On the Insurmountable Size of Truss-like Structures

Galileo postulated the existence of an insurmountable size for stone columns bearing a useful load as the size for which the structure is only able to resist its self-weight. Herein a method for the determination of the unsurmountable size for truss-like structures is shown, given the form of these structures and the ratio between the allowable stress and the specific weight of the material (the material structural scope). Three types of bars are considered: straight bars, with solid and hollow rectangular cross-section, and catenary bars with circular cross-section —a limit and theoretical case for estimating a meaningful upper bound of the structural scope—. An approximate rule to estimate the structural efficiency —here named GA rule— is shown, and is compared with numerical solutions using the proposed method.

Two-dimensional mesh optimization in the finite element method

The solution to the problem of finding the optimum mesh design in the finite element method with the restriction of a given number of degrees of freedom, is an interesting problem, particularly in the applications method. At present, the usual procedures introduce new degrees of freedom (remeshing) in a given mesh in order to obtain a more adequate one, from the point of view of the calculation results (errors uniformity). However, from the solution of the optimum mesh problem with a specific number of degrees of freedom some useful recommendations and criteria for the mesh construction may be drawn. For 1-D problems, namely for the simple truss and beam elements, analytical solutions have been found and they are given in this paper. For the more complex 2-D problems (plane stress and plane strain) numerical methods to obtain the optimum mesh, based on optimization procedures have to be used. The objective function, used in the minimization process, has been the total potential energy. Some examples are presented. Finally some conclusions and hints about the possible new developments of these techniques are also given.

Comparison of the seismic behaviour between three building tests, all based on a two storey model house

Integral Masonry System consisting of intersecting steel trusses alo ng each of the three dimensional directions of space on walls and slabs using any masonry material, had yet been backed up by the previous adobe test for seismic areas. This paper presents the comparison this last test and the adaptation of the IMS using h ollow brick. A prototype based on a two storey model house (6mx6mx6m) has being also built in two different scales in order to maximize the load and size of the shake table: the first one half size the whole building (3mx3mx3m) and the second, a quarter of the real size (3mx3mx6m). Both tests have suffered some mild to moderate damages while supporting the higher seismic action subjected by the shake table, without even fissuring the first test and with very few damages the second one. The thickness of the hollow brick wall and the diameter of the tree - dimensional truss reinforcement were scaled to the real size test in order to ascertain its great structural behaviour in relation to the previous structural model calculations. The aim of this study is to sum marize the results of the research collaboration between the ETSAM - UPM and the PUCP in whose laboratory these tests were carried out.

El lenguaje de la estructura: Mies van der Rohe y la construcción con acero y hormigón

John Summerson, en El Lenguaje Clásico de la Arquitectura, defiende que los órdenes clásicos que empleaban los arquitectos romanos para decorar sus edificios, no tienen una función estructural pero hacen expresivos a los edificios. Les hacen hablar. Arthur Schopenhauer afirma que “el destino de la Bella Arquitectura es poner de manifiesto la lucha entre el peso y la rigidez de los elementos estructurales”. Y Auguste Perret define la Arquitectura como “el arte de hacer cantar al punto de apoyo”. El objetivo de esta Tesis Doctoral es profundizar en la capacidad de expresión de la estructura. A través del estudio de las estructuras históricas, que se realiza en la primera parte de la Tesis, podemos concluir que existen tres categorías, tres maneras de expresión de la Estructura. Estructuras Vistas, que hablan, Estructuras Ocultas, que se esconden y Estructuras Ilusorias, que fingen. El Partenón y la Sainte Chapelle de París se estudian en el apartado de Estructuras Vistas. El Panteón, el Palacio de Carlos V, la Catedral de San Pablo en Londres, y otras arquitecturas renacentistas y romanas, en el apartado de Estructuras Ocultas. Y como Estructuras Ilusorias, la Alhambra (Dos Hermanas, Comares, el Patio de los Leones), Santa Sofía, y otras arquitecturas del barroco italiano. En la segunda parte de la Tesis se analiza la obra completa de Mies van der Rohe desde el punto de vista de esas tres categorías. Lo visto, lo oculto y lo ilusorio en las estructuras de Mies. Se estudia la evolución en la estructura de la casa, desde las primeras casas con Estructura Oculta de muro de ladrillo, hasta las últimas casas con Estructura Vista y columnas adelantadas, pasando por una etapa intermedia de casas con estructura mixta de muro de ladrillo en la que el acero comienza a hacer su aparición. Se analizan también seis soluciones estructurales en los Bloques y en las Torres: Estructura Vista reverberante, expresiva o inexpresiva vs Estructura Oculta con vestido horizontal, vestido reticular o vestido vertical. Y por último, se estudian las tres soluciones de Estructura Ilusoria que emplea Mies en sus Pabellones. La metodología de trabajo que se ha empleado se divide en cuatro apartados: El análisis bibliográfico; el análisis in situ de los edificios, que nos permite comprobar, por ejemplo, los efectos lumínicos de la columna acanalada del Partenón, o el efecto reverberante de las columnas de la Weissenhofsiedlung; el análisis crítico de planos y detalles constructivos, que nos lleva a concluir que la disposición de pantallas del Pabellón de Barcelona anula la lectura de la crujía estructural, y que la columna del restaurante Cantor se dispone con su alma perpendicular a la cercha, y no paralela, como cabría suponer si se quisiera aprovechar toda la capacidad portante del perfil en H; y por último, el análisis numérico y estructural, que nos lleva a confirmar el sobredimensionado de la Estructura del Patio de los Leones de la Alhambra o el sobredimensionado de la Estructura de la Casa Farnsworth. Lo que se confirma con esta Tesis Doctoral es que la Estructura es algo más, mucho más, que sólo transmisión de las cargas. ABSTRACT John Summerson, in The Classic Language of Architecture, argues that the classic orders used by Roman architects in the decoration of their buildings did not have a structural function, but made buildings expressive. They make them speak. Arthur Schopenhauer affirms that “the goal of Great Architecture is to highlight the struggle between the gravity and rigidity of structural elements”. And Auguste Perret defines Architecture as “the art of making the points of support sing”. The objective of this Doctoral Thesis is to examine the expressive capacity of structure. Following a study of historic structures in the first part of the thesis, we conclude that three categories exist, three ways of expressing Structure. Visible Structures that speak, Concealed Structures that are hidden and Illusory Structures that pretend. The Parthenon and the Sainte Chapelle in Paris are studied in the section on Visible Structures. The Pantheon, the Palace of Charles V, Saint Paul’s Cathedral in London, and other Renaissance and Roman architectures are dealt with in the Concealed Structures section. And, as examples of Illusory Structures, we focus on the Alhambra (The Hall of the Two Sisters, the Comares and the Court of the Lions), Saint Sophia, and other Italian Baroque architectures. In the second part of the Thesis the complete work of Mies van der Rohe is analysed from the perspective of these three categories. The visible, the concealed and the illusory, in the structures of Mies. We study how the structure of the house evolves, from the first houses with the Hidden Structure of the brick wall, to the later houses with Visible Structures and columns, via an intermediate phase of mixed-structure houses with brick walls, where steel first began to make its appearance. We also analyse six structural solutions in the Blocks and Towers: reverberant, expressive or inexpressive Visible Structure vs Concealed Structure with horizontal cladding, reticular cladding or vertical cladding. And finally, we look at the three Illusory Structure solutions that Mies employs in his Pavilions. The methodology employed is divided into four sections: a bibliographic analysis; an analysis in situ of the buildings, which allows us to test, for example, the lighting effects of the fluted column in the Parthenon, or the reverberant effect of the Weissenhofsiedlung columns; a critical analysis of plans and constructive details, which leads us to conclude that the arrangement of panels in the Barcelona Pavilion cancels out the structural centreline, and that the column in the Cantor restaurant is placed with its web perpendicular to the truss, and not parallel to it, as one might expect if one wanted to avail of all the load-bearing capacity of the H beam; and lastly, a numeric and structural analysis, which confirms the oversizing of the Court of the Lions structure in the Alhambra or the oversized structure of Farnsworth House. All of which confirms in this Doctoral Thesis that structure is something more, much more, than a mere conveyor of loads.

Structural efficiency variation with the problem size in some bending problems = Variación del rendimiento estructural con el tamaño del problema en algunos casos de flexión simple

Existe normalmente el propósito de obtener la mejor solución posible cuando se plantea un problema estructural, entendiendo como mejor la solución que cumpliendo los requisitos estructurales, de uso, etc., tiene un coste físico menor. En una primera aproximación se puede representar el coste físico por medio del peso propio de la estructura, lo que permite plantear la búsqueda de la mejor solución como la de menor peso. Desde un punto de vista práctico, la obtención de buenas soluciones—es decir, soluciones cuyo coste sea solo ligeramente mayor que el de la mejor solución— es una tarea tan importante como la obtención de óptimos absolutos, algo en general difícilmente abordable. Para disponer de una medida de la eficiencia que haga posible la comparación entre soluciones se propone la siguiente definición de rendimiento estructural: la razón entre la carga útil que hay que soportar y la carga total que hay que contabilizar (la suma de la carga útil y el peso propio). La forma estructural puede considerarse compuesta por cuatro conceptos, que junto con el material, definen una estructura: tamaño, esquema, proporción, y grueso.Galileo (1638) propuso la existencia de un tamaño insuperable para cada problema estructural— el tamaño para el que el peso propio agota una estructura para un esquema y proporción dados—. Dicho tamaño, o alcance estructural, será distinto para cada material utilizado; la única información necesaria del material para su determinación es la razón entre su resistencia y su peso especifico, una magnitud a la que denominamos alcance del material. En estructuras de tamaño muy pequeño en relación con su alcance estructural la anterior definición de rendimiento es inútil. En este caso —estructuras de “talla nula” en las que el peso propio es despreciable frente a la carga útil— se propone como medida del coste la magnitud adimensional que denominamos número de Michell, que se deriva de la “cantidad” introducida por A. G. M. Michell en su artículo seminal de 1904, desarrollado a partir de un lema de J. C. Maxwell de 1870. A finales del siglo pasado, R. Aroca combino las teorías de Galileo y de Maxwell y Michell, proponiendo una regla de diseño de fácil aplicación (regla GA), que permite la estimación del alcance y del rendimiento de una forma estructural. En el presente trabajo se estudia la eficiencia de estructuras trianguladas en problemas estructurales de flexión, teniendo en cuenta la influencia del tamaño. Por un lado, en el caso de estructuras de tamaño nulo se exploran esquemas cercanos al optimo mediante diversos métodos de minoración, con el objetivo de obtener formas cuyo coste (medido con su numero deMichell) sea muy próximo al del optimo absoluto pero obteniendo una reducción importante de su complejidad. Por otro lado, se presenta un método para determinar el alcance estructural de estructuras trianguladas (teniendo en cuenta el efecto local de las flexiones en los elementos de dichas estructuras), comparando su resultado con el obtenido al aplicar la regla GA, mostrando las condiciones en las que es de aplicación. Por último se identifican las líneas de investigación futura: la medida de la complejidad; la contabilidad del coste de las cimentaciones y la extensión de los métodos de minoración cuando se tiene en cuenta el peso propio. ABSTRACT When a structural problem is posed, the intention is usually to obtain the best solution, understanding this as the solution that fulfilling the different requirements: structural, use, etc., has the lowest physical cost. In a first approximation, the physical cost can be represented by the self-weight of the structure; this allows to consider the search of the best solution as the one with the lowest self-weight. But, from a practical point of view, obtaining good solutions—i.e. solutions with higher although comparable physical cost than the optimum— can be as important as finding the optimal ones, because this is, generally, a not affordable task. In order to have a measure of the efficiency that allows the comparison between different solutions, a definition of structural efficiency is proposed: the ratio between the useful load and the total load —i.e. the useful load plus the self-weight resulting of the structural sizing—. The structural form can be considered to be formed by four concepts, which together with its material, completely define a particular structure. These are: Size, Schema, Slenderness or Proportion, and Thickness. Galileo (1638) postulated the existence of an insurmountable size for structural problems—the size for which a structure with a given schema and a given slenderness, is only able to resist its self-weight—. Such size, or structural scope will be different for every different used material; the only needed information about the material to determine such size is the ratio between its allowable stress and its specific weight: a characteristic length that we name material structural scope. The definition of efficiency given above is not useful for structures that have a small size in comparison with the insurmountable size. In this case—structures with null size, inwhich the self-weight is negligible in comparisonwith the useful load—we use as measure of the cost the dimensionless magnitude that we call Michell’s number, an amount derived from the “quantity” introduced by A. G. M. Michell in his seminal article published in 1904, developed out of a result from J. C.Maxwell of 1870. R. Aroca joined the theories of Galileo and the theories of Maxwell and Michell, obtaining some design rules of direct application (that we denominate “GA rule”), that allow the estimation of the structural scope and the efficiency of a structural schema. In this work the efficiency of truss-like structures resolving bending problems is studied, taking into consideration the influence of the size. On the one hand, in the case of structures with null size, near-optimal layouts are explored using several minimization methods, in order to obtain forms with cost near to the absolute optimum but with a significant reduction of the complexity. On the other hand, a method for the determination of the insurmountable size for truss-like structures is shown, having into account local bending effects. The results are checked with the GA rule, showing the conditions in which it is applicable. Finally, some directions for future research are proposed: the measure of the complexity, the cost of foundations and the extension of optimization methods having into account the self-weight.