non-linear modelling of a geomesh - reinforced

NON-LINEAR MODELLING OF A GEOMESH REINFORCED EARTHEN WALL SUBJECTED TO DYNAMIC LOADING

Dissertation submitted as part requirement for the Degree of Master of Science in Earthquake and Civil Engineering Dynamics

By: Victor Hugo Portugal Quevedo

Supervisor: Professor Kypros Pilakoutas (UoS)

Co-supervisor: Professor Nicola Tarque (PUCP)

The University of Sheffield Department of Civil and Structural Engineering

September 2017

Declaration Statement

The author certifies that all materials produced in this dissertation entitled Non-linear numerical modelling of a geomesh-reinforced earthen wall subjected to dynamic loading is his own, except wherever clearly referenced to others.

Victor Hugo Portugal Quevedo

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Acknowledgements

Research in engineering is always a collective effort, which is why I would like to express my gratitude to all the students, research assistants and professors who helped me achieve my goal of doing interesting and useful research in structural engineering. First, I would like to give a very special thanks to Dr. Marcial Blondet and Dr. Nicola Tarque (both from the Pontifical Catholic University of Peru) for allowing me to have access to the experimental data of the DAI 113.0225 project, without which this dissertation would not have been possible. Furthermore, Dr. Tarque’s insight on earthquake and adobe related issues during our Skype meetings and constant e-mail conversations were extremely valuable for the outcome of this dissertation. Second, I would like to thank my supervisor, Kypros Pilakoutas, professor in the department of Civil and Structural Engineering at the University of Sheffield, whose expertise in seismic engineering was crucial in fully interpreting the numerical results I obtained regarding seismic performance and hysteretic behaviour of the overall adobe structure. Third, I extend my unending gratitude to Reyes García, postdoctoral research associate in the Concrete and Earthquake Engineering Research (CEER) group at the University of Sheffield, for guiding me every step of the way in this dissertation and for always encouraging me to give my best. Additionally, I would like to thank Fernando Cepero, PhD student in the department of Mechanical Engineering at the University of Sheffield, and Fabio Figueiredo, research fellow in the department of Civil and Structural Engineering at the University of Sheffield, for helping me master the finite element programme Abaqus/Standard. Without their help, my research would have been slowed down considerably. Lastly, I would like to thank all my MSc colleagues and friends with whom I shared countless hours of work and fun throughout the master’s programme. Their moral support and personalities fuelled me every step along the way of my work.

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Dedication

I dedicate this to my parents, for always supporting me and guiding me through life. To my brother and sister, for always encouraging me during my rough patches, to achieve all my objectives and for all the joy the bring into my life. To my grandfather, for showing the kind of person I want to be. To my grandmother, for always loving what I did in my career and supporting me during all my endeavours. To my godmother, for all those Sunday lunches across Lima and for always showing me the bright side of life in every possible situation. To all my family spread around the world, for showing me such kindness every time we met. To all my close friends in Peru and Austria, for proving me that friendship cares not about distance or time and for never letting me feel alone half a world away from home. To all my friends in Sheffield, for sharing a little bit of themselves with me and for making me feel at home half a world away from my own. To Peru, my country.

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Abstract

Adobe structures reinforced externally with geogrid mesh have become a viable alternative for improving the seismic vulnerability of adobe dwellings. The ductile properties of the geogrid provide the necessary deformability to the overall structure so that it may resist a moderate earthquake with certain damage and prevent loss of life. This practice has been studied thoroughly, on an experimental basis, in countries like Peru, but an adequate numerical model that simulates the combined cyclic and non-linear behaviour of the adobe and the geogrid reinforcement is still a work in progress. This research aims to improve a previous model of an externally-reinforced adobe wall where the geogrid was idealized as a linear-elastic material with an equivalent initial elastic modulus. Now, the new constitutive law for the geogrid will reproduce a non-linear elasto-plastic stress-strain relationship which will include appropriate unloading-reloading routes that would allow the model to simulate the real behaviour more accurately. These results will be useful for future studies in order to evaluate the seismic performance of reinforced earthen structures.

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Resumen El uso de geomallas como refuerzo en viviendas de adobe ante acciones sísmicas se ha convertido en una alternativa viable y económicamente asequible para poblaciones y regiones vulnerables alrededor de todo el mundo. Las propiedades dúctiles de la geomalla proveen a la estructura frágil la suficiente deformabilidad y confinamiento de manera que pueda resistir un movimiento sísmico con cierta cantidad de daño, pero sin colapso o pérdida de vidas. Esta práctica se ha estudiado extensivamente de manera experimental en países como Perú, pero un modelo adecuado que pueda simular el comportamiento tanto cíclico como no-lineal de la estructura híbrida aún se encuentra en desarrollo. Este trabajo de investigación tiene como objetivo mejorar un previo modelo numérico en Abaqus de un muro reforzado externamente con geomalla, donde esta última fue modelada como un material perfectamente elástico. Ahora, el nuevo modelo numérico incluirá propiedades elastoplásticas no-lineales de la geomalla, las cuales permitirían simular el comportamiento cíclico de manera más certera. Los resultados obtenidos en esta tesis serán de gran uso en futuros estudios, a manera de evaluar el desempeño sísmico de estructuras de adobe reforzadas con mayor facilidad.

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Table of Contents Declaration Statement .......................................................................................................... i Acknowledgements ............................................................................................................. ii Dedication ......................................................................................................................... iii Abstract ............................................................................................................................. iv Resumen ..............................................................................................................................v Table of Contents ............................................................................................................... vi List of Photographs ............................................................................................................ ix List of Figures ......................................................................................................................x List of Graphs ................................................................................................................... xii List of Tables ..................................................................................................................... xv Nomenclature....................................................................................................................xvi 1. Introduction ....................................................................................................................1 1.1. Aims ........................................................................................................................2 1.2. Objectives................................................................................................................2 1.3. Methodology and outline of the dissertation .............................................................3 2. Literature review .............................................................................................................6 2.1. Observed seismic response and damage in adobe buildings: Peru’s earthquake experience in Pisco in 2007 ..............................................................................................6 2.2. Experimental research on seismic strengthening of adobe dwellings .........................8 2.2.1.

Seismic reinforcement of adobe dwellings with bamboo cane..............................9

2.2.2.

Seismic reinforcement of adobe dwellings with electrically-welded steel mesh.. 11

2.2.3.

Seismic reinforcement of adobe dwellings with biaxial geogrid mesh ................ 12

2.3. Compressive and tensile behaviour of adobe .......................................................... 13 2.4. Tensile behaviour of geomesh reinforcement (biaxial geogrid) ............................... 17 2.4.1.

Hyperbolic function with modified Masing rule (Liu and Ling 2006) ................ 19

2.4.2.

Bounding surface model (Liu and Ling 2006) ................................................... 22

3. Cyclic pushover test of a geomesh-reinforced adobe wall .............................................. 24 3.1. Test specimen: geomesh-reinforced adobe wall ...................................................... 25 3.2. Testing programme and measuring instruments ...................................................... 27 3.3. Experimental results from cyclic pushover test of geomesh-reinforced adobe wall . 29 4. Material constitutive modelling ..................................................................................... 34 4.1. Adobe .................................................................................................................... 34 4.2. Geomesh reinforcement (biaxial geogrid)............................................................... 36

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4.2.1.

Analytical modelling of geomesh reinforcement................................................ 37

4.2.1.1. Calculation of initial elastic modulus (E0) ......................................................... 39 4.2.1.2. Calculation of ultimate tensile strength per meter (UTS) ................................... 41 4.2.1.3. Calibration of hyperbolic function curve ........................................................... 41 4.2.2.

Numerical calibration of geomesh-reinforcement constitutive model ................. 42

4.2.2.1. Modelling the equivalent geometry of geomesh-reinforcement layers ................ 45 5. Numerical modelling of geomesh-reinforced adobe wall ............................................... 53 5.1. Geometry of finite element model .......................................................................... 53 5.2. Material and section modelling .............................................................................. 54 5.3. Section modelling and meshing properties ............................................................. 55 5.4. Steps for analysis ................................................................................................... 56 5.5. Boundary and loading conditions ........................................................................... 58 5.6. Definition of types of analyses ............................................................................... 60 5.6.1.

Cyclic displacement scenario ............................................................................ 60

5.6.2.

Earthquake excitation scenario .......................................................................... 61

6. Calibration of numerical model with cyclic pushover experimental results ..................... 63 6.1. Influence of compressive and tensile damage factors on peak strength values during hysteresis cycles ............................................................................................................. 66 6.2. Influence of compressive and tensile fracture energy on peak strength values during hysteresis cycles ............................................................................................................. 69 6.3. Influence of peak compressive stress on peak strength values during hysteresis cycles 71 6.4. Limitations and inconsistencies of the evaluated FE models ................................... 72 6.4.1.

Cyclic locking................................................................................................... 73

6.4.2.

Precision of simulated strength magnitudes in phases 7, 8 and 9 ........................ 74

6.4.3.

Strength accuracy during negative hysteresis loops ........................................... 75

6.4.4.

Loading and unloading stiffnesses of FE models ............................................... 76

6.5. FE model chosen for evaluation of seismic performance ........................................ 78 7. Seismic performance of FE model subjected to scaled earthquake scenarios .................. 82 7.1. Modal analysis of FE model ................................................................................... 84 7.2. Seismic performance of FE model subjected to phase 1 of scaled earthquake signal (PGA=0.305g and PGD=28.91 mm) ............................................................................... 86 7.3. Seismic performance of FE model subjected to phase 2 of scaled earthquake signal (PGA=0.622g and PGD=53.71 mm) ............................................................................... 88 7.4. Seismic performance of FE model subjected to phase 3 of scaled earthquake signal (PGA=0.938g and PGD=78.51 mm) ............................................................................... 90 7.5. Seismic performance of FE model subjected to phase 4 of scaled earthquake signal (PGA=1.254g and PGD=103.31 mm) ............................................................................. 93 7.6. Seismic performance of FE model subjected to phase 5 of scaled earthquake signal (PGA=1.57g and PGD=128.11 mm) ............................................................................... 93 vii

8. Conclusions .................................................................................................................. 94 9. References .................................................................................................................... 97 10. Annexes ...................................................................................................................... 102 10.1. Annex 1: Hysteretic curves of evaluated FE models ............................................. 103 10.2. Annex 2: Material and numerical model properties of evaluated FE models ......... 109 10.3. Annex 3: Sets of damage factors used in FE models ............................................. 110 10.4. Annex 4: Time-histories for ground displacement and acceleration of scaled seismic signals based on Peruvian earthquake of May 1970 ...................................................... 112 10.5. Annex 5: Seismic map of Peru ............................................................................. 113

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List of Photographs

Photograph 2.1: Typical in-plane shear failure (left) and out-of-plane failure of façades in adobe buildings in Pisco after the 2007 earthquake (Source: Blondet et al. 2008b) ................6 Photograph 2.2: Vertical cracks along junctions of wall which failed in an out-of-plane manner (left) and diagonal cracks along the adobe wall related to in-plane shear failure (right) .........7 Photograph 2.3: Wooden beam used to tie up the ends of the vertical cane used as internal reinforcement (Source: Blondet et al. 2008a) ...................................................................... 10 Photograph 2.4: Installation of electrically-welded steel mesh reinforcement by nailing (left) and finished reinforced adobe structures with cement mortar covering the mesh (right) (Source: Blondet et al. 2008a) ............................................................................................ 11 Photograph 2.5: Geomesh-reinforced structure prior to shake table test (left) and after (right) (Source: Blondet et al. 2006) .............................................................................................. 12 Photograph 2.6: Uniaxial compression tests performed to determine peak compressive strength in an adobe brick assemblage (left) and in a single unit (right) (Source: Peralta and Torrealva 2009, left, and Illampas et al. 2014, right) ........................................................................... 15 Photograph 2.7: Real diagonal compression test performed on adobe assemblage (left) and description of the components of equation used to estimate ft (Source: Peralta and Torrealva 2009 and NTP E.080 Adobe 2006) ..................................................................................... 17 Photograph 3.1: Geomesh-reinforced adobe wall used in cyclic pushover test (Source: Blondet et al. 2005) ......................................................................................................................... 24 Photograph 3.2: Construction of adobe wall with alternated rows of bricks (Source: Blondet et al. 2005) ......................................................................................................................... 25 Photograph 3.3: Damage sustained during phase 6 (left). Close-up view of stepped cracks on the right side of the in-plane wall (right) (Source: Blondet et al. 2005)................................ 30 Photograph 3.4: Considerable crack growth during phase 8 (50 mm peak displacement) (Source: Blondet et al. 2005) .............................................................................................. 30 Photograph 3.5: Damage sustained after phase 9. Collapse was avoided. (Source: Blondet et al. 2005)............................................................................................................................. 31 Photograph 3.6: Damage sustained by the geomesh-reinforcement during phase 9 of the testing programme (100 mm peak displacement). Including some areas of great plastic deformation and rupture (Source: Blondet et al. 2005)............................................................................ 31 Photograph 4.1: Uniaxial tensile test performed on a sample of polypropylene geogrid BX1100 in the Structures Laboratory at PUCP (Source: Blondet et al. 2005) .................................... 38 Photograph 4.2: Bounding of geomesh-reinforcement layers through the adobe wall with the use of raffia strips at discrete locations (Source: Blondet et al. 2005) .................................. 45

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List of Figures

Figure 2.1: Arrangement of bamboo cane reinforcement both in the vertical and horizontal direction (Source: Blondet et al. 2008a) .............................................................................. 10 Figure 2.2: Stress-strain relationships of type A (a) and type B (b) geogrids (Liu and Ling 2006) ................................................................................................................................. 19 Figure 3.1: Sketch of dimensions of adobe wall .................................................................. 26 Figure 3.2: Frontal view of arrangement of geogrid layers in geomesh-reinforced adobe wall .......................................................................................................................................... 27 Figure 3.3: Lateral view of arrangement of geogrid layers in geomesh-reinforced adobe wall .......................................................................................................................................... 27 Figure 3.4: Position of LVDTs installed on adobe walls ..................................................... 28 Figure 4.1: Definition of inelastic compression curve of adobe using the CDP model (SIMULIA 2008) ............................................................................................................... 35 Figure 4.2: Definition of inelastic tensile softening curve of adobe using CDP model (SIMULIA 2008) ............................................................................................................... 36 Figure 4.3: Elastic properties of constitutive model for geomesh-reinforcement (Source: personal collection) ............................................................................................................ 43 Figure 4.4: Plastic properties of constitutive model for geomesh-reinforcement (Source: personal collection) ............................................................................................................ 43 Figure 4.5: Composite layup window in Abaqus/Standard .................................................. 44 Figure 4.6: Transformation of explicit geometry of geomesh-reinforcement into an equivalent sheet, with a constant reduced thickness ............................................................................. 46 Figure 4.7: Convergence study performed on equivalent geogrid sheet using approximate element sizes ranging from 100 mm (left) to 1.5 mm (right) ............................................... 47 Figure 5.1: Isometric (left) and frontal view (right) of the geometry of the FE model in Abaqus/Standard ................................................................................................................ 54 Figure 5.2: Assignment of composite layup sections in in-plane and perpendicular adobe walls (left) and mesh of GRAW (right) ........................................................................................ 55 Figure 5.3: Steps created in FE model ................................................................................ 57 Figure 5.4: Boundary and loading conditions created in FE model ...................................... 58 Figure 5.5: Boundary conditions associated with the step “Cyclic” ..................................... 59 Figure 5.6: Boundary conditions associated with the step “Earthquake” .............................. 60 Figure 5.7: Characteristic time and strain rates associated with diverse types of loading (Source: Lindholm 1971) ................................................................................................... 61 Figure 6.1: Comparison of effectiveness of confinement action with different varied confinement ratios in reinforced concrete elements (Source: Sulaiman M.F. et al. 2017) ..... 64 Figure 7.1: Mode of vibration #2 (left) which is a translational mode along the X-axis, and mode of vibration #1 (right), which is also a translational mode, but along the Z-axis ......... 85 Figure 7.2: Modes of vibration #16 (left) and #13 (right) were the second-most influential ones along the X-X and Z-Z axes respectively ............................................................................ 86 Figure 7.3: Residual plastic strains in the geomesh layer after phase 1 earthquake. Peak tensile strains were located around the corners of the window opening .......................................... 87 Figure 7.4: Tension damage in adobe continuum after phase 2 EQ ...................................... 89 Figure 7.5: Maximum stresses reached across the geomesh layers ...................................... 89 Figure 7.6: Tension damage sustained in the adobe continuum after phase 3 earthquake ..... 92

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Figure 7.7: Residual stresses in geomesh reinforcement after phase 3 earthquake................ 92 Figure 10.1: Map of past seismic events in Peru from 1960 to 2011 (Source: Instituto Geofísico del Perú) .......................................................................................................................... 113

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List of Graphs

Graph 2.1: Approximation of inelastic compressive stress-strain relation (Source: Illampas et al. 2014)............................................................................................................................. 14 Graph 2.2: Uniaxial compression test results from adobe assemblages of different historical landmarks across Peru (Vargas et al. 2016)......................................................................... 16 Graph 2.3: Uniaxial compression test results from adobe units of different historical landmarks across Peru (Vargas et al. 2016) ......................................................................................... 16 Graph 2.4: Approximation of monotonic and cyclic behaviour in type A geogrids using the hyperbolic function method with modified Masing rule ...................................................... 21 Graph 2.5: Stabilisation of strain hardening in geosynthetics materials after a certain amount of loading cycles (Source: Liu and Ling 2006) ................................................................... 21 Graph 2.6: Interaction of bounding lines with hardening parameters (Source: Liu and Ling 2006) ................................................................................................................................. 23 Graph 3.1: Comparison of GRAW and URAW cyclic behaviour ........................................ 32 Graph 4.1: Uniaxial testing of polypropylene geogrid #2 in the longitudinal (XMD) and transverse (MD) direction................................................................................................... 39 Graph 4.2: XMD nominal stress-strain values ..................................................................... 40 Graph 4.3: Isolation of near-zero values to obtain appropriate values for the calculation of E 0 .......................................................................................................................................... 40 Graph 4.4: Calculation of initial elastic modulus E0 through averaging of initial values of nominal stress-strain curve ................................................................................................. 41 Graph 4.5: Calculation of the value of UTS per meter......................................................... 41 Graph 4.6: Calibration of Hyperbolic function (with modified Masing rule) curve .............. 42 Graph 4.7: Calibration of the equivalent geomesh-reinforcement sheet thickness ................ 46 Graph 4.8: Convergence analysis performed on geogrid sheet with an equivalent thickness of 0.0985mm .......................................................................................................................... 48 Graph 4.9: Calibration of modified geogrid sheet thickness for a 100 mm approximate element size .................................................................................................................................... 49 Graph 4.10: Modified convergence analysis ....................................................................... 49 Graph 4.11: Simplification of unloading and reloading curves into linear expressions and compared to initial elastic linear curve................................................................................ 50 Graph 4.12: Comparison of real monotonic and cyclic tests of a polypropylene type A geogrid (Ling et al. 1998) ............................................................................................................... 51 Graph 4.13: Comparison of cyclic behaviour of geogrid considering different elastic moduli with real monotonic tensile behaviour ................................................................................ 52 Graph 6.1: Expected increase in peak compressive strength and deformation capacity of the adobe continuum ................................................................................................................ 65 Graph 6.2: Hysteretic curve of GRAW-COMP-#6 FE model which did not include damage factors either in compression or tension .............................................................................. 66 Graph 6.3: Hysteretic curve of FE model GRAW-COMP-#9. This FE model included tensile damage factors defined in Table 6.1 ................................................................................... 67 Graph 6.4: Comparison of hysteresis curves from FE models GRAW-COMP-#9 and #10 after change in compressive damage factors ............................................................................... 69 Graph 6.5: Influence of increase in tensile fracture energy from 0.10 N/mm (GRAW-COMP#25) to 0.125 N/mm (GRAW-COMP-#22) and 0.02 N/mm (GRAW-COMP-#20) .............. 70

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Graph 6.6: Influence of tensile fracture energy in GRAW when subjected to monotonic loading ............................................................................................................................... 71 Graph 6.7: Influence of increase of compressive fracture energy from 0.1426 N/mm (GRAWCOMP-#14) to 1.02 N/mm (GRAW-COMP-#18)............................................................... 71 Graph 6.8:Effect of increasing peak compressive strength in the adobe continuum from 0.40 MPa (GRAW-COMP-#12) to 1.0 MPa (GRAW-COMP-#13) ............................................. 72 Graph 6.9: Cyclic locking during cycles of peak displacements larger than 5 mm ............... 73 Graph 6.10: Increase of cyclic locking effect with increasing cycles of displacement .......... 74 Graph 6.11: Lack of precision of strength values from hysteresis loops of 20 mm onwards . 75 Graph 6.12: Difference in experimental strength magnitudes developed during positive and negative hysteresis loops .................................................................................................... 76 Graph 6.13: Lack of precision of magnitudes of unloading and reloading stiffnesses during the first 3 magnitudes of hysteresis loops. ................................................................................ 76 Graph 6.14: Depiction of unloading stiffness after only suffering deterioration from a single +50 mm hysteresis loop...................................................................................................... 77 Graph 6.15: Inability to adequately model damaged unloading and reloading stiffnesses during early hysteresis cycles ........................................................................................................ 78 Graph 6.16: Comparison of compressive curve chosen for FE model #32 (in yellow) to other compressive curve models used in some of the other FE models ......................................... 79 Graph 6.17: Tensile behaviour of chosen FE model (in light blue) ...................................... 79 Graph 6.18: FE model #32 showed the best fit for positive hysteretic cycles of 2.5 mm, 5 mm, and 10 mm ......................................................................................................................... 81 Graph 7.1: Horizontal ground acceleration time-history record from May 1970’s Peruvian earthquake, component N08W, registered in Lima, Peru ..................................................... 82 Graph 7.2: Scaled earthquake displacement signals based on May 1970’s Peruvian earthquake registered in Lima, Peru (phases 1 to 5) .............................................................................. 83 Graph 7.3: Comparison of modal frequencies and periods of vibration................................ 85 Graph 7.4: Time-history of simulated displacement of roof of GRAW relative to movement of the shaking table during phase 1 earthquake ....................................................................... 87 Graph 7.5: Time-history of simulated base shear of FE model during phase 1 earthquake ... 88 Graph 7.6: Time-history of simulated displacement of roof of GRAW relative to movement of the shaking table during phase 2 earthquake ....................................................................... 88 Graph 7.7: Time-history of simulated base shear of FE model during phase 2 earthquake ... 90 Graph 7.8: Time-history of simulated displacement of roof of GRAW relative to movement of the shaking table during phase 3 earthquake ....................................................................... 91 Graph 7.9: Time-history of simulated base shear of FE model during phase 3 earthquake ... 91 Graph 7.10: Time-history of simulated displacement of roof of GRAW relative to movement of the shaking table during phase 5 earthquake ................................................................... 93 Graph 7.11: Time-history of simulated base shear of FE model during phase 5 earthquake . 93 Graph 10.1: Hysteresis curves of FE models #6 and #9 ..................................................... 103 Graph 10.2: Hysteresis curves of FE models #10 and #10a ............................................... 103 Graph 10.3: Hysteresis curves of FE models #11 and #12 ................................................. 103 Graph 10.4: Hysteresis curve of FE model #10b ............................................................... 104 Graph 10.5: Hysteresis curve of FE model #10d ............................................................... 104 Graph 10.6: Hysteresis curves of FE models #13 and #14 ................................................. 105 Graph 10.7: Hysteresis curves of FE models #15 and #16 ................................................. 105 Graph 10.8: Hysteresis curves of FE models #17 and #18 ................................................. 105 Graph 10.9: Hysteresis curves of FE models #19 and #20 ................................................. 106 Graph 10.10: Hysteresis curves of FE models #22 and 24 ................................................. 106 Graph 10.11: Hysteresis curves of FE models #25 and #26 ............................................... 106 Graph 10.12: Hysteresis curves of FE models #27 and 28 ................................................. 107 xiii

Graph 10.13: Hysteresis curves of FE models #29 and #31 ............................................... 107 Graph 10.14: Hysteresis curves of FE models #33 and #34 ............................................... 107 Graph 10.15: Hysteresis curve of FE model #32 (chosen for seismic analysis) .................. 108 Graph 10.16: Scaled ground displacement and acceleration time-histories of simulation earthquakes ...................................................................................................................... 112

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List of Tables

Table 3.1: Cyclic pushover test phases ............................................................................... 28 Table 4.1: Geometric properties of BX1100 geogrid samples which were tested along the XMD and MD directions .................................................................................................... 38 Table 4.2: Rebar layers window in Abaqus/Standard .......................................................... 44 Table 4.3: Variation of the magnitude of elastic modulus within the geomesh-reinforcement material constitutive model properties ................................................................................ 51 Table 5.1: Arrangement of geomesh-reinforcement layers in FE model .............................. 56 Table 5.2: Example of tabular data used to define amplitude of cycles of displacement during the step “Cyclic” ................................................................................................................ 59 Table 6.1: Tensile damage factors defined for FE model GRAW-COMP-#9 ....................... 68 Table 6.2: Compressive damage factors defined in FE models GRAW-COMP-#9 (left) and #10 (right) .......................................................................................................................... 68 Table 6.3: Tensile damage factors chosen in final FE model #32 ........................................ 80 Table 6.4: Compressive damage factors chosen in final FE model #32 ................................ 80 Table 7.1: Modes of vibration considered in the analysis for X-X and Z-Z axes .................. 84 Table 10.1: Material and numerical properties of FE models evaluated in cyclic pushover analysis ............................................................................................................................ 109 Table 10.2: Sets #1 and #2 of tensile damage factors used in FE models ........................... 110 Table 10.3: Sets #1 and #2 of compressive damage factors used in FE models .................. 110 Table 10.4: Sets #3 and #4 of compressive damage factors used in FE models .................. 110 Table 10.5: Sets #5 and #6 of compressive damage factors used in FE models .................. 110 Table 10.6: Sets #7 and #8 of compressive damage factors used in FE models .................. 111 Table 10.7: Sets #9 and #10 of compressive damage factors used in FE models ................ 111 Table 10.8: Sets #11 and #12 of compressive damage factors used in FE models .............. 111

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Nomenclature

GRAW: Short for geomesh-reinforced adobe wall URAW: Short for unreinforced adobe wall FE: Short for finite element Mw: Moment magnitude in the Richter scale of a seismic event PGD (mm): Peak ground displacement of either earthquake or shake table signal PGA (g): Peak ground acceleration of either earthquake or shake table signal E0 (MPa): Initial elastic modulus of type A geogrid a0 (m/kN): Initial virtual stiffness of type A geogrid J0 (m/kN): Initial stiffness of type A geogrid e (mm): Rib thickness of geogrid b0 (m/kN): Half the value of virtual ultimate tensile strength b (m/kN): Virtual ultimate tensile strength UTS (kN/m): Ultimate tensile strength per meter T (kN/m): Tensile strength per meter of geogrid T (kN): Initial elastic modulus of type A geogrid f’c (MPa): Peak compressive strength of concrete fc (MPa): Peak compressive strength of adobe Gfc (N/mm): Compressive fracture energy of adobe ft (MPa): Peak tensile strength of adobe Gft (N/mm): Tensile fracture energy of adobe εcm (mm/mm): Inelastic compressive strain of adobe at which half the peak compressive value is reached during the softening curve xvi

εcp (mm/mm): Inelastic compressive strain of adobe at peak compressive stress dc (adimensional): Compressive damage factor dt (adimensional): Tensile damage factor Ed (MPa): Damaged elastic modulus JT (kN/m): Total stiffness Je (kN/m): Elastic stiffness Jp (kN/m): Plastic stiffness Jp+(kN/m): Constant plastic stiffness of tensile or compressive bounding line Jp- (kN/m): Constant plastic stiffness of compressive bounding line

T+ (kN/m): Tension values along the tensile bounding line T- (kN/m): Minimum tension values along the unloading (compressive) bounding line A (kN/m): Intercept values of the tensile bounding lines along the T-axis B (kN/m): Intercept values of the tensile bounding lines along the T-axis εel (mm/mm): Elastic strain εpl (mm/mm): Plastic strain hPL (adimensional): Hardening parameter for plastic stiffness during primary loading hUL (adimensional): Hardening parameter for plastic stiffness during unloading hRL (adimensional): Hardening parameter for plastic stiffness during reloading h0L (adimensional): Monotonic hardening parameter hkL (adimensional): Modified monotonic hardening parameter

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading

1. Introduction In Peru and many developing countries around the globe, adobe is viewed as a suitable construction material for simple housing structures suited for small families where other more resistant materials, like reinforced concrete or masonry, are not readily available. In fact, in most cases, there is simply no other alternative but to build with soil, since all other construction materials, such as reinforced concrete, fired clay bricks, wood, among others, have too high a cost (Blondet et al. 2011). Due to the convergence of the Nazca and South American tectonic plates, seismic activity is a part of everyday life and a crucial factor during structural design in Peru. This seismic hazard is contemplated in the current Peruvian seismic design code which divides the country into seismic regions based on geological and historical data of past earthquakes (see Annex 10). Depending on the region, peak ground accelerations may reach values of 0.45g, which is then amplified by the structure itself, reaching acceleration values of up to 1.50g during catastrophic events. Naturally, under these overwhelming conditions of seismic danger, unreinforced earthen structures fall incredibly short in terms of seismic performance, leading to the collapse of the structure and, consequently, to loss of life. So why still build with a material which provides structures with such underwhelming lateral resistance properties? Because, as said before, for most people that build with adobe, there is no other alternative. That is why, over the last decades, a lot of research has been conducted in developing new methods of seismic strengthening of adobe structures to provide security to its occupants during intermediate and severe seismic events. Many of said research has been pioneered by Peruvian academic and governmental institutions, such as the Pontifical Catholic University of Peru, the National University of Engineering and the Ministry of Housing (San Bartolomé et al. 2004), among others. Among these seismic strengthening methods, the one that stood out the most was one that used biaxial geogrid as an external mesh-reinforcement wrapped around the adobe structure. When tested in Blondet et al. (2005 and 2006), this reinforcement method provided sufficient ductility to the semi-brittle adobe structure for it to sustain damage in a controlled manner and avoid collapse even during an extreme simulated seismic event. In fact, Blondet et al. (2005) showed that enhancement in ductility allowed the structure to reach its peak strength even during very large displacements, which would have been impossible to achieve if unreinforced.

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Introduction Based on the success geomesh-reinforced adobe structures had during dynamic and quasistatic tests, it was deemed critical to attempt to create a numerical model which would allow to recreate such a ductile behaviour in an adobe structure, so that it may be used as a stepping stone towards the creation of a design guideline for geomesh-reinforced adobe structures. For if an adequate method is found to accurately model the geomesh-reinforced behaviour of adobe structures, their most important disadvantages could be corrected and more value could be placed upon its numerous advantages, among which are: •

Raw materials are very abundant and have a very low cost of extraction and processing

•

Simple and fast construction

•

Little expertise or supervision needed

•

Eco-friendly construction with near zero carbon footprint

•

Great durability

•

Good thermal properties

•

Compatibility of geomesh-reinforcement with the adobe material

Based on the information presented, the aims, objectives and methodology of this dissertation are described in the rest of this first chapter.

1.1. Aims The purpose of this dissertation is to acquire a better comprehension about the numerical modelling of geomesh-reinforced adobe structures so that it may represent seismic loading of real structures within reasonable accuracy and, subsequently, prove the effectiveness of geogrid mesh as a seismic reinforcement method. Specifically, this thesis work aims to accomplish the following: •

Show the advantages of geomesh-reinforcement as a seismic strengthening method in adobe dwellings obtained from empirical testing.

•

Evaluate the effectiveness of finite element modelling when evaluating the seismic performance of geomesh-reinforced adobe structures.

1.2. Objectives To achieve the proposed aims in sub-chapter 1.1, the following objectives were identified: •

Investigate the advantages and disadvantages of available reinforcement for adobe dwellings. Page 2

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading •

Extract and interpret the experimental results obtained from cyclic pushover tests of a geomesh-reinforced adobe wall to use them as a basis to calibrate the numerical results.

•

Define appropriate constitutive models for both the adobe bricks and geomeshreinforcement in Abaqus.

•

Create a finite element model that enables the simulation of cyclic behaviour of a geomesh-reinforced adobe wall in Abaqus.

•

Calibrate the numerical values with the experimental data obtained from the cyclic pushover test performed in Project DAI 113.0225.

•

Based on the calibration of the numerical results, define the main material mechanical properties that allow the finite element model to show good cyclic behaviour.

•

Evaluate the seismic performance of the calibrated finite element model when subjected to a regular and strong earthquake.

Each of the above-mentioned objectives represents a specific milestone along the elaboration of this dissertation. These objectives associate to a chapter within the dissertation, which describe all the work performed to reach them. The upcoming subchapter describes the methodology used during the research and modelling parts of this work.

1.3. Methodology and outline of the dissertation The fulfilment of the second aim of this work demanded a lot of research, analysis and FE modelling. As future research in this area is very important to fully comprehend how to model cyclic behaviour of combined materials, especially where there is little information about their inelastic behaviour, the methodology of this work is described in a way which makes replication of these conditions simple and direct. It consisted mainly of qualitative and quantitative analysis of the mechanical behaviour of 2 different materials which were combined to attempt to predict combined cyclic behaviour when subjected to cyclic loads. This process consisted of 3 parts: •

Acquisition and analysis of experimental data: Since the model had to be calibrated in order for its properties to reproduce results with reasonable accuracy, experimental test data was obtained from a cyclic pushover test performed on a geomesh-reinforced adobe wall as part of the DAI 113.0225 project (Blondet et al. 2005) at the Pontifical Catholic University of Peru. The author of this dissertation

Page 3

Introduction obtained the experimental test data with permission from Professor Nicola Tarque, a current professor in the faculty of Civil Engineering at PUCP. •

Investigation and analysis of current theory describing cyclic behaviour of both adobe and geogrid materials, as well as possible constitutive models for input in Abaqus/Standard.

•

Calibration of numerical model parameters with the acquired experimental data and seismic evaluation of the final FE model in Abaqus/Standard.

Based on the methodology and all the information stated throughout this chapter, it was decided that the organisation of the work presented in this document should be able to resemble the objectives presented in sub-chapter 1.2 in chronological order. This was because the information needed to achieve each chapter’s objective was presented in the previous section. Therefore, the outline of this dissertation is as follows. •

Chapter 2: Literature review. All the research regarding seismic performance of reinforced and unreinforced adobe dwellings, as well as mechanical properties of adobe and geomesh-reinforcement are presented in this chapter. The necessary theory needed to develop the constitutive material models of the finite element model in Abaqus was based on the information displayed in the literature review.

•

Chapter 3: Cyclic pushover test of a geomesh-reinforced adobe wall. This chapter contains all the information regarding the pushover cyclic test performed on a geomesh-reinforced adobe wall at the Pontifical Catholic University of Peru by Blondet et al. (2005). All the force-displacement data needed to calibrate the numerical results were extracted from project DAI 113.0225 (Blondet et al. 2005) and analysed in this section.

•

Chapter 4: Material constitutive modelling. Here, the constitutive material models are created and calibrated for use in Abaqus/Standard. These two material models were needed to create the complete numerical model, which is described in chapter 5.

•

Chapter 5: Numerical model of geomesh-reinforced adobe wall. All the work relative to creation of the numerical model in Abaqus/Standard is shown in this chapter. This includes the definition of geometric, section, material and mesh properties, as well as loading and boundary conditions necessary to simulate a cyclic lateral pushover test and an earthquake response.

•

Chapter 6: Calibration of numerical model with cyclic in-plane test results. The data obtained all the numerical analysis scenarios are evaluated in this section and the properties of the FE model calibrated based on the experimental results from chapter 3. Page 4


Chapter 7: Seismic performance of numerical model subjected to a real earthquake signal. Based on the calibrated numerical data obtained after calibration in chapter 6, the final FE model is used to evaluate the response of the structure after being subjected to a regular and extreme seismic event.

•

Chapter 8: Conclusions. This final chapter contains all the remarks and comments about the work process and the results obtained during this dissertation.

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Literature review

2. Literature review The effectiveness of numerical modelling in the prediction of the hysteretic and seismic response of a reinforced adobe structure depends on the designer’s understanding of all underlying topics. In this case, it was imperative that a thorough investigation of both empirical and theoretical knowledge about the mechanical and seismic behaviour of adobe structures, as well as geogrid monotonic and cyclic properties, was performed to be able to adequately explain seismic behaviour of a reinforced adobe structure, as well as model it numerically. This chapter includes all the background knowledge used to successfully create the GRAW numerical model in Abaqus/Standard.

2.1. Observed seismic response and damage in adobe buildings: Peru’s earthquake experience in Pisco in 2007 As stated earlier in this section, un-reinforced adobe structures have very poor seismic performance which, in cases of a severe shaking, lead to collapse and significant loss of life and property within areas of scarce economic resources. An example of such a severe case was the Pisco earthquake in Peru, which occurred on the 15th of August in 2007, reaching a moment magnitude Mw of 8.0 (Blondet et al. 2008b). The epicentre was located only 60 kms west of the city of Pisco, which is where the intensity of the earthquake was felt the most, destroying 80% of all buildings in the city and killing 593 people (INDECI 2008). This level of devastation at the infrastructure level was caused by lack of lateral in- and out-of-plane resistance of the adobe structures in combination with inadequate quality of material, labour force, wall connections and the liquefaction of the soil.

Photograph 2.1: Typical in-plane shear failure (left) and out-of-plane failure of façades in adobe buildings in Pisco after the 2007 earthquake (Source: Blondet et al. 2008b)

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Tensile capacity of the semi-brittle material is key in defining the level of resistance it can generate when subjected to lateral displacements. Since tensile stress is horribly low in the case of adobe, collapse of the structure begins when instability of separated adobe blocks within the wall start moving independently from each other due to accumulated tensile cracking along the mortar joints and/or bricks, be that either in the in-plane or out-of-plane directions. On the opposite side, adobe has a reasonable level of compressive strength, which makes it have satisfactory performance when only subjected to gravity loads. In the case of modern day adobe construction, people have started building more and more slender walls, increasing its vulnerability to overturning loads, especially in the out-of-plane direction. When analysing out-of-plane failure, the typical collapse sequence starts when vertical cracks start to appear at the junctions of the perpendicular walls (seen in Photographs 2.1 and 2.2). When these grow excessively, the wall becomes detached from the former and starts moving independently, which increases the stresses due to out-of-plane action and then collapses due to instability. The use of wooden beams as a tie, on top of the adobe walls, can limit the separation of fractured blocks of wall and mitigate their independent unstable behaviour relative to one another, reducing the overall probability of out-of-plane failure due to a higher confinement level. When shifting to the analysis of the failure mechanism during in-plane cyclic lateral loading of an adobe wall, the collapse sequence starts when diagonal-stepped cracks develop at the edges of its geometry and around the corners of openings, which then extend across the wall in a diagonal manner. The wall finally fails by shear, forming “X” or “V” diagonal stepped cracks across the area of the wall (shown in Photographs 2.1 and 2.2).

Photograph 2.2: Vertical cracks along junctions of wall which failed in an out-of-plane manner (left) and diagonal cracks along the adobe wall related to in-plane shear failure (right)

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Literature review As mentioned before, the disregard of the limits in slenderness in unreinforced (as well as reinforced) adobe walls increase both in-plane and out-of-plane failure, especially the latter. That is why the Peruvian national standard for adobe construction NTP E.080 (Normas legales 2006) and the national standard for seismic design NTP E.030 (Normas legales 2016) suggest limits in slenderness and interstorey drift in adobe structures. •

The Peruvian national standard for seismic design suggests a maximum vertical slenderness ratio λv of 8 and a value of 10 for maximum horizontal slenderness ratio λh (Normas legales 2016). In the case of interstorey drift, NTP E.030 establishes a maximum value of: 𝑑𝑚𝑎𝑥 = 0.005ℎ

•

Equation 2.1

Eurocode 8 (European Standard 2004), based on the class and structural type of building, suggests the following maximum interstorey drift of:

𝑑𝑚𝑎𝑥 =

•

0.005ℎ = 0.0125ℎ 0.4

Equation 2.2

The Peruvian national standard for adobe construction suggests a maximum vertical slenderness ratio λv of 8 and a value of 10 for maximum horizontal slenderness ratio λh (Normas legales 2016), where both used in combination must always satisfy the following inequality. 𝜆ℎ + 1.25𝜆𝑣 ≤ 17.5

Equation 2.3

Based on all the information stated in previous paragraphs of this sub-chapter, it is imperative to define design standards for seismic reinforcement of adobe structures based on all the research developed in the subject. This is the only way to reduce the loss of life and property.

2.2. Experimental research on seismic strengthening of adobe dwellings Based on the limitations unreinforced adobe structures present when subjected to moderate to strong earthquakes, several techniques for seismic strengthening, both for in- and out-ofplane actions, have been presented by the Pontifical Catholic University of Peru, one of the leaders in this subject. One example of the research put forth by this institution is the DAI 113.0225 project (funded by the directory for academic research of the university) which

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading performed a campaign of pushover cyclic tests to evaluate in-plane cyclic behaviour of internally and externally reinforced adobe walls with industrial materials, such as internal bamboo cane, internal plastic mesh, external electrically-welded steel mesh, external geogrid mesh, among others (Blondet et al. 2005). Another experimental campaign included the evaluation of the seismic performance of a natural scale adobe house reinforced with polymer mesh (Blondet et al. 2006). What seismic reinforcement in adobe structures basically does is provide the semi-brittle structure with enough deformability (ductility) to prevent collapse of the structure and consequently avoid loss of life during strong earthquakes. This enhancement in ductility allows the material to work in a micro-fissured state where the reinforcement takes on the tensile stresses developed during the seismic actions. This gives people inside enough time and warning during the onset of damage so that they may reach safety before the collapse of the structure. However, in order for such reinforcement to be effective, it must be continuous over the whole adobe structure, so that it may prevent the formation of damaged blocks and generate instability within the structure if block action develops. Also, in order for it to be completely integrated with the structure, a finishing layer of mud or other type of cover (like cement mortar) must be spread over the reinforcement (if external). Based on the above-mentioned information, and among all the types of reinforcement methods studied for adobe dwellings, 3 seismic strengthening methods which have been thoroughly studied and had its effectiveness proven are presented in the following subsections.

2.2.1. Seismic reinforcement of adobe dwellings with bamboo cane Bamboo cane reinforcement has been used to reinforce adobe structures in historical landmarks across Peru in places such as Chan Chan and Caral during pre-colonial ages; monuments which remain intact until today. The modern construction procedure (as described in Blondet et al. 2008a) is described as follows:

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Literature review

Figure 2.1: Arrangement of bamboo cane reinforcement both in the vertical and horizontal direction (Source: Blondet et al. 2008a)

The cane is placed vertically and spaced equidistantly along the development of the wall (see Figure 2.1). The base of each of the main bamboo cane vertical reinforcements must be anchored within the concrete foundation of the structure. The spacing between vertical canes is defined by 1.5 the width of the overall adobe wall. Additional to the vertical arrangement of reinforcement, crushed bamboo cane is placed every other row of adobe bricks and tied to the vertical canes, so that the wall is able to be confined thoroughly by forming an internal mesh. Lastly, the reinforcement is tied at the top to a wooden beam to ensure composite action (see Photograph 2.3).

Photograph 2.3: Wooden beam used to tie up the ends of the vertical cane used as internal reinforcement (Source: Blondet et al. 2008a)

This method of seismic reinforcement of adobe structures showed great response during a full-scale model subjected to ground displacements of a shaking table (Blondet et al. 1988) and during a cyclic pushover test which reached a maximum quasi-static displacement of

Page 10

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading 100 mm (Blondet et al. 2005). In all cases, considerable damage was sustained by the structure, but collapse was avoided completely (Blondet et al. 2008). However, despite of the clear advantages this type of seismic reinforcement provides, the availability of bamboo cane can prove to be troublesome, as significant quantity of decent quality cane is needed to adequately reinforce an adobe structure.

2.2.2. Seismic reinforcement of adobe dwellings with electrically-welded steel mesh Electrically-welded steel meshes have demonstrated to provide the adobe structure with great seismic performance during past earthquakes (San Bartolomé and Quiun 2015), such as the one of June 23 rd, 2001, which reached a moment magnitude Mw of 8.4. The meshes are wrapped externally around specific parts of the adobe structure, nailed to the walls at constant intervals and incorporated into the structure with cement mortar finish, as can be seen in Photograph 2.4. When a full-scale model reinforced with this method was subjected to a scaled earthquake displacement signal and the response was analysed in detail, the structure showed increased developed strength compared to an unreinforced structure and even other types of reinforced structures.

Photograph 2.4: Installation of electrically-welded steel mesh reinforcement by nailing (left) and finished reinforced adobe structures with cement mortar covering the mesh (right) (Source: Blondet et al. 2008a)

However, this addition in resistance came at a very steep price, which was that when the structure reached failure, the mechanism of collapse was brittle and sudden (no warning). An additional disadvantage that this type of reinforcement presents is its weakness to exposure to the elements. As the mesh consists of steel and if it is not adequately covered by cement mortar finishing layer, the mesh will start to succumb to corrosion, which will Page 11

Literature review reduce its durability in time. This is why the layer of finishing is needed. To mitigate corrosion in the reinforcement mesh. This is in some way contradictory, as it solves the problem of corrosion of the steel mesh on one side, but adds the problem of shrinkage and material incompatibility between adobe and cement on the other: adhesion levels between these two materials are usually low. Furthermore, electrically-welded wire mesh and cement are too expensive for low-income families, up to $200 to reinforce only one floor.

2.2.3. Seismic reinforcement of adobe dwellings with biaxial geogrid mesh Unlike the steel mesh, the geogrid layers do not provide increased strength during hysteretic cycles. Instead, they provide an excellent level of confinement which allows damaged blocks not to move independently from each other, avoiding structural instability due to block interaction. There, adobe structures reinforced with biaxial geogrid have outstanding hysteretic performance, as it enhances the structure’s ductility by at least 500% when the layers are positioned in strategic locations (Blondet et al. 2005). Blonde et al. (2006) also showed that this type of reinforced structure provided good dynamic response. In all cases, significant damage was sustained during very strong shaking of simulated ground displacements, but collapse was avoided completely (Blondet et al. 2005, 2006), as can be seen in Photograph 2.5. The construction process for integrating the geomesh-reinforcement into the adobe structure is as follows: Before the construction of the adobe wall, the layers need to be integrated in the interface between the bottom of the adobe wall and the concrete foundation, for them to be able to fully confine the extent of the adobe structure. After that, the geomesh-reinforcement is wrapped around both sides of the structure and tied to one another through raffia strips, which go directly through the wall at discrete interval across the area of the wall being reinforced.

Photograph 2.5: Geomesh-reinforced structure prior to shake table test (left) and after (right) (Source: Blondet et al. 2006)

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Furthermore, geogrids have a high durability when exposed to the elements, but is susceptible to UV light, which relaxes the material overt time, making it increasingly flexible. This can be avoided by providing the structure with a layer of mud mortar finish, which not only protects the geogrid layers from exposure, but helps reduce crack growth in most polymer-mesh-reinforced cases. Having said the above, it is necessary to mention that geogrid reinforcement may in some cases be somewhat expensive. However, the mechanical advantages it provides to lateral resistance of adobe structures, as well as its ease of transportation due to low weight per square meter, flexibility, compatibility to adobe and durability against corrosive conditions, geomesh-reinforcement is highly recommended for seismic strengthening and retrofitting of earthen structures (Blondet et al. 2008b). Provisions for strengthening of adobe constructions with geogrid meshes are present in Annex 1 of the E.080 Peruvian national standard for adobe construction (NTP E.080 2008).

2.3. Compressive and tensile behaviour of adobe As mentioned earlier, adobe has a very low compressive and tensile strength which vary considerably throughout samples due to the proportions and interaction of its main components. Adobe bricks are obtained by mixing soil (which is formed by clay and silt), sand and straw with water, mixture which is then moulded into prisms and then left to dry in the sun for a certain period of time. The exact proportions of clay, silt, sand and straw vary from country to country, therefore creating a large variation in values of compressive and tensile strength of the adobe bricks. Nevertheless, the proportions of these raw materials must be within certain a certain range in order for the adobe brick to fulfil its main structural purpose (Saroza et al. 2008). •

Clay needs to be present in sufficient amounts within the soil to create sufficient cohesion in the adobe brick for it to reach its peak compressive and tensile strength (Saroza et al. 2008). This relates directly to the clay-sand ratio. If this ratio is too low, the dried mixture will crumble under the loads applied (Saroza et al. 2008).

•

In relation with the above condition, the clay-sand ratio cannot surpass a certain threshold, as this would generate excessive volumetric shrinkage during the drying process and lead to excessive micro-fissures and to reduced mechanical capacities. This however may be controlled by adding more sand into the mixture, reducing the previous ratio (Saroza et al. 2008).

•

The amount of straw (or another appropriate organic fibre) is also very important because of two reasons: the presence of straw reduces the amount of volumetric Page 13

Literature review shrinkage during the drying process (Saroza et al. 2008) as well as enhances the fracture behaviour in tension (Blondet et al. 2011). Building on the information presented above, the determination of the main mechanical properties of adobe is troublesome because of two main reasons: one, there is still not much research available which can encompass all aspects of mechanical behaviour without the need of idealising material behaviour at some point, and second, the information present in the material that is available varies greatly from country to country. Therefore, only 3 sources are presented here to establish general empirical behaviour of adobe in compression and tension. First, one of the most studied material parameters within the research available is the peak compressive strength of adobe. It is worth mentioning that most research only focuses in obtaining the ultimate compressive strength for design, without describing inelastic behaviour with enough detail or at all (Peralta and Torrealva 2009 and Vargas et al. 2016). Only Illampas et al. (2014) presents an approximate description of the inelastic compressive stress-strain relation of adobe, shown in Graph 2.1. In order to obtain ultimate compression strength, uniaxial compression tests are performed on either adobe brick assemblages (Peralta and Torrealva 2009) or singular adobe units (prismatic or cylindric) (Illampas et al. 2014). Typical uniaxial compression tests (performed in Peralta and Torrealva 2009 and Illampas et al. 2014) are shown in Photograph 2.6: stress is applied at a constant load-rate (2 kN/min in Peralta and Torrealva 2009 and 0.1 MPa/s in Illampas et al. 2014) until the test specimen fails in compression.

Graph 2.1: Approximation of inelastic compressive stressstrain relation (Source: Illampas et al. 2014)

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Photograph 2.6: Uniaxial compression tests performed to determine peak compressive strength in an adobe brick assemblage (left) and in a single unit (right) (Source: Peralta and Torrealva 2009, left, and Illampas et al. 2014, right)

Having said that, Peralta and Torrealva (2009) tested 5 adobe brick assemblages (6 adobe units in assemblage, with overall dimensions of 115 x 210 x 620 mm) and obtained values for ultimate compressive strength ranging from 0.328 MPa to 0.521 MPa, which gave a mean ultimate compressive strength fc of 0.457 MPa (4.66 kg/cm2). The assemblages were prepared based on soil, sand and straw ratios suggested by the Peruvian national standard for adobe construction and laid to dry for 30 days before testing. Illampas et al. (2014) prepared prismatic and cylindric adobe specimens with varied soil, gravel, sand and straw ratios for compressive testing, which were oven-dried at 70°C after mixing and prior to testing. The values that were obtained varied greatly, as well as their shapes of stress-strain relation, depending on the geometric type. Lastly, Vargas et al. (2016) extracted 8 adobe assemblage specimens, as well as 16 adobe unit specimens, from historical landmarks across Peru, which were then subjected to uniaxial compression tests and obtained values ranging from 0.364 MPa to 0.756 MPa, and 0.434 MPa to 1.835 MPa, respectively. The results of these compression tests can be seen in Graphs 2.2 and 2.3. Additionally, Vargas et al. (2016) mentions the contribution of two other range values obtained from Vargas and Ottazzi (1981) and Silveira et al. (2012), which are shown in Graph 2.2.

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Literature review

Graph 2.2: Uniaxial compression test results from adobe assemblages of different historical landmarks across Peru (Vargas et al. 2016)

Graph 2.3: Uniaxial compression test results from adobe units of different historical landmarks across Peru (Vargas et al. 2016)

In case of the determination of peak tensile stress, Peralta and Torrealva (2009) and Vargas et al (2016) defined that the maximum tensile stress could be obtained through an indirect compression test. This test is defined as a diagonal compression test (shown in Photograph 2.7), which forces an assemblage of adobe bricks to fail in shear. The magnitude of the load at which the adobe assemblage fails is directly proportional to the tensile stress as follows: 𝑓𝑡 =

𝑃 , √2(𝑎𝑒𝑚 )

in MPa

Equation 2.4

Page 16

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Where P is the compressive load needed to achieve shear failure, a is the width of the assemblage in mm and em is the thickness of the assemblage. In Peralta and Torrealva (2009), the mean peak tensile stress ft calculated was 0.0415 MPa (=0.423 kg/cm2), whereas in Vargas et al. (2016), a range of values spanning from 0.01 MPa to 0.05 MPa was obtained.

Photograph 2.7: Real diagonal compression test performed on adobe assemblage (left) and description of the components of equation used to estimate f t (Source: Peralta and Torrealva 2009 and NTP E.080 Adobe 2006)

2.4. Tensile behaviour of geomesh reinforcement (biaxial geogrid) The development of research aimed at quantifying the time-dependent and independent monotonic and cyclic tensile behaviour of geosynthetics is relatively new, since most papers on the subjects, such as Bathurst and Cai (1994), Moraci and Montanelli (1997), Ling et al. (1998), Kongkitkul et al. (2004), Bathurst and Kailakin (2005), Liu and Ling (2006, 2007) and Cardile et al. (2017), have been published within the last three decades. This late bloom in research is mainly due to the increase in the use of biaxial geogrid to reinforce slopes, foundation soil and earthen structures, such as retaining walls (Ling et al. 1995), which has been increasing constantly during the last decades. In the midst of this increase, the introduction finite element programmes has made the need for accurate descriptions of the mechanical behaviour of geogrid reinforcement even more important, as the precision of the numerical data depends on a correct definition of the material model. Geogrids consist of polymeric material which can be extruded, bonded or woven into uniaxial or biaxial resisting meshes used to reinforce soils structures (Cardile et al. 2017). The polymeric material used to manufacture geogrids may be either high density polyethylene (HDPE), polypropylene (PP) or polyethylene terephthalate (PET). The first two are the most commonly used in manufacturing geogrids for all sorts of geotechnical

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Literature review engineering solutions, as mentioned before. However, all of them share a common general mechanical behaviour, which is highly nonlinear and sensitive to loading rate, creep deformations and temperature. Consequently, all the conditions which affect mechanical behaviour of geosynthetics in general fall into either of these two categories: •

Time-dependent properties: Geogrids of all polymeric materials, especially PET geogrids (Ling et al. 1998), are prone to viscoplastic deformations when subjected to high permanent stresses, such as might be present in embankments or retaining walls. Time of duration and magnitude of prestress affect the long-term response of polymeric materials, since its viscous properties make them prone to stress relaxation and creep deformations (Liu and Ling 2007), which have to be considered in all engineering solutions. Additionally, strain-rate also modifies monotonic and cyclic response of geogrids. Lastly, all of these properties are further affected if temperature is a considerable factor.

•

Time-independent properties: Instant response to a static or cyclic load, is invariant through time.

Having said that, normal time-independent mechanical response in geogrids can be viewed in terms of total strain which has elastic and inelastic strain components. This simplification is made because of the nature of the inelastic response of viscous-elastoplastic materials, such as polymeric geogrids, have compared to other types of materials, such as steel: Since the stress-strain relation does not have a clear definition of elastic and plastic behaviour, each strain increment is idealised to have an elastic and plastic component as is shown below: 𝑑𝜀 = 𝑑𝜀 𝑒𝑙 + 𝑑𝜀 𝑝𝑙

Equation 2.5

Furthermore, load-strain relationships of geogrids fall into 2 categories based on the manufacturing process of the geogrid, as well as the polymeric material used: type A or B geogrids. Type A geogrids represent most types of geogrids made of HDPE and PP and are defined by continuous softening of the stress-strain curve until it reaches a maximum stress value, which remains constant for a short range of elongation up to the rupture of the material under tensile loading (Liu and Ling 2006). Type B mechanical behaviour includes mostly PET geogrids and is characterised by first having a softening curve, followed by an increase in tangential stiffness until rupture (Liu and Ling 2006). Both types of tensile behaviour can be seen in Figure 2.2.

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Figure 2.2: Stress-strain relationships of type A (a) and type B (b) geogrids (Liu and Ling 2006)

Within the research and literature available, several methods were found which accurately approximated time-independent and dependent response of the constitutive model of both type A and B geogrids. Among them, the hyperbolic function with modified Masing rule and the bounding surface model were the most studied and used. The following sub-chapters explain the advantages and disadvantages of both methods.

2.4.1. Hyperbolic function with modified Masing rule (Liu and Ling 2006) This constitutive model includes different formulations for both monotonic and cyclic behaviour of either type A or B geosynthetics. It uses a hyperbolic function to approximate empirical values in monotonic and cyclic tests, which only requires few material constants to adequately create the model (Ling et al. 1995, Cai and Bathurst 1995 and Liu and Ling 2006). As it will be described in chapter 4, the uniaxial tensile test results of the geogrid reinforcement that was used to reinforce de adobe wall was a type A geogrid, which is why only the mathematical functions to model type A monotonic and cyclic behaviour was included in this sub-chapter. The hyperbolic function by itself can satisfactorily describe monotonic tensile behaviour of type A geosynthetics (Ling et al. 1995), as well as cyclic tensile behaviour after the Masing rule is applied (Cai and Bathurst 1995). The Masing rule is modified in order to simulate the accumulation of plastic strain in the model, which results in 3 different expressions to describe primary loading T,

𝑇=𝑎

𝜀 0 +𝑏𝜀

, in kN/m

Equation 2.6

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Literature review 𝑎0 = 1⁄𝐽 , in m/kN

Equation 2.7

0

𝑏 = 2𝑏0 =

1 𝑈𝑇𝑆

, in m/kN

Equation 2.8

unloading TU (𝜀−𝜀0 ) 1 ( )𝑎0 +𝑏0𝑈 |𝜀−𝜀0| 𝑘1

𝑇𝑈 = 𝑇0 +

, in kN/m

𝑏0𝑈 = 𝑏0 , in m/kN

Equation 2.9 Equation 2.10

and reloading TR during cyclic loading. 𝑇𝑅 = 𝑇0 + 𝑏𝑅 =

𝑏0 1−𝜀0

(𝜀−𝜀0) 𝑏 1 )𝑎0 + 𝑅 |𝜀−𝜀0| 𝑘2 2

, in kN/m

Equation 2.11

{1 + 𝑒 −𝑐(𝑁−1) }, in m/kN

Equation 2.12

(

These las two expressions, which depict unloading and reloading behaviour respectively, can capture the difference in unloading and reloading stiffness, variables which are normally different from the primary (monotonic) loading stiffness (Liu and Ling 2006). The main difference between these two expressions is their dependence to the number of load cycles. As can be seen in Equation 2.9 and 2.10, the unloading function is independent from the number of loading cycles N sustained up to the instant of unloading, whereas the reloading function’s virtual strength depends on the number of cycles N, which simulates the accumulation of plastic strains in the geosynthetics. The values for J0, UTS and c are obtained by calibrating the hyperbolic curve to the empirical monotonic and cyclic results available from uniaxial and cyclic tensile testing. An example of the accuracy of the approximation performed with the hyperbolic function is seen in Graphs 2.4 and 2.5.

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Graph 2.4: Approximation of monotonic and cyclic behaviour in type A geogrids using the hyperbolic function method with modified Masing rule

Despite the good precision and simplicity this model offers, the hyperbolic function fails to account for accurate depiction of strain hardening phenomenon which occurs with increasing hysteresis loops, as well as its stabilisation after a certain amount of loading cycles. This effect can be represented more accurately by the bounding surface model, which is explained in the next sub-chapter.

Graph 2.5: Stabilisation of strain hardening in geosynthetics materials after a certain amount of loading cycles (Source: Liu and Ling 2006)

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Literature review

2.4.2. Bounding surface model (Liu and Ling 2006) The bounding surface model requires the definition of a yielding line on the tensile side and an unloading line in the compression side1. These bounding lines, which according to Liu and Ling (2006) may be simplified to linear expressions, depict a kind of fully plastic range, by which plastic strain increases (or decreases when unloading) with a constant plastic modulus (Jp+ or Jp-). Since the tensile bounding line defines loading and reloading and the compressive bounding line defines unloading limits, the linear expressions are expected to be non-parallel (Liu and Ling 2006). The equations describing both bounding lines are listed below. 𝑇+ = 𝐴 + ̅̅̅ 𝐽𝑝+ 𝜀 𝑝𝑙 , in kN/m

Equation 2.13

𝑇− = 𝐵 + ̅̅̅ 𝐽𝑝− 𝜀 𝑝𝑙 , in kN/m

Equation 2.14

In addition to the bounding lines, a third expression defines the hardening parameters during primary loading, reloading and unloading, which allows the representation of approximation values in between the bounding lines. Since cyclic behaviour distinguishes 3 loading types, the hardening parameters of the plastic stiffness must also abide by that rule. The change in plastic stiffness in between the bounding lines is defined by the following hardening parameters, relative to a specific loading condition. ℎ 𝑃𝐿 = ℎ0𝐿 , for primary loading

Equation 2.15

ℎ 𝑈𝐿 = ℎ0𝐿 , for unloading

Equation 2.16

ℎ 𝑅𝐿 = ℎ0𝐿 + ℎ𝑘𝐿 √𝜀 𝑝𝑙 , for reloading

Equation 2.17

These hardening parameters, which change the value of the variable h below, define the change in plastic behaviour as defined by Liu and Ling (2006) below. Graph 2.6 shows the interaction of all the components mentioned until now. 𝑑𝜎

𝐽𝑝 = 𝑑𝜀𝑝𝑙 = 𝐽̅𝑝 + ℎ 𝛿

1

𝛿 𝑖𝑛 −𝛿

, in kN/m

Equation 2.18

A compression side is defined only as to make clear the location and behaviour of the unloading bounding line.

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Graph 2.6: Interaction of bounding lines with hardening parameters (Source: Liu and Ling 2006)

In general, change in overall stiffness in the bounding surface model can be obtained through the equation below. Since the value of elastic stiffness remains the same throughout the mechanical behaviour, the variation only comes from changes in the magnitude of plastic stiffness, which range is defined by the plastic stiffnesses of both the tensile and compression bounding lines. 1 𝐽𝑇

=

1 𝐽𝑒

1

+ , in kN/m

Equation 2.19

𝐽𝑝

After comparing both methods, it was decided that a hyperbolic function was more appropriate to approximate the monotonic and cyclic response of the geogrid which will be presented in chapter 3. Despite the fact that the bounding surface model presented more perks compared to the hyperbolic model in terms of accuracy when modelling cyclic behaviour of a geogrid, there was no cyclic test data available for the geomesh-reinforcement used to wrap the adobe wall in this project, which made the use of this model difficult. However, there was enough test data available to create an approximation with the hyperbolic function method.

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Cyclic pushover test geomesh-reinforced adobe wall

3. Cyclic pushover test of a geomesh-reinforced adobe wall As mentioned in previous chapters, great part of this work consisted in the creation of a numerical model which will be able to simulate, as closely as possible, the real cyclic and nonlinear behaviour of a geomesh-reinforced adobe wall subjected to in-plane cyclic loading. For this purpose, this project makes use of experimental results obtained from a cyclic pushover test of a geomesh-reinforced adobe wall, performed in Lima, Peru, at the Pontifical Catholic University of Peru (PUCP). This experiment was part of the DAI 113.0225 project (Blondet et al. 2005) which investigated the performance of different methods of seismic reinforcement for adobe dwellings to lateral cyclic displacements. Among them, the geomesh reinforcement option (Photograph 3.1).

Photograph 3.1: Geomesh-reinforced adobe wall used in cyclic pushover test (Source: Blondet et al. 2005)

This chapter talks about all the characteristics of the DAI 113.0225 project concerning the geomesh-reinforced adobe wall. The outline is as follows: •

Description of the test specimen. This part describes the geometric and material properties of the geomesh-reinforced adobe wall which was subjected to a testing programme of cyclic lateral loads.

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Testing programme and measuring instruments. All the test specimens within the DAI 113.0225 project were subjected to the same testing programme and results for displacements and lateral forces were measured specific positions.

•

Experimental results. This sub-chapter presents all the measurements obtained from the geomesh-reinforced adobe wall during testing and force-displacement curves are constructed. These results are evaluated and compared to results obtained from an equivalent un-reinforced adobe wall in order to see the effectiveness of the geomeshreinforcement in a real-case scenario.

3.1. Test specimen: geomesh-reinforced adobe wall The geomesh-reinforced adobe wall consisted of four main components: a concrete foundation, a concrete beam on top of the wall, 3 adobe walls and the geomesh reinforcement. The adobe walls consisted of an in-plane wall with a window opening, which included a lightweight timber lintel, and two perpendicular walls to simulate stiffness in the out-of-plane direction, all of which resembled an I-shaped wall. The in-plane wall was 2.36 m wide by 1.93 m tall and both perpendicular walls were 2.48 m wide by 1.93 m tall. All three walls had a thickness of 300 mm, which was equal to the length of one adobe brick.

Photograph 3.2: Construction of adobe wall with alternated rows of bricks (Source: Blondet et al. 2005)

The arrangement of bricks alternated the direction of the length of the bricks every row of units (shown in Photograph 3.2). The adobe units used to construct the test specimen had two sets of dimensions: 300 x 130 x 100 mm and 220 x 130 x 100 mm. The mixture used to fabricate the adobe units and mortar were made by a single person (to minimise variation

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Cyclic pushover test geomesh-reinforced adobe wall within batches of mixture), with the help of mechanical tools, and contained soil, coarse sand and straw at ratios of 5:1:1 and 3:1:1, respectively (Tarque 2011). Both the concrete foundation and top beam had roughly the same dimensions as the adobe walls. The top concrete beam’s purpose was to simulate typical gravity loads attributable to a regular roof structure. The adobe walls, concrete foundation and top beam weighed 87.6 kN, 31.4 kN and 16 kN respectively, weighing 135 kN in total. A sketch of the general dimensions of the adobe wall can be seen below in Figure 3.1.

Figure 3.1: Sketch of dimensions of adobe wall

Lastly, the geomesh-reinforcement used to wrap specific parts of the adobe walls during construction, varying the number of layers across the wall depending on the vulnerability of the area, was a BX1100 biaxial polypropylene geogrid (Tensar 2013). The arrangement of reinforcement was determined based on knowledge of seismic performance of unreinforced and unconfined masonry and adobe walls acquired through numerous experimental research studies performed at PUCP. As shown in the literature review, the common stepped “X” shear failure pattern in unreinforced masonry and adobe brick walls appears after excessive interconnection of tensile micro-cracks along the mortar interface (in some cases also present in the masonry units). These start around vulnerable areas of the wall, such as its corners and window openings, including the area around the lintel. Based on this knowledge, Figures 3.2 and 3.3 show the selected geomesh-reinforcement arrangement for the test wall.

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Figure 3.2: Frontal view of arrangement of geogrid layers in geomeshreinforced adobe wall

Figure 3.3: Lateral view of arrangement of geogrid layers in geomesh-reinforced adobe wall

3.2. Testing programme and measuring instruments The geomesh-reinforced adobe wall was subjected to a testing programme which was formed by 9 phases. Each phase consisted of alternating lateral displacements (alternating directions) with a characteristic peak value, as presented in Table 3.1, starting from 0.1 mm, in phase 1, up to 100 mm, in phase 9. Page 27


Table 3.1: Cyclic pushover test phases

These displacements were generated by a hydraulic actuator of 500 kN maximum capacity, which applied the load in push-pull cycles (Blondet et al. 2005). The lateral displacements with peak values of 0.1 mm and 0.5 mm during phase 1 and 2 were used to calibrate the instruments (which can be seen in Figure 3.4) installed both on the wall and actuator. Phases 3 through 9 were used to evaluate the walls’ lateral resistance to induced cyclic lateral displacement and to determine the failure pattern (Blondet et al. 2005).

Figure 3.4: Position of LVDTs installed on adobe walls

As seen in Figure 3.4, in order to capture the displacement values at diverse points on the wall, 17 LVDT measuring devices were installed at strategic places, as well as a load measuring instrument in the hydraulic actuator. The most important displacement values were the ones measured by LVDT #1, which captured the top-displacement history of the geomesh-reinforced adobe wall, which was placed on top of the concrete beam upon which the cyclic loads were applied.

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3.3. Experimental results from cyclic pushover test of geomesh-reinforced adobe wall The main results obtained from the cyclic tests were force-displacement curves, which showed the hysteretic behaviour of the geomesh-reinforced structure. Since phases 1 and 2 were only intended to calibrate the LVDT sensors present on the structure, the analysis was focused on the results stemming from phases 3 through 9, which included the hysteresis loops with 1.0 mm to 100.0 mm peak displacement. During phases 3 and 4, the structure remained essentially elastic and without any loss of stiffness, reaching a peak strength of approximately 35 kN during phase 4 (Graph 3.1). This behaviour is attributable to the mechanical properties of adobe: As mentioned in chapter 2, adobe is a low strength, semi-brittle material, which, for practical purposes, remains elastic up to around 90% of its peak compressive strength, from where it starts developing significant inelastic properties due to excessive micro-cracking within the adobe interface. This meant that the adobe walls were still within the elastic limit during the hysteresis loops of 1 mm and 2 mm, leaving both the adobe walls and geomesh-reinforcement without damage. The lack of damage in the reinforcement was mainly because the adobe structure was still within the elastic limit, which suggested that not enough deformation had taken place in the structure for the reinforcement to take part. The first hint of stiffness degradation was shown during the 5 mm hysteresis loop, corresponding to phase 5, which was due to the formation of the first stepped cracks around the corners of the in-plane wall, the base and top of the perpendicular walls and the area where the load was applied. Here, the peak compressive strength also remained around 35 kN, since cracking had not reached a critical stage, where it may have affected the ability of the structure to uniformly transfer stress along their interfaces (Graph 3.1). During phase 6, considerable damage started to accumulate around the same areas mentioned in phase 5 (shown in Photograph 3.3). Crack widths started to grow, for example, to up to 5.5 mm around the left side of the lintel and 4.0 mm on the right-hand side of the in-plane wall. In this case, accumulation of cracks caused the intervention of the geomeshreinforcement, which confined the adobe wall sufficiently enough to allow a slight increase in the base shear up to 39 kN (Graph 3.1).

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Photograph 3.3: Damage sustained during phase 6 (left). Close-up view of stepped cracks on the right side of the in-plane wall (right) (Source: Blondet et al. 2005)

Phases 7 and 8 (20 mm and 50 mm peak displacements respectively) presented continued damage accumulation around the before-mentioned locations, generating the growth of crack widths across the in-plane and perpendicular walls (shown in Photograph 3.4) and causing further confinement action by the geomesh-reinforcement, allowing the structure to displace to up to 50 mm without much loss of strength during the hysteresis loops (Graph 3.1). However, during this phase, the geomesh-reinforcement developed noticeable plastic deformation around the most critical areas in the perpendicular adobe walls.

Photograph 3.4: Considerable crack growth during phase 8 (50 mm peak displacement) (Source: Blondet et al. 2005)

Lastly, during phase 9 (100 mm peak displacement), the structure sustained great damage which caused relative sliding of in-plane blocks formed during the last 2 phases as well as

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading buckling of the perpendicular walls due to failure at both the base and top. Despite the level of damage sustained in the adobe walls, the geomesh-reinforcement managed to sufficiently confine the wall blocks formed during previous phases, preventing the collapse of the overall system (shown in Photograph 3.5). This caused the reinforcement to sustain extensive plastic deformation, even rupture, around critical areas such as the window opening and the base of the in-plane and lateral walls as shown in Photograph 3.6. Here, peak strength level of 39 kN was only reached at maximum peak displacement of 100 mm (Graph 3.1). This particularity was possible due to the level of confinement provided by the geomesh. As previous material degradation prevented the structure to reach peak strength at the 20 mm or 50 mm marks, the geomesh enabled the structure to deform enough to reach 39 kN.

Photograph 3.5: Damage sustained after phase 9. Collapse was avoided. (Source: Blondet et al. 2005)

Photograph 3.6: Damage sustained by the geomesh-reinforcement during phase 9 of the testing programme (100 mm peak displacement). Including some areas of great plastic deformation and rupture (Source: Blondet et al. 2005)

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The enhancement of ductility brought by the geomesh-reinforcement can be clearly seen in Graph 3.1, where the hysteresis curves of an identical adobe wall, but unreinforced, is compared to the results obtained from the geomesh-reinforced adobe wall. The results showed that the GRAW increased its ductility around 500% relative to the URAW, bearing in mind that the last stable hysteretic cycle of the latter structure was the ± 20 mm one (Blondet et al. 2005). Strength was not increased significantly, it even was lower during the ± 5 mm than the URAW, but it remained stable during all following hysteretic cycles, whereas strength in the unreinforced structure deteriorated with every passing cycle until it collapsed during the – 50 mm loop.

Graph 3.1: Comparison of GRAW and URAW cyclic behaviour

In a more detailed manner, due to the lack of confinement provided by active external reinforcement, the unreinforced structure was unable to develop sufficient ductility in order to reach peak base shear values after the ± 5 mm hysteresis loop. The deficit in ductility and strength in the unreinforced adobe wall became evident from phase 7 onwards. During the ± 20 mm hysteresis loop, the accumulated damage within the adobe walls impeded the development of the full peak strength of the wall (39 kN) achieved during phase 5. Furthermore, phase 8 of the unreinforced structure showed a sudden drop in strength around +30 mm, which was caused by severe damage of the adobe, which lead to excessive crack growth and instability due to sliding action of damaged wall blocks.

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading So, in summary, it has been proven that the use of biaxial geogrid as an external reinforcement in adobe structures was effective in enhancing overall stability and ductility during cyclic displacements. No significant strength increase provided, but the confinement action of the geomesh layers enabled the adobe structure to reach larger displacements, compared to the unreinforced version, without collapsing in the process. Next, chapters 4 and 5 describe the creation of an appropriate numerical model in Abaqus which will be calibrated based on these empirical results in chapter 6. Lastly, the seismic performance of the same numerical model will be evaluated by subjecting it to 5 scaled earthquake signals with different magnitudes.

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Material constitutive modelling

4. Material constitutive modelling This chapter presents all the work related to the analytical and numerical modelling of the constitutive models of both the adobe continuum and geomesh reinforcement layers. The adobe continuum was modelled with a Concrete Damaged Plasticity model in Abaqus/Standard (SIMULIA 2008), as suggested in Tarque (2011), whereas the geomeshreinforcement was modelled with an elastoplastic monotonic approach.

4.1. Adobe The constitutive model for the behaviour of adobe was built using the Concrete Damaged Plasticity (CDP) model present in Abaqus/CAE package (SIMULIA 2008) with the knowledge present in Lourenço (1996) and Tarque (2011). This model, which was initially intended for modelling the compressive, tensile, cracking and cyclic behaviour in concrete, was successfully used to model masonry structures in Lourenço (1996) and adobe structures in Tarque (2011). Therefore, it was decided that the FE modelling of the adobe continuum would be tackled using this constitutive model. As mentioned in chapter 2, adobe is a semi-brittle material with very low compressive and tensile strength. Since inelastic behaviour of adobe in compression and tension has not been researched in a quantitative aspect, a softening curve for both compression and tension was idealised as presented in Tarque (2011). In compression, adobe essentially behaves elastically up to around 80% to 90% of its peak compressive strength, from where it has a short hardening curve up to its peak compressive strength and then softens into a value of residual strength. In the case of Illampas et al. (2014), adobe cylinders of varying mixture conditions and dimensions were subjected to uniaxial compression tests. Aside from the geometrical properties, the main difference between the adobe specimens was the composition of the mixture. Illampas et al. (2014) included gravel when proportioning the mixture of adobe, which is not normally done in developing countries, such as Peru or Cuba (Blondet et al. 2005 and Saroza et al. 2008). The inclusion of gravel in the adobe mixture generated elevated compressive strength values in the specimens (relative to usual values)

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Based on the discussion above and in chapter 2, it was decided to set an initial value of 0.40 MPa as peak compressive strength in the CDP model, accompanied with an initial compressive fracture energy of 0.1426 N/mm, which determined the change in slope when strength started to decrease to a constant value at the top of hysteretic loops. These values were recommended in part by Tarque (2011) but were later changed during chapter 6 in order to find the appropriate set of magnitudes that could accurately describe hysteretic behaviour of a GRAW. The variables used to change the amount of compressive fracture energy were (also shown in Graph 4.1 below): •

Strain at peak compressive stress: εcp

•

Strain at half the peak compressive stress: εcm

•

Initial stress

•

Residual stress

Figure 4.1: Definition of inelastic compression curve of adobe using the CDP model (SIMULIA 2008)

After defining the compression curve, the initial peak tensile strength was set at 0.04 MPa. Tensile fracture energy determined the maximum magnitude of strength developed during cycles of displacement, since the stress able to be transmitted through the tensile crack was higher, more stress could be endured in tension overall. This increased maximum strength significantly. Since geomesh reinforcement in reality does not become active until the first micro-cracks appear in the mortar joins or adobe bricks, the initial values were set based on the recommendations for tensile behaviour set in Tarque (2011), which defined the cyclic behaviour of an unreinforced adobe structure.

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Figure 4.2: Definition of inelastic tensile softening curve of adobe using CDP model (SIMULIA 2008)

Finally, the elastic modulus of adobe was defined the same as in Tarque (2011), which was confirmed to be within the lower bound relative to the values of elastic moduli present in Vargas et al. (2016). This was to try to account for inferior quality material and unskilled labour force as well. Both peak compressive and tensile stresses were set within the lower boundaries of the investigated ranges of values for ultimate compressive and tensile behaviour. This was also bearing in mind the possibility of having had a poor construction process.

4.2. Geomesh reinforcement (biaxial geogrid) One of the main challenges at the beginning of this dissertation was selecting the most appropriate constitutive model for the geomesh reinforcement. Based on the theory of mechanical monotonic and cyclic behaviour of HDPE or polypropylene biaxial geogrids described in chapter 2 by Liu and Ling (2006, 2007), a suitable choice of constitutive model had to be chosen from the library of material models present in Abaqus. However, based on the current ones available, proper cyclic behaviour of a geogrid was impossible to achieve without extensive programming efforts, which is why its behaviour was simplified to a monotonic primary loading curve, with an initial elastic modulus equal to E0 (as described in chapter 2). This last value also described the unloading behaviour of the material in the numerical model. As to further explain this choice, proper cyclic behaviour consisted of different mathematical expressions for primary loading, unloading and reloading (Liu and Ling 2006), as shown in Page 36

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Graph 2.4. Even after simplifying the reloading and unloading curves to linear expressions, with constant elastic moduli, these values were not able to be incorporated in a material model. To define such diverse loading scenarios, further research must be done in creating an Abaqus/Standard subroutine (SIMULIA 2008) for cyclic-elastoplastic behaviour of highstrength-deformable polymers. Having said this, the elaboration of the constitutive model of the geomesh-reinforcement was elaborated based on the following assumptions: •

No significant prestress was present in the geomesh-reinforcement before and during cyclic displacements, which allowed creep effects to be negligible.

•

Loading rate was sufficiently fast in order to prevent stress relaxation of the material, which is why this effect was also considered negligible and non-pertinent to this dissertation.

This chapter consists of a two-step process: analytical modelling and numerical calibration, which are described in the following sub-chapters.

4.2.1. Analytical modelling of geomesh reinforcement As stated during sub-section 3.1, a BX1100 (Tensar 2013) polypropylene biaxial geogrid was used as an external reinforcement for the adobe wall. The manufacturer’s properties defined a minimum rib thickness of 0.76 mm and rib spacing of 30 mm and 40 mm in the longitudinal (XMD) and transverse direction (MD) respectively (Tensar 2013). To find out the response of the geogrid while in tension, two samples were taken and subjected to uniaxial tensile tests, which abided by ASTM D-5732-95 regulations (ASTM 1995), one for each direction of grid development of the geogrid. The geometric properties of the samples were chosen in compliance with the previously mentioned ASTM standard and are shown in Table 4.1. The results from uniaxial tensile tests were acquired from the testing campaign of seismic reinforcement for adobe buildings presented in project DAI 113.0225 (Blondet et al. 2005) and with permission from professor Nicola Tarque.

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Photograph 4.1: Uniaxial tensile test performed on a sample of polypropylene geogrid BX1100 in the Structures Laboratory at PUCP (Source: Blondet et al. 2005)

Table 4.1: Geometric properties of BX1100 geogrid samples which were tested along the XMD and MD directions

The initial results, which are shown below in Graph 4.1, indicate an increase in tensile force “T”, measured in kN, during elongation of the specimen, which was measured in mm. As can be clearly seen, there is some difference in the tensile properties along the longitudinal and transverse directions when looking at maximum strength and deformation: The MD monotonic curve showed some irregular behaviour starting from 12 mm elongation compared to typical behaviour of type A geogrids, as exhibited by the XMD curve. This may be attributed to measurement errors, but the most plausible cause is that, during the extrusion of the longitudinal ribs, the microstructure of the polymer in transverse ribs may have been affected, causing an irregular-shaped curve. Maximum tensile forces of 2.37 kN and 2.57 kN are reached in the XMD and MD directions, as well as maximum elongations of 32.32 mm and 29.45 mm, respectively. In regards for this dissertation and for practical purposes, tensile behaviour in the XMD and MD directions for this type of biaxial geogrid can be characterised as very similar and, therefore, the geogrid can be assumed as an isotropic material, instead of orthotropic.

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Graph 4.1: Uniaxial testing of polypropylene geogrid #2 in the longitudinal (XMD) and transverse (MD) direction

The analytical approximation of the BX1100 biaxial geogrid was performed using a hyperbolic function modified with Masing rule presented in Liu and Ling (2006), for which both the XMD and MD tensile characteristics were taken into consideration. For its construction, and as described in chapter 2, the following material properties had to be defined from the uniaxial tensile test curve: •

Initial elastic modulus E0. The coefficient a0 depends on the initial stiffness J0 of the tensile curve, which can be obtained by multiplying the initial elastic modulus by the thickness “e” of the geogrid.

•

Ultimate tensile strength per meter “UTS”. The value of the ultimate tensile strength per meter, which defines the magnitude of constant b0, depends on the maximum tensile strength developed during the uniaxial tensile test. Here, a modification had to be made so that the curve could fit the empirical data. This will be shown in subchapter 4.1.1.2.

4.2.1.1. Calculation of initial elastic modulus (E0) The initial elastic modulus E0 was calculated based on the uniaxial tensile test values from XMD monotonic curve, as it represented typical tensile behaviour in a type A geogrid. As the test results included very-near zero values during the very beginning of the experiment, it was important to isolate those values to obtain an adequate reading of the initial Young’s modulus. First, the tensile test data was transformed to nominal stress and strain to extract

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Material constitutive modelling values for elasticity moduli in MPa (shown in Graph 4.2) and then near-zero values were isolated to obtain a curve with constant, increasing values (shown in Graph 4.3).

Graph 4.2: XMD nominal stress-strain values

From these, differences in strain and stress were extracted from 7 sets of values so that their values of E may be averaged into an initial elastic modulus E0, which can be seen in Graph 4.4. Based on the calculations, E0 was set at the value of 5608.24 MPa. This value was converted to initial stiffness per meter J0, which equalled 4262.26 kN/m.

Graph 4.3: Isolation of near-zero values to obtain appropriate values for the calculation of E0

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Graph 4.4: Calculation of initial elastic modulus E0 through averaging of initial values of nominal stress-strain curve

4.2.1.2. Calculation of ultimate tensile strength per meter (UTS) The value for ultimate tensile strength per meter was extracted from 286 sets of values which marked the development of a strength plateau, starting at approximately 0.17 strain and ending at 0.27 strain along the XMD curve (see Graph 4.5). The 286 values of strength per meter were averaged and a value of 218 kN/m was obtained for UTS in the XMD direction.

Graph 4.5: Calculation of the value of UTS per meter

4.2.1.3. Calibration of hyperbolic function curve As said before, both XMD and MD tensile behaviours were considered when the hyperbolic function curve was calibrated. The previously mentioned value of 4262.26 kN/m was used for J0. For UTS, a modification had to be made in order for the hyperbolic function curve to fit both the XMD and MD tensile curves, as shown in Graph 4.6 and suggested by Ling and Page 41

Material constitutive modelling Liu (2006). The value used for this constant was 269.2 kN/m. As can be seen, the hyperbolic function curve mimics the tensile behaviour of the geogrid in both directions with reasonable accuracy: It shows a steady hyperbolic growth during all its development, not quite reaching a plateau within the shown strain values. This last characteristic is very similar to the tensile behaviour along the MD direction of the geogrid, which does not reach a strength plateau at all. The accuracy of this analytical model was proven after calculating the levels of variation in strength at different strain values. The maximum levels of variation of strength values from the analytical model was of 10%, during short ranges of strain and isolated instances, for both the XMD and MD directions. Nevertheless, the shape and maximum strength, which was of 2.36 kN at 32.32 mm, of the analytical model proved accurate for both types of geogrid tensile behaviour.

Graph 4.6: Calibration of Hyperbolic function (with modified Masing rule) curve

After the calibration of the hyperbolic function curve based on the uniaxial tensile test results for both the XMD and MD directions of the biaxial geogrid, the material model was created and calibrated in Abaqus/Standard.

4.2.2. Numerical calibration of geomesh-reinforcement constitutive model As mentioned at the beginning of chapter 4, both elastic and plastic properties were defined for the material model created for the geomesh-reinforcement in Abaqus/Standard. An elastic modulus of 5608.24 MPa was defined, which was equal to the initial elastic modulus E0 of the primary loading curve (monotonic curve) defined shown in Graph 4.6. Additionally, a Poisson’s ratio of 0.3 was defined, as suggested by Hussein and Meguid (2016) and Liu and Page 42

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Ling (2007). Plastic properties were defined through a tabular set defining the true stressstrain curve of the monotonic behaviour of the biaxial geogrid. The values of nominal stress and strain were converted to true magnitudes to account for the changes in area and length of the specimen during deformation. The formulas needed for such conversion are as follow. 𝜀𝑡𝑟𝑢𝑒 = ln(1 + 𝜀𝑛𝑜𝑚𝑖𝑛𝑎𝑙 )

Equation 4.1

𝜎𝑡𝑟𝑢𝑒 = 𝜎𝑛𝑜𝑚𝑖𝑛𝑎𝑙 (1 + 𝜀𝑛𝑜𝑚𝑖𝑛𝑎𝑙 )

Equation 4.2

For these terms to be accepted by Abaqus, an initial yield value of 1 MPa was assumed (SIMULIA 2008) for simplicity, which was consistent with the real monotonic behaviour of a polypropylene geogrid, since plastic strains develop basically as soon as deformation begins.

Figure 4.3: Elastic properties of constitutive model for geomesh-reinforcement (Source: personal collection)

Figure 4.4: Plastic properties of constitutive model for geomesh-reinforcement (Source: personal collection)

After the creation of elastoplastic material model, the next step was to determine appropriate section properties and mesh element type to include the geomesh-reinforcement in the FE model. To be able to incorporate reinforcement inside the FE model presented in chapter 5, 2 options were available in Abaqus/Standard: using sections with rebar or creating composite sections (SIMULIA 2008). •

The rebar option gave the option of defining reinforcement with defined spacing and area values, among other things.

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Table 4.2: Rebar layers window in Abaqus/Standard

When Abaqus/Standard processes the amount of reinforcement present within the section, it creates an equivalent reinforcement sheet with an equivalent thickness equal to 𝑒𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 =

𝐴𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡 𝑠

Equation 4.3

being s the spacing between discrete reinforcement ribs of geomesh, A the total area of reinforcement present within the spacing length and e the equivalent reinforcement thickness in sheet form, which runs along the length of the section. •

The composite section option enables the creation of plies of different materials which are perfectly bonded with each other. This option requires the reinforcement to be previously transformed into an equivalent geomesh sheet, which equivalent thickness is input as one of the composite layer variables, along material orientation and angle.

Figure 4.5: Composite layup window in Abaqus/Standard

In the end, since both the rebar and composite layup options consider the geomesh reinforcement as a sheet with an equivalent thickness, the composite layup option was chosen to model the reinforcement. It is very important to mention that using both methods convey a disadvantage, which was that the reinforcement was considered to be bonded perfectly with Page 44

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading the internal geometry, which in reality does not occur. In reality, both geomesh layers are tied through the adobe wall with raffia strips, which bond the reinforcement at discrete intervals along the adobe brick wall. One may get an approximation to perfect composite action when mortar is used to create a finishing layer on top of the geomesh reinforcement, but that method by itself is true to a certain extent, as the mortar is made from the same weak and brittle material as the adobe bricks. For the development of this work, it was worth overlooking this issue and establish a compromise in this particular subject. However, it is encouraged that further research be done in recreating real bonding conditions between the geomesh reinforcement and the adobe wall.

Photograph 4.2: Bounding of geomesh-reinforcement layers through the adobe wall with the use of raffia strips at discrete locations (Source: Blondet et al. 2005)

4.2.2.1. Modelling the equivalent geometry of geomesh-reinforcement layers As said in previous paragraphs, a composite layup section was used to model the hybrid behaviour of the geomesh-reinforced adobe wall. For this option to adequately represent strength provided by the geogrid layers, an equivalent thickness for a geogrid sheet-like section had to be calibrated based on the real uniaxial tensile behaviour of the biaxial geogrid used to wrap the adobe structure. This transformation can be seen in Figure 4.6.

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Figure 4.6: Transformation of explicit geometry of geomesh-reinforcement into an equivalent sheet, with a constant reduced thickness

As discussed in previous sub-chapters, a hyperbolic function modified with Masing rule (Liu and Ling 2006) was used to represent the idealised isotropic behaviour of the geomesh reinforcement, which was calibrated based on the XMD and MD tensile test data extracted from project DAI 113.0225 (Blondet et al. 2005). The numerical calibration of the equivalent thickness was done with a constant mesh size of 10.0 mm to maintain accuracy. The chosen equivalent thickness was 0.0985 mm, which closest depicted the real behaviour of the geogrid for both the real curve and the hyperbolic curve, as seen in Graph 4.7 below. The results from the numerical model with the previously mentioned equivalent thickness are represented by the green curve.

Graph 4.7: Calibration of the equivalent geomesh-reinforcement sheet thickness

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As can be seen, the numerical model presented lower strength values compared to the hyperbolic function curve. The green curve reached deficits of a maximum of 13.33% relative to the hyperbolic function during the initial part of the curve, which then dropped to near zero deficits around the final part of the curve. These variations were the unintended consequence of the change in geometry, from explicit form to equivalent sheet form, which affected strength values during the early onset of deformation of the geogrid sheet. Despite these variations initial strength values, the equivalent thickness was maintained at 0.0985 mm because its accurate behaviour prior to the rupture of the geogrid. This final value, however, suffered an increase after performing a mesh convergence study on the equivalent geogrid sheet. After this process, a convergence study was performed to determine the most appropriate mesh size to model the geomesh reinforcement. The same monotonic displacement analysis was run with approximate mesh element sizes of 100, 50, 20, 10, 2.5 and 1.5 mm to determine the mesh size at which the vertical reaction force converges.

Figure 4.7: Convergence study performed on equivalent geogrid sheet using approximate element sizes ranging from 100 mm (left) to 1.5 mm (right)

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Graph 4.8: Convergence analysis performed on geogrid sheet with an equivalent thickness of 0.0985mm

According to the convergence analysis shown in Graph 4.8, the value of vertical reaction force convergences at an approximate element size of 50 mm onwards. This is due to reduction of difference between the magnitudes of maximum strength from 1.17% to 0.12%. The lower limit for element size is set at 1.5 mm, as a mesh with smaller elements introduces excessive stiffness into the model. So, in order to appropriately model elastoplastic geogrid behaviour in the equivalent sheet, an approximate element size of 50.0 mm or smaller may be used. An approximate element size of 20.0 mm or 10.0 mm is suggested for jobs with regular to low computational cost. However, if computational cost is great, as it is with the model evaluated in this work, a greater mesh size than 50.0 mm may be used if the equivalent thickness of the geogrid sheet is modified, for it to be able to reproduce strength values that fall within the convergence analysis’ limits. Since the material model of the adobe continuum has been calibrated to an approximate element size of 100 mm, due to the great computational cost and stiffness issues of the model as a whole, the equivalent thickness of the geogrid sheet was adjusted in order to recreate converged values for an element size of 100 mm (see Graph 4.9).

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Graph 4.9: Calibration of modified geogrid sheet thickness for a 100 mm approximate element size

The value of the maximum reaction force for a 100 mm mesh size converged at an equivalent thickness of 0.0996 mm, which only differed in 0.052% and 0.17% relative to the converged magnitudes of force belonging to a 50 mm and 10 mm mesh size, respectively (see Graph 4.10). Based on these values, the final FE model properties of the geogrid-reinforcement sheet, assigned to the layers of the composite section defined earlier, were set at an approximate element size of 100 mm with a section width of 0.0996 mm.

Graph 4.10: Modified convergence analysis

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Material constitutive modelling After the convergence analysis and obtaining the equivalent thickness that was used in the geomesh-reinforcement layers in the FE model, a short comparative analysis was performed to attempt to simulate unloading and reloading behaviour by implementing different elastic moduli during cyclic loading of the equivalent geogrid sheet. The purpose of this analysis was to attempt to incorporate the unloading behaviour that real polypropylene geogrids present during a cyclic test. Graph 4.11 shows a linear approximation of the reloading and unloading elastic moduli compared to the initial elastic curve of the geogrid. The cyclic behaviour was approximated using the hyperbolic function modified with Masing rule from Liu and Ling (2006), as shown in chapter 2.

Graph 4.11: Simplification of unloading and reloading curves into linear expressions and compared to initial elastic linear curve

In general, as shown in chapter 2, during cyclic uniaxial tensile tests, the only property which is affected is the residual and accumulated plastic strain in the material, whereas the strength is not affected significantly (Ling et al. 1998). The comparison between experimental monotonic and cyclic behaviour of a polypropylene geogrid from tests performed by Ling et al. (1998) can be seen below in Graph 4.12.

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Graph 4.12: Comparison of real monotonic and cyclic tests of a polypropylene type A geogrid (Ling et al. 1998)

Following that same methodology, the hysteretic curves of the polypropylene geogrid with different initial elastic moduli were compared to one another and to the real monotonic XMD curve. The curves were obtained through a numerical simulation of cyclic displacements of the equivalent geogrid sheet in Abaqus/Standard. As the material model chosen for the polypropylene geogrid was an elastoplastic model which only describes monotonic behaviour without differentiating between initial, unloading and reloading elastic moduli, the magnitude of the elastic modulus was varied between the values of 5608.24 MPa and 3660.69 MPa, corresponding to the initial and unloading elastic moduli, in order to see their effect on the unloading behaviour during cyclic loading.

Table 4.3: Variation of the magnitude of elastic modulus within the geomeshreinforcement material constitutive model properties

The simulation results showed that a model with an elastic modulus equal to the unloading modulus would depict an unrealistic cyclic behaviour, especially during the first stages of loading, as it deviates from the monotonic curve excessively; approximately 30% to 50% of the total value of strength was lost during the initial cycles, when comparing the brown curve to the monotonic one. The blue curve showed appropriate cyclic behaviour and consistent Page 51

Material constitutive modelling with what was seen in real cyclic tests compared to monotonic test values, which was a slight increase in strength after several hysteretic loops relative to the monotonic values.

Graph 4.13: Comparison of cyclic behaviour of geogrid considering different elastic moduli with real monotonic tensile behaviour

Lastly, it is very important to mention that the chosen elastoplastic model in Abaqus/Standard had a very important disadvantage, which was that compressive behaviour could not be eliminated from the overall response, be that during monotonic or cyclic behaviour. Since the dimensions of the geogrid sheet were sufficient for it not to buckle during compressive loading, the sheet exhibited mirrored mechanical behaviour in compression, reaching similar maximum values as on the tensile side. For example, when unloading occurred after the last hysteresis loop (32 mm of positive displacement), the unloading curve should have flattened at around zero tensile strength. Instead, the unloading curve continued to drop linearly until it reached the yield function in the compressive zone, which was a mirror of the tension yield function. Compressive strength even reached higher values than its tensile counterpart, which was attributed to residual forces within the material after plastic deformation. This unrealistic compressive behaviour of the geogrid mesh was an unwanted effect of the material model used in Abaqus/Standard, which could not be eliminated due to restrictions in the FE programme (SIMULIA 2008).

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5. Numerical modelling of geomesh-reinforced adobe wall One of the main objectives of this dissertation was to create an adequate numerical model that would be able to simulate the response of a geomesh-reinforced adobe wall (presented in chapter 3) subjected to cyclic loading with reasonable accuracy. For that, the finite element (FE) programme Abaqus was used. Abaqus/Standard (SIMULIA 2008) allows the user to sketch complex geometry, assign the model appropriate material and geometric properties, loads, boundary conditions and analysis type through a graphic user interface and in script form. For this work, all the modelling was done using the graphic interface. The general properties of the model, which will be explained in detail over the next sub-chapters, are as follow: •

The geometry of the geomesh-reinforced was modelled explicitly with shell elements.

•

Four material types were defined: adobe, polypropylene, concrete and timber. The first two are modelled with elastoplastic properties, whereas the latter are modelled elastically for simplicity.

•

Composite sections were used to simulate the combined behaviour of a geomeshreinforced adobe section.

•

Shell elements of an approximate size of 100 mm were used in the model.

•

Three steps were created for analysis: initial, gravity, fundamental frequency, cyclic and earthquake.

•

Loading scenarios created: gravity loads.

•

Boundary conditions created: Fixed base, cyclic displacement and earthquake excitation.

•

Types of numerical analysis: Static implicit analysis and dynamic implicit analysis for the cyclic displacement and earthquake scenarios, respectively.

5.1. Geometry of finite element model The geometric layout and main properties were based on the properties of the same adobe wall present in Tarque (2011). As a macro-modelling technique was used to model the adobe wall (Tarque 2011), the geometry was modelled in an explicit manner, including all major material components and boundaries and considering the adobe material as a continuum.

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Numerical model of geomesh-reinforced adobe wall Since the main interest was in the results from lateral loading of the in-plane adobe wall, detailed numerical results across the thickness of the walls was not needed. Therefore, the base feature selected for modelling was a deformable and homogeneous 4-node shell with a uniform thickness of 300 mm (SIMULIA 2008). To re-create the appropriate geometry, datum planes were used to create the partitions in the planar shell and extrusion shells for the 3D geometry. The dimensions used were identical to the ones presented in chapter 3 and an adequate sectorisation of the geometry was created to properly assign the geomeshreinforcement, which can be seen in Figure 5.1.

Figure 5.1: Isometric (left) and frontal view (right) of the geometry of the FE model in Abaqus/Standard

5.2. Material and section modelling Based on the material types present in the geomesh-reinforced adobe wall, 4 material models were created: concrete, timber, adobe and polypropylene. The concrete was modelled as a very low strength concrete with internal steel reinforcement (only the weight was considered) in order to account for untrained labour force and inadequate proportioning of the concrete mixture. A density of 2400 kg/m3 (2.4x10-9 tonnes/mm3) was assigned, which corresponded to the density a normal-weight reinforced concrete material, with a characteristic compressive strength (fc) of 10 MPa, an elastic modulus of 26,000.0 MPa (British Standards Institution [BSi] 2004) and a Poisson’s ratio of 0.25 (Tarque 2011). The material used to model the wood lintel was lightweight timber, with an average density of 700 kg/m3 (7.0x10-10 tonnes/mm3), an elastic modulus of 15,000.0 MPa and a Poisson’s ratio of 0.15 (Tarque 2011). Both previously described materials were

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading modelled as perfectly elastic in the FE model, as the scope of this work did not include the evaluation of a detailed response of those materials. The polypropylene material model was intended to represent a BX1100 biaxial geogrid (Tensar 2013), as described in chapter 4. The constitutive model was chosen as elastoplastic, with a Young’s modulus of 5608.24 MPa (calculated in chapter 4) and a Poisson’s ratio of 0.3 (Hussein and Meguid 2016). As for the inelastic behaviour of the constitutive model of polypropylene, a yield function based on true stress and strain was used, as presented in chapter 4, which started at 1 MPa.

5.3. Section modelling and meshing properties As mentioned before, the general geometry of the model was created using planar and extrusion shell elements with a uniform thickness of 300 mm. Since the main intent of the model was to simulate the cyclic behaviour of a geomesh-adobe continuum, composite layup option was chosen to attempt the feat. This option enabled the creation of composite sections of geomesh-reinforcement and adobe within finite areas within the model, as shown in Figure 5.2. The geomesh layers were positioned on both faces of the shell elements with varying thickness, depending on the region in question.

Figure 5.2: Assignment of composite layup sections in in-plane and perpendicular adobe walls (left) and mesh of GRAW (right)

Based on the arrangement of geomesh-reinforcement presented in chapter 3 (Figures 3.2 and 3.3), a sectorisation of both in-plane and perpendicular walls was performed in order to

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Numerical model of geomesh-reinforced adobe wall assign the appropriate amount of reinforcement in each area. The variation in reinforcement layers was expressed by changing the thickness composite layers within specific sections. The geomesh-reinforcement thicknesses, for front and back of the section, are shown in Table 5.1 for each area presented in Figures 3.2 and 3.3:

Table 5.1: Arrangement of geomesh-reinforcement layers in FE model

Lastly, an approximate element size of 100 mm was selected to mesh the complete model. This size was selected after considering convergence studies performed for adobe and the geomesh reinforcement, as well as the computational cost associated with each mesh size, as presented in chapter 4.

5.4. Steps for analysis Contrary to other finite element programmes, Abaqus/CAE allows the user to divide the loading and boundary condition scenarios into steps, which are applied in sequence or simultaneously, depending on how one defines the dependency of each step (SIMULIA 2008). For this model 5 steps were created in order to obtain the needed data pertaining the cyclic pushover analysis (used for chapter 6) and seismic performance analysis (used for chapter 7): Initial, gravity, cyclic, earthquake and modal analysis.

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Figure 5.3: Steps created in FE model

•

Step 1 “Initial”: Abaqus requires an initial step to be defined in any model. This step contains all boundary and loading sets that must be applied at the start of time integration and will be propagated throughout the sequel steps (SIMULIA 2008). This condition propagated to all following steps and was present at time increment 0.

•

Step 2 “Gravity”: Following step 1, step 2 was defined to contain the gravity loads of the whole model and was defined as a general, static procedure. This step followed the step “Initial” and was applied through direct integration of one time-increment, as this loading scenario was very simple and detailed results were not of any particular interest. Initial and maximum time increments were defined as 1.0, allowing for a minimum time increment of 1.0x10-5. Additionally, the option to consider nonlinear geometry during the application of gravity loads was activated as a precautionary measure. A full Newton solution technique was used during this step of the analysis.

•

Step 3 “Cyclic”: This step was defined as a general, static procedure and was particularly important, since a lot of detail and accuracy was needed in order to calibrate the resulting hysteretic curves with the real values presented in chapter 3. Since large deformations were expected to develop in certain locations during the pushover cycles, the option for nonlinear geometry was activated and an automatic stabilisation factor (with dissipated energy fraction) of 0.0025 was used. Additionally, initial, maximum and minimum time increments of 0.01, 0.01 and 1.0x10-30 were defined, with a maximum number of 80,000.0 attempts at convergence. Lastly, a full Newton solution technique was used. This step was defined to follow step 2. If step 4 was intended to be executed, this step was suppressed.

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Numerical model of geomesh-reinforced adobe wall •

Step 4 “Earthquake”: This step was created to simulate an earthquake response to a seismic displacement signal. Therefore, step 4 was defined as a dynamic, implicit procedure. Same as in step 3, accuracy was of the utmost importance, which is why the initial, maximum and minimum time increments were assigned values of 0.01, 0.01 and 1.0x10-10, with a maximum number of 1’000,000.0 attempts at convergence. Lastly, the nonlinear geometry condition was activated and a full Newton solution technique was selected for analysis. This step was defined to follow step 2. If step 3 was intended to be executed, this step was suppressed.

•

Step 5 “Modal analysis”: This step was defined as a linear perturbation, frequency procedure and executed in parallel manner to steps 4 and 5. During the execution of either step 3 or 4, this step was suppressed. Following steps 1 and 2, the step “Modal analysis” was created to evaluate the mode shapes of the structure. By defining the amount of mode shapes that one wants to analyse, this type of analysis showed their influence within the model and determined which ones were participating the most. In this case, only the first 10 mode shapes were requested in order to determine the fundamental mode of vibration of the structure.

5.5. Boundary and loading conditions As described in sub-chapter 5.3, 5 steps were created to analyse both simulation scenarios, for which one loading condition and three boundary conditions were defined.

Figure 5.4: Boundary and loading conditions created in FE model

•

Fixed: This boundary condition (BC) was created to fix the base of the model to the floor, preventing motion in all 6 degrees of freedom (3 translational and 3 rotational). This BC was associated to the step “Initial” and was propagated to all other 4 steps.

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Gravity loads: This loading condition was defined as a gravity type load, for which the value and direction of gravity had to be defined. For this model, gravity value of -9,810.0 mm/s2 was defined along the Y-axis (2-axis in Abaqus), which was used in combination with the mass values (defined in sub-chapter 5.2) to establish the weight of all the model components. This loading condition was associated with the step “Gravity” and was propagated to all subsequent steps.

•

Pushover: The “Pushover” BC was defined as a lateral displacement, applied on one edge of the top concrete beam, associated with the step “Cyclic”. This BC included the definition of the value and direction of the applied displacement, which were +100 mm applied along the X-axis (or 1-axis in Abaqus). Additionally, the option of tabular amplitude was created which modified the absolute displacement value above, simulating a series of cycles of displacement (a series of pushes and pulls), with peak values ranging from 5 mm to 100 mm.

Figure 5.5: Boundary conditions associated with the step “Cyclic”

Table 5.2: Example of tabular data used to define amplitude of cycles of displacement during the step “Cyclic”

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Numerical model of geomesh-reinforced adobe wall

•

Earthquake: Finally, the “Earthquake” BC was defined as a lateral displacement applied at the base of the model and associated with the step “Earthquake”. Similar to “Pushover”, this BC used the peak ground displacement (PGD) of the earthquake as its maximum displacement along the X-axis in combination with a tabular amplitude option which defined the time-history of the earthquake as a ratio of the PGD over time.

Figure 5.6: Boundary conditions associated with the step “Earthquake”

5.6. Definition of types of analyses According to the outline of the dissertation stated in sub-chapter 1.3, chapters 6 and 7 required the model to be subjected to cyclic lateral displacements and an earthquake excitation, respectively. Since the earthquake signal and the cyclic displacements were applied along the X-axis of the model, or axis 1 in Abaqus (SIMULIA 2008), the two types of scenarios can be considered identical in essence. However, the main difference between the two scenarios, besides the location of the applied displacements, was the importance of inertial forces within the structural system.

5.6.1. Cyclic displacement scenario In the scenario where the model was subjected to cyclic lateral displacements, the frequency of the loading during all cycles was quite low, ranging from 0.0008 Hz to 0.004 Hz, which is associated with a low loading rate. This scenario was considered to be on the

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading verge of the static and quasi-static deformation. Under such low strain-rate conditions, inertial forces are not significant, and the system could be modelled with a static approach, neglecting the effect of inertial forces and damping forces. This difference is shown in Figure 5.7 below. In this case, the time integration method chosen was an implicit one.

Figure 5.7: Characteristic time and strain rates associated with diverse types of loading (Source: Lindholm 1971)

5.6.2. Earthquake excitation scenario The scenario involving earthquake excitations is a different story compared to the one described in 5.4.1. Here, inertial forces play a key role: Since frequencies within an earthquake signal are plentiful, different modes of vibration will amplify distinct types of structures. In general, the values of these frequencies fall within the range of low to regular strain-rate excitations, which classifies most types of earthquake excitations as intermediate strain-rate deformation. This means that inertial forces have to be included in the analysis due to their importance. Based on the conditions stated above, the analysis type chosen for this loading scenario was a dynamic implicit analysis. With this type of analysis, the system could be analysed dynamically, considering inertial, damping, deformation and ground forces applied on the model. An implicit time integration method was chosen.

For both scenarios, an implicit time integration method was chosen because of the advantages it presented. It enabled the model to be unconditionally stable and reduced the

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Numerical model of geomesh-reinforced adobe wall risk of convergence failure. The main disadvantage in using an implicit approach was that it implicated a higher level of computational cost because of the need of inverted stiffness matrices during calculations, but the gain in accuracy was worth the compromise in time, as equilibrium along the time continuum was assured. Compared to this option, an explicit time integration method only ensures equilibrium at discrete time intervals, loosing accuracy in the response of the overall system. Also, the increment size used in an implicit analysis allowed changes depending on the amount of instability present during the simulation, which was beneficial in preventing sudden crashes of the numerical model and enabled higher rates of convergence in low and intermediate strain-rate scenarios. Now, based on the information provided in chapters 3, 4 and 5, the FE model created was fitted to real hysteretic curve of pushover experiment while varying diverse material and model properties. After that analysis, the cyclic curve with the best material and FE properties was selected and subjected to several scaled earthquake signals.

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6. Calibration of numerical model with cyclic pushover experimental results After the definition of the material models selected for the adobe continuum and the geomesh-reinforcement, as well as the method of FE modelling in Abaqus/Standard presented in chapters 4 and 5, this part of the dissertation presents the calibration of the hysteresis curves obtained from by varying material and numerical properties of the model until the best fit was found. As described in chapter 3, the hysteresis curve from the cyclic pushover experiment showed that the geomesh-reinforcement provided necessary ductility to the overall adobe structure, enabling it to increase its strength by a small margin of approximately 11.4% (from 35 kN to 39 kN). The hysteresis loops were relatively stable and showed the deterioration of the adobe wall’s stiffness after each hysteretic cycle; deterioration which became evident after the 5 mm hysteresis cycle. There were several factors which were found to influence the overall hysteretic behaviour of the geomesh-reinforced adobe wall. Be that the development of strength during hysteretic loops, unloading behaviour (unloading slope), incurred damage in the structure, be that in tension or compression, among others. These factors were related directly with the properties of adobe compressive and tensile behaviour, as well as geomesh-reinforcement tensile behaviour, all of which could be simulated within the presented FE model through the adaptation of the following variables: •

Adobe peak tensile strength (ft): Peak tensile strength of the adobe defined the maximum strength developed by the overall reinforced structure. Since failure behaviour starts as soon as cracking in the continuum occurs, the higher the magnitude of ft the greater the strength the adobe structure was able to develop. However, another factor also greatly influenced that same ability.

•

Adobe tensile fracture energy (Gft): Tensile fracture energy also contributed to the maximum strength that the structure was able to develop, as it depended on the peak tensile stress as well. If the amount of stress able to be transmitted along deteriorating tensile cracks increases, so does the increase in overall strength of the adobe wall the greater the displacement becomes. This will become clear through the work presented in this chapter.

•

Adobe peak compressive strength (fc): Peak compressive strength of the adobe continuum does not affect overall lateral strength that much. However, it was Page 63

Calibration of numerical model with cyclic pushover experimental results important to define that property to be accurate with realistic values of peak compressive strength of adobe assemblages as shown in Peralta and Torrealva (2009). •

Adobe compressive fracture energy (Gfc): Compressive fracture energy is defined by the characteristics of the softening curve in the CDP model, which depended on the value of εcm, as well as the peak compressive strength. The magnitude of this variable had an effect on the slope of strength increase during reloading, as well as on the maximum value of strength reached after damage had occurred by surpassing peak tensile stress in critical locations of the structure.

Considering all the above-mentioned characteristics, some initial assumptions about confined behaviour of materials subjected to compressive and tensile stresses related to the GRAW were made before a calibration method was established. Based on that a priori knowledge, the identification of the most influencing variables and their values was made easier. Amongst these assumptions were the following: •

In general, confined materials develop a more ductile behaviour when subjected to compressive stresses, e.g. confined concrete, reaching higher values of deformation and peak compressive stress. Depending on the level of confinement and after reaching peak compressive stress, the softening curve will represent a controlled decrease in strength or the material will rupture when the maximum tensile stress of the confinement is reached. This behaviour can be extrapolated, to a certain extent, to geomesh-reinforced adobe. Since the geomesh-reinforcement was only bonded at discrete points throughout the adobe wall as seen in Photograph 4.2, it can be established that confinement was not as effective as it was idealised in the model. An example of effective confinement by reinforcement is seen in Figure 6.1 below.

Figure 6.1: Comparison of effectiveness of confinement action with different varied confinement ratios in reinforced concrete elements (Source: Sulaiman M.F. et al. 2017)

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Despite the difference in effectiveness of confinement of the geomesh-reinforcement in the natural-scale wall and the FE model, an increase in peak compressive strength of the adobe may be expected, as well as an increase in deformation capacity and reduction of damage-rate in compression, creating a more ductile behaviour (see Graph 6.1).

Graph 6.1: Expected increase in peak compressive strength and deformation capacity of the adobe continuum

•

In reality, tensile action of the geomesh-reinforcement is expected to be activated after certain damage in tension was sustained in the adobe wall. Since the FE model in this work was modelled to have a perfect bond between the adobe continuum and the geomesh-reinforcement on either side, tensile resistance exerted by the geogrid was anticipated to be activated as soon as tensile, and compressive, deformations occurred in the model.

•

Since previous calibrations in unreinforced adobe structures have already been attempted by Tarque (2011), any variation in tensile fracture energy was known to cause significant variations in strength during the cyclic loops. Even if the presence of geomesh-reinforcement helped the closing of tensile cracks, the amount of energy accumulated during displacement was expected to remain approximately the same, since the adobe continuum remained the same. Based on that, it was decided to leave the recommended magnitude of tensile fracture energy constant, as it was not expected to vary by a significant margin. However, variations in the fracture energy’s magnitude were performed in a manner to confirm the previously mentioned assumption.

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Calibration of numerical model with cyclic pushover experimental results In total, 50 models were created with different sets of magnitudes for the above-mentioned variables relative to the adobe behaviour both in compression and tension. Since uniaxial test values were available to model the behaviour of the geogrid reinforcement, the main effort during the evaluation of the FE models was in determining the appropriate values to adequately describe the behaviour of adobe during cyclic loading and while confined with geomesh-reinforcement. Out of the 50 models created, the ones with reasonable values were filtered and presented in the following sub-chapters.

6.1. Influence of compressive and tensile damage factors on peak strength values during hysteresis cycles As described in chapter 4, damage factors dc and dt define the level of damage that the adobe continuum has been subjected to during cyclic displacements. Depending on the magnitude and strain of activation, the hysteretic loops will have either lower or greater peak magnitudes of strength. This is directly true for tensile behaviour. Since tensile fracture energy and peak tensile stress define the magnitude of strength the adobe wall was able to develop, tensile damage factors reduced initial strength values acquired during the first hysteretic loops. For example, Graph 6.2 the hysteretic curve GRAW-COMP-#6 represents a FE model for which neither tensile nor compressive damage factors were defined, so that the effects of an undamaged material model may be evaluated.

Graph 6.2: Hysteretic curve of GRAW-COMP-#6 FE model which did not include damage factors either in compression or tension

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading For values of fc, ft, Gfc, Gft of 0.40 MPa, 0.04 MPa, 0.1206 N/mm and 0.01 N/mm, FE model GRAW-COMP-#6 reached peak strength values of 37 kN and 47 kN during the positive hysteresis loops of 2.5 mm and 5 mm, exceeding the real ones by 8.82% and 34.29%, respectively. As for their negative counter parts, simulation values reached -43 kN and -50 kN during the hysteretic loops of -2.5 mm and -5 mm, exceeding experimental strengths by 43.33% and 66.67%, respectively. The excessive strength values obtained for GRAWCOMP-#6 during the 5 mm hysteresis loops (positive and negative) were caused by the absence of tensile damage factors, which would have reduced the peak values of strength during the considered hysteretic loops, as well as all others that would have followed. The effect of appropriate tensile damage factors (shown in Table 6.1) was demonstrated through the results of FE model GRAW-COMP-#9, which are shown in Graph 6.3, effectively reducing simulated strength to much more accurate results. Peak simulated strengths of 39 kN and 36 kN were reached during the 5 mm and 10 mm positive hysteretic loops, which only exceeded experimental strength values by 11.43% and 2.86%, respectively. As for the negative cycles of -5 mm and -10 mm, simulated strength values reached -36 kN and -30 kN, which only exceeded experimental values by 20% and 3.23%, respectively.

Graph 6.3: Hysteretic curve of FE model GRAW-COMP-#9. This FE model included tensile damage factors defined in Table 6.1

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Calibration of numerical model with cyclic pushover experimental results

Table 6.1: Tensile damage factors defined for FE model GRAW-COMP-#9

The differences in simulated strength values from FE models GRAW-COMP-#6 and #9 prove the effectiveness of tensile damage factors within a cyclic loading scenario; by including the definition of damaged behaviour, strength values were able to be simulated with reasonable accuracy. In the case of compressive damage factors, their influence over general hysteretic behaviour of the simulated GRAW was not as notorious as the influenced exerted by the inclusion of tensile damage in the FE model. However, it was appropriate to define compressive damage in the FE model in order to simulate accurate compressive cyclic behaviour of adobe and attempt to obtain more accurate results. Graph 6.4 shows the difference in simulated strength values between the FE models GRAW-COMP-#9 and #10 after a change in compressive damage factors was defined at an inelastic strain value of 0.01. The reduction of compressive damage from 75% to 50% (shown in Table 6.2) in the FE model #10 generated a slight increase in strength during both positive and negative 10 mm and 20 mm hysteresis loops. The strength values from model #10 increased 1.3%, 5.14% and 3.51% during the positive 5 mm, 10 mm and 20 mm hysteresis loops, respectively. As for the negative hysteresis loops of 5 mm and 10 mm peak displacement values, model #10 increased its strength values by 2.78% and 5.06%, respectively.

Compressive damage factors - Set 1 Inelastic strain dc (mm/mm) 0.00 0.00 0.10 0.0008 0.75 0.01 0.90 0.02

Compressive damage factors - Set 2 Inelastic strain dc (mm/mm) 0.00 0.00 0.10 0.0008 0.50 0.01 0.90 0.02

Table 6.2: Compressive damage factors defined in FE models GRAW-COMP#9 (left) and #10 (right)

Since the change in compressive damage factors was at 0.01 inelastic compression strain, the positive 5 mm hysteresis loop was not affected by it, because not enough inelastic strain had been accumulated for significant amounts of the adobe continuum to have reached higher Page 68

Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading levels of compressive damage. However, its negative counterpart did suffer some changes, as its peak simulated strength reached higher values than #9. It is worth mentioning that the decrease in compressive damage did not affect the onset of the damaged loading curve present in both FE models during the start of the positive 20 mm hysteresis loop. It only increased the strength values by approximately 5%. Another important consequence of the reduction in compressive damage was the increase in reloading and unloading stiffness slopes in the FE model #10 relative to #9.

Graph 6.4: Comparison of hysteresis curves from FE models GRAW-COMP-#9 and #10 after change in compressive damage factors2

6.2. Influence of compressive and tensile fracture energy on peak strength values during hysteresis cycles As said at the beginning of this chapter, both compressive and tensile fracture energy play a role in defining hysteretic behaviour overall. The magnitude of tensile fracture energy, specifically, was directly proportional to the strength magnitudes achieved during simulation. This influence can be seen clearly in Graph 6.5, where tensile fracture energy was increased by 25% and 100%, from its initial value of 0.01 N/mm in FE model GRAWCOMP-#25. The effect on peak simulated strength reached was evident in the 10 mm hysteresis loop, where strength values increased by 25% and 43.75% from the initial magnitude achieved in 2

The big difference in magnitude between simulated and experimental strength during the 5mm hysteresis loops (positive and negative) were caused because of the absence of the 2.5 mm hysteresis loops. In some instances, to save computational time, some displacement cycles were skipped in order to analyse greater displacement cycles.

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Calibration of numerical model with cyclic pushover experimental results FE model #25. Based on the accuracy of the results obtained through these curves, a recommended range of 0.01 N/mm to 0.0125 N/mm was established for tensile fracture energy. This range provided the limitations on strength growth, which then was fine-tuned with appropriate magnitudes for damage factors and compressive fracture energy.

Graph 6.5: Influence of increase in tensile fracture energy from 0.10 N/mm (GRAWCOMP-#25) to 0.125 N/mm (GRAW-COMP-#22) and 0.02 N/mm (GRAW-COMP-#20)

Another way to show the effect of tensile fracture energy on the overall development of strength was to analyse the structure under monotonic loading. Graph 6.6 shows the results of a small difference in tensile fracture energy had over the undamaged adobe continuum. The variation in tensile fracture energy of 0.0025 N/mm (reduced from 0.01 N/mm in GRAW-COMP-#32c to 0.0075 N/mm in GRAW-COMP-#32d) caused a very sharp difference in strength development. Lastly, the influence of the magnitude of compressive fracture energy was not as evident as demonstrated for tensile fracture energy. Only slight increases in strength were observed after increasing the magnitude of compressive fracture energy, around 2.78% during the 5 mm, 10mm and 20 mm hysteresis loops, shown in Graph 6.7.

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Graph 6.6: Influence of tensile fracture energy in GRAW when subjected to monotonic loading

Graph 6.7: Influence of increase of compressive fracture energy from 0.1426 N/mm (GRAW-COMP-#14) to 1.02 N/mm (GRAW-COMP-#18)

6.3. Influence of peak compressive stress on peak strength values during hysteresis cycles As shown in chapter 2, the magnitudes of peak compressive and tensile stress of the adobe continuum was directly proportional to their respective fracture energies. The higher the value of peak stress, the greater the fracture energy. However, the effect caused by such

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Calibration of numerical model with cyclic pushover experimental results variations was not as evident as extending the fracture curve by decreasing the softening effect of both compressive and tensile behaviour. Graph 6.8 shows the effect that a 250% increase in peak compressive strength had on hysteretic behaviour of the GRAW. Even though the increase in compressive strength was significant (relative to the mechanical properties of adobe in compression), compressive fracture energy was only increased by 325%, from 0.1646 N/mm (GRAW-COMP-#12) to 0.5340 N/mm (GRAW-COMP-#13), compared to the magnitude obtained by increasing the value of εcm in FE model #18.

Graph 6.8:Effect of increasing peak compressive strength in the adobe continuum from 0.40 MPa (GRAW-COMP-#12) to 1.0 MPa (GRAW-COMP-#13)

Lastly, the evaluation of the effects caused by increasing peak tensile was not performed in this work, since this property does not vary by reinforcing adobe with geogrid. Normally, an increase in peak tensile stress is obtained by varying the ratio of soil, sand and straw present in the adobe, which was not part of the scope of this dissertation.

6.4. Limitations and inconsistencies of the evaluated FE models Even though reasonable results were obtained during the calibration process by several FE models, the hysteretic curves hinted to some important inconsistencies and limitations in the numerical models.

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6.4.1. Cyclic locking One of these evident inconsistencies was the bow-like loops which appeared at the end of the 10 mm, 20 mm, 50 mm and 100 mm hysteretic cycles. As shown in the background of Graph 6.9, the real hysteretic curve defined a plummet of strength immediately after the unloading part of the cycle had begun. However, most FE models showed an increase in strength as displacement was being reduced. A detailed analysis of this phenomenon pointed towards an incompatible behaviour of the material models for adobe and geomeshreinforcement used in Abaqus/Standard in combination when subjected to large3 cyclic displacements. The size of these loops grew larger after each phase (Graph 6.10), which indicated that after elastoplastic tensile action of the reinforcement during the displacement cycle, it immediately entered in compressive elastoplastic action due to the inability to avoid compressive behaviour, as stated in chapter 4 (SIMULIA 2008). The term cyclic locking was attributed to this type of occurrence. Unfortunately, this phenomenon was present in all FE models, which indicated that further research should be done to produce an adequate elastoplastic material model which can effectively eliminate compressive behaviour within the geomesh-reinforcement.

Graph 6.9: Cyclic locking during cycles of peak displacements larger than 5 mm

3

Large displacements relative to the drift capabilities of an unreinforced adobe wall. Those types of structures start developing damage along the mud mortar as soon as the top of the wall is subjected to 2.5 mm lateral displacements as shown by Blondet et al. (2005).

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Graph 6.10: Increase of cyclic locking effect with increasing cycles of displacement

6.4.2. Precision of simulated strength magnitudes in phases 7, 8 and 9 One of the main limitations of the use of the Concrete Damaged Plasticity model was that in hysteresis loops from ± 20 mm onwards, the simulated strength was not able to be approximated with precision to the real strength values obtained from the pushover cyclic test. Even though very low damage factors were used to model the cyclic compressive behaviour of adobe, the best approximations were only reached during the hysteresis cycles of ± 2.5 mm, ± 5 mm and ± 10 mm. In the best-case scenarios, as shown below in Graph 6.11, the deficit in strength relative to the experimental values were of 13 kN and of 9 kN during the positive and negative 20 mm hysteresis loops respectively, and of 11 kN during the positive 50 mm hysteresis loop. Out of all the differences in strength, the variation in percentage started from 27.3% at the negative 20 mm hysteresis loops. The rest of simulated strength values differed even more from the experimental values, which suggested that the FE models lost accuracy when hysteresis cycles grew larger than 10 mm. A probable theory that might have explained this behaviour was that not enough compressive fracture energy was used, which would have been interpreted as a very slow descent from the peak compressive strength. However, Graph ## showed that a tenfold increase in compressive energy did not have that much an effect on the overall ability of the model to develop adequate strength values from hysteresis loops of 20 mm onwards.

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading In order to determine precisely the reason why this deficit is only present in hysteresis loops greater than 10 mm, further research needs to be done which could complement the variation of factors and their effects studied in this dissertation.

Graph 6.11: Lack of precision of strength values from hysteresis loops of 20 mm onwards

6.4.3. Strength accuracy during negative hysteresis loops In addition to the cyclic locking phenomenon, the FE models were not able to accurately account for secondary damage sustained along the adobe wall which the experimental hysteresis curve is able to represent through reduced strength values during negative hysteresis cycles. As Graph 6.12 shows, the positive hysteresis loops were able to develop higher strength values (39 kN) because the adobe wall had not sustained any damage at the beginning of each phase. However, when the phase entered the negative displacement cycle, the wall had already been damaged to a certain extent which impeded the GRAW to reach the same strength values (-35 kN) as in their positive counterparts.

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No initial damage during new displacement cycles

Strength affected by incurred damage during positive cycle counterparts

Graph 6.12: Difference in experimental strength magnitudes developed during positive and negative hysteresis loops

6.4.4. Loading and unloading stiffnesses of FE models Furthermore, reloading and unloading stiffnesses could not be replicated in a satisfactory manner, as both reloading and unloading stiffnesses from numerical results were too great compared to empirical stiffnesses reached during the pushover analysis. This was especially true during the 5 mm and 10 mm hysteresis cycles, as is shown in Graph 6.13.

Graph 6.13: Lack of precision of magnitudes of unloading and reloading stiffnesses during the first 3 magnitudes of hysteresis loops.

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading However, during hysteresis cycles where considerable damage in the adobe material model had already occurred, some degree of accuracy was present during short periods of displacement ranges as shown in Graphs 6.14 and 6.15. In the case of Graph 6.15, which shows the curve of FE model #32, the initial behaviour of the loading and reloading stiffnesses during early hysteresis cycles was very peculiar: The loading slope during hysteresis loop 2.5 mm was very accurate. This was because neither the experimental nor the numerical model had suffered any damage. However, reloading stiffness during hysteresis loop of +5 mm did not capture deterioration properly. The same thing repeated itself during its negative counterpart, as well as in both 20 mm hysteresis loops.

Graph 6.14: Depiction of unloading stiffness after only suffering deterioration from a single +50 mm hysteresis loop

In order to establish an appropriate behaviour during unloading and reloading, an adequate material model should be created for the geomesh reinforcement using a subroutine and further research should be done regarding the effects of damage over more hysteretic cycles to see if the material model still reaches appropriate strength values but unloads with lower stiffness. The paradox is the following: if damage in the material model was increased, then strength values would have been lower in the 5 mm and 10 mm positive and negative loops if during the 2.5 mm loops some damage would have been defined. On the other hand, if the damage factor remained the same but strength was enhanced by increasing tensile peak stress for example, the accuracy during initial cycles, prior to damage, would have been off like the values obtained in curve #6.

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Graph 6.15: Inability to adequately model damaged unloading and reloading stiffnesses during early hysteresis cycles

After having explored all the possible combinations of material parameters in compression and tension, as well as damage factors and number of cyclic loops the FE models were subjected to, the FE model #32 was chosen for the seismic evaluation of the GRAW subjected to the 5 phases of the scaled 1970’s Peruvian earthquake.

6.5. FE model chosen for evaluation of seismic performance All the results of the cyclic pushover simulation of the most important FE models created are presented in Annex 1, as well as their respective material properties, in Annex 2. Based on the values for peak compressive and tensile stress discussed in chapters 2 and 4, it was decided that the best fit in terms of those variables was when using 0.40 MPa and 0.04 MPa as peak compressive and tensile stress in the Concrete Damaged Plasticity model. A short comparison was performed when compressive strength was varied from 0.40 MPa to 1.0 MPa and 0.6 MPa to see the effect on the hysteresis curves. However, no significant difference in behaviour was observed. Compressive fracture energy was set at 1.0178 N/mm, which gave the best fit in terms of strength. The overall compressive behaviour chosen for the final FE model is shown in Graph 6.16 as the yellow curve. Regarding tensile behaviour, the tensile fracture energy was set at 0.01 N/mm with a peak tensile stress value of 0.04 MPa (depicted in the light blue line in Graph 6.17).

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Graph 6.16: Comparison of compressive curve chosen for FE model #32 (in yellow) to other compressive curve models used in some of the other FE models

Graph 6.17: Tensile behaviour of chosen FE model (in light blue)

Furthermore, compressive and tensile damage played a big part in fine-tuning the strength values during the +2.5 mm, +5 mm and ±10 mm hysteresis cycles, which obtained excellent values as shown in Graph 6.18. In order to appreciate the differences taken in trying to finetune the strength values, Annex 3 contains all the set of compressive and tensile damage factors used during the simulations.

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Calibration of numerical model with cyclic pushover experimental results Tensile damage factors - Set 1 Crack displacement dt (mm) 0.00 0.00 0.80 0.25 0.90 0.50 0.95 0.80

Table 6.3: Tensile damage factors chosen in final FE model #32

When looking at the tensile damage factors during the simulations, it was decided to remain with the same set of damage factors as in set #1 for they gave the best fit for strength values during early cycles. Those values for tensile damage also estimated well the drop in strength after deterioration in tension, at least during the positive hysteresis loops. Set #1 of damage factors was based on the values used in Tarque (2011) to describe the cyclic damage when subjecting an unreinforced adobe model to short cyclic displacements. Furthermore, when evaluating the compressive damage factors to determine the best fit of the FE to the empirical curves, it was found that the less damage was defined, the values and shape reached during the positive strength plateaus in the 2.5 mm, 5 mm and 10 mm hysteresis loops were more accurate. For example, when using set #12, which was the set of compressive damage factors that was chosen for model #32, compared to using #9, which described more damage in a more progressive manner (but only reaching a maximum of 20% damage), the difference in curves showed that further hysteresis cycles did not reach the appropriate strength wanted after 1 cycle was done per peak displacement value.

Compressive damage factors - Set 12 Inelastic strain dc (mm/mm) 0.00 0.00 0.07 0.0005

Table 6.4: Compressive damage factors chosen in final FE model #32

Having defined all of the above values, it was important to mention that this model was the best fit within this part of the dissertation, but, as stated in the previous sub-chapter, it had some limitations in terms of development of strength and precision of unloading and loading stiffnesses.

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Graph 6.18: FE model #32 showed the best fit for positive hysteretic cycles of 2.5 mm, 5 mm, and 10 mm

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Seismic performance of numerical model subjected to scaled earthquake scenarios

7. Seismic performance of FE model subjected to scaled earthquake scenarios The purpose of this section of the dissertation is to evaluate the in-plane seismic performance of the final numerical model selected in chapter 6 when subjected to 5 scaled earthquake signals. The earthquake used as a basis for the seismic evaluation of the FE model was a Peruvian earthquake from May 31st 1970 of moment magnitude (M w) of 7.9 (RPP Noticias 2015). The earthquake’s hypocentre was located 64 kms beneath the surface, 44 kms southwest of the city of Chimbote, reaching intensities of up to XI on the Modified Mercalli Index (Tarque 2011) near the city of Yungay, which was one of the most affected zones. Overall ground movement during the seismic event lasted 45 seconds, causing the rupture of part of the Huascarán glacier (at an altitude of 5,000 m), which overlooked the city of Yungay. The avalanche of rock, ice and mud slid down the mountain valley and buried the whole city, killing 80,000 people, with an additional 20,000 who were never found, and injuring 140,000.

Graph 7.1: Horizontal ground acceleration time-history record from May 1970’s Peruvian earthquake, component N08W, registered in Lima, Peru

Since this is one of the most studied earthquake signals in Peru, it was of interest to subject the calibrated FE model to scaled versions of this event and evaluate its seismic performance at different values of peak ground displacement (PGD) and acceleration (PGA). The earthquake signal registered in Lima in Graph 7.1 was scaled to different intensities so as to have signals which represent frequent, intermediate and very strong earthquakes.

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading The original signal was filtered to discard frequencies lower than 0.15 Hz and then integrated twice in order to obtain the time-history of the ground displacement (Tarque 2011). The signals are presented in Annex 4, and their main characteristics are as follow: •

Phase 1 scaled earthquake signal: peak ground displacement was of 28.91 mm with a peak ground acceleration of 0.305g. The duration of the signal was of 30 seconds. This earthquake scenario was defined as a frequent earthquake.

•

Phase 2 scaled earthquake signal: peak ground displacement was 53.71 mm with a PGA of 0.622g. The duration of the signal was of 30 seconds. This earthquake was considered moderate.

•

Phase 3 scaled earthquake: PGD was 78.51 mm with a PGA of 0.938g. The duration of the signal was also 30 seconds. This signal was considered to be strong.

•

Phase 4 scaled earthquake: PGD was 103.31 mm and PGA was 1.254g. This earthquake signal was considered very strong and it lasted 30 seconds.

•

Phase 5 earthquake: PGD was 128.11 mm with a PGA of 1.57g. This earthquake scenario was labelled as severe or catastrophic and it also lasted 30 seconds.

The time-histories of the following field variables were evaluated in the FE model: relative displacement and base shear along the X-X direction. Additionally, a check for damage, stresses and plastic strains was made at the end of the seismic signal, for both the adobe and the geomesh layers.

Graph 7.2: Scaled earthquake displacement signals based on May 1970’s Peruvian earthquake registered in Lima, Peru (phases 1 to 5)

Lastly, as mentioned in chapter 2, a maximum interstorey drift was calculated based on the recommendations given by the Peruvian national standard for seismic design and Eurocode 8. Based on the expression for maximum interstorey drift allowed in masonry walls, for a wall height of 1.93 m and a class type of building of normal or IV (in case of the Eurocode): •

NTP E.030 defined a maximum interstorey drift of 13.15 mm.

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Seismic performance of numerical model subjected to scaled earthquake scenarios •

Eurocode 8 defined a maximum interstorey drift of 32.875 mm.

Each of the time-histories for relative displacement of the GRAW was compared to the recommended maximums above.

7.1. Modal analysis of FE model Prior to subjecting the FE model to the scaled earthquake signals, a short modal analysis of the overall structure was performed in order to identify the most influential modes of vibration of the structure both for the X-X and Z-Z axes. In order to find out the contributions of the mode shapes, a linear-elastic modal analysis was performed in Abaqus/Standard through a frequency-linear-perturbation step. Eurocode 8 specifies that, in a spatial modal analysis, the number of modes of vibration that have to be taken into account in the evaluation have to have a cumulative effective mass of at least 90% of the total mass in a certain direction. Therefore, it was found that 12 and 13 modes of vibration were needed to be evaluated in the X-X and Z-Z axes, respectively. However, modal shape 16 had an effective mass of 7.58% of the total mass of the model, which is why it was decided to extend the number of modes of vibration considered in the X-X axis to 16, instead of 12 (shown in Table 7.1 and graphically in Graph 7.3).

Table 7.1: Modes of vibration considered in the analysis for X-X and Z-Z axes

When considering the effective masses in the X-X axis, it was found that the most influential mode of vibration was the 2nd one, with 87.18% effective mass, which described a translational mode along the X-axis (shown in Figure 7.1). In the case of the Z-Z direction, the translational mode of vibration #1 was identified as the most influential one with 75.63% of effective mass (also shown in Figure 7.2). Additionally, the mode of vibration #16 and

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading #13 were the second-most important ones for the X-X and Z-Z directions respectively (see Figure ##).

Figure 7.1: Mode of vibration #2 (left) which is a translational mode along the X-axis, and mode of vibration #1 (right), which is also a translational mode, but along the Z-axis

Graph 7.3: Comparison of modal frequencies and periods of vibration

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Figure 7.2: Modes of vibration #16 (left) and #13 (right) were the second-most influential ones along the X-X and Z-Z axes respectively

Based on the results presented in Table 7.1 and Graph 7.3, the maximum period of vibration was approximately 0.08 seconds for both the first and second mode shape, which meant that the FE model, as in reality, was very stiff.

7.2. Seismic performance of FE model subjected to phase 1 of scaled earthquake signal (PGA=0.305g and PGD=28.91 mm) Overall, the response of the GRAW during phase 1 of the scaled earthquake was very stiff. This made perfect sense, since wall structures tend to have very high frequencies and therefore great rigidity. As can be observed in Graph 7.4, the relative displacement of the top of the GRAW reached a maximum value of 1.53 mm at 10.82s, which meant that very little to no degradation had taken place in the adobe continuum. However, the response also showed that most displacements were concentrated on the positive side, which hinted to the accumulation of some minor damage along the right side of the structure. This was confirmed when the FE model presented minor tensile damage in the adobe continuum at the base of the right perpendicular wall. Additionally, as was expected, the geomesh-reinforcement had some plastic damage across the in-plane wall, after the model was subjected to the phase 1 earthquake, which was consistent with usual tensile strain zones along the in-plane wall (see Figure 7.3). At the moment of peak displacement (10.82s), the maximum plastic strain reached in the geomesh-reinforcement layers was 0.001663, which was at the bottom right

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading corner of the window opening (see Figure 7.3). This amount of plastic strain meant that the geomesh-reinforcement was barely active during the phase 1 earthquake.

Graph 7.4: Time-history of simulated displacement of roof of GRAW relative to movement of the shaking table during phase 1 earthquake

Figure 7.3: Residual plastic strains in the geomesh layer after phase 1 earthquake. Peak tensile strains were located around the corners of the window opening

Furthermore, the maximum base shear developed was of -163.4 kN at 11.62s (seen in Graph 7.5), close to the time when the maximum displacement was reached. The development of base shear was consistent with the movement of the GRAW in general, especially during the first 12 seconds of the signal. After that, values for base shear went down to below 50 kN, which may signify some initial damage in the model after 12s. However, the relative displacement suffered in the GRAW was not significant enough to indicate any important damage to the structure.

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Graph 7.5: Time-history of simulated base shear of FE model during phase 1 earthquake

In summary, the FE model performed very well when subjected to the phase 1 earthquake signal. Relative displacement values remained mostly under 1.5 mm, which was considered satisfactory under both NTP E.030 and EC8 maximum interstorey drift standards for simple masonry structures. Additionally, the structure survived with only minor tensile damage in the adobe continuum on the bottom right perpendicular wall, which explained the slight shift of relative displacements to the positive side of the graph, indicating minute permanent displacements in the GRAW along +X-axis.

7.3. Seismic performance of FE model subjected to phase 2 of scaled earthquake signal (PGA=0.622g and PGD=53.71 mm) In this case, seismic performance of the FE model when subjected to the phase 2 earthquake was still within the acceptable scope in terms of peak relative displacements. As shown in Graph 7.6, the maximum displacement reached was 9.52 mm at 19.74s, which was 72.4% of the maximum allowable drift (13.15 mm) according to the Peruvian seismic code NTP E.030 and 28.96% of the value allowed by EC8.


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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Additionally, the overall displacement time-history indicated that permanent damage had been sustained at a noticeable level which caused a residual displacement of 3.24 mm on the positive side of the X-axis. This was confirmed when analysing the strain and damage patterns in the FE model in Abaqus/Standard. Figure 7.4 shows moderate tension damage on the left perpendicular wall, indicating a slight tilt of the wall towards the right side.

Figure 7.4: Tension damage in adobe continuum after phase 2 EQ

Figure 7.5: Maximum stresses reached across the geomesh layers

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Seismic performance of numerical model subjected to scaled earthquake scenarios Additionally, at the end of the phase 2 earthquake, the geomesh reached stresses to up to 125 MPa at the bottom corners of the in-plane wall, as well as at the bottom of the perpendicular walls. This meant that considerable stresses were already being exerted by the geomesh, controlling the tensile damage in the adobe continuum. Tensile stresses of approximately 30 MPa were reached in the geomesh around the window openings (see Figure 7.5).


Lastly, base shear reached a maximum value of 172.52 kN at 9.82s, shown in Graph 7.7. The development of base shear was consistent with the movement of the GRAW until approximately 12s when the shear reduced. Approximately around that moment the relative displacements started to shift to the positive side of the graph, which mean that tension damage had incurred, dissipating energy in the process and reducing accumulated base shear.

7.4. Seismic performance of FE model subjected to phase 3 of scaled earthquake signal (PGA=0.938g and PGD=78.51 mm) During phase 3 of the scaled earthquake, the GRAW reached a maximum relative displacement of 41 mm at 10.92s. This relative displacement of the roof of the GRAW surpassed the maximum allowable drift of Peru’s national standard by 212% and EC8’s limit by 24.7%. This sudden increase in drift was caused by excessive tensile damage in the adobe continuum of the perpendicular walls, which already started to deteriorate beyond 95% tensile damage. This can be seen in Figure 7.6 in red, indicating almost maximum deterioration. The intense damage was consistent with the drop in base shear as well. However, the maximum base shear developed during the phase 3 earthquake was considerably smaller than during phase 2. During phase 3, the maximum value was 124.84 kN, which was 27.64% smaller than in phase 2. This was the consequence of the presence of excessive damage since the early into the response, which kept the base shear low, but the damage in tension high, which dissipated energy.

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Lastly, the geomesh reinforcement prevented major damage from developing within the adobe continuum which was reinforced. As a consequence, reached stress levels of 187 MPa at the bottom of the perpendicular walls. Stresses around the window opening reached values of 110 MPa next to the corners and lintel, limiting the damage in tension in that area of the adobe continuum, where stresses only reached around 0.17 MPa around the top corners of the window opening.

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Figure 7.6: Tension damage sustained in the adobe continuum after phase 3 earthquake

Figure 7.7: Residual stresses in geomesh reinforcement after phase 3 earthquake

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7.5. Seismic performance of FE model subjected to phase 4 of scaled earthquake signal (PGA=1.254g and PGD=103.31 mm) Results from phase 4 were unfortunately not available as a programme error gave back absurd data.

7.6. Seismic performance of FE model subjected to phase 5 of scaled earthquake signal (PGA=1.57g and PGD=128.11 mm) During phase 5 of the scaled earthquake, the base shear in GRAW reached a maximum value of -186.01 kN, which was almost at the very beginning of the response. This hinted to a very stiff first response of the structure, force which was then dissipated through damage as the values for base shear dropped to 100 kN almost immediately and then to 50 kN. Maximum relative displacement was 87.71 mm, which by far exceeded recommended values of EC8 and the Peruvian standard. Specifically, by 167% and 567% in each case respectively. Graph 7.10 also pointed to residual values on the negative side, which hinted to excessive damage. Here, despite the very large displacements, the geogrid layer reached maximum values of 165 MPa, which was inconsistent and may have happened due to errors in the simulation.



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Conclusions

8. Conclusions The overall conclusions of this dissertation are divided per subject area so that further research can be built upon the comments. First, in summary the experimental results obtained in the pushover cyclic test from DAI 113.0225 showed the potential of geomesh-reinforcement in seismic strengthening of adobe structures. So, in order to further develop seismic strengthening methods and establish design guidelines, finite element modelling must keep up with new modelling techniques in order to appropriate simulate hysteretic behaviour in GRAWs. •

Comments and conclusions about the limitations of the FE model in relationship with the constitutive model of geogrid-reinforcement: o

Elastoplastic model does take compressive stresses. This gives an unwanted contribution of strength during analysis and alters overall behaviour somewhat.

o

Addition of strength during unloading created loops at the ends of the hysteresis cycles, which were denominated cyclic locking bows. This behaviour may come from combined behaviour of adobe and geomeshreinforcement models, as well as contribution of the composite geomeshreinforcement layers in compression.

o

In reality, compressive stresses do not develop in the reinforcement, since the geogrid normally buckles in compression. Explicit modelling in Abaqus allows the presence of in-plane buckling under compressive loading. However, explicit geometry could not be modelled within the timeconstraints of this MSc dissertation, but should be explored in further research.

o

Variations could have had the unintended consequence of the change in geometry, from explicit form to equivalent sheet form, which affected strength values during the early onset of deformation of the geogrid sheet.

•

Considerations during implicit static analysis: o

Models with low to none stability with dissipation energy had a low convergence rate. This is due to instabilities caused in step to step equilibrium, which in instability issues was not able to be achieved. A stabilisation factor of 0.00025 was used in all models so that convergence was able to be reached.

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading o

The simulation had limitations and simulation errors when the FE model was subjected to cyclic loads, followed by a monotonic one. This was done to try to predict maximum strength magnitudes after a few cycles of displacement, which inflicted some damage in the structure, allowing a more accurate depiction of real strength values during greater displacements. It seems that using a tabular amplitude to modify the base value of displacement only works if the cycles of displacement have approximately the same duration and displacements grow progressively, not abruptly. The sudden change from cyclic to monotonic loading caused the FE model to become overly flexible, generating very low strength values during the monotonic part of the displacement.

•

Considerations during implicit dynamic analysis: o

Model #32 which had a higher magnitude of compressive fracture energy and reduced damage factors in compression showed exceptional convergence rate during all seismic simulations. It seems that such a combination allows the model to develop large displacements without incurring in numerical instabilities.

o

During simulation, since there were many cycles, the geogrid material property allowed the top of the model to displace itself upwards, around 70 mm in the worst-case scenario. Either the geomesh material was strong enough to displace the concrete beam or there was an error in the simulation. If one does not take into consideration the influence of gravity during the earthquake, it makes sense that while adobe has lost practically all its tensile strength permanently, there’s little resistance to counteract the compressive accumulated stresses in the geomesh in pushing the structure upwards.

•

Conclusions about influence of material properties of adobe o

Tensile fracture energy, along with peak tensile stress, define the overall development of strength of the model. The greater the cracking energy allowed to be dissipated during the same amount of displacement, the higher the strength will be.

o

The FE model with the defined material properties as they were could not accurately predict the strength developed in hysteresis loops with a peak displacement value larger than 10 mm. It was thought that it was due to the amount of compressive fracture energy that the strength did not grow, but it did not make much difference. As of now, there would need to be further research to determine what caused this problem.

•

Conclusions about the seismic performance of the GRAW FE model: Page 95

Conclusions o

The FE model presented reasonable results during phases 1 through 3, giving results of displacements, stresses, damage and base shear consistent with what would normally happen in reality.

o

Phase 4 presented errors during simulation, which impeded the actual simulation to be performed.

o

Phase 5 probably gave out some erroneous data related to stresses and strains because they were not consistent with previous magnitudes in either displacement or stresses, especially in the geogrid mesh.

o

It is imperative that further research is done to adequately simulate severe seismic events.

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9.

References

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References Blondet, M., et al., (2011). Seismic resistant earthen construction: the contemporary experience at the Pontificia Universidad Católica del Perú. Informes de la Construcción [online]. 63(523), 41-50 (in Spanish). Caporale, A. et al (2015) Comparative micromechanical assessment of adobe and clay brick masonry assemblages based on experimental data sets. Composite Structures, 120, 208220. Cardile, G., et al., (2016). In-air tensile load-strain behaviour of HDPE geogrids under cyclic loading. In: Procedia Engineering. VI Italian Conference of Researchers in Geotechnical Engineering (CNRIG 2016) [online]. pp. 158(2016), 266-271. [Viewed 20 August 2017]. Available from: http://creativecommons.org/license/by-nc-nd/4.0/ Cardile, G., et al., (2017). Tensile behaviour of an HDPE geogrid under cyclic loading: experimental results and empirical modelling. Geosynthetics International [online]. 24(1),

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References San Bartolomé, A., et al., (2004). Effective system for seismic reinforcement of adobe houses. The 13th World Conference on Earthquake Engineering, August 1-6 2004, Vancouver, B.C., Canada. San Bartolomé, A. et al (2008) Performance of reinforced adobe houses in Pisco, Peru earthquake. In: The 14th World Conference on Earthquake Engineering, October 12-17, 2008. Beijing, China. San Bartolomé, A., and Quiun, D., (2015). Diseño de mallas electrosoldadas para el reforzamiento sísmico de viviendas de adobe típicas del Perú. Revista de la Facultad de Ingeniería UCV, 30(1), 71-80 (in Spanish). Saroza, B., et al., (2008). Study of the resistance to simple compression from adobe produced with various soils from Crescencio Valdés, Villa Clara, Cuba. Informes de la Construcción [online]. 60(511), 41-47. [Viewed 29 August 2017]. Available from: doi: 10.3989/ic.10.017 (in Spanish) Sulaiman, M.F., et al. (2017). A Review on Bond and Anchorage of Confined High-Strength Concrete. Journal of Structures [online]. 11, 97-109. [Viewed 28 August 2017]. Available from: doi: https://doi.org/10.1016/j.istruc.2017.04.004 Tarque, N., et al., (2010). The use of continuum models for analysing adobe structures. 15th World Conference on Earthquake Engineering, 15 WCEE 2012, Lisbon, Portugal. Tarque, N., et al., (2010). Seismic capacity of adobe dwellings. 14th European Centre for Training in Earthquake Engineering, 14th ECEE 2010, Ohrid, Macedonia. Tarque, S.N., (2011). Numerical modelling of seismic behaviour of adobe buildings. Ph.D. Thesis, Università degli Studi di Pavia. Tarque, N., et al., (2014). Numerical simulation of an adobe wall under in-plane loading. Earthquakes and Structures [online]. 6(6), 627-646. [Viewed 23 rd March 2017]. Available from: doi: http://dx.doi.org/10.12989/eas.2014.6.6.627 Tarque, N., et al., (2014). Numerical analyses of the in-plane response of unreinforced and reinforced adobe walls. 9th International Masonry Conference, 9th IMS, 2014, Guimarães. Tarque, N., et al., (2015). Masonry infilled frame structures: state-of-the-art review of numerical modelling. Earthquakes and Structures [online]. 8(1), 000-000. [Viewed 15 March 2017]. Available from: doi: http://dx.doi.org/10.12989/eas.2015.8.1.895

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Non-linear modelling of a geomesh-reinforced adobe wall subjected to dynamic loading Tensar International Corporation (2003). Product Specification Tensar Biaxial Geogrid [online]. Alpharetta, Georgia: Tensar International Corporation. [Viewed 15 June 2017]. Available from: www.tensarcorp.com Tensar International Corporation (2013). Product Specification Tensar Biaxial Geogrid [online]. Alpharetta, Georgia: Tensar International Corporation. [Viewed 15 June 2017]. Available from: www.tensarcorp.com Torres, A., (2012). Comportamiento sísmico del adobe confinado. Variable: Refuerzo horizontal. BSc Thesis, Pontificia Universidad Católica del Perú (in Spanish). Vargas, L., et al. (2016). Determinación de propiedades mecánicas de la mampostería de adobe, ladrillo y piedra en edificaciones históricas peruanas. BSc Dissertation, Pontifical Catholic University of Peru (in Spanish). Zou, C., et al., (2016). Creep behaviors and constitutive model for high density polyethylene geogrid and its application to reinforced soil retaining wall on soft foundation. Journal of Construction and Building Materials [online]. 114(2016), 763-771. [Viewed 5 July 2017]. Available from: doi: http://dx.doi.org/10.1016/j.conbuildmat.2016.03.194

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Annexes

10. Annexes Any complementary information pertinent to the development of this dissertation is presented in this chapter. The annexes are divided in 10 sections: •

Annex 1: Hysteretic curves of evaluated FE models

•

Annex 2: Material and numerical model properties of evaluated FE models

•

Annex 3: Sets of damage factors used in FE models

•

Annex 4: Results of modal analysis of FE model #32

•

Annex 5: Time-histories for ground displacement and acceleration of scaled seismic signals based on Peruvian earthquake of May 1970

•

Annex 6: Seismic map of Peru

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10.1. Annex 1: Hysteretic curves of evaluated FE models

Graph 10.1: Hysteresis curves of FE models #6 and #9

Graph 10.2: Hysteresis curves of FE models #10 and #10a


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Graph 10.4: Hysteresis curve of FE model #10b

Graph 10.5: Hysteresis curve of FE model #10d

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Graph 10.10: Hysteresis curves of FE models #22 and 24


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Graph 10.12: Hysteresis curves of FE models #27 and 28



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Graph 10.15: Hysteresis curve of FE model #32 (chosen for seismic analysis)

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10.2. Annex 2: Material and numerical model properties of evaluated FE models

Table 10.1: Material and numerical properties of FE models evaluated in cyclic pushover analysis

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10.3. Annex 3: Sets of damage factors used in FE models

Table 10.2: Sets #1 and #2 of tensile damage factors used in FE models

Table 10.3: Sets #1 and #2 of compressive damage factors used in FE models



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10.4. Annex 4: Time-histories for ground displacement and acceleration of scaled seismic signals based on Peruvian earthquake of May 1970

Graph 10.16: Scaled ground displacement and acceleration time-histories of simulation earthquakes

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10.5. Annex 5: Seismic map of Peru

Figure 10.1: Map of past seismic events in Peru from 1960 to 2011 (Source: Instituto Geofísico del Perú)

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