalent SDOF system
∑ (
) (
)
∑
( ⁄ ) (
⁄ )
Table 64: Equivalent SDOF capacity curve.
Fb (KN) dn (m) F* (KN) d* (m)
0 0.000 0 0.0000
5 0.000 4 0.0003
9 0.001 7 0.0007
14 0.001 11 0.0010
18 0.002 15 0.0014
23 0.002 19 0.0017
27 0.003 23 0.0021
32 0.003 26 0.0025
36 0.004 30 0.0032
41 0.005 34 0.0046
45 0.008 37 0.0068
45 0.010 38 0.0084
45 0.011 38 0.0094
47 0.014 39 0.0119
47 0.018 39 0.0152
46 0.022 38 0.0187
45 0.026 37 0.0220
45 0.030 37 0.0254
Idealized elasto-perfectly plastic force-displacement
Table 65: Idealized curve.
F* (KN) d* (m) F*/m* (m/s2)
0 0 0.00
0.0022 0.63
36 0.0034 0.97
36 0.0254 0.97
The resulting bilinear relation is illustrated in the following figure:
Figure 142: Capacity curves and bilinear representation of SDOF until drift limit of 0.5 %.
0
10
20
30
40
50
0.000 0.020 0.040Forc
e o
f t
SDO
F sy
ste
m (
F*)
Dispalcement of equivalent SDOF system (d*)
Capacity curve - SDOFBilinear approximationCapacity curve - MDOF
Appendixes
137
Period of SDOF
√
√
SDOF Target Displacement
[
]
[
]
For and
:
Figure 143: Capacity curve of SDOF and spectrum
MDOF Target Displacement
0
5
10
15
20
0.00 0.01 0.02 0.03 0.04
Acc
ele
rati
on
(Se
(T))
Displacement of equivalent SDOF system (d*)
Spectrum Capacity curve of SDOF
Appendixes
138
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.
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.
Figure 144: Convergence characteristics. – Case 1 (x)
Figure 145: Convergence characteristics. – Case 3 (x)
0
10
20
30
40
50
0 20 40 60 80
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear
0.000
0.001
0.010
0.100
1.000
10.000
0 20 40 60 80
Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
10
20
30
40
50
60
0 10 20 30 40
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.000
0.001
0.010
0.100
1.000
10.000
0 10 20 30 40Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
Appendixes
139
Figure 146: Convergence characteristics. – Case 1 (y)
Figure 147: Convergence characteristics. – Case 2 (Stiffness 0.01 N/mm3)
Figure 148: Convergence characteristics. – Case 2 (Stiffness 0.1 N/mm3)
Figure 149: Convergence characteristics. – Case 2 (Stiffness 0.1 N/mm3 at both ends)
0
50
100
150
200
250
0.00 0.10 0.20 0.30 0.40 0.50
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.000
0.001
0.010
0.100
1.000
10.000
0.00 0.20 0.40Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
10
20
30
40
50
0 5 10 15 20
Forc
e [
KN
]
Resultant displacements [mm]
Resultant base shear
Applied force
0.000
0.001
0.010
0.100
1.000
10.000
0 10 20
Dis
pla
cme
nts
var
iati
on
Displacements [mm]
0
10
20
30
40
50
0 10 20 30
Forc
e [
KN
]
Resultant displacements [mm]
Resultant base shear
Applied force
0.000
0.001
0.010
0.100
1.000
10.000
0 10 20 30
Dis
pla
cme
nts
var
iati
on
Displacements [mm]
0
10
20
30
40
50
0 10 20 30 40
Forc
e [
KN
]
Resultant displacements [mm]
Resultant base shear
Applied force
0.000
0.001
0.010
0.100
1.000
0 20 40
Dis
pla
cme
nts
var
iati
on
Displacements [mm]
Appendixes
140
Figure 150: Case 3 – Reduced stiffness.
Figure 151: Convergence characteristics. – Connection longitudinally (x)
Figure 152: Convergence characteristics. – Connection longitudinally (y)
Figure 153: Convergence characteristics. – Plank 40mm
0
10
20
30
40
50
60
0 10 20 30
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.000
0.001
0.010
0.100
1.000
10.000
0 10 20 30Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
20
40
60
80
0 5 10 15 20
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.000
0.001
0.010
0.100
1.000
10.000
0 5 10 15 20
Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
100
200
300
400
500
600
0 5 10 15 20 25
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear
0.000
0.001
0.010
0.100
1.000
10.000
0 5 10 15 20Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
10
20
30
40
50
60
0 10 20 30
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.000
0.001
0.010
0.100
1.000
10.000
0 10 20 30Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
Appendixes
141
Figure 154: Convergence characteristics. – Plank 80mm
Figure 155: Convergence characteristics for Steel frames. - Configuration 1
Figure 156: Convergence characteristics for Steel frames. - Configuration 2
Figure 157: Convergence characteristics for Steel frames. - Configuration 3
0
10
20
30
40
50
60
70
0 2 4 6
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear
0.001
0.010
0.100
1.000
10.000
0 10 20 30
Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
50
100
150
200
250
300
350
0 20 40 60
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.001
0.010
0.100
1.000
0 20 40 60Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
50
100
150
200
250
300
0 20 40 60
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.000
0.001
0.010
0.100
1.000
0 20 40 60Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
0
50
100
150
200
250
300
350
0 20 40 60
Forc
e [
KN
]
Dispalcement [mm]
Applied force
Resultant base shear0.001
0.010
0.100
1.000
0 20 40Dis
pla
cem
en
ts V
aria
tio
n
Displacement [mm]
Appendixes
142
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.
Figure 158: Energy variation at last steps of time history. - Case 1
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.
Figure 159: Energy variation at last steps of time history. - Configuration 1
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
3.65 3.85 4.05
Ene
rgy
vari
atio
n
Time (s)
0.0000
0.0001
0.0010
0.0100
0.1000
1.0000
10.0000
3.30 3.50 3.70 3.90
Ene
rgy
vari
atio
n
Time (s)
Appendixes
143
Appendix E: Case study drawings
Figure 160: Connections of timber beams to cavity walls at roof level.
Figure 161: Longitudinal connection of timber beams.
Appendixes
144
Figure 162: Building plans
References
145
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