Vulnerability analysis of timber frame walls under seismic loading

Spain| 2018 UNIVERSITAT POLITÈCNICA DE CATALUNYA Nilma del Coral Muñiz Acevedo Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Nilma del Coral Muñiz Acevedo Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Spain | 2018 Nilma del Coral Muñiz Acevedo Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS i DECLARATION Name: Nilma del Coral Muniz Acevedo Email: [email protected] Title of the Msc Dissertation: Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Supervisor(s): Luca Pelà, María Belén Jiménez, Elisa Poletti Year: 2018 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: Polytechnic University of Catalonia (UPC) Date: 23/07/2018 Signature: ___________________________ Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS iii ACKNOWLEDGEMENTS Firstly, I will like to thank my supervisors, Professor Luca Pelà and María Belén Jiménez for their guidance, support, and suggestions. My most sincere gratitude for the enthusiasm and motivation offered throughout this work. I would like to thank the financial support provided by the SAHC Consortium Scholarship. A special thanks goes to all my family and friends that from the distance always offered their unconditional motivation. I am grateful for the friends that shared this SAHC journey with me. A special thanks goes to that special one that shared every step of it. My greatest gratitude is to my family for all the support and words of motivation that made this experience possible. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme iv ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS v ABSTRACT In many regions of the world, the process of design and construction varies depending on local needs and conditions, availability of construction materials, and traditional construction techniques. Timber frames are an example of vernacular architecture with historical and cultural significance in many regions, like: Europe, Latin-America, North America and Oceania. The extensive diffusion of this type of construction has caused the development of several structural typologies and solutions. This master thesis studied three vernacular timber frame typologies in order to determine their resistance and behavior toward seismic actions. To obtain this aim, numerical analyses were performed by including variations in material, geometry, bracings and carpentry joints types. Nonlinear static (pushover) analysis following the lumped plasticity modelling approach were carried out to determine the structural capacity of the frames. The complex behavior of the carpentry connections was studied to carry out a modelling calibration based on previous experimental works. The results of these analyses contributed to understand the global behavior and the structural response of three timber frame typologies. A numerical model was developed for two traditional timber frame typologies from Valparaiso, Chile. The developed numerical model was based in the modelling calibration of two timber frames previously studied experimentally. The calibrated models were able to capture the stiffness of the timber frame walls, the deformation, the nonlinear behavior and the expected mechanisms that where described in previous experimental works. The analysis demonstrated the importance of the connections and their influence in the global behavior of the frame. It was understood, that the connections are the weakest location of the frame, where failure can occur. The work carried out in this study seek to search for a correct approximation to model timber frame structures in cases where there are no experimental studies that allow to obtain more accurate results from the numerical models. Since there are limited studies about the structural response of timber frame buildings under seismic loading, this master thesis seeks to contribute to their conservation and to motivate for their further research. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme vi ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS vii RESUMEN En muchas regiones del mundo, el proceso de diseño y construcción depende de las necesidades locales, la disponibilidad de materiales de construcción y de las técnicas constructivas tradicionales. Los entramados de madera son un ejemplo de arquitectura vernácula con un gran significado histórico y cultural en muchas regiones del mundo, como: Europa, América Latina, Norte América y Oceanía. La extensa difusión de este tipo de construcción a causado que se desarrollen diferentes tipologías estructurales. Esta tesis de maestría estudiará tres tipologías vernáculas de entramados de madera con el fin de determinar su resistencia y comportamiento ante un evento sísmico. Para obtener este objetivo diferentes configuraciones estructurales se estudiarán a través de análisis numéricos; incluyendo las diferentes variaciones en material, geometría, arriostramientos y uniones tradicionales de carpintería. El enfoque del modelo numérico consistirá en un análisis estático no lineal con plasticidad concentrada que proveerá la capacidad estructural del entramado, lo cual contribuirá al entendimiento de su comportamiento global y respuesta estructural. El complejo comportamiento de las conexiones de carpintería fue estudiado para llevar a cabo un modelo calibrado basado en experimentos previos. Los resultados de estos análisis contribuyeron al entendimiento del comportamiento estructural de las tres tipologías estudiadas. Un modelo numérico fue desarrollado para dos tipologías de entramados de madera de Valparaíso, Chile. Los modelos numéricos desarrollados fueron basados en la calibracion de dos entramados de madera previamente estudiados experimentalmente. Los modelos calibrados pudieron capturar la rigidez del entramado, la deformación, el comportamiento no lineal y el mecanismo esperado, según descrito en los experimentos previos. El análisis demostró la importancia de las conexiones y su influencia en el comportamiento global de la estructura, siendo esta la parte más frágil del entramado. El trabajo llevado a cabo en este estudio busca una aproximación correcta para modelar entramados de madera y que pueda ser aplicado a casos en donde no existe información experimental previa que permita obtener resultados mas exactos de los modelos numéricos. Ya que no hay una gran cantidad de estudios sobre la respuesta estructural de los entramados de madera ante carga sísmica, esta tesis de maestría busca contribuir a su conservación y a motivar a que se continúe desarrollando su estudio. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 6 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS from different experimental campaigns were used as a reference. They were selected in order to obtain a good calibration for the behavior of the timber frame in a numerical model. • Numerical Analysis After the numerical model is calibrated based in previous experiments, the next step is to create a numerical model of a real case study. Several numerical models were created starting from the elementary cell of the frame, then a shear wall and finally the modelling of two representative cases of timber frames. For all the models a nonlinear static analysis (pushover) was performed. 1.4 Outline of the Thesis The work is organized in the following chapters: Chapter 1 – Presents the introduction of the work, including the motivation, background, and the objectives of this thesis. Chapter 2 – Presents a detailed literature review on timber frame structures, including available experimental studies and numerical approaches. Chapter 3 – Provides a description of the calibration of the numerical models based on the use of beam elements, within the FEM theory, and with lumped plasticity. The FEM models are calibrated by comparison with experimental tests available in the literature. Chapter 4 – Provides a description of the typical structural configurations of timber frames of the city of Valparaiso, Chile, by considering the geometry of the members, the type of wood and the carpentry connections. Chapter 5 – Presents the numerical simulation of two representative real cases of timber frame shear walls of the city of Valparaiso, Chile. Chapter 6 – Presents the final remarks of the research, the principal conclusions derived from the study and recommendations for future works. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 7 CHAPTER 2 STATE OF THE ART 2.1 Timber Frame Structures: Historical and Technical Background The origin of timber frame structures probably dates back to the Roman Empire, since evidence of a half-timber frame construction from ancient Rome refered to as Opus Craticium, was found in the town of Herculaneum by archaelogists (Poletti, 2013). Figure 2.1 shows a building from the town of Herculaneum that was uncovered by archeologists after it was buried in the eruption of Vesuvius. Timber was also used in previous cultures, since it was identified that masonry was reinforced using timber elements in the palaces at Knossos in Minoan Crete suggesting that timber-laced masonry construction dates to the 1500 to 2000 B.C. (Langenbach, 2015). In Greece, timber was mainly used to reinforce and repair buildings after they were affected by earthquakes, but then it became part of the new buildings constructions techniques. Figure 2.1. c Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 8 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Timber frame constructions spread througthout the world, first by having an apperance in Turkey in the eighth century. The expansion continued to other countries including Britain, were the construction system is refered to as “half-timber”, France were is refered to as colombage, and Germany were is refered to as Fachwerk. Portugal, Italy, Spain, Scandinavia and India are other countries to where timber frame construction technologies spread and became part of their vernacular architecture. The English, German and French developed their houses based in timber frames and due to immigration, they brought this constructive system to countries in North, South and Central America. Due to the development and expansion of the timber frame construction around the world, is possible to find different configurations and typologies that share similarities in construction techniques and structural behavior. One of these similarities is their good structural behavior towards a seismic event. In many cases timber frame structures resulted to be more resistant than reinforced concrete buildings. For example, a collapsed reinforced concrete building next to a surviving timber frame building in Turkey after the 1999 earthquake is shown in Figure 2.2. A part of understanding the good behavior of timber frames, is to know the structural components that compose the frame. The general constitution of a timber frame constructive system is characterized by three main components: 1. Bare frame, composed of horizontal and vertical elements that resist the vertical load. 2. Bracings, mostly consisting of diagonal members that play a central role by resisting lateral forces. 3. Infill, composed of materials (stones, bricks, adobe, lath, plaster), that provide additional resistance. Timber, which is the main component of the frame, is what provides a good resistance capacity against horizontal loads, due to the great elastic properties of the material. However, the lateral resistance of a timber frame is not only provided by the wood, since the infill and the bracings also play an important role. The materials used as infill usually have a good compression resistance, which improves the resistance of the frame. The materials that compose a timber frame have good mechanical properties that combined create a good resisting system. However, the geometry of the frame and the connections between the structural members turn to be relevant, since scientific research have validated that the location of the failure often coincides with the connections. Figure 2.1. A collapsed reinforced concrete building next to a surviving timber frame building in Turkey after the 1999 earthquake. Photograph from: Adem Dog˘angün. (Langenbach, 2015). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 9 In historical timber frames, the connections between timber elements are usually carpentry joints. The type and characteristics of theses joints depend on the geographical location, traditional techniques and the age they were created (Branco, 2015). Common traditional carpentry joints that can be found in historical timber frames include: mortise and tenon joints, notched joints, lap joints, and scarf joints. Mortise and tenon joints are composed of two timber pieces, the mortise hole and the tenon tongue, that connect members that usually form an "L" or "T" type configuration, as shown in Figure 2.3(a). This kind of joint is mainly used when pieces connect at an angle between 45° to 90°, but when the angle is different from 90º, the nose of the tenon can be cut off and is called a skewed tenon (Branco, 2015), as shown in Figure 2.3(b). Figure 2.1. (a) Mortise and tenon joint. (b) Mortise and skeweed tenon. (Branco, 2015) Notched joints consist of joining two pieces, by compressing one piece into another through a “V” shape groove usually perpendicular to the length of the element that is going to be connected to, as shown in Figure 2.4(a). A tenon can be added to the notched joint to keep all the beams coplanar, as shown in Figure 2.4(b), but the notch is what creates the strength of the joint (Branco, 2015). Figure 2.1. (a) Notched joint. (b) Notched joint with tenon. (Branco, 2015) Lap joints can be classified in three types. One type is the full lap joint, were the joint is created without removing any material from the members that will be connected and is held in place by a pin, as shown in Figure 2.5(a). Another type is the half-lap joint, were the joint is created by removing material from both members, as shown in Figure 2.5(b). In this case usually half of the thickness of the members are removed. The last type is the dovetail-lap joint, were the joint is created by a tenon with a similar shape to the tail of a dove, as shown in Figure 2.5(c). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 10 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 2.1. (a) Full lap joint. (b) Half-lap joint. (c) Dovetail-lap joint. (Branco, 2015) Scarf joints are usually used when the required length of the members is not available, since this joint allows to connect members end to end, as shown in Figure 2.6. To secure the joint, usually pins are used. Scarf joints can be classified in different types based on their geometry, for example: (a) the halved-scarf joint, (a’) the lapped dovetail scarf joint, (b) the scarf joint, (c) the scarf joint with undersquinted ends, and (d) the bolt of lightning joint, all shown in Figure 2.6. Figure 2.1. Scarf joints. (Branco, 2015) The variation of the types of carpentry joints used, and other characteristics of timber frames like the geometry and materials created different typologies and constructive techniques over the years. Throughout history, timber frames resulted to be good and resistant structural systems. They spread throughout the world, especially in zones with seismic hazard. Today we can find interesting examples of historical timber frames, shown in Figure 2.7. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 11 Figure 2.1 Timber frame structures around the world. Base map exhibits the global seismic hazard of each region. (Vieux-champagne et al., 2014) A historical example of a timber frame building is the Gaiola. This construction typology was developed in Portugal after the 1755 Lisbon earthquake (Langenbach, 2015). Gaiola means cage and the name derived from the fact that the frame is composed of external masonry walls and an internal timber structure. This typology is also known as Pombalino since the Marquis of Pombal was the one that directed its development. The geometry of the frame consists of horizontal and vertical elements, and diagonal bracing members in an X shape, as shown in Figure 2.8. The usual carpentry joints between these elements are the connection by contact and the half-lap joints, shown in Figure 2.9. The infill is usually rubble or brick masonry, but it can also be composed of other materials like mud and hay, depending on their local availability. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 12 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 2.1. Pombalino building. (a) Structure of a whole building. (b) Example of a frontal wall. (Poletti, 2013) Figure 2.1 Carpentry joint connections: (a) Connection by contact. (b) Half-lap tee halving connection. Another historical example of a timber frame building is Casa Baraccata. This typology was developed in Italy after the 1783 Calabria earthquake (Langenbach, 2015). The name comes from the term “baracca” which was the name provided to the temporary timber houses that were built for refugees after seismic events. This typology was developed based in the construction methods already used in Portugal. The timber frame composition is similar to Pombalino, with the difference that the pillars are not composed of a unique timber element. However, their position is constant from the foundation to the roof. The proposal for the construction of Casa Baraccata was by Giovanni Vivenzio and it consisted of a construction by blocks. In this case he proposed three blocks with the idea that the central building had a higher height and the lateral ones act as buttresses (Poletti, 2013). The timber frame was embedded in the external masonry to act as a reinforcement, as shown in Figure 2.10. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 13 Figure 2.1 Casa Baraccata façade. (Poletti, 2013) The Quincha timber frame is an example of a construction technique found in historic buildings in Central and South America. The word quincha is derived from the Quechua word “khincha” which means a wall of branches (Quinn, 2017). The Quincha wall is composed of timber elements that create a frame, which is infilled with woven cane and mud, as shown in Figure 2.11. The geometry of the bare frame consists of beams, posts, and diagonal bracings that are connected by carpentry joints, usually mortise and tenon joints, and lap joints, shown in Figure 2.12. This technique dates back to 2600 BC, since a primitive form was found in Peru in the archeological site of Caral (Quinn, 2017). Timber frames with an infill of canes and mud were used in Central and South America since Pre-Hispanic times and it developed to different variations. The technique was usually used for the construction of rural houses and they were referred to by different names including bahareque in Costa Rica, pajareque in Honduras, vareque in Ecuador, pared francesa in Argentina, and cañizo in Spain. Figure 2.1. Quincha building: (a) Structure of a whole. (b) Example of a wall. (Quinn, 2017) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 14 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 2.1 Carpentry joint connections: (a) Lap joint connection. (b) Mortise and tenon connection. 2.2 Experimental Campaigns on Testing Timber Frame Structures Experimental works have been done previously to prove the effectiveness of historical timber frames when subjected to seismic loads. The experimental results have shown that the seismic response of the frame will depend of the wall geometry, the infill and type of connections. To study timber frame structures experimentally, they are subjected to monotonic and cyclic tests. Monotonic tests are performed by applying an increasing load to determine the ultimate capacity of the shear wall. While, cyclic tests are used to simulate the seismic action and obtain information about the shear resistance and global behavior of the system. These types of tests are useful to further develop analytical and numerical models that will contribute in the prediction of the seismic resistant behavior of these structures. Several experimental campaigns have studied real cases and replica specimens of infilled and bare timber frames typologies, like the following examples. 1The CNR Ivalsa in Trento, Italy, carried out an experimental campaign on a full-scale specimen of the Mileto masonry reinforced with timber framing (Borbone constructive system), shown in Figure 2.13(a) (Ruggieri and Zinno, 2015). The experimental test included cyclic tests by applying load increments to two frames, one with infill and a bare frame. 2The laboratory of Domaine University in France carried out an experimental campaign of a Haitian timbered masonry structure called ‘‘Kay peyi’’, shown in Figure 2.13(b) (Vieuxchampagne et al., 2014). In this campaign the frame was studied in three scales: the connections, the elementary cell and the shear wall, as shown in Figure 2.14. The tests performed included cyclic and monotonic tests. The advantage of the approach used in this study is that it provides an understanding of the behavior of the components of the wall, which results useful for further numerical investigations. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 15 Figure 2.2 (a) Borbone constructive system tested in CNR Ivalsa in Trento, Italy. (Ruggieri and Zinno, 2015) (b) Haitian timbered masonry structure tested in Domaine University in France. (Vieux-champagne et al., 2014) Figure 2.2. Three scale study of the Haitian timbered masonry structure. (Vieux-champagne et al., 2014) 3The University of Minho in Portugal carried out an experimental campaign to study the seismic performance of traditional half-timbered walls (Pombalino) with a real scale wall specimen in the laboratory, as shown in Figure 2.15(a) (Poletti, 2013). The experiment included cyclic tests for both, an infill and a timber frame specimen. The experimental procedure also included testing of the traditional timber connections. 4The Structural Engineering Laboratory of the University of Bath in UK carried out an experimental campaign to study a half-scale timber frame of the Quincha frame from Peru, as shown in Figure 2.15(b) (Quinn, 2017). The tests performed included a three-phase loading protocol: Phase 1: a stabilizing load cycle, Phase 2: a stiffness load cycle and Phase 3: a strength test. The tests were performed for a timber frame and for a frame with infill. The campaign also included the testing of traditional mortise and tenon connections in order to study their nonlinear behavior. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 22 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 2.2.2 (a) Stiffness test. (b) Strength test. (Quinn, 2017). Figure 2.2.2 Deflected shape of the timber frame. (Quinn, 2017). Quinn also studied the behavior of the connections and their characteristics. For the mortise and tenon connection, Quinn (2017) studied its rotational and translational stiffness. To determine the rotational stiffness a separate specimen of the connection was tested. The specimen was manufactured with a 60 mm square post and a 25 mm square tenon with 50 mm of length, as shown in Figure 2.25(a). The connection was tested by pushing the post horizontally inducing a moment until failure. Two specimens were tested, one with a tighter fit than the other. The obtained results are shown in Figure 2.25(b). Quinn (2017) also studied the possibility of determining the rotational stiffness based in the geometrical properties of the connection. To obtain a relation between geometry and rotational stiffness the component method was applied but modified depending on the center of rotation of the connection. The center of rotation marked with an X, in Figure 2.26(a), was determined by assuming that only the two portions of the tenon that are directly in contact with the walls of the mortise contribute to the stiffness. Using this assumption to modify the component method Quinn determined a realistic estimate to relate the length of the tenon and the rotational stiffness, as is shown in Figure 2.26(b). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 23 To determine the translational stiffness of the mortise and tenon connection a set of localized tests are required but were not included. Since there is no dowel to give tensile capacity to the mortice and tenon joint, a defining feature of the frames is the ability of the posts to move upwards (Quinn, 2017). Due to this behavior, Quinn (2017) assumed the translational stiffness based in the experimental data of vertical uplift measured for the frame, as shown in Figure 2.27. Figure 2.2.2 (a) Mortise and tenon tested. (b) Moment rotation curve. (Quinn, 2017). Figure 2.2.2 (a) Diagram showing portions of tenon contributing to stiffness. The cross marks the assumed center of rotation. (b) Variation in stiffness with length of tenon. (Quinn, 2017). Figure 2.2.2 Stress-strain relationship for tenons. (Quinn, 2017). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 24 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 2.3 Numerical Modelling of Timber Frame Structures The Finite Element Method (FEM) is a numerical tool useful in structural engineering and one of the most used approaches to perform numerical models of structures. Finite element analysis can be applied to simulate the behavior of an entire structure or of a single structural component. The complexity of the finite element model will depend on the type of study, considering the discretization of the physical object modeled and the division of the interconnected elements. A numerical model for a timber frame structure must consider the geometry of the frame, the material properties of the structural members, the loading and boundary conditions, and the behavior of connections. However, the method of analysis will depend on the purpose of the analysis and the expected output. Two main modelling approaches are the micro level model and the macro level model. Micro-models entail a high detail of analysis, but they require several input parameters and a high computational effort. On the other hand, macro-models represent a simplified approach that requires less input data and more affordable computations. Hybrid models are also a possibility when, for example, the purpose is to analyze in detail a specific structural element within a more complex structure (Costa et al., 2014). These considerations are important to select the best approach to be used in the development of a numerical model of a specific structural problem. 2.3.1 Micro Model A micro model is a small-scale model that provides a local behavior of the structure with a detailed level of discretization for all its components. This modelling approach is useful to determine failure mechanisms, but it requires a high computational effort. It is also useful to simulate separately the behavior of a basic component of the structure. The micro-modelling strategy can provide results with a high level of accuracy, but the amount of time needed to obtain the input parameters and the modelling computational effort causes this method to become less efficient in the application to some cases. An example of the application of micro modelling to timber frame structures was the model developed by Doudoumis (2010). The model consisted of a composite timber-masonry wall, as shown in Figure 2.28. In this case the behavior of the timber frame was simulated by inelastic elements with an elastoplastic behavior, and the masonry infill by shell elements with a pressure dependent material strength. The behavior of the connections was modeled by elastoplastic link elements and the interaction between the timber frame and the masonry infill by appliying the Coulomb’s law. Another application of micro modelling is the example of the model developed by Costa et al. (2014). This case consists of the anlaysis of a stone masonry structure from Azores, for which the masonry is reduced to its basic components: joints, blocks, and infill, as shown in Figure 2.29. The model assumed continum finite elements for the units and the mortar at the joints, and their interface was modeled by descontinuous elements that account for potential crack or slip. (Costa et al., 2014). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 25 Figure 2.3.1 Micro model of a composite timber-masonry wall. (Doudoumis, 2010). Figure 2.3.1 Micro-modelling of a masonry wall from Azores. (Costa et al., 2014). 2.3.2 Macro Model A macro model is a large-scale model that provides an overall response of the structure. This modelling approach results to be simple and with a low computational effort, providing a good balance between simplicity and accuracy. Due to this, macro models are commonly used to characterize the seismic structural response of historical buildings. For example, they are useful to estimate the lateral load capacity of timber frame buildings. A macro model can be applied by using a lumped plasticity or a distributed plasticity modelling approach. In the lumped plasticity model, nonlinear behaviors are assumed at the extremities of the structural element, while the body is modeled as an elastic part (Rahai and Nafari, 2013). However, the calibration of the inelastic element parameters will define the accuracy of the results. The distributed plasticity model assumes the nonlinear behaviors to occur at any element section. In this case the behavior of the section can be described in accordance with either fiber modeling approach or response curves reproducing the element behavior under reversible load (Rahai and Nafari, 2013). Several lumped plasticity models have been developed. For example, Lukic et al. (2018) developed a numerical model of a timber frame wall by first modelling the behavior of the connections based on an individual connection test, and then applying the calibrated nonlinearity to the modelling of the timber frame wall. The procedure consisted of calibrating the joints individually, as shown in Figure 2.30(a). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 26 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Then a two-dimensional numerical model of a timber frame wall was developed by applying the nonlinearities to the joints, as shown in Figure 2.30(b). The numerical model included beam-column elements for the horizontal and vertical members, truss elements for the diagonal bracings, and calibrated springs for the joints, which will control the global behavior of the frame. Ceccotti and Sandhaas (2010) also developed a numerical model based on the lumped plasticity modelling approach. In this case it was applied to a three-dimensional model of a X-Lam building, as shown in Figure 2.31. A similar procedure to the last case was used, but shear walls with X-Lam panels cannot be modelled with rotational springs, only with translational springs (Ceccotti and Sandhaas, 2010). The calibration of the springs was based in the results obtained from cyclic tests. Figure 2.3.2 (a) Half-lap connection with three behaviors (1 – shear, 2axial, 3rotational). (b) Numerical model with link elements. (Lukic et al., 2018) Figure 2.3.2 Numerical 3D model of X-Lam building. (Ceccotti and Sandhaas, 2010) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 27 CHAPTER 3 CALIBRATION OF NUMERICAL MODEL Considering the modeling approach of lumped plasticity described in Section 2.3.2, this chapter will explain the calibration process for two timber frame models that were previously studied by different experimental campaigns: Pombalino (Poletti, 2013) and Quincha (Quinn, 2017). The objective is to develop a numerical modeling approach that can be applied to similar typologies, since it is not always possible to obtain experimental results from a real case study. The model to be develop should be calibrated based on existing experimental results, reason why these previous studies will be considered. 3.1 Pombalino Timber Frame Calibration This first calibration consists in simulating the mechanical behavior of the Pombalino timber frame reported in the experimental work by Poletti (2013), and the analytical model done by Ciocci (2015). The timber frame wall is modeled by using the finite element analysis softwares DIANA FEA and SAP2000. To validate the accuracy of the softwares and the calibration, the numerical capacity curves obtained by both models are compared with the experimental capacity curve. Finally, the experimental collapse mechanism of the frame is studied and compared those provided by both softwares. The two-dimensional model considers the frame geometry, material properties, loading and boundary conditions. The geometry of the timber frame is composed of the elementary cell shown in Figure 3.1(a), with the dimensions described in Table 3.1. The material of the frame is Maritime Pine, modeled as a linear elastic isotropic homogeneous material with the properties described in Table 3.2. The loading conditions include a vertical load of 25 kN applied downwards at the three nodes were the top beam and the post intersect. An horizontal displacement of 0.1 m is applied at the left corner of the top beam. The boundary conditions include a restriction in horizontal and vertical directions at the nodes of the base. Also, a restriction in the horizontal direction is added at the node where the prescribed deformation is applied, as shownin Figure 3.1(b). For both modelling softwares, linear elastic beam elements are considered to represent the timber frame. However, the nonlinear behavior and finite stiffness of the carpentry joints is modeled by applying springs elements in DIANA FEA and concentrated hinges in Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 28 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS SAP2000. The carpentry joints of the Pombalino timber frame include: connection by contact and halflap tee halving connection, as described in Section 2.1. Figure 3.1. Elementary cell of the Pombalino timber frame: (a) Geometry. (b) Loading and boundary conditions. Table 3.1 Cross sectional dimensions of the elements of the frame. (Poletti, 2013) Elements Height (mm) Width (mm) Beams 160 120 Posts 80 120 Diagonals 80 120 Table 3.1 Material Characteristics of Maritime Pine. (Poletti, 2013) ft,0 15 MPa Gv 700 MPa ɣ 9 - ft,90 5 MPa v 0.3 Gft,0 70 Nmm/mm2 E0 11000 MPa ρ 590 kg/m3 Gft,90 50 Nmm/mm2 E90 5000 MPa αT 1 - Gfc,0 130 Nmm/mm2 fc,0 25 MPa αh 1 - Gfc,90 70 Nmm/mm2 fc,90 3 MPa β -1 - kp 0.001 - Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 29 The first step of the calibration is to analyze a rigid model and then create other models by adding the nonlinear behavior of the connections, as shown in Figure 3.2. The models were analyzed by: • MOD 0 – Rigid connections • MOD 1 – Hinged connections • MOD 2 – Semi-rigid connections between the diagonal elements and the main frame • MOD 3 – Semi-rigid connections between the diagonal elements and the main frame, and betweem the elements of the main frame Figure 3.1. Typical cell for each numerical model. (Ciocci, 2015). The first model analyzed is MOD 0 which considers all the connections as perfectly rigid. The second, MOD 1, considers a hinged connection between the diagonal and the frame by releasing all the rotational DOFs of each diagonal element at one end. The next models, MOD 2 and MOD 3, consider the connections as semi-rigid by introducing a stiffness value at the DOFs. The behaviors of the semi-rigid connections are introduced in the model by considering forcedeformation diagrams. The behavior of the connections by contact between the diagonals and the main frame are represented with axial and shear stiffness. While, the behavior of the half-lap connections between beams and posts of the main frame are represented with rotational stiffness. Ciocci (2015) developed an analytical model to compute the stiffnesses of the joints by applying the component method suggested by Drdácký et al. (1999), and Descamps (2009). The axial and shear stiffness of the connection by contact were calculated by applying the contribution of two stiffnesses: the stiffness provided by the contact area (𝑘𝑐) and the stiffness provided by the nail, as shown in Figure 3.3. The stiffness provided by the contact area was calculated by considering two contact areas, as shown in Figure 3.3(a) and (b); obtaining the values shown in Table 3.3. The stiffness provided by the nail was calculated based on the Eurocode equation for nails without pre–drilling by considering the extraction stiffness (𝑘𝑒) and the ultimate shear plane stiffness (𝑘𝑠𝑝), as shown in Figure 3.3(c); obtaining the values shown in Table 3.4. The rotational stiffness of the half-lap connection was also calculated by the application of the component method. In this case four contact areas were considered, as shown in Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 30 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 3.4(a). After performing the calculations of the component method the values shown in Table 3.5 were obtained by Ciocci (2015). Figure 3.1 (a) Contact areas of the connection by contact, (b-c) Axial and shear stiffness for the connection between the diagonals and the main frame. (Ciocci, 2015). Table 3.1 Stiffness Kc provided by the contact areas. (Ciocci, 2015). A1 (m2) Kc11 (kN/m) Kc12 (kN/m) Kc1 (kN/m) Kc1x (kN/m) Kc1y (kN/m) 6.84 x 10-4 1.15 x 105 6.20 x 104 4.02 x 104 2.85 x 104 2.85 x 104 A2 (m2) Kc21 (kN/m) Kc22 (kN/m) Kc2 (kN/m) Kc2x (kN/m) Kc2y (kN/m) 6.84 x 10-4 1.15 x 105 6.20 x 104 4.02 x 104 2.85 x 104 2.85 x 104 Table 3.1 Stiffness Ke and the stiffness Ksp provided by the nail. (Ciocci, 2015). Ke (kN/m) Kex (kN/m) Key (kN/m) Ksp (kN/m) Kspx (kN/m) Kspy (kN/m) 8.48 x 104 6 x 104 6 x 104 1.34 x 103 944 944 Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 31 Figure 3.1 (a) Contact areas of the half-lap connection. (b) Rotational stiffness of the connection. (Ciocci, 2015). Table 3.1 Rotational stiffness for the half–lap connection. (Ciocci, 2015) A1 (m2) K11 (kN/m) K12 (kN/m) K1 (kN/m) Z1 (m) 9.60 x 10-3 1.04 x 106 7.84 x 104 1.15 x 104 0.05 A3 (m2) K31 (kN/m) K32 (kN/m) K3 (kN/m) Z3 (m) 6.80 x 10-4 5.18 x 105 4.24 x 104 3.92 x 104 0.02 To develop the numerical models representing the different MODs, link elements with an assumed zero length were used to model the semi-rigid connections. Each link element is assumed to be composed of six separate springs, one for each deformational degree of freedom (u1, u2, r3) (Ciocci, 2015). Two link elements were considered. Link 1 is used to represent the connection between the diagonal elements and the main frame. While, Link 2 is used to represent the connection between beams and posts. For Link 1 the axial and shear stiffness are specified, while for Link 2 the rotational stiffness is defined. In MOD 2 a linear elastic force-deformation relationship is used for Link 1, as shown in Figure 3.5. While, in MOD 3 a linear and a nonlinear elastic force-deformation relationship is used for Link 1 and Link 2, as shown in Figure 3.6. For link 1, after reaching the maximum capacity, the load progressively decreased to a null value until the ultimate displacement (Poletti et al.,2016). The maximum capacity of the connection was determined by the analytical procedure by Ciocci (2015), previously explained, and the ultimate displacement was assumed to be the same of the half-lap connection. For Link 2, the results obtained for the half-lap connection test by Poletti (2013) were considered to assume a numerical tri-linear behavior force-displacement diagram, shown in Figure 3.7, with the corresponding tri–linear moment–rotation diagram obtained. Table 3.6 shows a summary of the values for the stiffness of the diagrams, used to calibrate the model. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 38 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 3.1 Hinges formation in each characteristic point of the Pombalino model by DIANA FEA. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 39 3.2 Quincha Timber Frame Calibration This second calibration consists in simulating the mechanical behavior of the Quincha timber frame reported in the experimental work by Quinn (2017). The timber frame wall is modeled by using the finite element analysis softwares DIANA FEA and SAP2000. To validate the accuracy of the softwares and the calibration, the numerical capacity curves obtained by both models are compared with the experimental capacity curve. Finally, the collapse mechanism of the frame is studied and compared for both softwares. The two-dimensional model considers the frame geometry, material properties, loading and boundary conditions. The geometry of the timber frame is composed of the elementary cell shown in Figure 3.15(a), with the dimensions described in Table 3.7. The material of the frame is Cypress, except for the low beam that is Sapelli. Both materials are modeled as linear elastic isotropic homogeneous with the properties described in Table 3.8. The loading conditions include a distributed load of 3.79 kN/m applied downwards at the top beam and a horizontal displacement of 0.1 m applied at the left corner of the top beam. The boundary conditions include a restriction in horizontal and vertical directions at the nodes of the base. Also, a restriction in the horizontal direction is added at the node where the prescribed deformation is applied, as shownin Figure 3.15(b). For both modelling softwares, linear elastic beam elements are considered to represent the timber frame. However, the nonlinear behavior and finite stiffness of the carpentry joints is modeled by applying springs elements in DIANA FEA and concentrated hinges in SAP2000. The carpentry joints of the Quincha timber frame include: mortise and tenon joints and lap joints, as described in Section 2.1. Figure 3.2 Elementary cell of the Quincha timber frame. (a) Geometry. (b) Loading and boundary conditions Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 40 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 3.2 Cross sectional dimensions of the elements of the frame. (Quinn, 2017) Elements Height (mm) Width (mm) Beams 60 80 Posts 60 80 Diagonal 90 30 Table 3.2 Material Characteristics of Sapelli and Cypress. (Quinn, 2017) Wood Sapelli Cypress E (kN/m2) 7.7E+06 6.4E+06 ρ (kg/m3) 400 390 v 0.3 0.3 The first step of the calibration process is to analyze a rigid model and then create other models by adding the nonlinear behavior of the connections. The models were analyzed by: • MOD 0 – Rigid connections • MOD 1 – Hinged connections • MOD 2 – Semi-rigid connections: rotational stiffness of the mortise and tenon • MOD 3 – Semi-rigid connections: rotational and translational stiffness of the mortise and tenon • MOD 4 – Semi-rigid connections: rotational and translational stiffness of the mortise and tenon, and translation of the lap joint. The first model analyzed is MOD 0 which considers all the connections as perfectly rigid. The second, MOD 1, considers hinged connections by releasing all the rotational DOFs at one end. The next models, MOD 2, MOD 3, and MOD 4 consider the connections as semi-rigid by introducing a spring stiffness value at the DOFs. The behaviors of the semi-rigid connections are introduced in the model by considering forcedeformation diagrams. The behavior of the mortise and tenon connections between the beams and the posts are represented with axial and rotational stiffness. While, the behavior of the lap connections between the diagonal and the beams of the main frame are represented with axial stiffness. The axial stiffness of the mortise and tenon connection, shown in Figure 3.16, was based in the experimental data of vertical uplift measured for the frame by Quinn (2017). The rotational stiffness of the mortise and tenon connection, shown in Figure 3.17, was based in the results obtained from the separate specimen of the connection that was tested by Quinn (2017). For the lap connection the stiffness provided by the Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 41 nail was calculated based on the Eurocode equation for nails without pre–drilling, obtaining the behavior shown in Figure 3.18, with a maximum capacity of 2.15 kN. Figure 3.2 Mortice and tenon connection translational stiffness. (Quinn, 2017) Figure 3.2 Mortise and tenon connection rotational stiffness. (Quinn, 2017) Figure 3.2 Lap connection capacity. (Quinn, 2017) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 42 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS The two-dimensional numerical model calibrated by SAP2000 considered linear elastic beam elements to model the timber frame and concentrated hinges to model the nonlinear behavior of the carpentry connections, as shown in Figure 3.19(a). The analysis consisted of a nonlinear static (pushover) analysis. The obtained capacity of the frame is represented by the force-displacement curve shown in Figure 3.20. The two-dimensional model calibrated by DIANA FEA considered the same assumptions, except that the nonlinear behavior of the carpentry connections was modeled by spring elements, as shown in Figure 3.19(b). The obtained capacity of the frame is represented by the force-displacement curve shown in Figure 3.20. The initial stiffness of the frame calibrated by SAP2000, resulted in 178 kN/m. While, for the model calibrated by DIANA FEA, resulted in 179 kN/m. The models resulted with a good initial stiffness approximation when compared with the initial stiffness of the experiment calculated by Quinn (2017), which resulted in 180 kN/m. However, similar to the previous case of modelling calibration, the maximum resistance of the frame resulted to be different, as seen in Figure 3.20. This discrepancy may be due to differences in the computational process of the softwares, since the input data utilized for both models was the same. Figure 3.2 Modelling of nonlinear connections: (a) Concentrated hinges in SAP2000. (b) Spring elements in DIANA FEA. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 43 Figure 3.2 Numerical load-displacement capacity curves developed by SAP2000 and DIANA FEA, compared with the experimental results of the Quincha frame tested by Quinn (2017). To complete and validate the calibration of the frame, the collapse mechanism was studied and compared for both softwares. The points were the curve shows a change in slope are studied in detail, since they will characterize the global behavior of the frame. For the model analyzed in SAP2000 the points (A-D) shown in Figure 3.21 were studied. The legend shown in Figure 3.22 characterizes the behavior of each hinge in order to understand when each is formed in the frame throughout the analysis. Point A occurs when nonlinear behavior starts to take place in the frame. At this point the hinges between the diagonal and the internal posts reach the yielding resistance along the flexural direction, as shown in Figure 3.23(a). At point B three of the hinges between the top beam and the posts reach yielding, as shown in Figure 3.23(b). At point C all the hinges between the top beam and the posts reach yielding, as shown in Figure 3.23(c). The rotational hinges that are generated between the top beam and the posts do not reach collapse. The final collapse mechanism is shown in Figure 3.23(d). Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 44 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 3.2 Pushover curve of Quincha model by SAP2000 with characteristic points of the global response. Figure 3.2 Hinges plastic deformation. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 45 Figure 3.2 Hinges formation in each characteristic point of the Quincha model by SAP2000. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 46 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS For the model analyzed in DIANA FEA the points (A-D) shown in Figure 3.24 were studied. The legend previously shown in Figure 3.22, characterizes the behavior of each hinge in order to understand when each is formed in the frame throughout the analysis. Point A occurs when nonlinear behavior starts to take place in the frame. At this point the hinges between the diagonal and the internal post, and between the diagonal and the top beam reach the yielding resistance along the flexural direction, as shown in Figure 3.25(a). At point B all the hinges of the connections of the diagonal reach yielding, as shown in Figure 3.25(b). The final collapse mechanism is shown in Figure 3.25(c). Comparing the collapse mechanism of the numerical models with the description of the experiment is concluded that the models correspond with the experimental observations by Quinn (2017). First, yielding of the connection between the diagonal and frame occur. The diagonal pulls downwards on the top plate, splitting the beam perpendicular to the grain and then gradual failure of top plate perpendicular to the grain occur. Visible inplane bending of diagonal was observed, as well as a brittel failure of diagonal in tension. Figure 3.2 Pushover curve of Quincha model by DIANA FEA with characteristic points of the global response. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 47 Figure 3.2 Hinges formation in each characteristic point of the Quincha model by DIANA FEA. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 54 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 4.2.2 Shear walls: (a) Typical shear wall with a door opening. (b) Typical shear wall with a window opening. Dimensions in cm. 4.3 Traditional Carpentry Joints Before the invention of nails or metal fasteners, timber structures constructions were characterized by the use of traditional carpentry joints. Valparaiso timber frames are characterized by these types of connections, which are a solution to joint timber elements without or with a reduced need of metal fasteners. If metal fasteners were added, their only function was to keep the elements together, but it did not provide any additional strength to the joint. These types of joints work by equilibrating forces transferring them from one element to another by friction or contact pressure. The timber frame from Valparaiso is characterized by three carpentry joints (Figure 4.8): • Mortise and tenon joints connecting the posts and beams • Notched joint connecting the diagonal with the central post • Connection by contact connecting the diagonal with the external posts Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 55 Figure 4.3 Joints location: A – Mortice and tenon joint, B – Nothced joint, C – Connection by contact. 4.3.1 Mortise and Tenon Mortise and tenon carpentry joints are semirigid connections classified as ensemble joints. They are composed of two timber pieces, the mortise hole and the tenon tongue, with the purpose of connecting the vertical and transversal elements of the timber frame. These joints connect members that usually form an "L" or "T" type configuration (Branco, 2015). The two timber elements are connected by the tenon that is formed at the end of a member and is inserted into a square or rectangular hole cut, the mortise. They are designed to transfer compression and they depend of the contact surface. The tenon has the function of providing resistance to the joint by transferring the load through the surface that surrounds the tenon. To guarantee this behavior the tenon must have a shorter length than the depth of the mortise (Arriaga, 2011). This type of carpentry joint is one of the most common and its technology continued to expand causing variations in geometry. For the case of Valparaiso, the tenon had a length of 50 mm and a cross section of 95 mm x 38 mm, while the mortise had the same cross section of the tenon. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 56 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 4.3.1 Mortise and tenon carpentry connection (a) Studs-sills connection. (b) Frontal and lateral view. (Jiménez, 2015) Figure 4.3.1 Mortise and tenon photographs (a) Studs-sills connection. (b) Girt-stud connection. (Jiménez, 2015) 4.3.2 Notched Connection The notched connection is a simple and common one. This type of joint is linked to the development of king post and similar frames (Branco, 2015). It consists of joining two pieces, one piece compressed in another through a notch (Arriaga, 2011). A notch is a “V” shape groove generally perpendicular to the length of the element that is going to be connected to (Figure 4.11). In Valparaiso timber frames this connection is used for joining the diagonal element with the internal post. In this case the notch is located in the post and is what creates the strength of the joint by transferring the load to the diagonal or central element. This connection is also used for joining the diagonal upper end with the external post. In this case the notch is located in the diagonal. For this type of connection, it is necessary to secure the pieces by nails, screws or ironworks. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 57 Figure 4.3.2 Notched carpentry joints (a) Upper diagonal-studs connection. (b) Central diagonal-stud connection. (Jiménez, 2015) Figure 4.3.2 Notched connection photographs (a) Upper diagonal-studs connection. (b-c) Central diagonal-stud connection. (Jiménez, 2015) 4.3.3 Connection by Contact The connection by contact is a simple connection usually used to joint diagonal timber elements with vertical and horizontal elements. In the case of Valparaiso timber frames this joint is used to connect the lower end of the diagonal element with the post and lower beam (Figure 4.13). The connection counts with two contact areas, one in contact with the post and another in contact with the lower beam. These contact areas are the ones that provides the strength to the joint. Another characteristic of this joint is that the connection requires the use of nails or metal fasteners to keep the joint together. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 58 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 4.3.3 Connection by contact (a) Diagonal and timber frame connection. (b) Frontal and lateral view. (Jiménez, 2015) Figure 4.3.3 Connection by contact photograph (Jiménez, 2015) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 59 CHAPTER 5 NUMERICAL ANALYSIS OF VALPARAISO TIMBER FRAME SHEAR WALLS 5.1 Valparaiso Timber Frame The following sections are focused into performing a numerical analysis of representative timber frame cases of the historical center of Valparaíso. The numerical analysis consisted of a nonlinear static analysis that considers the full load application of the dead and vertical loads, and a displacement control of the applied horizontal displacement. Other considerations including physical and mechanical properties were taken into account for the models. Several numerical analyses were performed: • Elementary cell • Window opening cell • Door opening cell • Shear walls – Case 1 and Case 2 5.1.1 Geometry and Materials According to Jiménez (2015) the geometry of the timber frame from Valparaiso is composed of posts, beams and diagonal elements with sections that could vary from 0.10 m x 0.10 m to 0.15 m x 0.15 m. For the numerical models it was assumed a cross section of 0.10 m x 0.15 m, being this the most typical for façade elements. The configuration of the frame varies depending of the type and quantity of openings, and the number of floors, but the posts are spaced from 0.40 m to 0.60 m, having an average length of 3.6 m, as explained in Section 4.2.1. The timber species most commonly used for traditional timber housing in Valparaíso in the studied period were the Chilean Oak and Oregon Pine, according to Jiménez (2015) and Jorquera (2016). However, these two materials have different mechanical characteristics. The Oak is denser than the Oregon Pine and has a higher elastic modulus as shown in Table 5.1. According to the structural inspections carried out by Jiménez (2015), Chilean Oak timber elements are commonly used to Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 60 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS configure façade frames and joist stories, while Oregon Pine elements to configure the internal partitions. Nevertheless, and due to the vernacular conditions of timber frames houses in Valparaíso, the configuration of the building can vary. The assumption of either of these wood species as the constitutive material of timber frames in Valparaíso would lead to considerably different structural behaviors. Since the shear walls to study are façade frames, only Oak was considered for the numerical models. Table 5.1.1 Material Properties of Chilean wood (Perez 1990 and INFOR 2010). Wood Specie Chilean Oak Oregon pine Modulus of elasticity [kN/m2] 1.232E+07 9.327E+06 Density [kg/m3] 492 344 Poisson Modulus 0.3 0.3 5.1.2 Restraint and Loading Conditions To simulate the base restraint conditions, for all the models, the nodes from the bottom beam are restrained in the vertical and horizontal directions. In order to apply the prescribed horizontal displacement for the analysis, the top beam is restrained in the horizontal direction. The loading conditions applied to the model include a vertical load and a prescribed horizontal displacement. The horizontal displacement was of 0.1 m, applied in increments for the nonlinear analysis. While the vertical load included the dead load of the beams, the joists and timber boarding of each floor. The total vertical load was calculated based on the structural survey carried out by Jiménez (2015). It was assumed that the internal walls are spaced at 4 m and that the slabs act in one direction. The loading guide for typical house loadings from evolution was used as a reference to obtain the floor loads shown in Table 5.2. The loading conditions varied for the two main representative cases of study that will be described in the following sections. For Case 1 the calculated tributary area was 22.84 m2, while for Case 2 was 223.34 m2. Due to this the loads to be consider per floor for each case resulted to be different. Finally, an occupancy load of 0.6 kN/m2 was considered for both cases. Table 5.3 show the loads to be consider per floor for each case. Table 5.1.2 Floor loads considerations. (evolution, n.d.) Joists (150mm x 50mm @ 450mm c/c) 0.12 kN/m2 Timber boarding 0.07 kN/m2 Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 61 Table 5.1.2 Distributed load to be consider per floor for each case of study. Case Case 1 Case 2 Joists 2.74 kN 2.80 kN Timber boarding 3.20 kN 3.27 kN Oak weight 0.89 kN 1.05 kN Occupancy 13.7 kN 14 kN Total distributed load 1.67 kN/m 1.46 kN/m 5.2 Application to Representative Cases of the Historical Center of Valparaiso The finite element software SAP2000 was used for the numerical models of timber frames from Valparaiso, since it provides a simpler application for modelling the nonlinear behavior of the carpentry joints. Before the model is analyzed it is calibrated to obtain the nonlinear behavior caused by the carpentry joints explained and described in the previous chapter. Since there are no previous experimental or analytical studies of the timber frames from Valparaiso, other experimental works were used as a reference for the calibration process. These are the Pombalino timber frame (Poletti, 2013) and the Quincha timber frame (Quinn, 2017), both described in Chapter 2 and explained for the calibration process in Chapter 3. Both frames have similarities with the timber frames from Valparaiso, specifically with the carpentry joint typologies. The Pombalino timber frame is used as a reference to relate the behavior of the connection by contact found in the timber frame from Valparaiso. However, the Quincha timber frame is used as a reference to relate the behavior of the mortise and tenon connection, and the notched connection. To model the connection by contact, the approach used by Ciocci (2015) in the analytical model of Pombalino was considered. To adjust the behavior of the joint to the case of Valparaiso, the component method was applied to obtain the appropriate contact stiffness by the following equation (5.1), 𝑘𝑐=𝐸𝛼√𝑏 𝑑 𝑐 (5.1) where, 𝑏 and 𝑑 are the dimensions of the contact area, 𝐸𝛼 is the modulus of elasticity according to the direction to the grain, and 𝑐 is a coefficient depending on the ratio between 𝑏 and 𝑑, and the coefficient of Poisson. The modulus of elasticity was evaluated according to equation (5.2), Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 62 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 𝐸𝛼=𝐸0∗𝐸90 𝐸0(𝑆𝑖𝑛 𝛼)2+𝐸90(𝐶𝑜𝑠 𝛼)2 (5.2) where 𝐸0 and 𝐸90 are the parallel and perpendicular moduli of elasticity and 𝛼 is the grain direction. Assuming that Oak has a strength class and characteristic values classified as D30 according to EN 338, 𝐸0 is 10 kN/mm2 and 𝐸90 is 0.64 kN/mm2. Finally, considering that 𝛼=71° , the modulus of elasticity resulted in 7.10E+05 kN/m2. To obtain the contact stiffness a contact area of 0.0237 m2 was considered and the stiffness resulted in 9.117E+04 kN/m. Finally, to obtain the yielding force and the yielding displacement equations 5.3 and 5.4 were applied considering 𝑓90 =8 N/mm2. 𝐹𝑦 resulted in 189.72 kN and 𝑑𝑦 resulted in 0.00208 m. By assuming an ultimate displacement of 0.05 m the force-deformation relation, shown in Figure 5.1, was obtained. 𝐹𝑦=𝐴𝑐∗𝑓90 (5.3) 𝑑𝑦=𝐹𝑦 𝑘 (5.4) Figure 5.2 Force-deformation for the connection by contact. To model the mortise and tenon connection, the approach proposed by Quinn (2017) for the Quincha timber frame model was considered. To adjust the behavior of the joint to the case of Valparaiso the relation proposed by Quinn between the tenon geometry and the rotational stiffness, shown in Figure 5.2, was used. Considering a tenon length of 50 mm the rotational stiffness resulted in 8 kNm/rad and the moment-rotation relation, shown in Figure 5.3, was obtained. For the translational stiffness the Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 63 stress-strain relation described by Quinn was assumed for the behavior of the mortise and tenon connections, as shown in Figure 5.4. Figure 5.2. Variation in stiffness with length of tenon. (Quinn, 2017) Figure 5.2 Moment-rotation diagram for the mortise and tenon connection. Figure 5.2 Stress-strain relationship for the mortise and tenon connection. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 70 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 5.2.4Case 1: Montealegre #138. (Jiménez, 2015) Figure 5.2.4 Case 1 Montealegre 138 (a) main façade. (b) modelling idealization. (Jiménez, 2015) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 71 Figure 5.2.4 Case 1 Montealegre 138: numerical model. Figure 5.2.4 Case 1 Montealegre 138: hinges. Figure 5.2.4 Case 1 Montealegre 138: releases. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 72 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 5.2.4 Case 1 Montealegre 138: numerical pushover curve. The second case of study (Case 2) is Almirante Montt #491: The house from Almirante Montt #491, shown in Figure 5.21 and 5.22, is characterized by three storeys configured with timber frames. For this study, the main façade timber frame wall will be analyzed. The composition of the wall includes three storeys with six window openings at each floor. The frame is composed of beams, posts and diagonal bracings, as shown in Figure 5.23(b). To develop the numerical model, the same characteristics and assumptions used for the simple models (elementary cell, door opening and window opening) were considered. Figure 5.24 shows the setup of the numerical model including the loading and boundary conditions. Figures 5.25 and 5.26 show the location of the application of the hinges for modeling the nonlinear behavior of the connections, and the releases considered. The obtained pushover curve of the shear wall is shown in Figure 5.27. The curve shows that nonlinear behavior starts to take place when the lateral load reaches 8 kN and that the maximum load at 0.1 m displacement is 22 kN. Figure 5.2.4 Case 2: Almirante Montt #491. (Jiménez, 2015) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 73 Figure 5.2.4 Case 2: Almirante Montt #491. (Jiménez, 2015) Figure 5.2.4 Case 2: Almirante Montt 49 (a) main façade. (b) modelling idealization. (Jiménez, 2015) Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 74 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 5.2.4 Case 2 Almirante Montt 49: numerical model. Figure 5.2.4 Case 2 Almirante Montt 49: hinges. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 75 Figure 5.2.4 Case 2 Almirante Montt 49: releases. Figure 5.2.4 Case 2 Almirante Montt 49: numerical pushover curve. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 76 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 77 CHAPTER 6 CONCLUSIONS 6.1 Summary In order to represent the behavior of a traditional timber frame wall, two-dimensional models were developed following the lumped plasticity modelling approach and taking into account the nonlinearities produced by the carpentry connections. The analysis performed was a nonlinear static (pushover) analysis. Two experimental campaigns were studied in order to perform a valid modelling calibration that can be applied to a timber frame wall without previous studies. Two modeling softwares, SAP2000 and DIANA FEA, were used in order to validate the calibration. Finally, two cases of timber frames from Valparaiso, Chile, were studied and analyzed. The models were able to capture the stiffness of the timber frame walls, the deformation, the nonlinear behavior and the expected mechanisms that where based in the calibrations of Pombalino and Quincha frames. However, a more detailed study is needed to obtain more accurate results. 6.2 Outcomes of the Study The analysis performed on timber frame walls demonstrated the importance of the connections and their influence in the global behavior of the frame. It was understood, that the connections are the weakest location of the frame, where failure can occur. Based on the research performed for this study, it was concluded that the behavior of the connections will depend of the type of joint, the constructive technique, the geometry, the wood properties, and their location in the frame. The connections behavior was modeled by a calibration process based in previous experimental campaigns. The outcome of the numerical calibration resulted with a good approximation compared to the experimental results. Since the model calibration was performed by two softwares, SAP2000 and DIANA FEA, part of the outcome of this study is to identify the advantages and disadvantages for their application in each software. SAP2000 resulted to provide a more user-friendly approach for the modelling of the nonlinear behavior of the carpentry connections, by using concentrated nonlinear Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Title of the thesis Erasmus Mundus Programme 78 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS hinges. Meanwhile, in DIANA FEA the nonlinearities were represented by spring elements that needed to consider assumptions like: the size of the spring, their working direction, and the way the elements that were connected are tied. The application in this case can result more time consuming and less accurate. It was concluded that the selection of the software to be used for modeling is very important and it will depend of the expected outcome. An advantage of SAP2000 is the simplicity and time efficiency in the case of applying the connections behavior to the model. However, an advantage of DIANA FEA is that it provides the required tools to model the interaction between different materials. This will be useful to further develop a numerical model that not only considers the nonlinearity of the connections, but also the interaction between timber and masonry. The calibration of the models resulted to be a very important step in this study, since the numerical models of typical typologies of timber frames form Valparaiso were based on this calibration. The outcome of this study was to contribute to the development of a numerical model of a real case study for which there is no previous experimental research. The purpose for the development of these models was to create a methodology that can be applied to similar cases where there is no previous information about the structural behavior of the frame. The work carried out seek to search for a correct approximation to model timber frame structures in cases where there are no experimental studies that allow to obtain more accurate results from the numerical models. Is important to mention that results obtained for the cases of Valparaiso are qualitative results, since only the geometry of the real case study was considered, and the mechanical behavior of the joints was assumed based in experimental studies of similar cases. A more critical analysis of the mechanisms of this specific typology and a more detailed calibration of the connections is needed to obtain more accurate results. It was noted that the capacity obtained from the models of the walls of Valparaiso resulted low when compared with the behavior of other timber frames like for example Pombalino. The behavior of the wall can be improved if other detailed studies are carried out for the calibration of the model. 6.3 Suggestions for Future Research This topic results very interesting and with a high possibility of extension for research. The numerical calibrations and analyses performed in this study are a contribution to the analysis of traditional timber frame walls. One of the further developments could be to perform a detailed study of the timber frames from Valparaiso and from the joints that characterize the frame by performing experimental works. This could lead to less qualitative and more accurate results for the mechanisms of the walls. Another interesting option could be to develop more detailed analytical studies of the carpentry joints, in order to obtain a more accurate calibration of the model. Vulnerability Analysis of Timber Frame Walls Under Seismic Loading Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 79 The timber frame typologies studied in this thesis are characterized by having infill and interaction with masonry walls. However, for this study only the timber frame was considered. It could be interesting and useful to develop a numerical model where the interaction between the two materials (timber and masonry) is included. The infill provides additional stiffness to the frame and there are many variations of the materials used, depending of the typology and region. An interesting further research could be to study how the variations of materials, based on their mechanical characteristics, affect the stiffness and global behavior of a wall. Another possibility of development is to study the seismic vulnerability of the timber frame. For this purpose, the following procedure is suggested: (a) Study and analyze different structural typologies of timber frames. (b) Apply the spectrum capacity method to determine the seismic response of the structure subjected to different seismic scenarios. (c) Calculate the fragility curves to obtain a probabilistic study of the levels of damage of the structure, considering different seismic demands.