Numerical analysis of the reinforced arch method
Abstract
Masonry arches, vaults and domes are architectural elements diffusely present in a great part of the human construction heritage. Brought to its maximum magnificence from the Roman genius, and then spread all over the World throughout History, especially in the Occidental and Mediterranean countries, the arch has become one of the most recognizable mark of our architectural legacy. The Reinforced Arch Method is a rather recent strengthening technique for masonry arches and vaults that entails the application of steel post-tensioned cables at their extrados or intrados, in order to enhance capacity and ductility. Many experimental tests have been performed on this technique, but numerical modelling has not been yet fully deepened. Indeed, by means of numerical Finite Element models, the present work mainly aims at confirming the effectiveness of this procedure. At the same time, it envisages calibrating and validating the numerical tools, by comparing their results with the ones derived from an experimental campaign on scale models, performed by Prof. Lorenzo Jurina at the Polytechnic University of Milan. Macro-models, based on smeared crack material, and micro-models, based on elastic blocks with hinge opening, have been produced and tested. A comparative study on the results obtained with both procedures has been performed, underlying their specific qualities and drawbacks, especially regarding their capability to capture the collapse phenomenon and the interaction between the original structure and the strengthening system.
Full text
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS I DECLARATION Name: Alberto Bonetto Email: [email protected] Title of the Msc Dissertation: Numerical Analysis of the Reinforced Arch Method Supervisor(s): Prof. Pere Roca, Prof. Lorenzo Jurina Year: 2016/17 I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. I hereby declare that the MSc Consortium responsible for the Advanced Masters in Structural Analysis of Monuments and Historical Constructions is allowed to store and make available electronically the present MSc Dissertation. University: Universitat Politècnica de Catalunya Date: July 14, 2017 Signature:
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme II ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS III ACKNOWLEDGEMENTS I would like to thank my supervisors, Professors Pere Roca and Lorenzo Jurina for their availability and all the help and suggestions they provided me during the entire dissertation period in Barcelona. Thanks also to the lecturers of the Master Programme in Padova, especially Professor Climent Molins and Enrico Garbin for their illuminating lessons. Thanks to the Consortium, for the scholarship supplied. A special thanks to my parents for all the support they gave me during all the Master’s period, and to all of my classmates, my friends and all the people that supported me during the whole dissertation time, especially Vera, Marco, Sara, Alicia and Armando.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme IV ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS V ABSTRACT Masonry arches, vaults and domes are architectural elements diffusely present in a great part of the human construction heritage. Brought to its maximum magnificence from the Roman genius, and then spread all over the World throughout History, especially in the Occidental and Mediterranean countries, the arch has become one of the most recognizable mark of our architectural legacy. The Reinforced Arch Method is a rather recent strengthening technique for masonry arches and vaults that entails the application of steel post-tensioned cables at their extrados or intrados, in order to enhance capacity and ductility. Many experimental tests have been performed on this technique, but numerical modelling has not been yet fully deepened. Indeed, by means of numerical Finite Element models, the present work mainly aims at confirming the effectiveness of this procedure. At the same time, it envisages calibrating and validating the numerical tools, by comparing their results with the ones derived from an experimental campaign on scale models, performed by Prof. Lorenzo Jurina at the Polytechnic University of Milan. Macro-models, based on smeared crack material, and micro-models, based on elastic blocks with hinge opening, have been produced and tested. A comparative study on the results obtained with both procedures has been performed, underlying their specific qualities and drawbacks, especially regarding their capability to capture the collapse phenomenon and the interaction between the original structure and the strengthening system.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme VI ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS VII RESUMEN Los arcos, bóvedas y cúpulas de mampostería son elementos arquitectónicos ampliamente presentes en el patrimonio construido de la humanidad. Fueron llevados a su máximo esplendor por los genios Romanos, y su uso se propagó por todo el mundo a lo largo de la historia. Especialmente en occidente y en los países mediterráneos el arco se ha convertido en uno de los símbolos más reconocidos del legado de la arquitectura. El método del arco reforzado es una técnica de refuerzo estructural para arcos y bóvedas relativamente reciente que implica la utilización de cables postensados en el intradós o extradós, con el fin de aumentar la resistencia y la ductilidad. Se han realizado muchas campañas experimentales con esta técnica, sin embargo, no se ha desarrollado aún en profundidad el modelado numérico. Por lo tanto, el objetivo de este trabajo es confirmar la eficacia del método del arco reforzado mediante el uso de modelos numéricos de elementos finitos. Al mismo tiempo, contempla la calibración y validación de las herramientas numéricas, comparando los resultados con los obtenidos en la campaña experimental en modelos a escala, realizada por el Prof. Lorenzo Jurina de la Universidad Politécnica de Milán. Se han elaborado y probado macro-modelos, basados en ‘smeared crack material’, y micro-modelos, basados en bloques elásticos con apertura de rótulas. Además, se ha realizado un estudio comparativo de los resultados obtenidos con ambos procedimientos, destacando tanto sus cualidades específicas como sus desventajas, poniendo especial atención a su capacidad de capturar correctamente el fallo de la estructura y la interacción entre la estructura original y el sistema de refuerzo.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme VIII ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS XV Figure 43 – Ring (Limit State) output configuration. .............................................................................. 36 Figure 44 – Original unreinforced arch geometry, load at 1/6 span, collapse configurations. From top clockwise: experimental, static and kinematic one. .......................................................... 38 Figure 45 – Comparison of collapse configurations for load at 1/6 of the span. Top: experimental. Bottom: equivalent depth models: static limit analysis (right) and kinematic one (left). ... 40 Figure 46 – Collapse configurations of arch loaded at 1/6 of the span, obtained with both limit analysis approaches (kinematic on the left, static on the right), in the case of equivalent density for obtaining the same capacity of the experimental arch. .................................................... 40 Figure 47 – Equivalent confinement radial load due to post-tensioned cables, applied at the extrados (left) or at the intrados (right). R, r: the radius of the arches, T: cable post-tension. ........ 42 Figure 48 – Comparison of experimental and limit analysis results for the reinforced arch. ................ 43 Figure 49 – Percentage errors on collapse load computed with limit analysis to the experimental results. .............................................................................................................................. 44 Figure 50 – RING sensitivity to depth (left) and depth increment (right). .............................................. 45 Figure 51 – Micro-model, unreinforced arch (82 mm depth), load at 1/6 of the span, linear geometry, deflection of loaded point vs load applied. ........................................................................ 48 Figure 52 – Micro-model, unreinforced arch (82 mm depth), load at 1/6 of the span, non-linear geometry, deflection of loaded point vs load applied. ....................................................... 49 Figure 53 - Micro-model, unreinforced arch (82 mm depth), load at 1/2 of the span, linear geometry, deflection of loaded point vs load applied. ........................................................................ 49 Figure 54 - Micro-model, unreinforced arch (82 mm depth), load at 1/2 of the span, non-linear geometry, deflection of loaded point vs load applied. ....................................................... 50 Figure 55 – Top: micro-model unreinforced, 82 mm depth, non-linear geometry, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x200 amplification). Bottom: experimental collapse configuration. ................................................................... 50 Figure 56 - Top: micro-model unreinforced arch, 82 mm depth, non-linear geometry, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x1000 amplification). Bottom: experimental collapse configuration. ................................................................... 51 Figure 57 – Arch with equivalent radial reinforcement at the extrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied. .................................................... 53 Figure 58 - Arch with equivalent radial reinforcement at the extrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied. .................................................... 53 Figure 59 - Top: micro-model with equivalent radial reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x30 amplification). Bottom: experimental collapse configuration............................................. 54 Figure 60 - Top: micro-model with equivalent radial reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x100 amplification). Bottom: experimental collapse configuration............................................. 55
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme XVI ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 61 – Arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied. .................................................... 56 Figure 62 - Arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied. .................................................... 56 Figure 63 - Top: micro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x200 amplification). Bottom: experimental collapse configuration. ............................................ 57 Figure 64 - Top: micro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x50 amplification). Bottom: experimental collapse configuration. ............................................ 58 Figure 65 – Details of the cable/wood interface in the experimental arch. (Giglio, 2008). .................... 59 Figure 66 - Cable axial force at the end of the application of the post-tension (left) and at collapse (right). (Micro-model, physical cables, φ=5°, deformation x20). ....................................... 61 Figure 67 – Shear stress in the interface at the end of the application of the post-tension (left) and at collapse (right). (Micro-model, physical cables, φ=5°, deformation x20). ......................... 61 Figure 68 – Normal stress in the interface at the end of the application of the post-tension (left) and at collapse (right). (Micro-model, physical cables, φ=5°, deformation x20). ......................... 61 Figure 69 – Three point chosen for investigating the interaction cable/arch. ........................................ 62 Figure 70 – Friction angle φ = 0.6°. Shear stress trend. ....................................................................... 62 Figure 71 - Friction angle φ = 5°. Shear stress trend. ........................................................................... 63 Figure 72 - Friction angle φ = 20°. Shear stress trend. ......................................................................... 64 Figure 73 - Friction angle φ = 30°. Shear stress trend. ......................................................................... 65 Figure 74 – Micro-model, load at ½ span, physical modelling of cables at the extrados, 294 N posttension, 90 mm depth: comparison of capacity curve with different friction angles cable/arch. ......................................................................................................................... 66 Figure 75 – Micro-model with physical reinforcement at the extrados (90 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. ........................ 67 Figure 76 - Arch with physical cable at the extrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. .......... 68 Figure 77 - Arch with physical cable at the extrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. .......... 68 Figure 78 - Left: micro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x50 amplification). Right: experimental collapse configuration. ...................................................................... 69
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS XVII Figure 79 - Micro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, collapse deformed configuration (x50 amplification). Left: axial force in the cable, right: interface normal stress. ........................................................................................... 69 Figure 80 - Micro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, collapse deformed configuration (x50 amplification). Left: shear stress at the interface, right: shear displacements. ............................................................................... 69 Figure 81 - Left: micro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x70 amplification). Right: experimental collapse configuration. ...................................................................... 70 Figure 82 - Micro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, collapse deformed configuration (x70 amplification). Left: axial force in the cable, right: interface normal stress. ........................................................................................... 70 Figure 83 - Micro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, collapse deformed configuration (x70 amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 70 Figure 84 – Cable at the extrados: micro-models results and comparison with previous results and experimental ones. ............................................................................................................ 70 Figure 85 - Micro-model with physical reinforcement at the intrados (100 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. ........................ 71 Figure 86 - Arch with physical cable at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. ......... 72 Figure 87 - Arch with physical cable at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. ......... 72 Figure 88 - Left: micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x50 amplification). Right: experimental collapse configuration. ...................................................................... 73 Figure 89 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x50 amplification). Left: axial force in the cable, right: interface normal stress. ........................................................................................... 73 Figure 90 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x50 amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 73 Figure 91 - Left: micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x300 amplification). Right: experimental collapse configuration. ...................................................................... 74
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme XVIII ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 92 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, collapse deformed configuration (x300 amplification). Left: axial force in the cable, right: interface normal stress. ............................................................................................ 74 Figure 93 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, collapse deformed configuration (x300 amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 74 Figure 94 – Physical cable reinforcement at the intrados: micro-models results and comparison with previous results and experimental ones. .......................................................................... 74 Figure 95 – Micro-models with anchored cables at the extrados results: force vs vertical displacement at the loading point. Linear and non-linear geometry analysis of no post tension and 294 N post-tension case. ......................................................................................................... 75 Figure 96 - Micro-models with anchored cables at the intrados results: force vs vertical displacement at the loading point. Linear and non-linear geometry analysis of no post tension and 294 N post-tension case. ......................................................................................................... 76 Figure 97 – Anchored cables at the extrados micro-model: cable tension trend at ¼ of the semicircumference. ................................................................................................................... 77 Figure 98 - Anchored cables at the intrados micro-model: cable tension trend at ¼ of the semicircumference. ................................................................................................................... 77 Figure 99 – Anchored cables at the extrados micro-model, no post-tension, step 300, deformation scale x10. From top downwards: compressive stress between blocks, normal stress at cable/arch interface, cable tension, shear stress at cable/arch interface. ........................ 78 Figure 100 – Capacity curves obtained with macro-models, loaded at half span, with different values of E and ft. Load vs loaded point deflection at half span. ..................................................... 81 Figure 101 - Unreinforced arch, load at 1/2 of the span. Top: macro-model (82 mm depth, ft = 0.01 MPa, E = 2000 MPa), collapse hinged mechanism, x15 deformation scale, principal crack strains. Bottom: experimental arch collapse configuration. .............................................. 82 Figure 102 - Capacity curves obtained with macro-models loaded at 1/6 of the span. Left: model Κ (ft=0.01 MPa). Right: model Λ (ft = 0.0015). Load vs loaded point deflection at 1/6 of the span. .................................................................................................................................. 83 Figure 103 - Macro-models of the unreinforced arch, loaded at 1/6 of the span. Comparison of capacity obtained with different values of tensile strength. ............................................... 83 Figure 104 – Unreinforced arch, load at 1/6 of the span. Top: macro-model results (82 mm depth), collapse hinged mechanism, x30 deformation scale, principal crack strain: ft = 0.0015 MPa (left), ft = 0.01 MPa (right). Bottom: experimental arch collapse configuration. ....... 83 Figure 105 – Macro-model results: arch with equivalent radial reinforcement at the extrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied. ........... 84 Figure 106 - Macro-model results: arch with equivalent radial reinforcement at the extrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied. ........... 85
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS XIX Figure 107 - Top: macro-model with equivalent radial reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, principal crack strain at collapse deformed configuration (x20 amplification). Bottom: experimental collapse configuration............................................. 85 Figure 108 - Top: macro-model with equivalent radial reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, principal crack strain at collapse deformed configuration (x60 amplification). Bottom: experimental collapse configuration............................................. 86 Figure 109 - Macro-model results: arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied. ........... 87 Figure 110 - Macro-model results: arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied. ........... 87 Figure 111 – Top: macro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, principal crack strain at collapse deformed configuration (x20 amplification). Bottom: experimental collapse configuration............................................. 88 Figure 112 - Top: macro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, principal crack strain at collapse deformed configuration (x10 amplification). Bottom: experimental collapse configuration............................................. 89 Figure 113 - Macro-model with physical reinforcement at the extrados (90 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. ........................ 91 Figure 114 - Macro-model results: arch with physical cable at the extrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. .......................................................................................................................... 91 Figure 115 - Macro-model results: arch with physical cable at the extrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. .......................................................................................................................... 92 Figure 116 – Left: macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, crack strains at collapse deformed configuration (x40 amplification). Right: experimental collapse configuration. ................................................................................ 92 Figure 117 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, collapse deformed configuration (x25 amplification). Left: axial force in the cable, right: interface normal stress. ........................................................................................... 92 Figure 118 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, collapse deformed configuration (x25 amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 93 Figure 119 - Left: macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, crack strains at collapse deformed configuration (x40 amplification). Right: experimental collapse configuration. ................................................................................ 93
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme XX ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 120 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, collapse deformed configuration (x40 amplification). Left: axial force in the cable, right: interface normal stress. ............................................................................................ 93 Figure 121 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, collapse deformed configuration (x40 amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 93 Figure 122 - Macro-model with physical reinforcement at the intrados (90 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. ........................ 94 Figure 123 - Macro-model results: arch with physical cable at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. .......................................................................................................................... 95 Figure 124 - Macro-model results: arch with physical cable at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. .......................................................................................................................... 95 Figure 125 - Left: macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, crack strains at collapse deformed configuration (x amplification). Right: experimental collapse configuration. ................................................................................. 96 Figure 126 - Macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x40 amplification). Left: axial force in the cable, right: interface normal stress. ................................................................................. 96 Figure 127 - Macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x40 amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 96 Figure 128 - Left: macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, crack strains at collapse deformed configuration (x amplification). Right: experimental collapse configuration. ................................................................................. 97 Figure 129 - Macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, collapse deformed configuration (x amplification). Left: axial force in the cable, right: interface normal stress. ............................................................................................ 97 Figure 130 - Macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, collapse deformed configuration (x amplification). Left: shear stress in the interface, right: shear displacements. ............................................................................... 97 Figure 131 – Unreinforced arch: comparison between micro and macro-models results. .................... 99 Figure 132 – Equivalent radial load at the extrados reinforced arch: comparison between micro and macro-models results. ....................................................................................................... 99
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS XXI Figure 133 - Equivalent radial load at the intrados reinforced arch: comparison between micro and macro-models results...................................................................................................... 100 Figure 134 – Arch with modelling of reinforcement cable at the extrados: comparison between micro and macro-models results. ............................................................................................. 100 Figure 135 - Arch with modelling of reinforcement cable at the intrados: comparison between micro and macro-models results. ............................................................................................. 100
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme XXII ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS XXIII LIST OF TABLES Table 1 – Semi-circular arch tests setups. (Giglio, 2008). .................................................................... 22 Table 2 – Collapse load for sliding cables at the extrados. (Giglio, 2008). ........................................... 24 Table 3 – Collapse load for sliding cables at the intrados. (Giglio, 2008). ............................................ 24 Table 4 – Cable elongation at maximum external loading for each prestress level (load at ¼ of the span). (Giglio, 2008). ........................................................................................................... 28 Table 5 – Hinge opening at maximum external loading for each prestress level (load at ¼ of the span). (Giglio, 2008). ...................................................................................................................... 28 Table 6 – Parameters of masonry due to total strain crack material in the macro-models. .................. 31 Table 7 – Interface modelling: Coulomb friction parameters to set. ...................................................... 33 Table 8 – Linear elastic material properties. ......................................................................................... 34 Table 9 – Unreinforced arch, exact geometry: limit analysis results compared to experimental ones, for all the load positions. ........................................................................................................... 37 Table 10 – Equivalent models determined by limit analysis. Results and comparison with experimental ones. .................................................................................................................................... 39 Table 11 – Collapse hinged configurations: experimental arch and kinematic analysis with original and reduced depths. ................................................................................................................... 41 Table 12 – Comparison of collapse load for different post tension levels between experimental results and limit analysis with original geometry arch. Cable at the extrados. ............................... 43 Table 13 - Comparison of collapse load for different post tension levels between experimental results and limit analysis with original geometry arch. Cable at the intrados. ................................ 43 Table 14 – Equivalent depths, computed by static limit analysis, for obtaining the same collapse load of experiments in the reinforced arch cases (load at half span). ........................................ 44 Table 15 – sensitivity analysis of ultimate load to the depth of the arch in RING. ................................ 45 Table 16 – Wooden blocks linear elastic parameters. *Equivalent density has been used to keep constant the overall weight of the experimental arch. ......................................................... 47 Table 17 – Coulomb friction interface parameters between blocks. ..................................................... 48 Table 18 – Unreinforced arch: analysis setup details. .......................................................................... 48 Table 19 – Micro-model unreinforced, 82 mm depth, numerical results compared. ............................. 51 Table 20 – Geometrical parameters and density of reinforced arches with radial load equivalent to 294 N of post-tension. ................................................................................................................ 52 Table 21 – Micro-models with equivalent radial load at the extrados, analysis setup. ......................... 52 Table 22 – equivalent radial load reinforcement at the extrados: micro-models results and comparison with experimental ones and limit analysis. .......................................................................... 54 Table 23 - equivalent radial load at the intrados reinforced models, analysis setup. ........................... 55 Table 24 - Equivalent radial load reinforcement at the intrados: micro-models results and comparison with experimental ones. ....................................................................................................... 58
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme XXIV ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 25 – Cables and interface parameters. *Gap opening depends on the position of reinforcement: in the case of reinforcement at the extrados it is activated, in the case of at the intrados it is not. ....................................................................................................................................... 60 Table 26 – Free sliding cable at the extrados, reinforced micro-model, analysis setup. ...................... 67 Table 27 - Free sliding cable at the extrados, reinforced micro-model, analysis setup. ....................... 71 Table 28 – Anchored cables reinforced arch micro-models, analysis setup. ........................................ 75 Table 29 – Parameters of arch body in the macro-models. *These parameters are variable, and they have been object of a deeper analysis to determine better the valuable intervals. ............ 79 Table 30 – Unreinforced macro-models loaded at half span: parameters setups for determining E and ft influence on the vertical load capacity. ............................................................................. 80 Table 31 - Unreinforced macro-models’ analysis setups for determining E and ft influence on the vertical load capacity. .......................................................................................................... 80 Table 32 – Macro-models of the unreinforced arch, loaded at half span. Comparison of capacity obtained with different values of tensile strength and Young modulus. .............................. 81 Table 33 - Unreinforced macro-models loaded at 1/6 of the span: parameters setups. ....................... 82 Table 34 - Macro-models with equivalent radial load at the extrados, analysis setup. ......................... 84 Table 35 - equivalent radial load reinforcement at the extrados: macro-models results and comparison with experimental and limit analysis ones. .......................................................................... 86 Table 36 - Macro-models with equivalent radial load at the intrados, analysis setup. .......................... 87 Table 37 - Equivalent radial load reinforcement at the intrados: macro-models results and comparison with experimental and limit analysis ones. .......................................................................... 89 Table 38 - Macro-models with cables modelling at the extrados, analysis setup. ................................ 90 Table 39 – Cables at the extrados modelling: macro-models results and comparison with previous ones and experimental ones. .............................................................................................. 94 Table 40 - Macro-models with physical cables at the intrados, analysis setup. .................................... 94 Table 41 - Cable at the intrados: macro-models results and comparison with previous results and experimental ones. .............................................................................................................. 97
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 7 2.2.1 Finite Element Models Finite Element analysis, in masonry structures, is much diffused nowadays. This approach is most of the time based on a continuum solid modelling. Depending on the detail of modelling, micro-modelling and macro-modelling can be distinguished, as explained above for the cases a) and c) by Lourenço, (1996). A typical drawback in masonry structures macro-modelling (Roca et al., 2010) is that the smeared crack constitutive law, normally used in commercial programs, very often do not allow to localize the damage pattern as in the reality. In fact, in unreinforced masonry usually cracks form in highly confined areas of panels, while this kind of models tend to show a diffused crack pattern. This problem has been overcome through a novel crack-tracking algorithm (Clemente et al., 2006), that allows to localize better the damage path (Figure 4). Figure 4 – Mallorca cathedral: comparison between smeared crack model analysis and crack-tracking one, in case of transversal response under seismic load (Clemente et al., 2006). Micro-modelling is a less diffused approach, due to its difficulty to handle. It is more used for analysing special problems with localized phenomena that especially involve interaction between blocks and joints. Concerning masonry arches and vaults, several examples of FE analysis are available. Lourenço (2002) analyses a masonry arch loaded at ¼ of the span, comparing macro-modelling FEA results with limit analysis one. Both physical and geometrical non-linear approaches are carried out, and even two different values of tensile strength are considered. From the comparison of the results obtained, it can be stated that by increasing the tensile resistance the safety factor raises, with a peak-like response curve, but the residual value is not affected by these parameters. Since limit analysis encompasses no tensile resistance, models with ft =0 showed a better accordance with theoretical results, without the presence of peaks. Finally, adding the contribution of non-linear geometry does not affect much the final response (Figure 5).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 8 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 5 – Collapse mechanism obtained and comparison between results of a FE masonry arch model, loaded at span quarter (Lourenço, 2002) Similar analyses have been carried out by Kumar and Bhandari (2005), through the comparison of experimental results on masonry arches loaded at the span quarter, and an 80 8-noded 2D elements FE model. The constitutive law included crushing and cracking, and its parameters were calibrated through the experimental values previously obtained. Even in this case, experimental 3D arches and 2D FE models show a good accordance in terms of capacity curve, as shown in Figure 6. Figure 6 – Masonry arch 2D modelling: crack pattern and comparison of the experimental and analytical loaddeflection curve at loading point (Kumar and Bhandari, 2005). Betti et al. (2008), modelled a single-span masonry barrel arch bridge, loaded vertically at ¼ of the span, using a 2D plane strain elements model, including adaptive unilateral friction-contact interface elements. This means that when in any point between blocks the stress state is not admissible, the corresponding joint is substituted by unilateral contact interface, that allows crack opening. In this case, infill was also modelled. The results obtained showed good accordance with the experimental ones (Figure 7).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 9 Figure 7 – Collapse mechanism of infilled arch, with adaptive contact-friction interfaces (Betti et al., 2008). 3D models can also be considered, implying the use of tetra-hexa brick elements. Milani and Lourenço (2012) analysed a skew single span masonry arch, tested at Bolton Institute, UK, and a multi span bridge of five semi-circular arches, both loaded by an eccentric vertical load. This was carried out by means of 3D FE models with rigid eight-noded parallelepiped elements, with non-linear interfaces. This type of FE modelling has shown results in compliance with limit analysis ones and with the pre-existing results (Figure 8), obtained on the same example. In particular, it is much suggested for unusually shaped structures, such as skew arches, that need a 3D approach to be modelled and to well interpret their behaviour. Figure 8 – Bolton Institute skew arch. Degraded interface patch, obtained through the non-linear homogenized FE code. (Milani and Lourenço, 2012). 2.2.2 Discrete Element Models Typically, Discrete Element Models are based on the separate representation of the mechanical behaviour of the units and the interaction between them (Lemos, 2007). They usually comply with these assumptions: - Rigid blocks with deformability concentrated at joints. Nevertheless, deformable blocks formulations are also available;
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 10 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS - Interaction between blocks is represented through point contacts, or edge-to-edge contacts, with no attempt to get a continuous distribution of stress through contact surfaces; - Separation of blocks and large displacements analysis are typically allowed. By definition, the Discrete Element Method applies to a computational approach only if it permits finite rotations and displacements of discrete units, and is able to compute new contacts between elements as the analysis proceeds (Roca et al., 2010) Constitutive behaviour of contacts can be rigid, with forced no overlapping between blocks, or soft, allowing finite axial and shear contact stiffness. In the latter, stress depends on the relative displacement between blocks. Several 2D programs, implementing DE models, are available, such as UDEC (Itasca, Minneapolis, MN, USA), 3DEC (Cundall, 1988; Hart et al., 1988) and CICE (Hocking et al., 1992). DE models have been largely used for masonry arches modelling, in particular by Gilbert and Melbourne (1995), who investigated the problem of ring separation between multi-ring block arches (Figure 9). Figure 9 – Discrete Element multi-ring arches failure analysis, Gilbert and Melbourne (1995). A more recent type of DE model is also available, combining FE and DE: the Discontinuous Deformation Analysis (DDE). This method is based on an assumed deformation field, within distinct domains of ordinary shapes. These type of model, derived from rock mechanics, allows to perform analysis of rigid block motion and block deformation, simultaneously. A DDA analysis of a masonry arch, with which the back-fill represented by a system of deformable blocks (Figure 10), created by a random pattern of joints, was carried out by Bicanic et al. (2001).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 11 Figure 10 - DDA of Edinburgh University model arch bridge (Bicanic et al., 2001). Another approach for discontinuous modelling, suitable for arches infill, is the use of Particle Flow models (PFC). These models are based on the discretization of materials through spherical elements, in contact one to another. Thavalingam et al. (2001) analysed the case of an experimental infilled masonry bridge model, tested at Cambridge University, using three different models: Finite Element (DIANA), PFC and DDA. The results are shown in Figure 11, also in a graph for comparing the capacity curve obtained. Figure 11 – Different 2D modelling approaches of an infilled masonry bridge. Clockwise from bottom-left: PFC model, FE model (DIANA), DDA model, comparison of results, (Thavalingam et al., 2001). 2.2.3 Alternative Approach: One Dimensional Elements To discretise masonry arches, a possible alternative to Finite Element models, based on 2D or 3D elements, is the usage of one-dimensional beam elements. This procedure is rather unusual, because
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 12 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS of the predominance of FE programs, even though it shows many positive outcomes, such as low computing demand and the possibility to display directly the value of internal forces. Molins and Roca (1998) proposed a curved members elements model with variable cross section and non-linear material behaviour (elastoplastic under shear and compression and linear elastic-perfectly brittle in tension, with Mohr-Coulomb-like dependence of shear resistance from compressive stress). Moreover, non-linear geometry condition is included. The results obtained on a masonry arch, loaded at quarter-span, are presented in Figure 12. These results have been compared with limit analysis and laboratory tests ones, showing the reliability of these kind of models. Figure 12 – Collapse configuration of a masonry arch and comparison between numerical model results (Non- Linear Material Analysis and Non-Linear Material and Geometry Analysis) experimental and limit analysis one. (Molins and Roca, 1998).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 13 3. THE REINFORCED ARCH METHOD Masonry arch is one of the most diffused architectonical element in historical structures, since the Roman period. Most of the heritage buildings and constructions present nowadays, built before the advent of reinforced concrete at the end of the 19th century, are based on the presence of arched and vaulted structures. Arches are especially suitable to withstand vertical uniform loads, because of their ability to work only in compression, that is particularly suitable for masonry, which typically has a relevant compressive strength, but a poor tensile capacity. For this reason, arches can suffer from the presence of horizontal loads (e.g. seism) or non-uniform vertical ones (Figure 13), especially if the amount of compression between the arch element is low, because of the uprising of tensile stresses. These aspects typically cause hinges opening, that can lead to relevant permanent deformations or even to the collapse of the structure, when the maximum number of hinges is reached and the mechanism is activated. Figure 13 – Deformed masonry arch in the roman Milvio bridge in Rome. (Giglio, 2008). 3.1 Features of the R.A.M. Strengthening Technique The Reinforced Arch Method is a rather recent strengthening technique, developed by Prof. Lorenzo Jurina at Polytechnic University of Milan (IT), based on the application of steel post-tensioned cables at the extrados or the intrados of masonry arches and vaults (Jurina, 2012). This technique improves the response of the structure because of: - Providing tensile resistance on one side of the structure, so as to contrast the opening of some hinges, preventing the full formation of the complete mechanism;
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 14 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS - Applying an additional state of uniform compression that is able to re-centre the thrust line and therefore increases the geometrical safety factor of the arch or the vault; - Improving the resistance to shear failure between blocks (sliding), due to the increment of compression inside the structure. All of these aspects have been proving, through both experimental tests and actual interventions on real structures (Figure 14), the great suitability of this procedure for the consolidation of historical masonry structures. In fact, tests have shown that both resisting capacity and ductility are significantly improved. Figure 14 – Examples of strengthening interventions through the RAM: a masonry arch at Villa Borromeo, Senago (MI), IT, (Jurina, 2003) and a ribbed vault in St. Caterina church in Lucca, IT, (Jurina, 2014). The RAM can also be interpreted as both a passive and an active strengthening technique. The first definition is due to the passive capability of providing extra strength, connected with the high tensile resistance of the steel cables. This is aimed at suppling the typical extremely low value of this parameter in masonry. This aspect is the principle of most of the strengthening techniques used nowadays, even the most recent ones, such as FRP and FRCM application. Nevertheless, the RAM has the peculiar characteristic of being also an active technique. In fact, the post-tensioning of cables induces suddenly an additional compressive state to the masonry, that increases immediately the safety of the structure, by re-centring the line of thrust. The effect of post-tension is, in other words, an active confinement applied to the arch. If we consider a semi-circular arch, the action induced at the structure is a constant radial distributed load, that depends on the arch radius (Figure 15). The active attitude of the RAM entails also the possibility to re-calibrate the post-tension in case of loss of tension, or when the conditions of the structure require it. The permanent presence on the cables of anchors for tensioning allows that. This is a highly remarkable aspect, since in other strengthening procedures this possibility
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 15 is not granted. These characteristics make this technique particularly suitable for a possible incremental approach to the strengthening intervention, that requires the possibility to rearrange the amount of reinforcement by checking the response of the structure through direct measuring or even monitoring systems. The incremental approach is highly cutting-edge in strengthening of historical structures: this makes the RAM an updated procedure, despite it encompasses the usage of mainly traditional materials, such as steel. Moreover, the absence of bonding, besides its low invasiveness, permits also sliding between steel cables and the underlying structure during external loading (e.g. seism): this avoids unwanted debonding failure, which usually happens in other strengthening systems that require a certain amount of static shear resistance between the original structure and the reinforcement. In fact, the RAM transfers the confinement load mainly in the radial direction. The presence of shear is connected only with friction between cables and masonry, but usually sliding is allowed. This permits also a redistribution of forces between structure and reinforcement during external loading. Figure 15 – Interaction forces, between arch and steel cables, applied at the extrados (A) or at the intrados (B), induced by post-tensioning. (Jurina, 2012). As shown in Figure 15, this technique can be applied at the extrados or at the intrados of the arch. The first case is the most common one, since is the less aesthetically invasive when the visible part of the structure is the intrados, as in most of the cases (e.g. arch-bridges, church vaults, etc.). The second case, instead, is applied mostly when the extrados is not accessible, or it is too complex or invasive for the rest of the structure (e.g. infilled vaults that need a deep dismantling to reach the extrados). Obviously, when the cables are applied at the intrados, proper anchorage must be realized to transfer the radial load at the masonry structure, while in the case of cables at the extrados the load is transferred by simple contact. In the case of relatively small arches and vaults, the anchoring of cables is carried out through the application of steel devices in the lateral masonry walls. In the case of arch bridges, instead, the anchors can be applied even by means of micro-piles in the surrounding soil. Often cables are not in direct contact with masonry, but, especially when the entity of the post-tension is high, a layer of mortar or other devices (Figure 16) can be applied to create the contact, to distribute better the
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 16 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS confinement action. In some cases, as in the one of cables at the intrados, even some small pulley can be fixed to the masonry, to let the cable slide (Figure 14). Figure 16 – Monastero degli Ovietani di Nerviano (MI): devices for redistributing the confinement due to cable tension (Jurina, 2002). One of the main features of the RAM, even in comparison with other strengthening procedures, is its remarkable reversibility. This is probably the most distinguishing feature of this type of technique. In fact, the invasiveness of the interventions, to be carried out to apply the steel cables, is very low. The most damaging practise is limited to some drilling on the masonry units, in the case of application of cables at the intrados, or the casting of some mortar to prepare the surfaces of the arch/vault. Therefore, the permanent damage to the original structure is almost null. All the steel devices can be relatively easily dismantled, bringing again the arch to its original conditions. Also, the steel devices can be easily substituted in case of damage or replaced with better ones in case of the improvement of the technology available.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 23 Figure 23 – Collapse mechanisms for arches loaded at ¼ of the span, with the formation of four hinges: cables at the extrados (top) and cables at the intrados (bottom). (Giglio, 2008). In the case of sliding cables, therefore with a constant confinement force applied to the arch (neglecting significant curvature variations due to the deformations), results showed a great improvement of the load capacity of the arch (Table 2 and Table 3), in both the reinforcement arrangements (at the intrados and at the extrados). The increase of the capacity has revealed a quasi-linear dependence on the entity of the cable force, that can be noticed in Figure 24 and Figure 25. In these figures is highlighted the decrease of the capacity when the load position moves towards the cable centre, saving a slight increase in at half span. In most of the cases, it was noticed a slightly higher improvement of the capacity in the case of the reinforcement at the intrados, as visible in Figure 26.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 24 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 2 – Collapse load for sliding cables at the extrados. (Giglio, 2008). Table 3 – Collapse load for sliding cables at the intrados. (Giglio, 2008). Figure 24 – Sliding cables, reinforcement at the extrados: collapse load vs load position (left), collapse load vs cable force. (Giglio, 2008). 100 200 300 400 500 600 051 23 10 8 6 7 20 108 59 37 31 26 31 39 152 86 59 51 51 53 59 188 112 85 75 74 80 79 231 139 108 98 97 106 98 258 160 126 119 118 126 196 434 284 235 220 230 241 294 603 411 341 333 335 364 Load position [mm] Cables force [N] Collapse load [N] - sliding cables at the extrados 100 200 300 400 500 600 051 23 10 8 6 7 98 267 163 119 108 109 137 196 452 315 244 235 247 259 294 704 486 363 344 366 431 Collapse load [N] - sliding cables at the intrados Cables force [N] Load position [mm]
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 25 Figure 25 - Sliding cables, reinforcement at the intrados: collapse load VS load position (left), collapse load VS cable force. (Giglio, 2008). Figure 26 – Sliding cables, collapse load VS application point: comparison of results from reinforcement at the extrados and the intrados, for different cable force (from top left, clockwise: 98.1 N, 196.2, 294.3 N). (Giglio, 2008).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 26 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Experiments with the reinforcement cables anchored to the supporting table have also been performed. In the previous cases, in fact, the cables could slide freely along the arch circumference, during the external loading phase. This permitted to theoretically maintain a uniform confinement force all along the arch during the whole test (neglecting the variation due to curvature changes, as stated above). In this second type of tests, instead, the cable was constrained at both its ends, to provide further rigidity at the whole system. These tests were performed on models of the same geometry described above, with the only addition of anchors underneath the base, to block the cables sliding (Figure 27). Figure 27 – Details of overall tests setup (left) and anchors detail (right). (Giglio, 2008). The plan of the tests entailed the application of a punctual load at two positions: ¼ and ½ of the span, with the application of 4 levels of post-tension: 98 N (10 kgf), 196 N (20 kgf) or of 294 N (30 kgf). The reinforcement cables were applied only at the extrados. The collapse was not reached in any test, showing a dramatic increase of the capacity of the arch. The loading phase was carried out by increasing of 392 N (40 kgf) stepwise the punctual load, until 2354 N (240 kgf), and then unloading with the same load step size. In every step, the amount of displacement in some significant points of the arch was measured, so as to enable the possibility to trace some load/displacements curves (Figure 29 and Figure 30). The deformed shape of the arch was also monitored graphically, by means of special pens connected to the system, able to follow the arch deformation and trace lines on a vertical paper support (Figure 28). With these traces, it was also possible to measure the cable elongation (Table 4) and the hinges opening at the point of maximum load applied (Table 5), for the case loaded at ¼ of the span.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 27 Figure 28 – Details of the traces of the deformed shapes taken. (Giglio, 2008). For the sake of brevity, only results concerning vertical displacements of the cases of no post-tension and 294 N are reported. Both load positions have been considered. Figure 29 – Anchored cables tests, load at 1/4 of the span. Top: Configuration at maximum load applied, no posttension case. Bottom: load/vertical displacements curves at loaded point, no post-tension case (left) and 294 N post-tension case (right). (Giglio, 2008).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 28 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 30 - Anchored cables tests, load at 1/2 of the span. Top: Configuration at maximum load applied, no posttension case. Bottom: load/vertical displacements curves at loaded point, no post-tension case (left) and 294 N post-tension case (right). (Giglio, 2008). Table 4 – Cable elongation at maximum external loading for each prestress level (load at ¼ of the span). (Giglio, 2008). Table 5 – Hinge opening at maximum external loading for each prestress level (load at ¼ of the span). (Giglio, 2008).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 29 4. MODELS USED Numerical models have been used for the aim of this work. They have been used to simulate the experimental results presented above. The results obtained have been double-checked also in comparison with limit analysis, both with static and kinematic approaches, by means of computer programs. The tools used are: - Finite element macro-models; - Finite element micro-model, with interfaces; - Kinematic limit analysis with a commercial program (Ring); - Static limit analysis with an Excel spreadsheet. 4.1 Finite Element Models Finite element models have been used for simulating the experimental campaign described in the previous chapter. Macro-models entail the application of one only material for the arch body, based on masonry materials characteristics, calibrated so as to capture the results from the wooden blocks experiments. Even micro-models, with perfectly elastic blocks and opening interfaces, have been produced. All the Finite Element models have been developed through the software Diana FEA. The elements used for discretizing the arch body are 4 or 8-noded plane stress elements (Q8MEM or CQ16M), with two DOF per node (Figure 31). Plane stress elements permitted to assign the real thickness of 100 mm to the arch. These plane elements present, in the regular setup, 2x2 Gauss integration points. The number of elements used to discretize the arch depth varies from 12 to 16, for a size of about 6 to 8 mm, depending on the case analysed. Figure 31 – Q8MEM plane element (left) and CQ16M plane element (right). (DIANA FEA User’s Manual Release 10.1, 2016). All the models have in common the application of gravity load as a volume force in the -Y direction. The external incremental load was applied in terms of an imposed vertical displacement in -Y direction, at the centre of the depth, for different span fractions. Supports were applied as horizontal and vertical translational constraints at the impost (Figure 32). The analysis carried out is a non-linear static one, even in some cases with non-linear geometry (Total or Updated Lagrange). First the gravity was applied in 4 steps, then also the vertical incremental displacements were applied in a variable number of steps,
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 30 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS depending on the case analysed. Newton-Raphson method was used for solving the equilibrium at any load step, with a maximum number of iteration variable between 50 and 200. Control of convergence was applied in terms of energy, force and displacement, with tolerances respectively of 10-4 ÷ 10-2, 10- 2, 10-2. Figure 32 – Typical finite element discretization, load and supports used. 4.1.1 Macro-Models - Unreinforced The macro-models used present on only material on the overall body of the arch. It has been modelled with the Total Strain crack approach (Vecchio and Collins, 1993), based on the smeared crack model, suitable especially for masonry and concrete. This mainly because, despite trying to simulate the behaviour of a rigid-block and dry-joints experimental model, this paper looks forward at the application of the RAM to the real masonry arches case. The total strain crack model is based on the assumption that the stress is evaluated in the directions which are given by cracks. Rotating cracks approach has been chosen: this means that the stress–strain relations are evaluated in the principal directions of the strain vector. The constitutive stress-strain law is different in compression and in tension, and unloading is secant. Concerning the tensile behaviour, linear-crack energy approach has been chosen, while for the compressive one, the parabolic is used (Figure 33). Figure 33 – Stress-strain paths chosen. (DIANA FEA User’s Manual Release 10.1, 2016).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 31 The value of the parameters used, listed in Table 6, are presented in the following chapters. These values were chosen in order to accomplish the peculiar case of dry joints (no tensile strength) and very resistant blocks (infinite compressive strength). Table 6 – Parameters of masonry due to total strain crack material in the macro-models. Since the external load, consisting on imposed vertical displacements, was applied at the centre of the arch depth, to avoid unwanted local cracking, a limited area around the loading point was modelled with a higher tensile strength. 4.1.2 Macro-Models – Reinforced Two main types of reinforcement have been applied: - confinement as equivalent radial load; - actual modelling of steel cables as reinforcement. The first case simply consists on the addition of an equivalent radial load to the unreinforced macromodel described above (Figure 34 and Figure 35). This load depends on the entity of cables-post tension and the curvature of the cables. Figure 34 –Model with equivalent radial load reinforcement at the extrados. Parameter Notation Unit SI Young Modulus E N/m2 Poisson Ratio υ- Density ρ kg/m3 Tensile Strength ftN/m2 Tensile Fracture Energy GfN/m Residual Tensile Strength ff_res N/m2 Compressive Strength fcN/m2 Compressive Fracture Energy GcN/m Residual Compressive Strength fc_res N/m2 Compressive Linear Tensile
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 32 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 35 - Model with equivalent radial load reinforcement at the intrados. As stated above, cables were also modelled. The cables were introduced in the phased analysis after the application of the gravity load, and the gradually post-tensioned by means or simple punctual force (Figure 36). Figure 36 – Macro-model with physical cables at the extrados (notice the application of post-tension). These macro-models entail the use of one dimensional elements to discretise the cables. The element used to ensure compatibility with the linear plane ones, are 2-noded enhanced 2D trusses (L4TRU). These are essentially 2D truss elements, with the addition, compared to the regular ones, of the degrees of freedom perpendicular to the truss axis (Figure 37). The integration is direct (1 point). The use of truss elements, instead of cable ones, does not affect the results, since compression was never obtained in any result. Figure 37 – L4TRU enhanced 2D truss elements (DIANA FEA User’s Manual Release 10.1, 2016).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 39 Table 10 – Equivalent models determined by limit analysis. Results and comparison with experimental ones. The validity of the hypothesis on the impossibility of the cross-section of the arch to react in its entire surface is confirmed. This is also supported by the distribution of hinges obtained with the equivalent models, that is much more coherent with the one of the experimental arch. As is can be noticed in Figure 45, limit analysis shows that there is no hinge forming at the loading point, as it was observed in in Figure 44. The collapse configuration is completely different from the one obtained with limit analysis of the arch with the original geometry, but it highly resembles the experimental one (Figure 45). Therefore, in the unreinforced numerical models have been shaped with the equivalent depths computed with the back-analysis explained here. This means that the actual geometry has not been modelled, since it has been showing to lead to much more conservative results than the limit analysis ones. Typically, the experimental models show a higher capacity than the one computed with limit analysis, due to the presence of a minimum tensile strength. In this case happens the opposite, due to the absence of tensile strength in the physical model, and the presence of some unknown imperfections that reduce arch cross-section and the global capacity. Depth [mm] Density [kg/m3] Clear span [mm] Static (Excel) Kinematic (RING) Experiment Error Static Error Kinematic 1/12 82 732 1218 46 52 51 -10% 2% 1/6 82 732 1218 21 21 23 -6% -5% 1/4 79 759 1221 10 10 10 -3% -1% 1/3 82 732 1218 8 9 8 2% 9% 5/12 82 732 1218 6 6 6 8% 9% 1/2 82 732 1218 6 6 7 -6% -6% Load position (span fraction) Equivalent model data Collapse load [N] Comparison 0 10 20 30 40 50 60 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Collapse Load [N] Load position - span ratio Experimental Static (Excel) Kinematic (RING)
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 40 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 45 – Comparison of collapse configurations for load at 1/6 of the span. Top: experimental. Bottom: equivalent depth models: static limit analysis (right) and kinematic one (left). Furthermore, in favour of the assumption concerning the reduced effective depth, it has been tried to vary the original density of 600 kg/m3, that could have been affected by small errors, keeping the original geometry, including the depth of 100 mm. This means that the overall weight is not kept equal to the original one anymore. Results from limit analysis has shown, for the sample case of punctual load at 1/6 of the span, that the equivalent density to obtain the experimental collapse load, should be of 3 kg/m3 for the static approach, and 3.16 kg/m3 in case of the kinematic one. This means that the error on the density should be about 46 ÷ 49 %, that is not plausible. Moreover, the collapse hinged configuration would be considerably different from the real experimental one (Figure 46). Figure 46 – Collapse configurations of arch loaded at 1/6 of the span, obtained with both limit analysis approaches (kinematic on the left, static on the right), in the case of equivalent density for obtaining the same capacity of the experimental arch. In the further numerical analysis, for the sake of brevity, the cases of load applied at 1/6 and 1/2 of the span has been analysed. This is a convenient choice, since these two cases present the same equivalent depth of 82 mm (Table 11) for obtaining the experimental behaviour.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 41 Experimental arch Collapse configuration Original geometry Kinematic analysis Reduced depth Kinematic analysis Table 11 – Collapse hinged configurations: experimental arch and kinematic analysis with original and reduced depths. 1/12 span Load: 51 N 1/6 span Load: 23 N 1/4 span Load: 10 N 1/3 span Load: 8 N 5/12 span Load: 7 N 1/2 span Load: 6 N Depth: 100 mm Load: 98 N Depth: 82 mm Load: 52 N Depth: 100 mm Load: 43 N Depth: 100 mm Load: 26 N Depth: 100 mm Load: 19 N Depth: 100 mm Load: 16 N Depth: 100 mm Load: 18 N Depth: 82 mm Load: 21 N Depth: 79 mm Load: 10 N Depth: 82 mm Load: 9 N Depth: 82 mm Load: 6 N Depth: 82 mm Load: 6 N
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 42 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 5.2 Reinforced Arch Limit analysis has been applied also to the cases of reinforced arches. The kinematic analysis was not possible in this case, since the program Ring does not include the option of such reinforcement, so only the Excel spreadsheet for graphic statics has been used. In fact, this tool provides the possibility to find the line of thrusts including a uniform radial confinement load, applied at the intrados or at the extrados of the arch. The confinement radial load was calculated with the simple relation involving the curvature, explained in Figure 47. Figure 47 – Equivalent confinement radial load due to post-tensioned cables, applied at the extrados (left) or at the intrados (right). R, r: the radius of the arches, T: cable post-tension. The cases of punctual load at 1/6 and half span have been analysed, for all the post-tension levels. Same problem arose in unreinforced case has been rising also in the reinforced one: the resistance calculated with limit analysis is sensitively higher than the one obtained experimentally, for low posttension levels (Figure 48). Nevertheless, it increases with the post-tension level (Table 12 and Table 13), approaching the experimental values for the maximum ones, especially in the case of cable at the intrados. Therefore, the same equivalent depth used for analysing the unreinforced case cannot be applied for the reinforced arch cases. Not even a unique depth can be established for all the post-tension levels. This has been interpreted as a tendency of the cables presence to restore the original depth, the more post-tension confinement is applied (Figure 49).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 43 Table 12 – Comparison of collapse load for different post tension levels between experimental results and limit analysis with original geometry arch. Cable at the extrados. Table 13 - Comparison of collapse load for different post tension levels between experimental results and limit analysis with original geometry arch. Cable at the intrados. Figure 48 – Comparison of experimental and limit analysis results for the reinforced arch. Experimental Limit Analysis Error Experimental Limit Analysis Error 20 59 72 23% 31 48 53% 39 86 102 18% 53 78 46% 59 112 130 16% 80 106 32% 78 139 158 13% 106 133 26% 98 160 183 15% 126 162 29% 196 284 318 12% 241 310 28% 294 411 451 10% 364 452 24% Load at 1/6 span Load at 1/2 span Post tension [N] Collapse Load [N] - Cables at the Extrados Experimental Limit Analysis Error Experimental Limit Analysis Error 98 163 186 14% 137 160 16% 196 315 324 3% 259 305 18% 294 486 456 -6% 431 440 2% Load at 1/6 span Load at 1/2 span Post tension [N] Collapse Load - Cables at the Intrados 0 100 200 300 400 500 0 100 200 300 400 Collapse Load [N] Post-Tension [N] Cables at the Extrados - 1/6 span Experimental Limit Analysis 0 100 200 300 400 500 0 100 200 300 400 Collapse Load [N] Post-Tension [N] Cables at the Extrados - 1/2 span Experimental Limit Analysis 0 100 200 300 400 500 600 0 100 200 300 400 Collapse Load [N] Post-Tension [N] Cables at the Intrados - 1/6 span Experimental Limit Analysis 0 100 200 300 400 500 0 100 200 300 400 Collapse Load [N] Post-Tension [N] Cables at the Intrados - 1/2 span Experimental Limit Analysis
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 44 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 49 – Percentage errors on collapse load computed with limit analysis to the experimental results. The values of the equivalent depths have been computed through the Excel spreadsheet for graphic statics, with the same procedure described for the unreinforced arch, for each post-tension level. For this back-analysis, the case of load at half span has been chosen. This because the static limit analysis tool is more reliable for load positions closer to the mid-point, than for the ones closer to the impost, because of the arch discretization. It must be underlined that the value of the equivalent depth computed is affected from a rounding error (1 mm) that is not always totally negligible. This is the reason why, presenting the numerical results, both experimental and limit analysis results will be included. Nevertheless, the results obtained confirm the increase of the effective depth with the increment of the post-tension rate (Table 14). Table 14 – Equivalent depths, computed by static limit analysis, for obtaining the same collapse load of experiments in the reinforced arch cases (load at half span). For the sake of brevity, in the next sections concerning numerical analysis, only the cases with maximum post-tension force will be presented, therefore the models will have 90 mm of depth for the case with the cables at the extrados, and 100 mm (full depth) for the ones with the cables at the intrados. 0 10 20 30 40 50 60 0 100 200 300 400 Limit Analysis Error [%] Post-Tension [N] Cables at the Extrados Load at 1/6 span Load at 1/2 span -10 -5 0 5 10 15 20 0 100 200 300 400 Limit Analysis Error [%] Post-Tension [N] Cables at the Intrados Load at 1/6 span Load at 1/2 span Cables at the Extrados Cables at the Intrados 20 85 39 85 59 87 78 88 98 88 91 196 89 92 294 90 100 Equivalent Depth [mm] Post tension [N] 80 85 90 95 100 105 0 100 200 300 400 Equivalent depth [mm] Post-Tension [N] Cables at the Extrados Cables at the Intrados
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 45 The results of this back-analysis for determining the equivalent depth, performed by attempts with Ring, are affected from an approximation error. In fact, the sensitivity of the arch capacity to the geometrical depth is significant. Therefore, the approximation of ±0.5 mm could propagate in a not negligible error on the limit load. To prove this, a model representing the experimental unreinforced arch, with a punctual load at half span, has been produced in RING. Variating the depth from 80 mm to 90 mm, and applying consequently the equivalent density, for keeping the overall weight constant, the capacity of the arch has been computed (Table 15). Table 15 – Sensitivity analysis of ultimate load to the depth of the arch in RING. Figure 50 – RING sensitivity to depth (left) and depth increment (right). In Figure 50 is shown a relative increment of load capacity 9 times greater than the relative increment of depth (to an original value of 80 mm). Depth [mm] Density [kg/m3]Span [mm] Rise [mm] Ultimate Load [N] 80 750 1220 610.0 5.3 81 741 1219 609.5 5.9 82 732 1218 609.0 6.5 83 723 1217 608.5 7.0 84 714 1216 608.0 7.6 85 706 1215 607.5 8.2 86 698 1214 607.0 8.8 87 690 1213 606.5 9.3 88 682 1212 606.0 9.9 89 674 1211 605.5 10.5 90 667 1210 605.0 11.2
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 46 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 47 6. NUMERICAL ANALYSIS: MICRO-MODELS 6.1 Hypothesis and Arch Body Material Parameters The micro-models produced are aimed at simulating as faithfully as possible the actual condition of the experimental arches, in terms of stiff wooden units, simply in contact one to each other, with no tensile strength at their interface. Therefore, linear elastic material has been chosen for modelling the blocks, with material parameters as equal as possible to the ones of the beechwood used for shaping the voussoirs (Table 16). Wood is a composite orthotropic material, nevertheless the behaviour inside the block is not relevant for our analysis, because no compressive failure has happened during the experiments. The collapse behaviour observed, that these models aim at capturing, is the rotational hinges opening. Then, the only fundamental material parameter in this case is the density, but also the elastic ones have been applied carefully. In the reinforced cases, the cable weight has been neglected in all the analysis carried out. Table 16 – Wooden blocks linear elastic parameters. *Equivalent density has been used to keep constant the overall weight of the experimental arch. Most of the capacity and the deformability of the model are due to the presence of interface elements. The values of the parameters of the Coulomb friction material assigned at these elements are aimed at preventing shear sliding, and providing no tensile resistance to hinges opening. Moreover, the normal elastic stiffness of the interface has been chosen relatively high, so as to minimise blocks superpositions, simulating as much as possible a rigid contact, while the shear one has been chosen reasonably, compared with the ones suggested for masonry (Lourenço, 2002), in absence of more specific data (Table 17). Notation Value Unit E15000 MPa υ0.3 - 82 mm depth models ρ732 kg/m3 90 mm depth models ρ667 kg/m3 100 mm depth models ρ600 kg/m3 Density* Wooden Blocks - Elastic parameters Parameter Young Modulus Poisson's ratio
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 48 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 17 – Coulomb friction interface parameters between blocks. 6.2 Unreinforced Arch - Results As stated above, for the unreinforced arch a depth of 82 mm has been computed with limit analysis to comply the experimental results. Equivalent density is therefore 732 kg/m3. For the two load positions concerned, both linear and non-linear geometry analysis (Total Lagrange approach) have been performed, for sake of comparison. Analysis setup is reported in Table 18. Table 18 – Unreinforced arch: analysis setup details. Figure 51 – Micro-model, unreinforced arch (82 mm depth), load at 1/6 of the span, linear geometry, deflection of loaded point vs load applied. Parameter Notation Value Unit Normal stiffness Kn10000 N/mm3 Shear stiffness Ks2000 N/mm3 Cohesion c 0 MPa Friction angle φ1.55 rad Dilatancy angle ψ0 rad Gap opening Tensile strength ft0 MPa Blocks Interface - Parameters Linear Friction Element Type 4-noded Element Size 6 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01 0 5 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Load [N] Deflection [mm] 1/6 span - Linear Geometry Numerical Experimental Static Limit Analysis
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 55 Figure 60 - Top: micro-model with equivalent radial reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x100 amplification). Bottom: experimental collapse configuration. Also the case of reinforcement at the intrados has been taken into consideration. In this case the depth computed by limit analysis, for the case concerned of 294 N post-tension (30 kg), is the full one of 100 mm. Therefore, the models represent the exact geometry and density (600 kg/m3) of the real experimental arch. In these cases, non-linear geometry analysis has been performed, to get a better convergence. Analysis setup is reported in Table 23. Table 23 - Equivalent radial load at the intrados reinforced models, analysis setup. Element Type 4-noded Element Size 8 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01 Non-Linear Geometry Updated Lagrange
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 56 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 61 – Arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied. Figure 62 - Arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied. 0 100 200 300 400 500 600 0 0.05 0.1 0.15 0.2 0.25 0.3 Load [N] Deflection [mm] 1/6 Span Load - Reinforcement at the Intrados Numerical Experimental Limit Analysis 0 100 200 300 400 500 0 0.05 0.1 0.15 0.2 0.25 Load [N] Deflection [mm] 1/2 Span Load - Reinforcement at the Intrados Numerical Experimental Limit Analysis
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 57 Figure 63 - Top: micro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x200 amplification). Bottom: experimental collapse configuration.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 58 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 64 - Top: micro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x50 amplification). Bottom: experimental collapse configuration. Table 24 - Equivalent radial load reinforcement at the intrados: micro-models results and comparison with experimental ones. Even in this case, the results obtained with the micro-model analysis are in good accordance with the experimental ones (Table 24), with slightly lower values, but they faithfully reproduce the limit analysis ones (Figure 63, Figure 64). It must be reminded that once again results concerning arches loaded at half span have a slightly better coherence with the experimental ones (Figure 62) than the one loaded at 1/6 of the span (Figure 61), for the same reasons of the previous cases, explained above. Numerical Experim. Limit Analysis to Experim. to L. Analysis 1/6 464 486 452 -4.4% 2.8% 1/2 437 431 431 1.6% 1.6% Load position (span fraction) Collapse load [N] Error
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 59 6.4 Modelling of Steel Cables Reinforcement 6.4.1 Reinforcement Parameters One of the main aims of this work is to investigate the behaviour of the steel cable reinforcement (Figure 65), especially in terms of interaction with the arch, during the external loading. Modelling this interaction is not straightforward, since the parameters governing the interaction are not known in general. It is not known also the actual influence of these parameters on the capacity of the arch and on the overall behaviour of the structure. To simulate this interaction, a model based on Coulomb friction has been chosen. The same interface elements used for the block contact interface have been applied to the cable/arch interfaces, but with completely different parameters. For the sake of brevity, not all the unknown parameters of this interaction mechanism can be fully investigated, so some of them have been estimated with reasonable values. The main one, that has been deeply investigated in the following paragraph, is the friction angle φ. In Table 25 are reported all the parameters involved in these analysis, with their qualitative uncertainty rate: known, estimated or variable. These parameters have been studied to simulate as faithfully as possible the experimental cases concerned. Nevertheless, in possible further developments involving actual masonry structures, they would need a more accurate calibration, considering the materials in contact. The density of the cables has been set as null, so as to neglect their weight, as in the previous analysis. The value of the Young modulus of the cables has been chosen as a reasonable one considering this kind of device. In fact, in the case of cables, usually it is sensitively lower than the common one of pure steel, due to non-elastic deformation phenomena. The interface friction surface width is another source of uncertainty, and it has been estimated as a reasonable vale of 0.1 mm. Nevertheless, its influence can be, at least partially, transferred to the friction angle variability, that has been investigated, since the friction force involves both these parameters in the same formula derived from Coulomb’s law (shear stress is integrated in the width). Figure 65 – Details of the cable/wood interface in the experimental arch. (Giglio, 2008).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 60 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 25 – Cables and interface parameters. *Gap opening depends on the position of reinforcement: in the case of reinforcement at the extrados it is activated, in the case of at the intrados it is not. 6.4.2 Influence of Cable/Arch Friction Angle Through micro-models, the influence of friction angle on the behaviour of the overall reinforced structure has been studied. For the sake of brevity, the only case of the arch loaded at half span, reinforced at the extrados with a post-tension of 294 N (30 kg), has been investigated, assuming that the results can be extended to all the other cases. The features of the models are the one explained in 4.1.2 and 4.1.4. After the application of gravity load at the unreinforced arch, the cable has been added and consequently the post-tension applied in 10 steps. Then, increments of the external load have been applied. The main advantage of these models is the possibility to investigate the state of stress of the cable and its interaction forces with the arch. As an example, it is reported in Figure 66 the distribution of the axial force all along the steel cable, at the end of the application of the post-tension force, for the case of friction angle φ = 5°. As it can be easily noticed, at the end of the application of the post-tension, the presence of shear stress at the interface (Figure 67) creates a state of non-uniform distribution of the tension inside the cable. The cables should be free to slide, therefore the normal stress (the actual confinement effect) remains constant along the semi-circumference (Figure 68). Notation Uncertainty Class Value Unit E Estimated 196000 MPa υEstimated 0.3 - ρKnown 0 kg/m3 Notation Uncertainty Class Value Unit d Known 2 mm A Known 6.18 mm2 t Estimated 0.1 mm Parameter Notation Uncertainty Class Value Unit Normal stiffness KnEstimated 10000 N/mm3 Shear stiffness KsEstimated 100 N/mm3 Cohesion c Estimated 0 MPa Friction angle φVariable 0.6° - 30° Deg Dilatancy angle ψEstimated 0° Deg Gap Opening* Tensile strength ftKnown 0 MPa Young Modulus Poisson's ratio Density Steel Cable - Elastic Parameters Cable/Arch Interface - Parameters Linear Friction Parameter Steel Cable - Geometric Parameters Parameter Cable Diameter Cables Section Area Friction Surface Width
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 61 Figure 66 - Cable axial force at the end of the application of the post-tension (left) and at collapse (right). (Micromodel, physical cables, φ=5°, deformation x20). Figure 67 – Shear stress in the interface at the end of the application of the post-tension (left) and at collapse (right). (Micro-model, physical cables, φ=5°, deformation x20). Figure 68 – Normal stress in the interface at the end of the application of the post-tension (left) and at collapse (right). (Micro-model, physical cables, φ=5°, deformation x20). The aim of this sensitivity analysis is to determine a sufficiently low value of the friction angle, that permits the actual sliding of the cables and guarantees a low shear stress at the interface. Moreover, the effect itself on the arch capacity has been investigated. Four values of friction angle have been considered: 0.6°, 5°, 20°, 30°. The interaction cable/arch has been studied in three representative points of the arch, shown in Figure 69: - Close to the impost (A); - At about 45° (B); - Close to the maximum height point (C). The points A and C have not been chosen at the very base and tip to avoid disturbance due to local phenomena, such as stress concentrations or changes of sign of the parameters (Figure 69).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 62 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 69 – Three point chosen for investigating the interaction cable/arch. Figure 70 – Friction angle φ = 0.6°. Shear stress trend.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 63 Figure 71 - Friction angle φ = 5°. Shear stress trend. It can be noticed that, increasing the friction angle from 0.6° to 5°, during all the post-tensioning phase the cable slides, since the value of shear stress is always equal to the limit one, in all the three point considered (Figure 70 and Figure 71). During external loading, instead, at the points A and B, the phase of adherence becomes longer, proportionally to friction angle increase. In both cases, in A and B, the shear stress inverts its sign, due to the inversion of the relative displacement cable/arch from the posttensioning phase to the external loading one. This can be notice also in Figure 66, from the direction of the displacement of the cable ends, relatively to the arch extrados, in the two phases: during posttensioning they move downwards, then during external loading they move upwards, causing also the inversion of the friction force direction. Nevertheless, at point C the cable always slides. In addition, during the first phase, it can be noticed that the limit value of the shear stress always increases linearly. This is due to the constant increment of the post-tension applied, whose consequence is the constant increment of the radial normal stress and therefore of the shear resistance, due to Coulomb’s friction law.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 64 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 72 - Friction angle φ = 20°. Shear stress trend. By further incrementing the friction angle to 20° and 30°, the tendency described is confirmed, and becomes more evident (Figure 72 and Figure 73). In point A, the adherence phase increases. Even, in B and in C full adherence is reached during all the second stage. Also during the post-tensioning phase adherence is enhanced. In both the friction angle cases adherence is obtained even for point C, while in A and B sliding is still present. This is due to greater relative displacement cable/arch activated during the first phase in the points closer to the impost (A and B), compared to the one near the symmetry axis (C). To sum up, as expected, increasing the friction angle adherence is enhanced.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 71 Table 27 - Free sliding cable at the extrados, reinforced micro-model, analysis setup. Also in this case, are reported the axial force distribution, the normal confinement stress and the shear stress at the interface cable/arch, soon after the post-tension application phase (Figure 85). Even in this case, the normal stress at interface, if integrated along the width of the interface (0.1 mm), gives the value of the equivalent confinement, here with a slight variability along the circumference. Equivalent Radial Load→ p = T / R = 294 N / 600 mm = 0.49 N/mm Confinement Effect from Cable Modelling→ p’ = σR x t = 4.83÷4.91 MPa x 0.1 mm = 0.483÷0.491 N/mm T: cable traction, R: external radius, σR: radial normal stress, t: interface width. Figure 85 - Micro-model with physical reinforcement at the intrados (100 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. Even in this case, the capacity curves are reported in comparison with the ones computed for the cases concerning equivalent radial load reinforcement, treated in 6.3. It must be underlined that, for the case of cables modelled at the intrados, for load position at ½ span, non-linear geometry has been applied, to get a better convergence. Element Type 8-noded Element Size 8 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 72 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 86 - Arch with physical cable at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. Figure 87 - Arch with physical cable at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 1.2 Load [N] Deflection [mm] 1/6 Span Load - Reinforcement at the Intrados Radial Load Cables Limit Analysis Experimental 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Load [N] Deflection [mm] 1/2 Span Load - Reinforcement at the Intrados Radial Load Cables Limit Analysis Experimental
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 73 Figure 88 - Left: micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, norm of displacements at collapse deformed configuration (x50 amplification). Right: experimental collapse configuration. Figure 89 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x50 amplification). Left: axial force in the cable, right: interface normal stress. Figure 90 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x50 amplification). Left: shear stress in the interface, right: shear displacements. Even in the case of cables at the intrados, the results show a good accordance, in terms of capacity (Figure 86 and Figure 87), with the ones derived from the equivalent load reinforcement (Figure 94), in both the load positions (Figure 88 and Figure 91). Again, the tension of the cable is keeping rather uniform around the post-tension value (293÷297 N). The shear stress value at the interface are also relatively low (Figure 90 and Figure 93) (0.05÷0.09 MPa) and the normal stress keeps constant (radial confinement effect) (Figure 89 and Figure 92). The amount of shear relative displacements remains between 0.13 mm and 0.80 mm.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 74 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 91 - Left: micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, norm of displacements at collapse deformed configuration (x300 amplification). Right: experimental collapse configuration. Figure 92 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, collapse deformed configuration (x300 amplification). Left: axial force in the cable, right: interface normal stress. Figure 93 - Micro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, collapse deformed configuration (x300 amplification). Left: shear stress in the interface, right: shear displacements. Figure 94 – Physical cable reinforcement at the intrados: micro-models results and comparison with previous results and experimental ones. 6.4.4 Anchored Cables Models Results Micro-models have been used to investigate also the behaviour of the reinforced arch in the case of cables anchored at the ends. These cases showed a considerable increase of the capacity of the system, in the experimental tests, as explained in 3.2. For the sake of brevity only the case loaded at half span has been studied, in the cases of reinforcement applied either at the extrados or at the intrados. Cable Model. Radial Confinement Experim. Limit Analysis to Radial Confinement to Experim. to L. Analysis 1/6 468 464 486 452 0.7% -3.7% 3.5% 1/2 436 437 431 431 -0.3% 1.2% 1.2% Load position (span fraction) Collapse load [N] Error
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 75 Concerning the cable force, only the case with no post-tension and 294 N have been modelled. These models do not claim to capture the deformability of the system, but their aim is just to confirm the overall behaviour of the reinforcement, in terms of capacity and internal force distribution. The material parameters used for the blocks, the cables and the interfaces are the same of the ones used for the sliding cables cases, explained in the previous paragraph, in Table 25, with friction angle again φ = 0.6°. Both linear and non-linear geometry analysis have been performed. Analysis setup is reported in Table 28. Table 28 – Anchored cables reinforced arch micro-models, analysis setup. Figure 95 – Micro-models with anchored cables at the extrados results: force vs vertical displacement at the loading point. Linear and non-linear geometry analysis of no post tension and 294 N post-tension case. Element Type 8-noded Element Size 8 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01 Non-Linear Geometry Total Lagrange 0 2000 4000 6000 8000 10000 12000 14000 0246810 12 Load [N] Deflection [mm] Cables at the Extrados LG Nt = 0 N NLG Nt = 0 N NLG Nt = 294 N LG Nt = 294 N Experimental Max. Load
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 76 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 96 - Micro-models with anchored cables at the intrados results: force vs vertical displacement at the loading point. Linear and non-linear geometry analysis of no post tension and 294 N post-tension case. It is evident also from the numerical results that the anchored cables are able to provide a much higher capacity, in terms of limit load and deformability. The capacity curve shows a linear behaviour in the case of linear geometry, and a quasi-linear one in the non-linear geometry analysis. In the first case, the analysis has been stopped without reaching the collapse, while in the second one a brittle collapse was reached, due to bucking phenomena. Nevertheless, as stated before, these results do not have a quantitative aim, since the numerical models do not take into account any material non-linearity. Even failures due to sliding between blocks is neglected. Therefore, these results only confirm the relevant amplification of the capacity of the arch, even largely overcoming the maximum load applied in the experimental tests of 2354 N, but they are not able to give a reliable indication of the new value of the collapse load. Moreover, it can be noticed that applying post-tension at the cables does not affect much the overall behaviour, since the improvement is rather negligible compared to the effect of the cable anchoring itself. In fact, in Figure 95 and Figure 96, it can be noticed that the post-tensioned case curve lays just slightly above the not-post-tensioned case one. Concerning the cable tension, in Figure 97 and Figure 98 is evident the difference between the cable tension trend in the anchored case and the sliding one, in a point chosen at ¼ of the semi-circumference. In the first one, the force is constantly increasing, while in the second one it remains substantially constant at the post-tension value. In Figure 99 it can be noticed that, at an intermediate load step (300), the values of the tension of the cable is relevantly high, but rather uniform, likewise the one of radial interface stress. Nevertheless, shear stress is still relatively low, due to the extremely low friction angle chosen. This means that the great increase of overall strength is not due to friction increase, but to the increase of the confinement effect (one order of magnitude). 0 2000 4000 6000 8000 10000 12000 14000 0246810 12 Load [N] Deflection [mm] Cables at the Intrados LG Nt = 0 N NLG Nt = 0 N NLG Nt = 294 N LG Nt = 294 N Experimental Max. Load
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 77 Figure 97 – Anchored cables at the extrados micro-model: cable tension trend at ¼ of the semi-circumference. Figure 98 - Anchored cables at the intrados micro-model: cable tension trend at ¼ of the semi-circumference. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 050 100 150 200 250 300 Cable Tension [N] Load Step Cables at the Extrados Nt = 0 N Nt = 294 N Nt = 294 N Sliding Cables 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 050 100 150 200 250 300 Cable Tension [N] Load Step Cables at the Intrados Nt = 0 N Nt = 294 N Nt = 294 N Sliding Cables
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 78 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 99 – Anchored cables at the extrados micro-model, no post-tension, step 300, deformation scale x10. From top downwards: compressive stress between blocks, normal stress at cable/arch interface, cable tension, shear stress at cable/arch interface.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 79 7. NUMERICAL ANALYSIS: MACRO-MODELS 7.1 Hypothesis and Arch Body Material Parameters The aim of this document is to investigate the behaviour of the RAM, also by means of macro-models. In fact, usually these kinds of models are more convenient for masonry structures, since they permit to produce a uniform material pattern, with averaged parameters of the elements composing masonry, including, in one only material, units and joints behaviour. This typically leads to a lower modelling effort and a lower computational time. The most diffused yield criteria used to simulate masonry non-linear behaviour are the ones based on the total-strain crack model, which includes smeared cracks patterns. In this case the subject of modelling is the experimental semi-circular arch, that entails no tension resistant joints and wooden-blocks. Therefore, the intrinsic behaviour of the elements and of the overall arch is rather different from the one of masonry structures. Nonetheless, it has been assumed that, once assessed coherent values of the parameters involved, macro-models can be a possible valuable alternative to micro-models. The parameters listed in Table 6, as in the previous case concerning micromodels’ interface between cable and arch, have various levels of uncertainty. Some of them are known and some of them are unknown. What is important, when trying to define the unknown ones, is to understand their actual influence on the results desired. For instance, a reliable deformability analysis is not the scope of this study, but what is important is to obtain faithful results in terms of load capacity. To sum up, the macro-models’ parameters have been calibrated to obtain equivalent models in terms of load capacity, while in micro-models the goal was to get the most faithful model compared with the experimental one. In Table 29 are reported the parameters involved in the smeared-crack model applied at the arch body, with their values and their level of uncertainty. Table 29 – Parameters of arch body in the macro-models. *These parameters are variable, and they have been object of a deeper analysis to determine better the valuable intervals. The most influencing parameters have been studied, in particular their effect on the resisting mechanism and the final value of load capacity. They are the Young modulus, the tensile strength and the fracture energy. Theoretically, the tensile strength should be set as null, but in this smeared crack model Notation Uncertainty Class Value Unit E Unknown 2000 MPa υUnknown 0.25 Unreinforced ρKnown 732 kg/m3 Extrados Reinf. ρKnown 667 kg/m3 Intrados Reinf. ρKnown 600 kg/m3 ftUnknown 0.001÷0.1 MPa GfUnknown 0.0001÷0.01 N/mm ft_res Unknown 0 MPa fcUnknown 100 MPa GcUnknown 160 N/mm fc_res Unknown 0 MPa Residual Compressive Strength Density Tensile Strength* Tensile Fracture Energy* Residual Tensile Strength Compressive Strength Compressive Fracture Energy Parameter Young Modulus* Poisson Ratio
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 80 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS problems due to incorrect failure arise, causing divergence. Therefore, a low but not null value of this parameter is needed. Compressive strength has been chosen extremely high to avoid any possibility of damage in compression. The value of fracture energy has been calculated with reasonable formulations concerning masonry, found in literature. Gf [N/mm] = 0.1 ft [N/mm2] Gc [N/mm] = 1.6 fc [N/mm2] Once again, the density has been calculated depending on the depth of the arch concerned, to keep constant the overall weight. 7.2 Unreinforced Arch – Results To understand the influence of the Young modulus and the tensile strength on the arch capacity, 4 macro-models have been produced, with different values of these parameters. As a sample, the unreinforced arch loaded at half span has been chosen, therefore the one with 82 mm of depth and 732 kg/m3 of density. Concerning the Young modulus, two values have been considered: one reasonable for masonry (2000 MPa) and one as close as possible to the one of the wood used to shape the wedges of the experimental model (15000 MPa). As regards tensile strength, instead, three values have been considered: 0.0015 MPa, 0.01 MPa and 0.1 MPa (Table 30). Values of the tensile strength, lower than the ones chosen, induced unwanted severe cracking in the tangent direction, that led to early divergence, as will be shown more exhaustively in the next chapters. Analysis setup is reported in Table 31. Table 30 – Unreinforced macro-models loaded at half span: parameters setups for determining E and ft influence on the vertical load capacity. Table 31 - Unreinforced macro-models’ analysis setups for determining E and ft influence on the vertical load capacity. Model ft [MPa] Gf [MPa] E [MPa] Α0.1 0.01 2000 Β0.01 0.001 2000 Γ0.0015 0.00015 2000 Δ0.01 0.00015 15000 Element Type 4-noded Element Size 7 mm Steps increment of Displacement 0.002 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 87 Table 36 - Macro-models with equivalent radial load at the intrados, analysis setup. Figure 109 - Macro-model results: arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied. Figure 110 - Macro-model results: arch with equivalent radial reinforcement at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied. Element Type 4-noded Element Size 8 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Load [N] Deflection [mm] 1/6 Span Load - Reinforcement at the Intrados Numerical Experimental Limit Analysis 0 100 200 300 400 500 600 0 0.5 1 1.5 2 2.5 3 3.5 4 Load [N] Deflection [mm] 1/2 Span Load - Reinforcement at the Intrados Numerical Experimental Limit Analysis
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 88 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 111 – Top: macro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, principal crack strain at collapse deformed configuration (x20 amplification). Bottom: experimental collapse configuration. A shown in Figure 109, Figure 110, Figure 111 and Figure 112, the results are in good accordance with the experimental and the limit analysis ones. The increase of tensile strength in the case loaded at ½ of the span causes a peak-like behaviour, clearly visible on the capacity curve trend. After the peak, the curve converges anyway from above to the experimental value of capacity.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 89 Figure 112 - Top: macro-model with equivalent radial reinforcement at the intrados, 100 mm depth, load at 1/2 of the span, principal crack strain at collapse deformed configuration (x10 amplification). Bottom: experimental collapse configuration. Table 37 - Equivalent radial load reinforcement at the intrados: macro-models results and comparison with experimental and limit analysis ones. The numerical results show, in general, an optimal accordance with the limit analysis ones (Table 37). Compared with the experimental ones they better fit in the case of load applied at half span, while for the load at 1/6 of the span they show a slightly lower value. This difference can be ascribed both at the uncertainty of the spreadsheet for graphic statics on the calculation of the depth (performed with the load at half span) and to the tensile strength value choice. Numerical Experim. Limit Analysis to Experim. to L. Analysis 1/6 446 486 452 -8% -1% 1/2 435 431 431 1% 1% Load position (span fraction) Collapse load [N] Error
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 90 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 7.4 Modelling of Free Sliding Steel Cables Reinforcement - Results Macro-models have been used also to simulate the cases with physical cable modelling. The value of the friction angle has been taken again as φ = 0.6°, so as to enable free sliding of the cables, as demonstrated with micro-models in 6.4.2. The other parameters involved, concerning the cable and the sliding interface behaviour, are the same as the ones used for micro-model analysis, reported in Table 25. As stated above, unlike the cases of micro-modelling, 4-noded plane stress elements have been used. Therefore, the elements used for the discretization of the steel cables are 2D linear enhancedtruss, and even interface ones are of the same order (linear). The same cases as before have been considered: - Cable at the extrados (90 mm depth), 294 N of prestress, load at 1/6 and half of the span; - Cable at the intrados (100 mm depth), 294 N of prestress, load at 1/6 and half of the span. Even in this case, when the cables were applied at the extrados, debonding has been allowed (gap opening of the interface), while with cables at the intrados, the interfaces worked both in tension and in compression. In all the cases, linear geometry analysis has been performed. Concerning the arch body material, the same properties of the cases of the macro-models with the equivalent radial load reinforcements, explained in 7.3, have been applied. Analysis setup is reported in Table 38. Table 38 - Macro-models with cables modelling at the extrados, analysis setup. Again, the normal stress at interface, if integrated along the width of the interface (0.1 mm), gives almost exactly the value of the equivalent confinement (Figure 113). Equivalent Radial Load → p = T / R = 294 N / 695 mm = 0.423 N/mm Confinement Effect from Cable Modelling → p’ = σR x t = 4.23 MPa x 0.1 mm = 0.423 N/mm (maximum) T: cable traction, R: external radius, σR: radial normal stress, t: interface width. Element Type 4-noded Element Size 6 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 91 Figure 113 - Macro-model with physical reinforcement at the extrados (90 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. Figure 114 - Macro-model results: arch with physical cable at the extrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Load [N] Deflection [mm] 1/6 Span Load - Reinforcement at the Extrados Radial Load Cables Limit Analysis Experimental
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 92 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 115 - Macro-model results: arch with physical cable at the extrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. Figure 116 – Left: macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, crack strains at collapse deformed configuration (x40 amplification). Right: experimental collapse configuration. Figure 117 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, collapse deformed configuration (x25 amplification). Left: axial force in the cable, right: interface normal stress. 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Load [N] Deflection [mm] 1/2 Span Load - Reinforcement at the Extrados Radial Load Cables Limit Analysis Experimental
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 93 Figure 118 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/6 of the span, collapse deformed configuration (x25 amplification). Left: shear stress in the interface, right: shear displacements. The results show a good accordance with the previous ones (Figure 114, Figure 115, Figure 116 and Figure 119), despite a slightly lower value in the case loaded at half span (Table 39). Nevertheless, the validity of this kind of modelling is confirmed, but still the value of the tensile strength could be affecting the results. The contours of cable tension and radial load show, again, a rather constant distribution along the arch (Figure 117 and Figure 120). Shear stress remains low (Figure 118 and Figure 121) Figure 119 - Left: macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, crack strains at collapse deformed configuration (x40 amplification). Right: experimental collapse configuration. Figure 120 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, collapse deformed configuration (x40 amplification). Left: axial force in the cable, right: interface normal stress. Figure 121 - Macro-model with physical reinforcement at the extrados, 90 mm depth, load at 1/2 of the span, collapse deformed configuration (x40 amplification). Left: shear stress in the interface, right: shear displacements.
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 94 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 39 – Cables at the extrados modelling: macro-models results and comparison with previous ones and experimental ones. As in all the previous cases, even reinforcement at the intrados has been analysed. The model features are the same of the one used in the previous analysis, but with 100 mm depth and steel cable reinforcement at the intrados, with no possibility to debond at the interface with the arch (Figure 122). To avoid the problems of unwanted tangential cracking, again the case loaded at ½ of the span needed a higher value of the tensile strength (0.03 MPa). Analysis setup is reported in Table 40. Table 40 - Macro-models with physical cables at the intrados, analysis setup. Equivalent Radial Load → p = T / R = 294 N / 600 mm = 0.49 N/mm Confinement Effect from Cable Modelling → p’ = σR x t = 4.84 MPa x 0.1 mm = 0.484 N/mm T: cable traction, R: external radius, σR: radial normal stress, t: interface width. Figure 122 - Macro-model with physical reinforcement at the intrados (90 mm depth): end of the application of post-tension (294 N). Top: shear stress interface distribution (left) and normal stress interface distribution. Bottom: axial load in the reinforcement (left) and vertical displacements of the arch body (right). Deformation scale x200. Cables Radial Confinement Experim. Limit Analysis to Radial Confinement to Experim. to L. Analysis 1/6 387 395 411 389 -2% -6% 0% 1/2 350 361 364 360 -3% -4% -3% Load position (span fraction) Collapse load [N] Error Element Type 4-noded Element Size 7 mm Steps increment of Displacement 0.001 mm Iterative Method Full Newton-Raphson Convergence Control Energy, Force, Displacements Tolerance 0.01
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 95 Figure 123 - Macro-model results: arch with physical cable at the intrados (294 N post-tension), load at 1/6 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. Figure 124 - Macro-model results: arch with physical cable at the intrados (294 N post-tension), load at 1/2 of the span, deflection of loaded point vs load applied, comparison with equivalent load case. 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Load [N] Deflection [mm] 1/6 Span Load - Reinforcement at the Intrados Radial Load Cables Limit Analysis Experimental 0 100 200 300 400 500 600 0 0.5 1 1.5 2 2.5 3 3.5 4 Load [N] Deflection [mm] 1/2 Span Load - Reinforcement at the Intrados Radial Load Cables Limit Analysis Experimental
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 96 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 125 - Left: macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, crack strains at collapse deformed configuration (x amplification). Right: experimental collapse configuration. Figure 126 - Macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x40 amplification). Left: axial force in the cable, right: interface normal stress. Figure 127 - Macro-model with physical reinforcement at the intrados, 100 mm depth, load at 1/6 of the span, collapse deformed configuration (x40 amplification). Left: shear stress in the interface, right: shear displacements. Even in this last case, involving reinforcement at the intrados, the results show a good accordance with the previous ones, especially the ones coming from equivalent radial load reinforcement (Figure 123, Figure 124, Figure 125, Figure 128 and Table 41). This confirms the equivalence of these two procedures, and also the fact that the cables are not affected from significant friction phenomena. This is proved from the contours of the cable tension and of the radial interaction force (Figure 126, Figure 129), that both remain substantially constant all along the arch boundary. Even in this case, shear stress is low, confirming the sliding cables phenomenon (Figure 127 and Figure 130).
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 103 9. CONCLUSIONS 9.1 Summary Arches, vaults and domes constitute a smart way to use masonry in constructions, because of their ability to work mainly in compression, given its poor tensile strength. This is why, still nowadays, many arched structures of the ancient times are still standing, even after centuries or millennia from their construction, and have become part of our architectural heritage. Nevertheless, arched structures have always been suffering from problems connected with non-uniform vertical loads, such as horizontal actions, differential settlements and pointed vertical loads. This led to the application of different types of strengthening devices throughout history. The Reinforced Arch method is a recent consolidation technique, that consists on the application of steel post-tensioned cables at the intrados or at the extrados of the structure. It is a rather novel procedure, but it can be classified also as a traditional one because of its basic working principles, its elevate compatibility, its simple materials and its easiness to apply. It has been largely tested by Professor Lorenzo Jurina at the Polytechnic University of Milan, with more than 500 experimental tests performed. 9.2 Conclusions The present work aimed at confirming the effectiveness of this strengthening technique, through the use of finite element numerical models. Experiments on collapse of wooden blocks scaled roman arches were numerically reproduced and simulated. The validity of the results obtained have been confirmed by comparison both with the experimental data and limit analysis ones. After a calibration phase by means of limit analysis, to compensate geometrical problems connected with the reduction of the effective cross-section of the arch, numerical analyses have been performed. Two types of models have been considered and compared: micro-models, modelled with elastic blocks and hinge opening, and macro-models, that have been provided of a smeared crack yield criterion, suitable for masonry structures. All of the results obtained, in terms of vertical load capacity, have been compared, highlighting pros and cons of any modelling technique. Results showed good accordance with the experimental ones, and even more with the ones derived from limit analysis. Therefore, in general it can be stated that both micro and micro-models can be used to simulate the effect of this strengthening method. Nevertheless, micro-models showed a more reliable behaviour all along the loading path, while macro-models showed peaks with no physical significance, due to the presence of tensile strength. Reinforcement has been modelled in different ways: applying an equivalent radial confinement or actually modelling the post-tensioned cables, showing basically an equivalence in the results. Moreover, a more detailed analysis, based on micro-modelling, has been performed to investigate the influence of friction angle between cables and arch on the global behaviour and the capacity of the structure. This analysis underlined the transition from a sliding behaviour of cables to an adherence one, proportional
Numerical Analysis of the Reinforced Arch Method Erasmus Mundus Programme 104 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS to the increase of this parameter. This was accompanied by an increase of the overall capacity. With the same models, also the cases of anchored cables have been studied, confirming the great increase of capacity highlighted in the experimental campaign, if compared with the cases of not anchored ones. Concerning macro-modelling, a sensitivity analysis on tensile strength of masonry has been performed, highlighting the influence of this parameter on the final capacity. This uncertainty on the value of tensile strength represents one of the main drawbacks of this type of models. Therefore, in general, it can be stated that micro-models are more reliable and better simulate the behaviour of such reinforced structures. 9.3 Future Development The numerical tools can be further enhanced in terms of material definition of both the arch body and the reinforcement, especially for simulating the behaviour of real masonry arches. Possible future improvements can be carried out by assigning a non-linear material at the arch blocks in micro-models, to capture damage in compression. Also, non-zero tensile strength at the interfaces (mortar joints) can be added. Moreover, concerning the reinforcement modelling, a yield criterion for the cables’ steel can be added, to capture possible post-elastic effects, in particular for obtaining a reliable capacity curve in the cases of anchored cables. Finally, non-fixed anchors can be introduced, to simulate potential displacements during the arch loading. All this treatise can be extended also to barrel vaults, by means of 3D models, to investigate the effect of diffusion of stresses in the third dimension, and perform a parametrical analysis for the optimization of the strengthening technique.
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