Behavior of reinforced concrete frame structures with decoupled infill walls under earthquake loading

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POLITECNICO DI MILANO

School of Civil, Environmental and Land Management Engineering

Master of Science degree in Civil Engineering - New Structures Track - Final Thesis

BEHAVIOR OF REINFORCED CONCRETE FRAME STRUCTURES

WITH DECOUPLED INFILL WALLS UNDER EARTHQUAKE

LOADING

By:

SANTIAGO FLORES CALVINISTI Matr. 903454

Supervisor: Dr. Marko Marinković and Prof. Vitomir Racic ACADEMIC YEAR 2019/2020 

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ACKNOWLEDGMENT

First of all, I would like to express my sincere gratitude to Dr. Marko Marinković, for the dedication and support he has given me through out the development of my thesis. Special thanks to Prof. Vitomir Racic for motivating me to take this Erasmus program in Belgrade. To my family, specially my parents and my brother, who have given me their unconditional support during my whole journey. Finally to all my friends who I know I can always count on.

Santiago Flores Calvinisti Belgrade, 2020

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ABSTRACT

Infill reinforced concrete walls are widely used in civil engineering construction. Although they are normally considered as non structural elements, it has been proven through many studies that there is actually an interaction between the infill walls and the RC concrete frames that changes the behavior of the whole structure under earthquake loading.

This thesis is based on the experimental tests done by Marinković (2018) designed to study the seismic behavior of a single bare frame and see how this behavior is affected when a traditional infill system wall and an innovative decoupled infill system (INODIS) are construted inside this frame. This innovative system has an elastomer that is installed along the infill-frame interface in order to decouple them.

Three single frames (bare frame, traditional infill frame and innovative infill frame) were modeled using a commercial software SAP2000. This models were calibrated with Marinković (2018) test results. After validation, they were used to study the seismiv behavior of frame structures with different infill wall configurations.

2D frame models were analyzed, consisting of different heights and different infill wall distributions. The results showed that the innovative system is able to minimize the effects of an infill masonry wall inside a frame, indicating that there is an effective decoupling between the infill-frame. Also the soft storey effect caused by the lack of walls at the ground level is greatly reduced in comparison with the traditional infill,.

Finally a 3D structure was numerically modeled and subjected to a response spectrum with different PGA, the three different systems (bare, traditional and innovative system) results were compared and analyzed. Also different infill arrangements were made to study the effects of uneven infill wall distribution in the first floor. Results showed that the traditional infill system with unsymmetrical infill distribution presented torsional effects and also soft storey mechanism. These are greatly reduced by employing an innovative system.

Keywords: infilled frame, INDODIS, SAP2000 modeling, RC frame behavior. 

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ASTRATTO

I muri di tamponamento realizzati con mattoni e malta sono ampiamente utilizzati nelle costruzioni di ingegneria civile. Sebbene siano normalmente considerati elementi non strutturali, è stato dimostrato in molti studi che esiste effettivamente un'interazione tra le pareti di tamponamento e la struttura in calcestruzzo armato, la quale modifica il comportamento dell'intera struttura sottoposta ad azione sismica.

Questa tesi si basa sui test sperimentali condotti da Marinković (2018) progettati per studiare il comportamento sismico di un singolo telaio e vedere come questo è influenzato dalla presenza di una parete di tamponamento costruita con metodo tradizionale e un’ altra costruita con un sistema innovativo disaccoppiato (INODIS). Questo sistema innovativo ha un elastomero che viene installato lungo l'interfaccia struttura-parete così da disaccoppiare le strutture.

Tre telai (telaio solo di calcestruzzo armato, telaio con tamponamento tradizionale e telaio con tamponamento innovativo) sono stati modellati utilizzando un software commerciale SAP2000. Questi modelli sono stati calibrati con i risultati degli esperimenti di Marinković (2018). Dopo la convalida, sono stati impiegati nello studio del comportamento sismico di strutture intelaiate con differenti configurazioni delle pareti di tamponamento.

Sono stati analizzati i modelli di telai 2D, costituiti da diverse altezze e diverse distribuzioni delle pareti di tamponamento. I risultati hanno mostrato che il sistema innovativo è in grado di minimizzare gli effetti di una parete di tamponamento all'interno di un telaio, indicando che esiste un efficace disaccoppiamento tra il telaio e il muro. Anche l'effetto piano debole causato dalla mancanza di pareti a livello del suolo è notevolmente ridotto rispetto al tamponamento tradizionale.

Infine, una struttura 3D è stata modellata numericamente e soggetta a uno spettro di risposta con PGA diversi, i tre diversi sistemi (telaio solo di calcestruzzo armato, tradizionale e innovativo) sono stati confrontati e analizzati. Sono stati inoltre presi diversi configurazioni di tamponamento per studiare gli effetti della distribuzione irregolare delle pareti nel primo piano.

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I risultati hanno mostrato che il sistema di tamponamento tradizionale con distribuzione di muri non simmetrici presentava effetti torsionali e anche un meccanismo a piano debole. Questi sono notevolmente ridotti impiegando un sistema innovativo.

Parole chiave: telaio con muri di tamponamento, INDODIS, modellazione SAP2000, comportamento del telaio in calcestruzzo armato. 

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TABLE OF CONTENT

Acknowledgment i

Abstract ii

Astratto iii

Table of Content v

List of Figures vii

List of Tables xiii

1. Introduction 1

1.1. Objectives of the research ...6

1.2. Methodology ...6

2. Literature Review 7 2.1. Introduction ...7

2.2. Failures modes ...8

2.2.1. Failure in masonry infill panels ...8

2.2.1. Failure in concrete frame ...13

2.3. Masonry infill modeling ...17

2.3.1. Micro-models ...17

2.3.2. Macro-models ...18

2.4. Equivalent diagonal strut ...19

2.4.1. Single strut models ...19

2.4.2. Multiple strut models ...27

2.4.3. Properties ...30

2.4.4. Cyclic loading behavior ...38

2.5. Infill wall behavior improvement ...47

3. Experimental Testing 54 3.1. Introduction ...54

3.1.1. Mechanical properties of the materials ...55

3.1.2.1. Concrete ...55

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3.1.2.1. Steel reinforcement ...56

3.1.2.1. Masonry assembly ...56

3.1.2.1. Elastomer ...56

3.1.2. Experimental tests ...57

3.1.2.1. Test A: Bare frame ...58

3.1.2.2. Test BI: Traditional infill ...60

3.1.2.3. Test DIO: INODIS system ...61

3.1.2.4. System contribution ...64

4. Numerical simulations 65 4.1. Description of bare frame model ...68

4.2. Description of infill strut model ...74

4.3. Description of elastomer modeling ...80

4.4. Results of the calibrated models ...82

4.4.1. Bare frame model results ...83

4.4.2. Infill strut model results ...84

4.4.3. Elastomer model results ...86

5. Analysis of a 2D structural frame 89 5.1. Introduction ...89

5.2. Description ...90

5.3. Results ...93

5.3.1. Low rise building ...96

5.3.2. Medium rise building ...104

5.3.3. High rise building ...112

6. Analysis of a 3D structural frame 120 6.1. Results ...123

6.1.1. Analysis along X axis ...132

6.1.2. Analysis along Y axis ...142

7. Conclusions 152

Bibliography 154

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LIST OF FIGURES

Figure 1.1. (a) and (b) damages caused by L’Aquila earthquake in 2009, interaction between

column and infill (Ricci et al., 2011) ...2

Figure 1.2. (a) infill diagonal failure; (b) horizontal sliding; and (c) corder crushing (Ricci et al., 2011) ...3

Figure 1.3. (a) and (b) soft storey mechanism collapse (Ricci et al., 2011) ...4

Figure 2.1. Shear distribution in infill panel (Crisafulli, 1997) ...8

Figure 2.2. Diagonal cracking (Crisafulli, 1997) ...9

Figure 2.3. Stepped cracking of masonry (Crisafulli, 1997) ...9

Figure 2.4. Failure due to masonry corner crushing (Crisafulli, 1997) ...10

Figure 2.5. Flexural cracking of infill panel (Crisafulli, 1997) ...11

Figure 2.6. Flexural behavior of panel a) stress concentration due to rocking, b) crack development in tension zone (Pacheco, 2016). ...11

Figure 2.7. Shear sliding (Crisafulli, 1997) ...12

Figure 2.8. Shear failure along a mortar joint: a) pure mode; b) mixed / hybrid mode (Pacheco, 2016) ...12

Figure 2.9. Flexural collapse mechanism (Crisafulli, 1997) ...13

Figure 2.10. Flexural collapse mechanism with shear sliding (Crisafulli, 1997) ...14

Figure 2.11. Tension failure (Crisafulli, 1997) ...14

Figure 2.12. Bar anchorage failure (Crisafulli, 1997) ...15

Figure 2.13. Shear failure (Crisafulli, 1997) ...16

Figure 2.14. Beam-Column joint failure (Crisafulli, 1997) ...16

Figure 2.15. Modeling strategies: (a) masonry example; (b) detailed micro-modeling; (c) simplified micro-modeling; (d) macro-modeling (Lourenço et al., 1995). ...18

Figure 2.16. Equivalent truss mechanisms (Crisafulli, 1997) ...19

Figure 2.17. Equivalent diagonal strut (Asteris et al. 2011) ...21

Figure 2.18. Variation of w/dm as a function of 𝜆h (Crisafulli, 1997) ...24

Figure 2.19. Variation of w/dm with respect to the relative stiffness (a) 𝜆h and (b) 𝜆* (Tarque et al., 2015) ...26

Figure 2.20. Multiple strut models (Crisafulli, 1997) ...28

Figure 2.21. Multiple strut model used by Thiruvengadam (1985) (Tarque et al., 2015) ...29

Figure 2.22. Multiple strut model used by Crisafulli and Carr (2007) (Tarque et al., 2015) 30 .. Figure 2.23. Trilinear lateral force proposed by Panagiotakos and Fardis (1994) (Sassun et. al., 2015). ...33

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Figure 2.24. Force-displacement relationship for diagonal strut (Decanini et al., 2004) ...34

Figure 2.25. Force-displacement relationship for diagonal strut (Dolsek and Fajfar, 2007) 36 ... Figure 2.26. Force-displacement relationship for diagonal strut (Liberatore et al., 2018) ...37

Figure 2.27. Hysteresis model proposed by Klingner and Bertero (1978) (Di Trapani et al., 2014) ...39

Figure 2.28. Hysteresis model proposed by Doudoumis and Mitsopoulou (1986) (Di Trapani et al., 2014) ...40

Figure 2.29. Hysteresis model proposed by Klingner and Bertero (1978) (Di Trapani et al., 2014) ...41

Figure 2.30. Strength envelope and hysteretic curve (Chrysostomou, 1991) ...41

Figure 2.31. Hysteresis model proposed by Reinhorn et al. (1995), takes into account (a) Wen-Bout Model; (b) Stiffness and strength degradation; (c) Slip lock model; (d) integrated model (Reinhorn et al., 1995) ...42

Figure 2.32. Experimental and computed hysteresis curve (Combescure et al., 1996) ...43

Figure 2.33. General characteristics of the hysteresis behavior model (Crisafulli, 1997) ...44

Figure 2.34. (a) Reloading curve parameters; (b) reloading curve elastic modulus (Crisafulli, 1997) ...44

Figure 2.35. Hysteretic model (Rodrigues et al., 2008) ...46

Figure 2.36. Solution for infill behavior improvement (Marinković, 2018) ...47

Figure 2.37. (a) connecting beam between primary and secondary (Braga et al., 2011) ...48

Figure 2.38. Infill wall with sliding joints (Preti et al., 2012) ...50

Figure 2.39. Details of the proposed system: 1.C- shape units; 2. mortar bed-joints; 3. sliding joints; 4. clay units; 5. interface joints; 6. shear key; 7. plaster (Morandi et al., 2016) ...50

Figure 2.40. Infill configuration with horizontal ...52

Figure 3.1. INODIS system layout (Marinković, 2018) ...55

Figure 3.2. Experimental stress-strain curves for the elastomers under cyclic loading (Marinković, 2018) ...57

Figure 3.3. Load protocol for Test A (Marinković, 2018) ...59

Figure 3.4. Hysteretic curve Test A (Marinković, 2018) ...59

Figure 3.5. Load protocol for Test BI (Marinković, 2018) ...60

Figure 3.6. Hysteretic curve Test BI (Marinković, 2018) ...61

Figure 3.7. Load protocol for Test DIO (Marinković, 2018) ...62

Figure 3.8. Hysteretic curve Test DIO, phase 1 (Marinković, 2018) ...63

Figure 3.9. Hysteretic curve Test DIO, phase 3 and 4 (Marinković, 2018) ...63

Figure 3.10. Infill contribution (Marinković, 2018) ...64

Figure 4.1. Infill contribution (Federal Emergency Management Agency, 2000) ...66

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Figure 4.2. Stress-strain curve for monotonic loading (Mander et al., 1988) ...67

Figure 4.3. (a) concrete stress-strain curve; (b) steel strep-strain curve ...69

Figure 4.4. Infilled frame dimensions and reinforcement (Marinković, 2018) ...70

Figure 4.5. (a) column section A, 250 x 250mm; (a) column section B, 250 x 250mm; (a) beam section A, 250 x 450mm; and (a) beam section B, 250 x 450mm ...71

Figure 4.6. (a) column section A, mesh = 15x15; (a) column section B, mesh = 15x15; (a) beam section A, mesh = 10x15; and (a) beam section B, mesh = 10x15 ...72

Figure 4.7. Fiber hinge location in the bare frame model ...73

Figure 4.8. (a) Concrete hysteresis model; and (b) Degrading hysteresis model (Computers & Structures, Inc., 2016) ...73

Figure 4.9. Macro modeling of traditional infill in SAP2000 ...75

Figure 4.10. Pivot hysteresis model (Computers & Structures, Inc., 2016) ...79

Figure 4.11. (a) elastomer model; (b) and (c) corner detail with a vertical and a horizontal link model simulating the elastomer and the diagonal link simulating the strut 80 Figure 4.12. Elastomer force-displacement curve based on experimental results by Marinković (2018) ...81

Figure 4.13. Takeda hysteresis model (Computers & Structures, Inc., 2016) ...82

Figure 4.14. Comparison between the numerical model pushover curve and the experimental hysteresis envelope by Marinković (2018) of the bare frame ...83

Figure 4.15. Comparison between the numerical model time history and the experimental hysteresis by Marinković (2018) of the bare frame ...83

Figure 4.16. Proposed and experimental force-displacement curve ...84

Figure 4.17. Comparison between the numerical model pushover curve and the experimental hysteresis envelope by Marinković (2018) of the infilled frame ...85

Figure 4.18. Comparison between the numerical model time history and the experimental hysteresis by Marinković (2018) of the infilled frame ...85

Figure 4.19. Pushover curve of the numerical model and the experimental hysteresis envelope by Marinković (2018) of the infilled frame with elastomers ...86

Figure 4.20. Numerical model time history and experimental hysteresis by Marinković (2018) of the infilled elastomeric frame, Phase 1 ...87

Figure 4.21. Numerical model time history and experimental hysteresis by Marinković (2018) of the infilled elastomeric frame, Phase 3-4-5 ...87

Figure 5.1. Typical plan floor for all buildings ...90

Figure 5.2. Infill wall configuration for low rise building (Abdelaziz, 2019) ...91

Figure 5.3. Infill wall configuration for medium rise building (Abdelaziz, 2019) ...91

Figure 5.4. Infill wall configuration for high rise building (Abdelaziz, 2019) ...91

Figure 5.5. (a), (c) and (e) response spectrum of 0.1g, 0.3g and 0.5g respectively; (b), (d) and (f) time history of 0.1g, 0.3g and 0.5g respectively. ...92

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Figure 5.6. (a) Periods of the first, second and third modes; (b) ratio between model and bare frame period for the first, second and third mode. ...96 Figure 5.7. (a) Pushover analysis curve; and (b) maximum base shear ...97 Figure 5.8. Linear elastic response spectrum results with a 0.1g PGA: (a) model’s period on

the response spectra curve; (b) absolute displacement at each storey; and (c) inter storey drift ...98 Figure 5.9. Non linear time history results with a 0.1g PGA: (a) absolute displacement at

each storey; and (b) inter storey drift ...99 Figure 5.10. Linear elastic response spectrum results with a 0.3g PGA: (a) model’s period on

each storey; and (b) inter storey drift ...103 Figure 5.14. (a) Periods of the first, second and third modes; (b) ratio between model and

bare frame period for the first, second and third mode. ...104 Figure 5.15. (a) Pushover analysis curve; and (b) maximum base shear ...105 Figure 5.16. Linear elastic response spectrum results with a 0.1g PGA: (a) model’s period on

each storey; and (b) inter storey drift ...111 Figure 5.22. (a) Periods of the first, second and third modes; (b) ratio between model and

bare frame period for the first, second and third mode. ...112 Figure 5.23. (a) Pushover analysis curve; and (b) maximum base shear ...113

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Figure 5.24. Linear elastic response spectrum results with a 0.1g PGA: (a) model’s period on the response spectra curve; (b) absolute displacement at each storey; and (c)

inter storey drift ...114

Figure 5.25. Non linear time history results with a 0.1g PGA: (a) absolute displacement at each storey; and (b) inter storey drift ...115

Figure 5.26. Linear elastic response spectrum results with a 0.3g PGA: (a) model’s period on the response spectra curve; (b) absolute displacement at each storey; and (c) inter storey drift ...116

Figure 5.28. Linear elastic response spectrum results with a 0.5g PGA: (a) model’s period on the response spectra curve; (b) absolute displacement at each storey; and (c) inter storey drift ...118

Figure 6.1. (a) bare frame; (b) fully infilled frame, either traditional or decoupled; (c) corner building, either traditional or decoupled; and (d) fully open ground floor structure, either traditional or decoupled ...121

Figure 6.2. Ground floor of: (a) bare frame and open ground floor; (b) fully infilled frame; (c) corner building ...122

Figure 6.3. (a) Periods of the first, second and third modes; (b) ratio between model and bare frame period for the first, second and third mode. ...126

Figure 6.4. Modal participation per model: (a) mode 1; (b) mode 2; and (c) mode 3 ...127

Figure 6.5. 1st mode deformed shape: (a) to (g) model 1 to model 7, respectively ...128

Figure 6.6. 2nd mode deformed shape: (a) to (g) model 1 to model 7, respectively ...129

Figure 6.7. 3rd mode deformed shape: (a) to (g) model 1 to model 7, respectively ...130

Figure 6.8. Linear elastic response spectrum results, model’s period on the response spectra curve: (a) 0.1g PGA; (b) 0.3g PGA; and (c) 0.5g PGA ...131

Figure 6.9. (a) Pushover analysis curve; and (b) maximum base shear ...132

Figure 6.10. Linear elastic response spectrum results with a 0.1g PGA: (a) and (c) absolute displacement at each storey along the X and Y direction, respectively; and (b) and (d) inter storey drift along the X and Y direction, respectively ...133

Figure 6.11. Non linear time history results with a 0.1g PGA along the X direction: (a) absolute displacement at each storey; and (b) inter storey drift ...134

Figure 6.12. Non linear time history results with a 0.1g PGA along the Y direction: (a) absolute displacement at each storey; and (b) inter storey drift ...135

Figure 6.13. Linear elastic response spectrum results with a 0.3g PGA: (a) and (c) absolute displacement at each storey along the X and Y direction, respectively; and (b) and (d) inter storey drift along the X and Y direction, respectively. ...136

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Figure 6.14. Non linear time history results with a 0.3g PGA along the X direction: (a) absolute displacement at each storey; and (b) inter storey drift ...137 Figure 6.15. Non linear time history results with a 0.3g PGA along the Y direction: (a)

absolute displacement at each storey; and (b) inter storey drift ...138 Figure 6.16. Linear elastic response spectrum results with a 0.5g PGA: (a) and (c) absolute

displacement at each storey along the X and Y direction, respectively; and (b) and (d) inter storey drift along the X and Y direction, respectively. ...139 Figure 6.17. Non linear time history results with a 0.5g PGA along the X direction: (a)

absolute displacement at each storey; and (b) inter storey drift ...140 Figure 6.18. Non linear time history results with a 0.5g PGA along the Y direction: (a)

absolute displacement at each storey; and (b) inter storey drift ...141 Figure 6.19. (a) Pushover analysis curve; and (b) maximum base shear ...142 Figure 6.20. Linear elastic response spectrum results with a 0.1g PGA: (a) and (c) absolute

displacement at each storey along the Y and X direction, respectively; and (b) and (d) inter storey drift along the Y and X direction, respectively. ...143 Figure 6.21. Non linear time history results with a 0.1g PGA along the Y direction: (a)

absolute displacement at each storey; and (b) inter storey drift ...144 Figure 6.22. Non linear time history results with a 0.1g PGA along the X direction: (a)

absolute displacement at each storey; and (b) inter storey drift ...151

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LIST OF TABLES

Table 2.1. Equivalent strut width calculation (Tarque et al., 2015) ...26

Table 2.1. Equivalent strut width calculation (continued) (Tarque et al., 2015) ...27

Table 2.2. Coefficient values K1 and K2 (Noh et al., 2017) ...34

Table 2.3. Empirical equations for infill backbone curve (Huang and Burton, 2020) ...38

Table 2.4. Suggested limits for empirical constants (Crisafulli, 1997) ...45

Table 3.1. Reinforcement tests results (Marinković, 2018) ...56

Table 3.2. Mean values of the test results for the masonry assembly (Marinković, 2018) 56 .... Table 3.3. Elastomer static and dynamic elastic modulus (Marinković, 2018) ...57

Table 3.4. Tests performed (Marinković, 2018) ...58

Table 3.5. Load phases for DIO test (Marinković, 2018) ...61

Table 4.1. Material properties for concrete and reinforcement steel (Marinković, 2018) ...69

Table 4.2. Calibration parameters for rebar hysteresis model ...74

Table 4.3. Material properties for masonry unit and interaction (Marinković, 2018) ...76

Table 4.4. Calculation of the dimensionless value 𝜆h ...76

Table 4.5. Calculation of the strut width bm ...77

Table 4.6. Values of the shear strength and bed joint shear strength ...77

Table 4.7. Calculation of the failure modes ...77

Table 4.8. Calculation of the lateral strength Hmfc ...78

Table 4.9. Calculation of the lateral strength Hmfc ...78

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1. INTRODUCTION

Masonry materials, such as stone and bricks, have a huge variety of applications in construction of civil engineering structures. Its use date back to the first civilizations that populated the Earth. The ruins of Jericho (Middle East, 7,350 BC), the pyramids of Egypt (2,500 BC), the pyramids built by the Mayan civilization (500 AD), the Great Wall of China (200 BC to 200 AD), are just a few examples of constructions that testify to the use and durability of this type of material (Jan Fiala et al., 2019).

Masonry walls, called infill walls, are not only used for structural purposes, but also to divide spaces, to provide fire protection, for sound insulation, as well as purely architectural elements. They have been utilised for its color, shape, texture, availability, durability, for its thermal insulation capacity and low cost, compared to other materials.

Although widely used, it is complicated to include all variables into consideration when modeling an infill wall. This is due to the fact that there are many solutions for filling a frame: homogenous and non-isotropic, partially hollow and full, mortar characteristics, etc. In addition, the lack of a simple analytical calculation model that envelopes said variables makes this process cumbersome. For this reason, infill walls are rarely and often inadequately included in structural models.

Currently structural analysisis carried out by assuming only the self weight of the infill walls and neglecting their horizontal load bearing capacity. The practical approach to consider the infill wall as a non structural element and to consider it only in the calculation of the modal mass, would be correct if the infill had not interacted with the frame. However, after a major earthquake events, observed damage in infill walls indicate the presence of the interaction. Moreover, it implies resistance of walls to horizontal dynamic loads. It has been shown that the infill walls significantly increase the structure behavior such as rigidity, period, weight, irregularities and energy dissipation capacity (Abrahamczyk et al. 2019; Furtado et al. 2019).

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Even though infill walls absorb certain portion of the earthquake, they are brittle elements and thus tend to be the first elements to present damages and crack. After this happens, the reinforced concrete frame will take the earthquake loads and the infill walls will only continue as a vertical load. This modifies the natural period and stiffness of the structure and may cause undesirable responses like torsion and soft storey effect (Marinković, 2018).

The column-infill interaction has been reported many times throughout the years and many cases were found after the earthquake in L’Aquila 2009. As seen in Figure 1.1a and b, there are partial infill walls in contact with adjacent columns. This interaction causes a reduction in the slenderness of the column and thus creates a higher concentration of shear stresses in the column that finally made them fail (Ricci et al., 2011).

Figure 1.1. (a) and (b) damages caused by L’Aquila earthquake in 2009, interaction between column and infill (Ricci et al., 2011)

Many authors (Ricci et al., 2011; Paulay and Priestley, 1992; Crisafulli, 1997; El-Dakhakhni et al., 2003) have classified the different failure mechanisms in walls into: (i) horizontal sliding due to joint failure; (ii) infill diagonal failure; and (iii) infill corner crushing. These failures can be seen in Figure 1.2.

(a) (b)

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Bull Earthquake Eng (2011) 9:285–305 297

Fig. 8 Shear failure of: (a) rectangular, (b) circular columns, (c) column adjacent to partial infilling panels, (d) squat column adjacent to basement level concrete walls

With regard to the rectangular column in Fig.8a, whose section is 30× 100 cm2_, belong-ing to a 1980s’ buildbelong-ing, shear failure is evident, involvbelong-ing the top end section. Transverse reinforcement has hoop spacing of approximately 15–20 cm, and is definitely under-designed with respect to column section size, that is, with respect to the inertia of the section, thus leading to premature shear failure of the element. The brittle failure mechanism is highlighted by the crushing of the concrete within the reinforcement and the complete opening of the third and fourth hoops from the top end of the element.

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Fig. 8 Shear failure of: (a) rectangular, (b) circular columns, (c) column adjacent to partial infilling panels, (d) squat column adjacent to basement level concrete walls

With regard to the rectangular column in Fig.8a, whose section is 30× 100 cm2_, belong-ing to a 1980s’ buildbelong-ing, shear failure is evident, involvbelong-ing the top end section. Transverse reinforcement has hoop spacing of approximately 15–20 cm, and is definitely under-designed with respect to column section size, that is, with respect to the inertia of the section, thus leading to premature shear failure of the element. The brittle failure mechanism is highlighted by the crushing of the concrete within the reinforcement and the complete opening of the third and fourth hoops from the top end of the element.

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Figure 1.2. (a) infill diagonal failure; (b) horizontal sliding; and (c) corder crushing (Ricci et al., 2011)

In their study authors concluded that the elements that were mainly damaged during the earthquake were the infill panels. Although these were considered in structural analysis as nonstructural elements, he mentioned that there was a clear interaction between the column-infill interface and that it should be taken to account to avoid premature brittle failure (Ricci et al., 2011; Paulay and Priestley, 1992; Crisafulli, 1997; El-Dakhakhni et al., 2003)

Ricci et al. (2011) mentioned that this interaction can increase the global stiffness and spectral acceleration demand causing an increase in base shear and finally collapse. Unequal distribution of the infills along the height and/or plan can cause irregularities and undesirable effects. Some examples can be seen in Figure 1.3, he reported that the structure in Figure 1.3a had a discontinuity of the infills in the second level, which is the one that failed. He also reported that in the structure shown in Figure 1.3b, there was an irregular distribution of the infills in the second floor with respect to the others due to garage entrances; this structure had a soft storey mechanism collapse.

(b)

(a) (c)

302 Bull Earthquake Eng (2011) 9:285–305

Fig. 12 Infill panel failures: diagonal cracking (a), (b) and corner crushing (c) mechanisms

Fig. 13 External infill panel failures without connection between layers (a) and with ineffective connection (b)

the lower beam by means of a small pawl. This constructive solution leads to a decrease in the interaction mechanism between RC frame and external infill panel, for both in-plane and out-of-plane seismic forces. The low efficacy of the restraint applied to the external panel, coupled with the ineffective or completely absent connections between the two layers, leads to damage limited in most cases to the external infill panel which can easily show an out-of-plane failure due to seismic action in both directions, as can be observed in Fig.13b. Neither local nor global interaction effects between infills and the RC structure are negligible. As was previously emphasized, local interaction between infill panel and adjacent columns can

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Figure 1.3. (a) and (b) soft storey mechanism collapse (Ricci et al., 2011)

Infills may provide a high energy dissipation mechanism, thus reducing the demand on frames and also the inter storey drift but may induce a soft storey mechanism if there is an irregular distribution in plan and height. This contribution of the infill panels depends on different parameters like: distribution of the walls in plan and elevation, stiffness, parameters, height/ length ratio, location of openings and many others (Tarque et al., 2015).

There have been several authors (Decanini et al., 2004 ; Mohamed et al., 2019 ; Kaushiket al., 2006) that have modeled RC structure and compared its seismic behavior to one of a masonry infilled frame. All of them have concluded that infill walls significantly increase the maximum base shear during an earthquake, it causes a reduction in the period and after infill failure, columns may be subjected to a higher concentration of stresses. This shows that the infill walls should be included at early stages of design as it affects the frame elements and if not considered, these might fail during a seismic event.

These past sudies have made it clear that there is a need to include the infill walls when designing and analyzing a structure. The problem that has arised is standardizing a way in which infill walls should be numerically modeled in order to analyze a structure. For example Tarque et al., (2015) commented that in the design codes, the contribution of the infills to the frame and the interaction between them is not considered adequately. Mostly because the behavior has not been standardized due to its complexity and that too many different cases should be considered due to the amount of variables present. He also commented that there are

(a) (b)

Fig. 14 Soft storey mechanism examples in L’Aquila: (a) Via Porta Napoli, (b) Via Dante Alighieri (Pettino)

lead to (i) a reduction in the effective height of the column, an increase in shear demand and a consequent brittle failure of the column when the infill panel partially occupies the frame bay; (ii) shear demand concentration at the end of the column and consequent brittle failure when diagonal compression is applied by the panel on the RC element.

As a global phenomenon infill-structure interaction increases global stiffness of the com-plex system and consequently spectral acceleration demand. Besides, it can represent a source of irregularity in plan or elevation (e.g. pilotis) when the infill distribution is irregular.

Some particular cases of structural failure after the L’Aquila event and mainly caused by irregularities in plan or elevation are reported in Fig.14. The first structure (see Fig.14a) was situated in the centre of L’Aquila (Via Porta Napoli); it was irregular in elevation and the second storey had an evident discontinuity in terms of infill distribution; in the left wing of the building there was a sort of porch. In terms of building collapse, damage was concentrated on the second storey, leading to complete failure of the upper levels.

The structure proposed in Fig. 14b was placed in the residential zone of Pettino (Via Dante Alighieri), close to L’Aquila centre. The building has an irregular shape in plan, similar to a T; infill distribution makes the structure irregular in elevation due to the presence of garage entrances. The building showed a soft-storey mechanism at the first storey that can

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Fig. 14 Soft storey mechanism examples in L’Aquila: (a) Via Porta Napoli, (b) Via Dante Alighieri (Pettino)

lead to (i) a reduction in the effective height of the column, an increase in shear demand and a consequent brittle failure of the column when the infill panel partially occupies the frame bay; (ii) shear demand concentration at the end of the column and consequent brittle failure when diagonal compression is applied by the panel on the RC element.

As a global phenomenon infill-structure interaction increases global stiffness of the com-plex system and consequently spectral acceleration demand. Besides, it can represent a source of irregularity in plan or elevation (e.g. pilotis) when the infill distribution is irregular.

Some particular cases of structural failure after the L’Aquila event and mainly caused by irregularities in plan or elevation are reported in Fig.14. The first structure (see Fig.14a) was situated in the centre of L’Aquila (Via Porta Napoli); it was irregular in elevation and the second storey had an evident discontinuity in terms of infill distribution; in the left wing of the building there was a sort of porch. In terms of building collapse, damage was concentrated on the second storey, leading to complete failure of the upper levels.

The structure proposed in Fig.14b was placed in the residential zone of Pettino (Via Dante Alighieri), close to L’Aquila centre. The building has an irregular shape in plan, similar to a T; infill distribution makes the structure irregular in elevation due to the presence of garage entrances. The building showed a soft-storey mechanism at the first storey that can

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some indications in Eurocode 8, for example regarding irregularities and other recommendations for the column-infill interaction. However they state that nonlinear analysis are crucial to determine the real failure patterns caused by infills.

Marinković (2018) also gave some remarks from the Eurocode 8 regarding the infill walls in frames. He commented that detailed design rules are not present, even though there is clear evidence that infill wall collapse may be present during earthquake due to in and out of plane loading.

Looking at Eurocode 8 (EN 1998-1, 2004) it indicates that if the infills are in contact with the frame but no additional connection is used, if they were considered at the beginning as non-structural elements and if they are built after the concrete has hardened; then additional considerations should be taken into account. It also indicates that plan irregularities should be avoided at all costs and that seismic action effects in columns should be increased if irregularities in elevation exist. Also a spacial model should be employed to analyze the structure and that infills should be included in it, disregarding those with significant opening like a door or a window (EN 1998-1, § 4.3.6).

Due to this lack of standardization and also the negative effects that might be caused by infill walls, there have been some proposed solutions in order to reduce this effects of the infill/frame interaction. One of these was studied by Marinković (2018), where he subjected a single infilled frame with a decouples system to cyclic and monotonical loading. This system gave promising results as it can deocuple effectively the infill from the wall and thus having the structure behave almost as a bare frame.

However the lack of a defined a clear method to model this type of structures in commercial structural analysis programs makes it hard to implement this solution. Also, a proper simple macro model approach that is able to capture the behavior of an infill wall with this innovative system has not ben proposed yet. So this research was focused on not only modeling and analysing the seismic behavior of infill and the innovative system but also proposing a macro model for the appropriate simulation of the innovative system.

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1.1. Objectives of the research

The present work focuses on implementing and validating a model of a frame with an infill wall, based on an experimental tests from Marinković (2018), subjected to gravitational loads and lateral loads. Part of the process of implementation and validation is based on adequately defining the mechanical and geometrical properties of the different walls tested in his experimental tests.

The following are the objectives of this thesis work:

i. To develop numerical models, simulate a bare frame, an infilled frame and a decoupled infilled frame and validate each with the laboratory test results by Marinković (2018). ii. Select a macro-elements to model an equivalent diagonal that is able to capture the

infilled wall frame behavior in the experimental tests by Marinković (2018).

iii. Compare the behavior of a traditional infilled wall system with the INODIS system (Innovative Decoupled Infill System), made from an elastomeric material, which has the capacity to isolate the behavior of the infilled wall from the surrounding frame.

iv. Study the soft story effect on different infill wall arrangement systems in a two- and three-dimension structure.

1.2. Methodology

First a literary review will be made to understand the behaviors and characteristics that should be taken into account when modeling infill walls in a frame system. Afterwards, the experimental tests by Marinković (2018) will be modeled and calibrated in order to capture the cyclic and monotonic behavior them. In total three models were calibrated which consist of: (i) bare frame; (ii) traditional infilled frame; and (iii) infilled frame with innovative system. Then, after obtaining similar results to the experimental ones, more complex cases will be modeled to analyze the behavior of a multistorey structure with infill walls subjected to an earthquake. Within these models, the soft storey effect will be studied while using a traditional infill system or an innovative decoupling one.  

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2. LITERATURE REVIEW

2.1. Introduction

There has been many studies indicating that the interaction of the infill walls with the surrounding frame can cause beneficial or disadvantagous effects: increase the structure behavior such as rigidity, period, weight, irregularities and energy dissipation capacity. Even though the masonry wall has a brittle behavior, the combined behavior with a surrounding frame can increase the ductility of the overall structure as the walls function as an energy dissipation mechanism. Besides, the infill walls reduce lateral deformation due to its high stiffness (Crisafulli, 1997).

Unfortunately the dissipation is only achievable at low or medium displacements, so quick stiffness degradation is present. For this and other reasons, through time, masonry infill walls were considered as non-structural elements. Still, when using them in structures, there is a modification of the natural period and thus there might be an increase or decrease on seismic forces; there is a increase on shear stresses on vertical elements due to wall-column interaction; irregularities in plan may cause undesirable torsional response; and formation of soft-stores mechanisms may arise (Crisafulli, 1997).

When a frame is subjected to a lateral load, the infill panel comes into contact with the column and beams, creating a shear concentration in the four corner as shown in the following Figure 2.1. When the frame is subjected to additional loading, crack start to appear along the frame elements and the panel interface, causing a separation of in infill except on opposite compressive corners (Figure 2.1b). It is for these reasons that many research are based on a diagonal compressive strut element. At the edges of the compressive strut, there is a concentration of stresses along the beam and column elements, which depend on its stiffness, that may surpass each elements maximum strength. As the lateral loads continues to increase, different types of failures modes may arise.

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Figure 2.1. Shear distribution in infill panel (Crisafulli, 1997)

2.2. Failures modes

Mann and Mäuller (1982) conducted multiple tests on masonry panels. They tried to simulate the effect of a horizontal load on a infilled structure with a typical load condition. The experiment consisted on panels initially loaded with uniform vertical compression 𝜎0 and then an increasing lateral load was applied at the upper beam. Experimental tests demonstrated that there are different types of in-plane failure mechanisms that can show up depending on the combined actions of vertical, flexural and shear load (Quinteros, 2014)

2.2.1. Failure in masonry infill panels

There are three main types of masonry infill in-plane panel failure, the first one is due to shear, the second one is due to bending and the third one due to compression.

Failure mode due to shear can be regularly seen in masonry walls subjected to a combination of vertical and lateral loads, this is the most common type. The main peculiarity of this failure mode is that it begins in the center of the panel and spreads towards the corners where it can take two different kind of schemes, these are: (i) straight cracking and (ii) stepped cracking. Strong mortar is an important factor since it makes the difference between this two modes. (Pacheco, 2016).

Straight cracking arises when the tensile strength of the brick units is exceeded. This failure is characterized by having diagonal cracks in the panel along the individual units and in the

(a) (b)

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mortar joints as seen in Figure 2.2. It has to be mentioned that this kind is not necessarily a failure mode because the infill is still confined inside the frame, it is more considered as a serviceability limit state (Ghiassi et al., 2012; Crisafulli, 1997)

Figure 2.2. Diagonal cracking (Crisafulli, 1997)

Alternatively, stepped cracking is characterized by the horizontal and vertical failure along mortar joints, it occurs under the same conditions as diagonal cracking, but with the uniqueness of weak mortar joints, as seen in Figure 2.3. This is one of the most recurrent failures that happens in infill walls, as a result of shear or lateral forces in the plane (Ghiassi et al., 2012; Crisafulli, 1997)

Figure 2.3. Stepped cracking of masonry (Crisafulli, 1997)

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Failure mode due to compression can follow two different kinds of mechanisms as a result of the stresses the infill wall is subjected to, these can be: (i) failure of the corners and (ii) failure of the middle portion of the infill (Crisafulli, 1997).

Corner crushing, Figure 2.4, is caused by very high compression stresses, that are developed in the corner regions due to the deformation of the frame and separation between said frame and the infill. This kind of behavior is seen in very flexible frames, where the frame-panel contact area is reduced, thus causing a concentration of compressive stresses in the edge regions. It might also be presented when low-quality masonry or hollow bricks are used for the construction of the wall or panel enclosed by weak joints and strong elements (Crisafulli, 1997).

Figure 2.4. Failure due to masonry corner crushing (Crisafulli, 1997)

Failure mechanism due to compression failure of the diagonal strut is characterized by the crushing of the central part of the panel, is linked to buckling of slender masonry wall due to out-of-plane deformations. Similar to diagonal cracking, after cracks start appearing tensile stresses located on the diagonal are alleviated. This behavior makes the masonry between the cracks be axially loaded and thus a failure mechanism occurs (Crisafulli, 1997; Asteris et al., 2011).

In some of the instances where the bending effect prevail and the frame columns are not strong, bending cracks may appear in the tension zone of the infill wall due to the low tensile strength as seen in Figure 2.5 (Crisafulli, 1997).

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Figure 2.5. Flexural cracking of infill panel (Crisafulli, 1997)

Even though this failure is usually described by a rocking mechanism, the effect is normally characterized by the crushing of the masonry in the compressed zone due to stress concentrations (Figure 2.6a). The rocking involves development of horizontal cracks in the tension zone along the columns and mortar. These cracks decrease in length along the height of the column as seen in Figure 2.6b (Pacheco, 2016).

Figure 2.6. Flexural behavior of panel a) stress concentration due to rocking, b) crack development in tension zone (Pacheco, 2016).

Shear sliding failure is characterized by the sliding of a bed joint and is related to weak mortar beds as seen in Figure 2.7 (Asteris et al., 2011). This kind of mode does not usually occur exclusively or as a pure failure mode, as it would require walls with a very low aspect ratio and subjected to low axial loads infrequent conditions in practice (Pacheco, 2016).

40 Los resultados del trabajo de Petry y Beyer (2013) no son aplicables aquí puesto que, por un lado, su tipología de partida, la de edificio de muro portante de MNR con piso de losa de concreto, difiere de la de edificio de muro portante de MNR con vigas-dintel. Por otro lado, las paredes propias de los edificios de MNR modernos de Suiza (ladrillo hueco y un mortero comercial de cemento) no son equiparables con las tradicionales de ladrillo macizo del Eixample-Barcelona. No obstante, constituye un trabajo de interés por la categorización de las condiciones de borde en los pilares que establece (con base en un factor de longitud efectiva) y el cómo aborda el estudio de la influencia de dichas condiciones en el comportamiento sísmico de los edificios de MNR (Fig. 3.5b).

3.1.1.2 Falla por flexión: volteo (rocking)

La falla por flexión de un pilar está vinculada al alcance de la resistencia compresiva de la mampostería (fu) en la sección extrema de dicho pilar (Fig. 3.6a). El equilibrio en el estado

último de esfuerzos en flexocompresión asume las siguientes hipótesis: 1) una distribución rectangular de esfuerzos en la parte comprimida de la base del pilar y 2) el desprecio de la resistencia a la tracción de las juntas de mortero horizontales (Magenes y Calvi 1997). En consecuencia, se deduce la capacidad de resistencia a flexión (Mup):

2 1 2 up u D t p p M f

κ

  = _ − _   (3.1)

donde, D es la longitud del pilar; t es el espesor del pilar; p=P/A es el esfuerzo normal sobre el pilar; fu es la resistencia compresiva de la mampostería y κ=0.85 es el coeficiente

de transformación en distribución rectangular (equivalente) de tensiones.

Figura 3.6 Flexocompresión en pilares: a) condición de esfuerzos para cálculo de resistencia a flexión; b) agrietamiento por flexión y zona efectiva de corte

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Figure 2.7. Shear sliding (Crisafulli, 1997)

Results of experimental done by Abrams and Shah (1992) also shown that a pure horizontal failure of the mortar bed, as shown in Figure 2.8a, is not really common. Instead the failure is presented more as a mixed one between shear cracking and horizontal sliding, Figure 2.8b. Typically these failure initializes with a rucking mechanism phase without reaching the crushing of the contact surface, Figure 2.6b. Subsequently a failure mechanism along the mortar bed is normally presented as one of the two final cracking pattern modalities shown in Figure 2.8b. Each of these two modalities are characterized by: 1) an extension of one of the cracks of the tensioned zone due to rocking, or 2) a transitional phase from a stepped crack to a horizontal one (Pacheco, 2016).

Figure 2.8. Shear failure along a mortar joint: a) pure mode; b) mixed / hybrid mode (Pacheco, 2016)

47 y Calvi 1994; Magenes et al. 2008). En el modelo propuesto en este trabajo, al igual que en la mayoría de los existentes en la literatura (v.g. Magenes y Della Fontana 1998; Magenes 2000; Lagomarsino et al. 2013), se ha optado por ignorar este modo de falla.

Figura 3.9 Deslizamiento en la junta horizontal: a) modo puro; b) modo mixto/híbrido

Precisamente, los resultados de campañas experimentales (Abrams y Shah 1992; Magenes y Calvi 1992; Magenes y Calvi 1994; Bosiljkov et al. 2003; ElGawady et al. 2005) han evidenciado que el deslizamiento en la junta horizontal puede presentarse con relativa frecuencia, pero como un modo de falla mixto/híbrido. Las formas típicas en que se presenta son las esquematizadas en la figura 3.9b. El aspecto común de estas fallas es su fase inicial de grieta debidoa al mecanismo de volteo (rocking) que se ha desarrollado sin llegar al aplastamiento en el otro extremo de la base de la pared. Luego, el deslizamiento en la junta suele presentarse siguiendo una de las dos modalidades de patrón de agrietamiento final mostrados en la figura 3.9b. Cada una de estas dos modalidades se caracterizan por: 1) una extensión de una de las grietas por flexión, o 2) una fase de transición de agrietamiento diagonal escalonado.

Magenes y Calvi (1997), que en campañas experimentales previas habían detectado este modo de falla (Magenes y Calvi 1992; Magenes y Calvi 1994), lo corroboraron con simulación numérica utilizando elementos finitos no lineales. Además, comprobaron que la fórmula propuesta por el Eurocódigo 6 (CEN 1994) para la resistencia a corte de paredes de mampostería evalúa aceptablemente la resistencia de la pared a un modo de falla mixto con deslizamiento en la junta horizontal. Dicha fórmula incorpora como variable una longitud no agrietada efectiva (D’), que considera una porción en la base de la pared agrietada por flexión. Típicamente, esta longitud no agrietada se calcula despreciando la resistencia a tracción de la junta horizontal de mortero y asumiendo una distribución simplificada (constante o lineal) de los esfuerzos de compresión. Magenes y Calvi (1997) mejoraron la calidad predictiva de la fórmula del Eurocódigo 6 (CEN 1994), incluyendo la relación de corte αVen la deducción de D’:

'

3

1

2

V

D

P

β



α



=

_

−

_





(3.9)

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2.2.1. Failure in concrete frame

It is important to study the failures produced in the frame of the infill wall as well. Frames with infills are usually designed as a bare frame, without considering the particular aspects resulting from the interaction between the panel and the frame. The most important failure modes are: (i) flexural collapse mechanism, (ii) failure due to axial loads, (iii) shear failure of the columns, (iv) beam-column joint failure (Crisafulli, 1997).

Flexural collapse mechanism occurs in the region of maximum bending moment (Figure 2.9), that is the top and bottom of the column. This type is not considered as a collapse mechanism yet because the infill wall hasn’t actually failed and the frame can still act as a pinned frame. Normally the collapse mechanism occurs when the infill wall fails (Crisafulli, 1997).

Figure 2.9. Flexural collapse mechanism (Crisafulli, 1997)

When weak mortar is present in the infill, there is an alteration of the bending moment diagrams due to shear sliding of the infill wall (Figure 2.10). Same as the flexural collapse mechanism, plastic hinges form at the top and the bottom of the columns but not on the beam, instead there is one located at mid column height (Crisafulli, 1997).

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Figure 2.10. Flexural collapse mechanism with shear sliding (Crisafulli, 1997)

Failure due to axial loads are normally uncommon because columns are design to have great compressive strength. Nevertheless, recurrent loading and unloading cycles may cause buckling in longitudinal reinforcement hence developing a failure. Also, this effect causes the column to be in tension, creating cracks along the element. Two different failure modes can result: (i) Yielding of longitudinal reinforcement and (ii) bar anchorage failure (Crisafulli, 1997).

After cracks have already appeared in the column section and as lateral displacement continues, yielding of the longitudinal reinforcement may occur, causing considerable axial deformation on the columns. As for the masonry walls, they are subjected to rocking as explained before, the effect can be observed in Figure 2.11 (Crisafulli, 1997).

Figure 2.11. Tension failure (Crisafulli, 1997)

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Bar anchorage failure occurs when there is not enough bonding between the reinforcement bars and the concrete; or when not enough development length is given. This kind of failure produces a rocking mechanism of the entire frame and infill, Figure 2.12 (Crisafulli, 1997).

Figure 2.12. Bar anchorage failure (Crisafulli, 1997)

Mix interaction between the column and the infill wall may cause shear failure in the columns. This failure can appear in the contact zone (Fig. 2.13) between the infill and the column after separation occurs due to stress concentrations, where shear forces are higher. It occurs after cycles of loading and unloading where columns present cracks due to tensile forces, and is also tightly associated on the amount of shear reinforcement on the column (Crisafulli, 1997).

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Figure 2.13. Shear failure (Crisafulli, 1997)

Because a frame and wall system is subjected to high demand of rotation and shear. Beams must be design in order to be strong enough to withstand the loads but also ductile to dissipate energy. An increment on the reinforcement compromises its ductile and may form collapse mechanisms due to strength and stiffness deterioration. Cracks form from the interior of the corner towards the exterior and come in pairs, as seen in Figure 2.14 (Crisafulli, 1997; Patnaik et al., 2012).

Figure 2.14. Beam-Column joint failure (Crisafulli, 1997)

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2.3. Masonry infill modeling

Considering the orthotropic nature of masonry and that it behaves highly in a non-linear way, it can only be studied through numerical analysis techniques which considered non-linear characteristics of the material. Due to advances in technology, finite element models have been a great tool to facilitate the analysis of masonry structures and structural analysis in general. But being masonry a composite material, made up of individual units joined by mortar, it is more complicated to apply this kind of analysis. This is because the accuracy of the results is remarkably susceptible to the suitable selection of materials’ constitutive models as well as on different parameters involved, which are commonly obtained through experimental tests. A good way to calibrate the model and to check the success of the modeling is by comparing its results to the ones obtained through experimental tests (Guzmán et al., 2002; Lourenço et al., 1995). There are two different approaches that can be followed when modeling a infill wall through finite element modeling: the micro-modeling and the macro modeling. These approaches use different theories with different levels of complexity and computational cost, all of them characterized by different theoretical foundations and levels of detail. In general terms, micro-modeling is the one that assumes the masonry as a heterogeneous material, generally giving more accurate results; while the macro-modeling assumes the masonry as a homogenous material (Quinteros et al., 2014)

2.3.1. Micro-models

It is a masonry infill modeling technique where the infill is represented as a discontinuous collection of units or bricks linked by joints in their real position. The mortar and the individual units are defined by continuum elements while the interface between these two is defined by discontinuous. Characteristics such as the Young’s modulus, Poisson’s ratio, inelastic properties and others are also considered. After implementing all this characteristics and behavior, it is possible to represent combine mixed actions between the units, mortar and interface; and it makes it possible to closely examine its behavior (Lourenço et al., 1995).

It is probably the finest tool for understanding the actual behavior of masonry. One benefit of its application is that it can reproduce different failure mechanisms like cracking or sliding. This

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kind of models are normally used to study the local behavior of masonry structures. Considering it uses a notable discretization of elements, its application carries a very high computational cost with respect to the scale for which it is employed. In addition, to adjust the individual models, experimental tests have to be performed and compared to the micro-model in order to obtain accurate results. There are two types of micro models

• Detailed micro-models: It is a more refined approach, mortar units and joints are discretized by continuous finite elements, Figure 2.15 (b);

• Simplified micro-models: elements of continuous behavior represent the masonry units while the behavior of the mortar joints and junction between the units and the mortar are represented by discontinuous elements, Figure 2.15 (c) (Asteris, 2015; Quinteros, 2014; Pacheco 2016).

Figure 2.15. Modeling strategies: (a) masonry example; (b) detailed micro-modeling; (c) simplified micro-modeling; (d) macro-modeling (Lourenço et al., 1995).

2.3.2. Macro-models

It is a modeling technique where the masonry is taken as a homogenous and anisotropic material. This method is suitable when the dimensions of the structure are big enough, where there are uniform stresses across the infill and where the global behavior is relevant. For this reason, this kind of modeling has more of a practical application considering it requires a lower

The proposed models were developed in the doctoral thesis project of the first author and have been implemented in the DIANA finite element code, which has been used in all the analyses.

2 Approaches towards computational modeling of masonry structures

Only recently the masonry research community began to show interest in sophisticated numerical tools as an opposition to the prevailing tradition of rules-of-thumb and empirical formulae. The fact that little importance has been attached to numerical aspects is confirmed by the absence of any well established models. The difficulties in adopting existing numerical tools from more advanced research fields, namely the mechanics of concrete, rock and composite materials, are hindered by the particular characteristics of masonry. Masonry is a composite material that consists of units and mortar joints, see Fig. la. A comprehensive analysis of masonry, hereby denoted detailed micro-modeling, must then include a representation of units, mortar and the unit / mortar interface, see

Fig. lb. In this case units and mortar in the joints are represented by continuum elements whereas the unit-mortar interface is represented by discontinuous elements. The Young's modulus, Poisson's ratio and, optionally, inelastic properties of both unit and mortar are taken into account. The interface represents a potential crack / slip plane with initial dummy stiffness to avoid inter-penetration of the continuum. This enables the combined action of unit, mortar and interface to be studied under a magnifying glass. Such a representation of masonry leads to large memory and time requirements and a simplified micro-modeling of masonry will be preferably used here, see Fig. Ie. In this case expanded units are represented by continuum elements whereas the behavior of the mortar joints and unit-mortar interface is lumped in discontinuous elements. Each joint, consisting of mortar and the two unit-mortar interfaces, is lumped into an "average" interface

Unit (brick, block, etc)

Bed joint

Perpend or head joint

(a) "Unit" "Joint"

\

j '\ iI, --:Ir ----:Ir---___ JI __ ----:Ir---___----:Ir---___ J __ _ (e) Unit _Mortar (b) Composite

/

_,_, ________ J L _______ _ ---, r - - - , r---I r---I I I ___ J L ________ ...J L __ _ - - - , r - - - -_,_, (d) Interface Unit/mortar

Fig. 1. Modeling strategies for masonry structures: (a) masonry sample; (b) detailed micro-modeling; (c) simplified micro-modeling; (d) macro-modeling.

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computational cost without compromising on the global precision level. These studies have given rise to a technique called equivalent diagonal strut that, although it can adequately represent the overall behavior of the infill wall in structures, they do not show local behavior even in a linear range (Quinteros, 2014).

In this technique bricks, mortar and its interface are represented by the same element (Fig. 2.15d). This type of discretization is usually encompassed by a homogenization techniques, which basically consist in replacing all the complex geometries and properties into a simplified geometry that contains the global behavior, so it becomes a homogenous and anisotropic continuum (Quinteros, 2014).

2.4. Equivalent diagonal strut

2.4.1. Single strut models

One of the most known macro-models for infill walls is the equivalent diagonal strut model, based on the effect of stress concentration in a diagonal zone of the walls, as a result of the application of lateral loads. This deformation results in the formation of a compression strut between two opposite corners of the frame as seen in Figure 2.16 (Zuñiga, 2005)

Figure 2.16. Equivalent truss mechanisms (Crisafulli, 1997)

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Polyakov (1960) was one of the first that suggested the use of an equivalent diagonal strut based on the results on his diagonally loaded in compression. Holmes (1961), taking an idea from Polyakov, proposed an equivalent pin-jointed diagonal strut with the mechanical characteristics of the wall, setting an equivalent width w of 1/3 of the length of the diagonal.

where d is the length of the equivalent diagonal and w is the equivalent strut width. From a practical perspective, the equivalent diagonal was easily employed and was able capture the behavior of the infill, that is why different researchers have tried to develop different procedures to calculate the properties of the diagonal. Despite obtaining very good results in terms of forces, the deflections alway showed results lower than the real ones (Asteris et al., 2011; Olusola, 2008).

One year later, Smith (1962) developed tests on masonry infills and found out that the actual equivalent width w was dependent on the diagonal’s length, more specifically when it was in between 0.10 to 0.25 of the length (Asteris et al. 2011).

Later Stafford-Smith (1966) proposed a dimensionless parameter 𝜆h dependent on the relative

stiffness of the wall and the frame. They based themselves on analytical equations that correlate the contact zone of an infill and a frame. The 𝜆h factor is defined as

where Ew is the masonry panel’s modulus of elasticity; EcIc is the column’s flexural rigidity, tw is

panel thickness of the infill panel; h is the column’s height between beams’ centerlines; hw is the

height of the infill; 𝜗 is the angle between the horizontal and the diagonal of the wall as seen in Figure 2.17, given by (Asteris et al., 2011; Olusola, 2008).

(2.1) w = d 3 (2.2) λh= h 4 Ew_4EIwsin 2θ cIchw (2.3) θ = tan−1 ( hw Lw)