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Dottorato di Ricerca in Ingegneria delle Strutture e del recupero edilizio e urbano - IX ciclo N. S. Presentazione del lavoro di tesi Analisi non lineare di pareti murarie sotto azioni orizzontali: modellazione a telaio equivalente Fisciano, 6 Maggio 2011 Dipartimento di Ingegneria Civile – Università degli Studi di Salerno Dottorando: Ing. Riccardo Sabatino Tutor: Prof. Vincenzo Piluso Co-Tutor: Prof. Gianvittorio Rizzano

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All exact science is dominated by the idea of approximation Bertrand Russell PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling Science is organized knowledge Herbert Spencer

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Chapter 1: Introduction Chapter 1: Introduction Chapter 2: FEM modelling Chapter 2: FEM modelling Chapter 3: Masonry Buildings Modelling Strategies Chapter 3: Masonry Buildings Modelling Strategies Chapter 4: Mechanical Behaviour of masonry panels Chapter 4: Mechanical Behaviour of masonry panels Chapter 5: Matrix Analysis of structures Chapter 5: Matrix Analysis of structures Chapter 6: The FREMA code Chapter 6: The FREMA code Outline PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Chapter 1: Introduction Chapter 1: Introduction Chapter 2: FEM modelling Chapter 2: FEM modelling Chapter 3: Masonry Buildings Modelling Strategies Chapter 3: Masonry Buildings Modelling Strategies Chapter 4: Mechanical Behaviour of masonry panels Chapter 4: Mechanical Behaviour of masonry panels Chapter 5: Matrix Analysis of structures Chapter 5: Matrix Analysis of structures Chapter 6: The FREMA code Chapter 6: The FREMA code Outline PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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First Stone Dwellings (8350 b.C. – Jericho, Tell-es-Sultan) Introduction Fresco found in Rekhamaras Tomb (1500 b.C. – Egypt) Djoser Pyramid (2600 b.C. – Egypt) PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Performance-based Earthquake Engineering Non-linear static procedures (NLP) Introduction Non-linear Static Analysis PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Strategies for Modelling Masonry Buildings Introduction FEM models Simplified models Very accurate prediction Very accurate prediction Any kind of structure may be analysed Any kind of structure may be analysed X Time-consuming X Amount of input data X High Analytical Skills required Suitable for professional purposes Suitable for professional purposes Quick analyses Quick analyses X Regular geometry needed X Simplifications need to evaluated VS PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Chapter 1: Introduction Chapter 1: Introduction Chapter 2: FEM modelling Chapter 2: FEM modelling Chapter 3: Masonry Buildings Modelling Strategies Chapter 3: Masonry Buildings Modelling Strategies Chapter 4: Mechanical Behaviour of masonry panels Chapter 4: Mechanical Behaviour of masonry panels Chapter 5: Matrix Analysis of structures Chapter 5: Matrix Analysis of structures Chapter 6: The FREMA code Chapter 6: The FREMA code Outline PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Accurate Modelling: mesoscale model Blocks are modelled using continuum elements, while mortar and brick-mortar interfaces are modelled by means of nonlinear interface elements (Lourenço & Rots, 1996). Solid and interface elements account for large displacements, while only interface elements represent cracks in mortar and bricks. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Material model Multi-surface nonassociated plasticity Elastic response Elastic stiffness Mortar joints Yield functions F 1 - F 2 Plastic potentials Q 1 - Q 2 A novel 2D nonlinear interface element PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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A novel 2D nonlinear interface element Material model Nonassociated plasticity Nonassociated plasticity Elastic response Elastic stiffness penalty factor Brick interface Yield function F 1 Plastic potential Q 1 PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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A novel 2D nonlinear interface element Material properties t - tensile strength C - cohesion G f,II - mode II fracture energy C - compressive strength G f,I - mode I fracture energy G f,I uzuz t - friction angle <0 G f,I I u x(y) c tan G C - crushing energy c uzuz GcGc PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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A novel 2D nonlinear interface element Work-softening plasticity Evolution of the material parameters with W pl1 - plastic work related to F 1 W pl2 - plastic work related to F 2 Evolution of the surfaces PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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A novel 2D nonlinear interface element Traction deformation response tension-compression cyclic behaviour shear cyclic behaviour tension shear PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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In-plane behaviour Vermeltfoort AT, Raijmakers TMJ (1993) J4DJ5D p v =0.3 MPa mortar interface brick interface PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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In-plane behaviour Vermeltfoort AT, Raijmakers TMJ (1993) J4DJ5D Wpl1 Wpl2 p v =0.3 MPa p v =2.12 MPa dynamic analysis PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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In-plane behaviour Mesh assessment Mesh refinement Number of integration points over the interface PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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In-plane behaviour Vermeltfoort AT, Raijmakers TMJ (1993) Wpl1 Wpl2 PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting: Influence of spandrels Numerical Modelling (STRAUS7): 144 panels Panels have been obtained by assembling a reference module whose geometry (piers, spandrels) has been properly varied. The parameter, ratio between the shear stiffness of piers and spandrels, has been used to take into account the panels geometry. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Panels have been modelled by assuming two limit schemes: infinite stiffness spandrels and unreinforced spandrels. The parameter represents the expected improvement of relative shear strength achievable by means of spandrels retrofitting. Retrofitting: Influence of spandrels PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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The comparison shows a significant improvement in the field <1.5, i.e. for weak spandrels. In such field the average strength improvement can be estimated as: Retrofitting: Influence of spandrels PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting Then the panels 1-1 and 4-4 have been extensively investigated by considering the retrofitting approach suggested in Italian Building Code. For each panel, the weak spandrel ( =0.70) and the strong spandrel ( =5.35) schemes have been analysed. Three different kinds of reinforcements have been taken into account: Injection Grouts; Reinforced plasters; Ring beams / Jack Arch. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting The shear resistance of the panels has been evaluated by means of a non-linear static analysis. Both the unreinforced and the reinforced wall have been analyzed. The improvement deriving from the reinforcement has been summarized into the parameter where T is the reinforced wall resistance, T nc the unreinforced wall resistance. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting: Wall 1-1 For weak spandrels walls, the spandrel improvement gives the same results of injection grouts/reinforced plasters. For strong spandrels walls, best improvements have been achieved with injection grouts/reinforced plasters. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting: Wall 4-4 PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting [Modena et al.] The expected improvement gets the same order of magnitude of data available in literature PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Retrofitting: Parete 4-4 For the wall 4-4, further numerical simulations have been performed, by assuming the reinforcement (reinforced plaster/injection ) applied to 1 to 4 storeys. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Chapter 1: Introduction Chapter 1: Introduction Chapter 2: FEM modelling Chapter 2: FEM modelling Chapter 3: Masonry Buildings Modelling Strategies Chapter 3: Masonry Buildings Modelling Strategies Chapter 4: Mechanical Behaviour of masonry panels Chapter 4: Mechanical Behaviour of masonry panels Chapter 5: Matrix Analysis of structures Chapter 5: Matrix Analysis of structures Chapter 6: The FREMA code Chapter 6: The FREMA code Outline PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Simplified Models PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Chapter 1: Introduction Chapter 1: Introduction Chapter 2: FEM modelling Chapter 2: FEM modelling Chapter 3: Masonry Buildings Modelling Strategies Chapter 3: Masonry Buildings Modelling Strategies Chapter 4: Mechanical Behaviour of masonry panels Chapter 4: Mechanical Behaviour of masonry panels Chapter 5: Matrix Analysis of structures Chapter 5: Matrix Analysis of structures Chapter 6: The FREMA code Chapter 6: The FREMA code Outline PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Description of the model Main features 1. Displacement Control approach NLP 2. Global and local equilibrium 3. Spread plasticity approach 4. Quick Analysis and Easy Post-processing Piers (H eff after Dolce, 1991) Spandrels Rigid Offsets Equivalent Frame Model PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Piers - Constitutive Laws Generalized Uniaxial Compressive Stress-Strain Relationship A=2, B=-1, C=2 [Hendry, 1998] A=6.4, B=-5.4, C=1.17 [Turnšek-Čačovič, 1980] Accurate Moment-Curvature Accurate Moment-Curvature [After Tomaževič, 1999] PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Piers - Flexural Behaviour normalised neutral axis normalised axial force normalised bending moment Cross-section Equilibrium Equations PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Moment-Curvature relationship workflow D, t,, u u cr M YES NO END PhD Dissertation Talk – Fisciano, 6 th May 2011

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Piers - Shear Behaviour [After Anthoine, Magenes, Magonette, 1994] Experimental Behaviour Model Ultimate drift u = 0.4% H eff [Italian Building Code] PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Piers - Shear Behaviour PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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i =V i /K i min V u i el K i+1 =K i *V u /V i NO YES el =V u /K el K i+1 =K el Shear-strain relationship workflow PhD Dissertation Talk – Fisciano, 6 th May 2011

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Collapse condition when the desired value of drift (set by the user) is attained (Italian Building Code suggets = for shear collapse) Shear-strain relationship workflow PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Spandrels - Shear Behaviour Experimental Behaviour Model Residual Strength = 0.25 [Magenes and Della Fontana, 1998] PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Spandrels - Flexural Behaviour Experimental Behaviour Model [After Calderoni et al., 2008] PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Spandrels - Flexural Behaviour Proposed formulations for M u – [Italian Building Code, 2008] 1.Stress-block approach (same equation of piers) 2. If no tensile-resistant element is present M u =0 PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Spandrels - Flexural Behaviour Proposed formulations for M u [Schubert & Weschke, 1986] Take into account an equivalent strut provided with a tensile strength f tu Take into account an equivalent strut provided with a tensile strength f tu f tu is the minimum between two collapse mechanisms: f tu is the minimum between two collapse mechanisms: a) bricks failure b) bed joints failure a) b) PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Spandrels - Flexural Behaviour Spandrels M-N Limit Domain [Cattari and Lagomarsino, 2008] = ratio between tensile strength f tu and compressive strength = ratio between tensile strength f tu and compressive strength Constitutive Law Improvement of rocking resistance, also with low (or zero) values of N. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Chapter 1: Introduction Chapter 1: Introduction Chapter 2: FEM modelling Chapter 2: FEM modelling Chapter 3: Masonry Buildings Modelling Strategies Chapter 3: Masonry Buildings Modelling Strategies Chapter 4: Mechanical Behaviour of masonry panels Chapter 4: Mechanical Behaviour of masonry panels Chapter 5: Matrix Analysis of structures Chapter 5: Matrix Analysis of structures Chapter 6: The FREMA code Chapter 6: The FREMA code Outline PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Masonry Panels – Anthoine, Magonette and Magenes (1998) Low Panel High Panel Cross-Section: 100 x 25 cm 2 Low panel high: 135 cm High panel high: 200 cm Normal Load: 150 kN The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Pavia Door Wall – Calvi and Magenes (1994) The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Pavia Door Wall – Calvi and Magenes (1994) The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Pavia Door Wall – Calvi and Magenes (1994) The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Catania Project - Investigation on the seismic response of two masonry buildings (2000) Via Martoglio 2D Wall Equivalent Frame model: 128 elements, 81 nodes, 219 DOF The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Model 1: Masonry, NO R.C. Ring Beams Via Martoglio 2D Wall The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Model 2: Masonry, Elastic R.C. Ring Beams (E=20,000 MPa) Via Martoglio 2D Wall The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Model 3: Masonry, Elastic R.C. Ring Beams (E=4,000 MPa) Via Martoglio 2D Wall The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Catania Project - Investigation on the seismic response of two masonry buildings (2000) Preliminary validation of the model Via Verdi Building Wall 1 Wall 2 Wall 3 PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Via Verdi – Wall 1 The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Via Verdi – Wall 2 The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Via Verdi – Wall 3 The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Mallardo et al. (2008) – Palazzo Renata di Francia The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Mallardo et al. (2008) The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) Two-storey, 7-bay masonry wall Two-storey, 7-bay masonry wall Two lateral load distributions considered: Two lateral load distributions considered: 1.Uniform (ACC) F= {1.00; 0.59} 2.Inverse Triangular (LOAD) F= {1.00; 1.19} The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) – 7B_Uniform Proposed Model SAP 2000 Discrete FEM model The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) – 7B_Inverse Triangular Proposed Model SAP 2000 Discrete FEM model The FREMA Code PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) Mesh Refinement =Log(N c / x) PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) – 7B_Inverse Triangular Mesh Refinement PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) – 7B_Uniform Mesh Refinement PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Salonikios et al. (2003) Time-cost Analysis =Log(N c / x) PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Time-cost Analysis PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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This dissertation deals with the seismic behaviour of masonry structures; This dissertation deals with the seismic behaviour of masonry structures; The first part of the work is aimed at understanding the potentialities of very accurate FEM model in predicting masonry panels seismic response; the panels simulated by means of ADAPTIC showed a very good prediction of the experimental results, both in terms of force-displacement curve and in terms of cracks path. The first part of the work is aimed at understanding the potentialities of very accurate FEM model in predicting masonry panels seismic response; the panels simulated by means of ADAPTIC showed a very good prediction of the experimental results, both in terms of force-displacement curve and in terms of cracks path. A further application of simplified (homogeneous) FEM models has been performed on masonry panels, aiming at evaluating the influence of spandrels reinforcement on the overall resistance; in the same application some reinforcement techniques have been applied considering the Italian Building Code approach; A further application of simplified (homogeneous) FEM models has been performed on masonry panels, aiming at evaluating the influence of spandrels reinforcement on the overall resistance; in the same application some reinforcement techniques have been applied considering the Italian Building Code approach; Conclusions PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Conclusions In the second part of the dissertation, a novel equivalent frame model has been developed. The main features of the model have been discussed, by highlighting the main features of the proposed model (displacement control approach, accurate moment-curvature for piers behaviour, spandrels behaviour); In the second part of the dissertation, a novel equivalent frame model has been developed. The main features of the model have been discussed, by highlighting the main features of the proposed model (displacement control approach, accurate moment-curvature for piers behaviour, spandrels behaviour); A validation and application of the model has been carried out comparison with experimental tests and accurate numerical simulations A validation and application of the model has been carried out comparison with experimental tests and accurate numerical simulations The comparison showed a good agreement between the proposed model and both experimental and numerical results, showing that FREMA code is a reliable tool for performing the non-linear static analysis of masonry panels. The comparison showed a good agreement between the proposed model and both experimental and numerical results, showing that FREMA code is a reliable tool for performing the non-linear static analysis of masonry panels. PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Thank you very much! PhD Dissertation Talk – Fisciano, 6 th May 2011 R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

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Dottorato di Ricerca in Ingegneria delle Strutture e del recupero edilizio e urbano - IX ciclo N. S. Presentazione del lavoro di tesi Analisi non lineare di pareti murarie sotto azioni orizzontali: modellazione a telaio equivalente Fisciano, 6 Maggio 2011 Dipartimento di Ingegneria Civile – Università degli Studi di Salerno Dottorando: Ing. Riccardo Sabatino Tutor: Prof. Vincenzo Piluso Co-Tutor: Prof. Gianvittorio Rizzano

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