Análisis de parámetros de fendas para la estimación de las propiedades mecánicas de vigas de madera de Pinus sylvestris L. Analysis of parameters for assessing drying fissures to estimate their influence on the mechanical properties of Pinus sylvestris L. wooden beams

El principal objetivo de este estudio es evaluar la influencia de las fendas de secado en las propiedades mecánicas de vigas de madera. Para esto, se utilizan 40 vigas de Pino silvestre (Pinus sylvestris L) de 4200 mm de longitud y 150x200 mm de sección que fueron ensayadas según norma EN 408. Las fendas se registran detalladamente atendiendo a su longitud y posición en cada cara de la viga, y midiendo el espesor y la profundidad cada 100mm a lo largo de la viga. Solo el 10% de la muestra es rechazada por las fendas, según los criterios establecidos por la norma española de clasificación visual UNE 56544. Para evaluar la influencia de las fendas en las propiedades mecánicas, se usan tres parámetros globales basados en el área, el volumen o la profundad de la fenda, y dos locales basados en la profundidad máxima y la profundidad en la zona de rotura. Además se determina la densidad de las piezas. Estos parámetros se comparan con las propiedades mecánicas (tensión de rotura, módulo de elasticidad y energía de rotura) y se encuentra escasa relación entre ellos. Las mejores correlaciones se encuentran entre los parámetros relacionados con la profundidad de las fendas, tanto con el módulo de elasticidad como con la tensión de rotura. The aim of this study is the evaluation of the influence of drying fissures on the mechanical properties of timber beams. For that purpose, 40 sawn timber pieces of Scots pine (Pinus sylvestris L.) with 150x200 mm in cross-section and 4200 mm in length have been tested according to EN 408, obtaining MOR and MOE. The fissures were registered in detail measuring their length and position in each face of the beam, and the thickness and depth every 100 mm in length. Only 10 % of the pieces were rejected because fissures, according to UNE 56544 Spanish visual grading standard. To evaluate the influence of fissures in mechanical properties three global parameters: Fissures Area Ratio or ratio between the area occupied by fissures and the total area in the neutral axis plane of the beam; Fissures Volume Ratio or ratio between volume of fissures and the total volume of the beam; Fissures Average Depth and two local parameters were used: Fissures Maximum Depth in the beam, and Fissures Depth in the broken zone of the beam. Also the density of the beams was registered. These parameters were compared with mechanical properties (tensile strength, elasticity modulus, and rupture energy) and the relationship between them had not been founded. The best relationship was founded between the elasticity modulus y the tensile strength with the parameters which included the depth of the fissures.

Evaluation of the mechanical properties of self compacting concrete using current estimating models. Estimating the modulus of elasticity, tensile strength, and modulus of rupture of self compacting concrete

This study includes an analysis of the applicability of current models used for estimating the mechanical properties of conventional concrete to self-compacting concrete. The mechanical properties evaluated are: modulus of elasticity, tensile strength, and modulus of rupture. An extensive database which included the dosifications and the mechanical properties of 627 mixtures from 138 different references, was used. The models considered are: ACI, EC-2, NZS 3101:2006 (New Zealand code) and the CSA A23.3-04 (Canadian code). The precision in estimating the modulus of elasticity and tensile strength is acceptable for all models; however, all models are less precise in estimating the modulus of rupture.

Evaluation of the influence of visual parameters on wave transmission velocity in sawn chestnut timber

Non-destructive, visual evaluation and mechanical testing techniques were used to assess the structural properties of 374 samples of chestnut (Castanea sativa). The principal components method was applied to establish and interpret correlations between variables obtained of modulus of elasticity, bending strength and density. The static modulus of elasticity presented higher correlation values than those obtained using non-destructive methods. Bending strength presented low correlations with the non-destructive parameters, but there was some relation to the different knot ratios defined. The relationship was stronger with the most widely used ratio, CKDR. No significant correlations were observed between any of the variables and density.

Caracterización de la madera de Castanea sativa Mill. de procedencia española

La madera de castaño de procedencia española no está presente como material estructural en la norma europea EN 1912, y por lo tanto no está asignada a ninguna clase resistente como sí lo están otras especies españolas. Por ello, en el presente trabajo se realizó una caracterización de madera aserrada de castaño de Asturias con fines estructurales, realizando los ensayos según la norma UNE EN 408:2004, y calculando sus valores característicos según la norma UNE EN 384:2010. Los valores obtenidos permitieron asignar una clase resistente de las establecidas en la norma UNE EN 338:2010. Fueron evaluadas un total de 260 probetas de dos secciones (40x100 y 40x150 mm) siendo asignada una clase resistente D24, resultando la densidad y la resistencia a flexión los parámetros limitantes y observándose un valor de módulo de elasticidad superior al correspondiente a dicha clase. La relación entre los módulo de elasticidad longitudinal y transversal obtenidos experimentalmente fue de 10 aproximadamente, mientras que la normativa establece, como valor genérico, 16. Se observó que la singularidad de la madera que provocó un mayor porcentaje de rotura fue la presencia de nudos, presentando estas probetas un valor de resistencia significativamente menor. Chestnut timber from Spain is not included as a structural timber in the European standard EN 1912 nor is it assigned to any strength class like other Spanish species. Therefore, a characterization of structural chestnut timber from Asturias was made according to the UNE EN 408:2004, and the characteristic values were calculated according to the UNE EN 384:2010. The values obtained allowed the assignment of a strength class according to UNE EN 338:2010. 260 samples of two sections (40x100 and 40x150 mm) were tested and a D24 strength class was assinged. Density and bending strength were the limiting parameters, and the value of modulus of elasticity was higher than the values asssigned to D24. The relationship between modulus of elasticity and the shear modulus obtained was approximately equal to 10, while this value in the UNE EN 384:2010 is set at 16. The characteristic of the wood which caused a higher percentage of failure was the presence of knots, resulting in significantly lower resistance values.

Polypropylene fiber-reinforced self-compacting concrete for the reconstruction of the cathedral of La Laguna, Canary Island, Spain

High performance materials are needed for the reconstruction of such a singular building as a cathedral, since in addition to special mechanical properties, high self compact ability, high durability and high surface quality, are specified. Because of the project’s specifications, the use of polypropylene fiber-reinforced, self-compacting concrete was selected by the engineering office. The low quality of local materials and the lack of experience in applying macro polypropylene fiber for structural reinforcement with these components materials required the development of a pretesting program. To optimize the mix design, performance was evaluated following technical, economical and constructability criteria. Since the addition of fibers reduces concrete self-compactability, many trials were run to determine the optimal mix proportions. The variables introduced were paste volume; the aggregate skeleton of two or three fractions plus limestone filler; fiber type and dosage. Two mix designs were selected from the preliminary results. The first one was used as reference for self-compactability and mechanical properties. The second one was an optimized mix with a reduction in cement content of 20 kg/m3and fiber dosage of 1 kg/m3. For these mix designs, extended testing was carried out to measure the compression and flexural strength, modulus of elasticity, toughness, and water permeability resistance

Clasificación visual estructural de la madera de pino radiata procedente de Cataluña

Se han clasificado visualmente, según la norma española UNE 56544; 150 piezas de madera aserrada de pino radiata procedentes de Cataluña (España) con una sección de 80x150 mm y una longitud de 2.500 mm. Esta norma establece dos calidades visuales (ME-1 y ME-2) para piezas con grueso menor o igual que 70 mm y una única calidad visual (MEG) para gruesos mayores de 70 mm. Con el fin de conocer el efecto frontera entre la pequeña y la gran escuadría se clasificaron las piezas según ambas categorías. Se determinó la tensión de rotura en flexión, el módulo de elasticidad en flexión y la densidad de cada pieza mediante el ensayo de acuerdo con la norma UNE-EN 408 siguiendo los ajustes definidos en la norma UNE-EN 384. Un 53 % de las piezas se clasificaron como MEG con una resistencia característica a flexión de 28 N/mm2 y un valor medio del módulo de elasticidad de 9.900 N/mm2. Por otra, un 42 % de las piezas se clasificaron como ME-2 y sólo un 3 % como ME-1; en este caso las propiedades mecánicas de la calidad ME-2 no se alejan mucho de las de la calidad MEG. El elevado porcentaje de piezas rechazadas (47 % frente a la MEG) se debe principalmente al defecto de alabeo responsable de un 39 % de rechazo. Si no se considera el alabeo en la clasificación el porcentaje de rechazo desciende al 17 % y las propiedades mecánicas de la calidad MEG presentan un valor característico de resistencia a flexión de 21 N/mm2 y un módulo de elasticidad de 9.100 N/mm2. A sample of 150 pieces of radiata pine sawn timber with 180x120 mm in cross-section and 2.500 mm in length sourcing from Catalonia (Spain) were visually graded according to Spanish standard UNE 56544. This standard establish two visual grades (ME-1 and ME-2) for timber pieces with thickness equal or less than 70 mm, and one grade (MEG) for thickness bigger than 70 mm. In order to know the border effect between small and larger cross-section the pieces were graded in both categories. The modulus of rupture, modulus of elasticity and density were obtained by test according to EN 408 standard and the adjustments of EN 384. A 53 % of pieces were classified as MEG with a characteristic bending strength of 28 N/mm2 and a mean value of MOE of 9.900 N/mm2. On the other hand, a 42 % of pieces were classified as ME-2 and only a 3 % as ME-1; in this case the mechanical properties of ME-2 are not far from MEG properties. The high percentage of rejected pieces (47 % vs MEG) is mainly due to the twist defect with a 39 % of rejected pieces. If twist is not considered in the grading process the reject percentage decreases to 17 % and the mechanical properties of MEG grade present a characteristic value of bending strength of 21 N/mm2 and a MOE of 9.100 N/mm2.

Clasificación de madera estructural de gran escuadría de Pinus sylvestris l. mediante métodos no destructivos

Esta Tesis Doctoral aborda el estudio de algunas técnicas no destructivas para la clasificación de madera de pino silvestre (Pinus sylvestris L.) de procedencia española y de gruesa escuadría para uso estructural. Para la estimación del módulo de elasticidad y de la resistencia se han aplicado técnicas basadas en la propagación de una onda a través de la madera: onda de ultrasonidos (Sylvatest) o de impacto (Microsecond Timer) en dirección longitudinal, o vibración en dirección longitudinal y transversal (PLG). Para la estimación de la densidad se han utilizado métodos puntuales basados en el penetrómetro (Pilodyn) y en la resistencia al arranque de un tornillo. Las variables obtenidas han sido relacionadas con los resultados de la clasificación visual y con las propiedades de la madera determinadas mediante ensayo mecánico. Además, se ha estudiado la influencia de la humedad de la madera en la velocidad de propagación de la onda para definir factores de corrección a los equipos comerciales utilizados en esta Tesis Doctoral. La muestra de estudio está formada por 244 piezas procedentes de El Espinar, Segovia, con dimensiones nominales 150 x 200 x 4000 mm (218 piezas) y 100 x 150 x 3000 mm (26 piezas). De todas las piezas se tomaron datos de dimensiones, contenido de humedad y clasificación visual según la norma UNE 56544. En las primeras 218 vigas se aplicaron las técnicas de ultrasonidos, onda de impacto y vibraciones, se determinó la densidad de cada pieza completa y se ensayaron según la norma UNE-EN 408 para obtener el módulo de elasticidad global (en todos los casos) y local (en un porcentaje), así como de la tensión de rotura. Se extrajeron tres rebanadas para los ensayos puntuales y para el cálculo de la densidad. En las otras 26 piezas se repitieron los ensayos (transmisión de onda, vibración y clasificación visual) durante el proceso de secado natural, desde que la madera se encontraba húmeda (en torno al 40 %) hasta la humedad de equilibrio higroscópico (en torno al 9%). Respecto a la clasificación visual no se han observado diferencias significativas entre la calidad MEG o las rechazadas. Se han estudiado las consecuencias del secado (principalmente las deformaciones) y no se ha encontrado justificación para que estos defectos penalicen la clasificación. Para la densidad, el mayor R2 obtenido ha sido de un 47% a partir del uso combinado de los dos equipos puntuales (penetrómetro y arranque de tornillo). Para el módulo de elasticidad y la tensión de rotura, la mejor relación se ha obtenido a partir de la técnica de vibración longitudinal, con unos coeficientes de 79% y un 52% respectivamente. Se ha estimado que el aumento de un punto porcentual en el contenido de humedad de la madera produce una pérdida de velocidad de onda del 0,58% para Sylvatest y Microsecond Timer, y del 0,71% para PLG. Estos valores son generalizables para un rango de humedades entre 9 y 25 %. Abstract This Doctoral Thesis approach the study of some non-destructive techniques as a classification method for structural use of Scots pine wood of Spanish origin with large cross section. To estimate the modulus of elasticity and strength have been used techniques based on the propagation of a wave through the timber: ultrasonic wave (Sylvatest) or stress wave (Microsecond Timer) in longitudinal direction, or vibration in longitudinal and transversal direction (PLG). Local probing methods have been applied to estimate the density, based on penetrometer (Pilodyn) and the screw withdrawal resistance meter. The different variables obtained were compared with the results of the visual grading and the values of the properties of the wood determined by the standardized test of the pieces. Furthermore, the influence of the moisture content of the wood on the velocity of propagation of the waves through the timber has been analyzed in order to establish a correction factor for the commercial devices used in this Doctoral Thesis. The sample tested consists of 244 pieces from El Espinar, Segovia, with nominal dimensions 150 x 200 x 4000 mm (218 pieces) and 100 x 150 x 3000 mm (26 pieces). Data collection about dimensions, moisture content and visual grading according to the UNE 56544 standard were carried out on all the pieces. The first 218 pieces were tested by non destructive techniques based on ultrasonic wave, stress wave and vibration, the density was measured on each piece and bending test according to the UNE-EN 408 standard was carried out for calculating the global modulus of elasticity (all the pieces) and the local one (only a representative group), as well as the bending strength. Three slices were removed for implementing the local probing and to calculate the density. In the other 26 pieces the tests (wave transmission, vibration and visual grading) were repeated during the natural drying process, from wet timber (around 40 % moisture content) up to the equilibrium moisture content (around 9%). Regarding the visual grading no significant differences were observed between MEG or rejected pieces. The effects of drying (deformations) have been studied, and justification for the specification hasn't been found. To estimate the density, the greater R2 obtained was 47% by using both penetrometer and screw withdrawal. For the modulus of elasticity and bending strength, the best relationship has been found with the longitudinal vibration, with coefficients of 79% and 52% respectively. It has been estimated that an increase of a point of the moisture content of the wood produces a decrease on the velocity obtained from ultrasonic or stress wave of 0,58%, and 0,71 % for the one obtained from vibration. Those values can be generalized for a range of moisture content from 9 to 25 %.

Mechanical properties of self-consolidating concrete using conventional concrete models

The objective of this study is to analyze the applicability of current models used for estimating the mechanical properties of conventional concrete to self-consolidating concrete (SCC). The mechanical properties evaluated are modulus of elasticity, tensile strength,and modulus of rupture. As part of the study, it was necessary to build an extensive database that included the proportions and mechanical properties of 627 mixtures from 138 different references. The same models that are currently used for calculating the mechanical properties of conventional concrete were applied to SCC to evaluate their applicability to this type of concrete. The models considered are the ACI 318, ACI 363R, and EC2. These are the most commonly used models worldwide. In the first part of the study, the overall behavior and adaptability of the different models to SCC is evaluated. The specific characterization parameters for each concrete mixture are used to calculate the various mechanical properties applying the different estimation models. The second part of the analysis consists of comparing the experimental results of all the mixtures included in the database with the estimated results to evaluate the applicability of these models to SCC. Various statistical procedures, such as regression analysis and residual analysis, are used to compare the predicted and measured properties. It terms of general applicability, the evaluated models are suitable for estimating the modulus of elasticity, tensile strength, and modulus of rupture of SCC. These models have a rather low sensitivity, however, and adjust well only to mean values. This is because the models use the compressive strength as the main variable to characterize the concrete and do not consider other variables that affect these properties.

Estudio de hormigones reciclados no estructurales fabricados con árido reciclado mixto : propiedades mecánicas y expansión debida al contenido de sulfatos

El objetivo principal de este trabajo de investigación es estudiar las posibilidades de utilización del árido reciclado mixto para un hormigón reciclado en aplicaciones no estructurales, justificando mediante la experimentación la validez para esta aplicación, tanto del árido reciclado como del hormigón reciclado. Esta tesis se centró en los aspectos más restrictivos y limitativos en la utilización de los áridos mixtos en hormigón reciclado, basándose tanto en la normativa internacional existente como en los resultados obtenidos en los estudios bibliográficos consultados. La primera tarea realizada fue la caracterización completa de las propiedades del árido reciclado mixto, recogiendo especialmente los siguientes aspectos: granulometría, contenido de finos, absorción y densidades, composición del árido reciclado, índice de lajas, coeficiente de Los Ángeles, partículas ligeras y contenido de sulfatos. De este estudio de los áridos reciclados, se han destacado relaciones entre las propiedades. Las diferentes correlaciones permiten proponer criterios de calidad de un árido reciclado mixto para un hormigón reciclado. Se ha elegido un árido reciclado mixto entre los estudiados, de características límite admisibles, para obtener resultados conservadores sobre el hormigón reciclado fabricado con él. En una segunda etapa, se ha realizado un estudio de dosificación completo del hormigón reciclado, evaluando la consistencia del hormigón en estado fresco y la resistencia a compresión del hormigón en estado endurecido y se ha comparado con las mismas propiedades de un hormigón convencional. Se ha analizado la capacidad de absorción del árido conseguida con los métodos de presaturación empleados y en función de su estado de humedad, para poder evaluar las relaciones agua/cemento totales y efectivas del hormigón. Se ha estudiado el efecto de estos dos parámetros tanto en la consistencia como en la resistencia del hormigón reciclado. Finalmente, se ha estudiado el hormigón fabricado con un 50% y 100% de una partida de árido reciclado mixto de calidad admisible y se han ensayado las siguientes propiedades: consistencia, resistencia a compresión, resistencia a tracción indirecta, módulo de elasticidad dinámico, cambios de longitud, porosidad abierta y microscopía. Para analizar el efecto de los sulfatos, se han añadido artificialmente cantidades de yeso controladas en el hormigón reciclado. Se fabricaron hormigones con dos tipos de cemento, un cemento CEM I 42,5 R con elevado contenido de C3A, que debería dar lugar a expansiones mayores y un cemento con adiciones puzolánicas CEM II A-P 42,5 R, que atenuaría el comportamiento expansivo en el hormigón. Los resultados finales indican que la utilización del árido reciclado mixto en proporciones de hasta un 50%, permiten cubrir la gama de resistencias más exigentes dentro del hormigón no estructural. El contenido de sulfatos puede variar desde un 0,8% hasta un 1,9%, según el tipo de cemento y la proporción de sustitución del árido natural por árido reciclado mixto. Tanto en el caso del árido reciclado como en el hormigón, se ha realizado un estudio comparativo entre el conjunto de datos recopilados en la bibliografía y los obtenidos en este estudio experimental. En varias propiedades del hormigón reciclado, se han comparado los resultados con las fórmulas de la Instrucción EHE-08, para establecer unos coeficientes de corrección a aplicar a un hormigón reciclado con fines no estructurales. The main objective of this investigation work is to study the possibilities of using recycled mixed aggregate for a recycled concrete in non structural applications, justifying by means of experimentation both the validity of the recycled aggregate and recycled concrete. This thesis focused on the most restrictive and limiting aspects in the mixed aggregate use in recycled concrete, on the basis of the international standards as well on the results obtained in the bibliographic studies consulted. The first task achieved was the complete charcaterization of the mixed recycled aggregate properties, specially the following aspects: grain size analysis, fines content, absorption and densities, recycled aggregate composition, flakiness index, Los Angeles coefficient, lightweight particles and sulphate content. From this study, correlations between the properties were highlighted. The different correlations make possible to propose quality criterions for recycled mixed aggregate in concrete. Among the recycled aggregates studied, one of acceptable characteristics but near the limits established, was chosen to obtain conservative results in the recycled concrete made with it. In a second step, a complete recycled concrete mix design was made, to evaluate concrete consistency in the fresh state and concrete compressive strength in the hardened state and its properties were compared to those of a control concrete. The aggregate absorption capacity was analized with the presaturation methods achieved and in function of its state of humidity, to evaluate the total and effective water/cement ratios. The effect of these two parameters, both in consistency and compressive strength of recycled concrete, was studied. Finally, the concrete made with 50% and 100% of the elected recycled mixed aggregate was studied and the following concrete properties were tested: consistency, compressive strength, tensile strength, dynamic modulus of elasticity, length changes, water absorption under vacuum and microscopy. To analize the effect of sulphate content, some controlled quantities of gypsum were artificially added to the recycled concrete. Concretes with two types of cement were made, a cement CEM I 42,5 R with a high content of C3A, that would lead to major expansions and a cement with puzzolanic additions CEM II A-P 42,5 R that would lower the expansive behaviour of concrete. The final results indicate that the use of mixed recycled aggregate in proportions up to 50% make possible to cover the overall demanding strengths within the non structural concrete. Sulphates content can range between 0,8% and 1,9%, in function of the type of cement and the proportion of natural aggregate replacement by mixed recycled one. Both in the case of recycled aggregate and concrete, a comparative study was made between the data coming from the bibliography and those obtained in the experimental study. In several recycled concrete properties, the results were compared to the formulas of Spanish Instruction of Structural Concrete (Instruction EHE-08), to establish some correction coefficients to apply for a non structural recycled concrete.

Mechanical actuator for biomimetic propulsion and the effect of the caudal fin elasticity on the swimming performance

This paper presents a mechanical actuator for the biomimetic propulsion of swimming devices and the experimental study of the effect of the caudal fin elasticity on the overall performance. The design of the proposed drive allows the DC motor to operate at constant speed, so all the power of the motor is spent only for the motion of the caudal fin. A prototype of the actuator, in which the caudal fin serves as a driving element, is manufactured and tested in both laboratory and natural conditions. The swimming speed, the thrust efficiency and the maneuverability are evaluated for caudal fins with different stiffness. The caudal fin whose rigidity varies relative to both vertical and horizontal cross-section, exhibits the best performance. The achieved results also confirm that the proposed actuator could be of great interest to applications in the field of underwater operation, ocean investigation and environmental protection.

Boundary elements in potential and elasticity theory

A general theory that describes the B.I.E. linear approximation in potential and elasticity problems, is developed. A method to tread the Dirichlet condition in sharp vertex is presented. Though the study is developed for linear elements, its extension to higher order interpolation is straightforward. A new direct assembling procedure of the global of equations to be solved, is finally showed.

Quality grading of cork stoppers based on porosity, density and elasticity.

An objective control method for grading cork stoppers is presented using a cork stopper quality index based on porosity, density and elasticity, these being the properties which have the greatest influence on the closure capacity of the stopper. The elasticity of the cork stopper is measured through the relaxation ratio, which is defined by the relationship between the relaxation force exerted by the cork in the bottleneck and the compressive force exerted by a caliper to fit the stopper in the bottle. The relaxation ratio, defined in this way, represents the part of the compression force which is applied to the stopper on insertion and which is recovered in the form of the relaxation force to achieve closure. The calculation of the relaxation ratio involves the measurement of the relaxation force of the fitted stopper. This force has been measured rigorously and precisely using a device developed in the Cork Laboratory at the INIA-CIFOR and which is presented for the first time in this paper.

Variation throughout the tree stem in the physical-mechanical properties of the wood of Abies alba Mill. from the Spanish Pyrenees

This study analyses the variation of main physical-mechanical properties of wood along the longitudinal and radial directions of the tree for Abies alba Mill. growing in the Spanish Pyrenees. Small clear specimens were used to study the properties of volumetric shrinkage (VS), density (?), hardness (H), bending strength (MOR), modulus of elasticity (MOE), maximum compressive strength parallel to the grain (MCS) and impact strength (K). Several models of properties variation in the longitudinal and radial directions were analyzed. Main trends of variation of properties throughout the tree stem were identified although none of them could be fitted to predictive statistical models. Along the longitudinal direction, the properties studied followed a downward trend from the base to the crown, which was not significant in all cases, indicating that no differences in quality existed. Throughout the radial direction the trend is upward for the first 40-50 growth rings, after which it slopes downwards, more gently at first until rings 70-75 and then more steeply. This behaviour is related to variation in wood structure from the pith to the bark, depending on whether the wood is juvenile, sapwood or heartwood, and to wood maturity and microfibril angle. Authors encourage carrying further studies on other populations of A. alba in the Spanish Pyrenees to check if the trends found in this study apply to other provenances.

The p-Adaptive Biem Approach for Two-Dimensional Elasticity Analysis

This paper presents the development and application of the p-adaptive BIEM version in elastostatics. The basic concepts underlying the p-adaptive technique are summarized and discussed. Some Pascal pseudocodes which show the way how such a technique can be implemented easily in microcomputers are also provided. Both the applicability and the accuracy of the method proposed here are illustrated through a numerical example.

Optimal boundary geometry in an elasticity problem: a systematic adjoint approach

In different problems of Elasticity the definition of the optimal gcometry of the boundary, according to a given objective function, is an issue of great interest. Finding the shape of a hole in the middle of a plate subjected to an arbitrary loading such that the stresses along the hole minimizes some functional or the optimal middle curved concrete vault for a tunnel along which a uniform minimum compression are two typical examples. In these two examples the objective functional depends on the geometry of the boundary that can be either a curve (in case of 2D problems) or a surface boundary (in 3D problems). Typically, optimization is achieved by means of an iterative process which requires the computation of gradients of the objective function with respect to design variables. Gradients can by computed in a variety of ways, although adjoint methods either continuous or discrete ones are the more efficient ones when they are applied in different technical branches. In this paper the adjoint continuous method is introduced in a systematic way to this type of problems and an illustrative simple example, namely the finding of an optimal shape tunnel vault immersed in a linearly elastic terrain, is presented.