Edge function method applied to thin plates resting on Winkler foundation

The Edge Function method formerly developed by Quinlan⁽²⁵⁾ is applied to solve the problem of thin elastic plates resting on spring supported foundations subjected to lateral loads the method can be applied to plates of any convex polygonal shapes, however, since most plates are rectangular in shape, this specific class is investigated in this thesis. The method discussed can also be applied easily to other kinds of foundation models (e.g. springs connected to each other by a membrane) as long as the resulting differential equation is linear. In chapter VII, solution of a specific problem is compared with a known solution from literature. In chapter VIII, further comparisons are given. The problems of concentrated load on an edge and later on a corner of a plate as long as they are far away from other boundaries are also given in the chapter and generalized to other loading intensities and/or plates springs constants for Poisson's ratio equal to 0.2

Modelling the Effect of Climate Change on Environmental Pollution Losses from Dairy Systems in the UK

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High-strain composites and dual-matrix composite structures

Most space applications require deployable structures due to the limiting size of current launch vehicles. Specifically, payloads in nanosatellites such as CubeSats require very high compaction ratios due to the very limited space available in this typo of platform. Strain-energy-storing deployable structures can be suitable for these applications, but the curvature to which these structures can be folded is limited to the elastic range. Thanks to fiber microbuckling, high-strain composite materials can be folded into much higher curvatures without showing significant damage, which makes them suitable for very high compaction deployable structure applications. However, in applications that require carrying loads in compression, fiber microbuckling also dominates the strength of the material. A good understanding of the strength in compression of high-strain composites is then needed to determine how suitable they are for this type of application.

The goal of this thesis is to investigate, experimentally and numerically, the microbuckling in compression of high-strain composites. Particularly, the behavior in compression of unidirectional carbon fiber reinforced silicone rods (CFRS) is studied. Experimental testing of the compression failure of CFRS rods showed a higher strength in compression than the strength estimated by analytical models, which is unusual in standard polymer composites. This effect, first discovered in the present research, was attributed to the variation in random carbon fiber angles respect to the nominal direction. This is an important effect, as it implies that microbuckling strength might be increased by controlling the fiber angles. With a higher microbuckling strength, high-strain materials could carry loads in compression without reaching microbuckling and therefore be suitable for several space applications.

A finite element model was developed to predict the homogenized stiffness of the CFRS, and the homogenization results were used in another finite element model that simulated a homogenized rod under axial compression. A statistical representation of the fiber angles was implemented in the model. The presence of fiber angles increased the longitudinal shear stiffness of the material, resulting in a higher strength in compression. The simulations showed a large increase of the strength in compression for lower values of the standard deviation of the fiber angle, and a slight decrease of strength in compression for lower values of the mean fiber angle. The strength observed in the experiments was achieved with the minimum local angle standard deviation observed in the CFRS rods, whereas the shear stiffness measured in torsion tests was achieved with the overall fiber angle distribution observed in the CFRS rods.

High strain composites exhibit good bending capabilities, but they tend to be soft out-of-plane. To achieve a higher out-of-plane stiffness, the concept of dual-matrix composites is introduced. Dual-matrix composites are foldable composites which are soft in the crease regions and stiff elsewhere. Previous attempts to fabricate continuous dual-matrix fiber composite shells had limited performance due to excessive resin flow and matrix mixing. An alternative method, presented in this thesis uses UV-cure silicone and fiberglass to avoid these problems. Preliminary experiments on the effect of folding on the out-of-plane stiffness are presented. An application to a conical log-periodic antenna for CubeSats is proposed, using origami-inspired stowing schemes, that allow a conical dual-matrix composite shell to reach very high compaction ratios.

Estudio comparativo de diferentes soluciones de fachada y cubierta para un edificio residencial

[ES]En el proyecto se analizan diferentes soluciones de fachada y cubierta de un edificio residencial, buscando la eficiencia energética y la economía. Para ello, partiendo de un caso concreto, se estudian las cargas térmicas del edificio existente y de otras dos soluciones posibles. En estas además, se utilizan materiales sostenibles, con el fin de hacer el menor impacto medioambiental posible. Se ha analizado también la rentabilidad de estas soluciones, para comprobar cuál es la más adecuada desde un punto de vista económico.

Propuesta de catálogo para la selección de rodamientos de vuelco en aplicaciones cuasiestáticas

[ES]La propuesta de catálogo realizada en este proyecto, cuyo objetivo es convertirse en el catálogo de la empresa de rodamientos Iraundi S.A., constará de dos partes. La primera de ellas será una parte teórica e informativa que incluirá toda la información necesaria sobre la manipulación, montaje, mantenimiento y selección del tipo de rodamiento. Para ello se ha recabado información tanto del propio catálogo de Iraundi S.A., como de varios de los fabricantes de rodamientos más importantes del momento, como son INA, GOIMO, ROTHE-ERDE, y SKF. La segunda parte del catálogo lo formaran las tablas y gráficas de selección de rodamiento. Estas gráficas han sido renovadas con respecto al catálogo existente, y se ha tratado de conseguir unas gráficas que sean fáciles de interpretar, y que de manera sencilla se puedan obtener los tamaños y tipos de rodamiento adecuados, dependiendo de la aplicación de funcionamiento y de las fuerzas radiales, axiales y de momento que estos vayan a soportar. Éstas gráficas han sido calculadas exclusivamente para este proyecto, y validadas mediante la aplicación de elementos finitos, de los que finalmente se han obtenido unas gráficas que simplifican la selección óptima del rodamiento, y que tienen en cuenta el esfuerzo axial, radial, y momento que soportará el rodamiento, así como las fuerzas que serán capaces de soportar los tornillos empleados para el anclaje.

SCARA motako robot industrialaren simulazio birtuala eta interakzioa

[EU]Gradu amaierako lan honetan, Industria Ingenieritza graduan zehar emandako zenbait irakasgaietan landutako kontzeptuak oinarri bezala hartuta SCARA robot industrialaren analisia egin da. Lau askatasun gradu dituzten robot hauek oso erabiliak dira industrian beraien lan ziklo azkar, karga handiak jasateko kapazitate, errepikortasun eta aplikazio ugariengatik. Proiektua, bi zati nagusitan banandu da: SCARA motako robot baten analisi zinematikoa, robotaren irudikapena eta ibilbideen sorkuntza Matlab programa erabiliz. Adept Cobra e-Vario 600, SCARA motako prototipo errealarekin interakzioa. Proiektuan zehar garatuko diren edukien egitura ondorengoa izango da: Lehenik eta behin, proiektuko lehen atalean proiektua kokatuta dagoen testuinguruaren azalpena emango da, baita ere testuinguru horrek gaur egun, sail desberdinetan, duen garrantzia. Proiektuko bigarren atalean, lana egiterakoan proposatu diren helburuak eta proiektuak eskaintzen dituen onurak adieraziko dira. Bestetik, proiektuko hirugarren atalean, gaiaren egoeran, SCARA robotera heldu arte robotek historian zehar izan duten garapena eta hauen sailkapen desberdinak azalduko dira. Laugarren atalean, metodologian, proiektu honetan garatuko diren bi zati nagusien deskribapen zehatza emango da. Azkenik, proiektuko azken atalean, lana burutzeko bete behar izan diren ataza bakoitzaren deskribapena eta iraupena, aurrekontua, arriskuen analisia eta proiektu honetatik atera ditugun ondorioak ematen dira.

Speculation-Aware Resource Allocation for Cluster Schedulers

Real-time demand response is essential for handling the uncertainties of renewable generation. Traditionally, demand response has been focused on large industrial and commercial loads, however it is expected that a large number of small residential loads such as air conditioners, dish washers, and electric vehicles will also participate in the coming years. The electricity consumption of these smaller loads, which we call deferrable loads, can be shifted over time, and thus be used (in aggregate) to compensate for the random fluctuations in renewable generation.

In this thesis, we propose a real-time distributed deferrable load control algorithm to reduce the variance of aggregate load (load minus renewable generation) by shifting the power consumption of deferrable loads to periods with high renewable generation. The algorithm is model predictive in nature, i.e., at every time step, the algorithm minimizes the expected variance to go with updated predictions. We prove that suboptimality of this model predictive algorithm vanishes as time horizon expands in the average case analysis. Further, we prove strong concentration results on the distribution of the load variance obtained by model predictive deferrable load control. These concentration results highlight that the typical performance of model predictive deferrable load control is tightly concentrated around the average-case performance. Finally, we evaluate the algorithm via trace-based simulations.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.

Oxygen isotope ratios in coexisting minerals of regionally metamorphosed rocks

The O¹⁸/O¹⁶ ratios of coexisting minerals from a number of regionally metamorphosed rocks have been measured, using a bromine pentafluoride extraction-technique. Listed in order of their increasing tendency to concentrate O¹⁸, the minerals analyzed are magnetite, ilmenite, chlorite, biotite, garnet, hornblende, kyanite, muscovite, feldspar, and quartz. The only anomalous sequence detected occurs in a xenolith of schist, in which quartz, muscovite, biotite, and ilmenite, but not garnet, have undergone isotopic exchange with surrounding trondjemite.

With few exceptions, quartz-magnetite and quartz-ilmenite fractionations decrease with increasing metamorphic grade determined by mineral paragenesis and spatial distribution. This consistency does not apply to quartz-magnetite and quartz-ilmenite fractionations obtained from rocks in which petrographic evidence of retrogradation is present.

Whereas measured isotopic fractionations among quartz, garnet, ilmenite, and magnetite are approximately related to metamorphic grade, fractionations between these minerals and biotite or muscovite show poor correlation with grade. Variations in muscovite-biotite fractionations are relatively small. These observations are interpreted to mean that muscovite and biotite are affected by retrograde re-equilibration to a greater extent than the anhydrous minerals analyzed.

Measured quartz-ilmenite fractionations range from 12 permil in the biotite zone of central Vermont to 6.5 permil in the sillimanite-orthoclase zone of southeastern Connecticut. Analyses of natural assemblages from the kyanite and sillimanite zones suggest that equilibrium quartz-ilmenite fractionations are approximately 8 percent smaller than corresponding quartz-magnetite fractionations. Employing the quartz-magnetite geothermometer calibrated by O'Neil and Clayton (1964), a temperature of 560°C was obtained for kyanite-bearing schists from Addison County, Vermont. Extending the calibration to quartz-ilmenite fractionations, a temperature of 600°C was obtained for kyanite-schists from Shoshone County, Idaho. At these temperatures kyanite is stable only at pressures exceeding 11 kbars (Bell, 1963), corresponding to lithostatic loads of over 40 km.

Imperfection Insensitive Thin Shells

The buckling of axially compressed cylindrical shells and externally pressurized spherical shells is extremely sensitive to even very small geometric imperfections. In practice this issue is addressed by either using overly conservative knockdown factors, while keeping perfect axial or spherical symmetry, or adding closely and equally spaced stiffeners on shell surface. The influence of imperfection-sensitivity is mitigated, but the shells designed from these approaches are either too heavy or very expensive and are still sensitive to imperfections. Despite their drawbacks, these approaches have been used for more than half a century.

This thesis proposes a novel method to design imperfection-insensitive cylindrical shells subject to axial compression. Instead of following the classical paths, focused on axially symmetric or high-order rotationally symmetric cross-sections, the method in this thesis adopts optimal symmetry-breaking wavy cross-sections (wavy shells). The avoidance of imperfection sensitivity is achieved by searching with an evolutionary algorithm for smooth cross-sectional shapes that maximize the minimum among the buckling loads of geometrically perfect and imperfect wavy shells. It is found that the shells designed through this approach can achieve higher critical stresses and knockdown factors than any previously known monocoque cylindrical shells. It is also found that these shells have superior mass efficiency to almost all previously reported stiffened shells.

Experimental studies on a design of composite wavy shell obtained through the proposed method are presented in this thesis. A method of making composite wavy shells and a photogrametry technique of measuring full-field geometric imperfections have been developed. Numerical predictions based on the measured geometric imperfections match remarkably well with the experiments. Experimental results confirm that the wavy shells are not sensitive to imperfections and can carry axial compression with superior mass efficiency.

An efficient computational method for the buckling analysis of corrugated and stiffened cylindrical shells subject to axial compression has been developed in this thesis. This method modifies the traditional Bloch wave method based on the stiffness matrix method of rotationally periodic structures. A highly efficient algorithm has been developed to implement the modified Bloch wave method. This method is applied in buckling analyses of a series of corrugated composite cylindrical shells and a large-scale orthogonally stiffened aluminum cylindrical shell. Numerical examples show that the modified Bloch wave method can achieve very high accuracy and require much less computational time than linear and nonlinear analyses of detailed full finite element models.

This thesis presents parametric studies on a series of externally pressurized pseudo-spherical shells, i.e., polyhedral shells, including icosahedron, geodesic shells, and triambic icosahedra. Several optimization methods have been developed to further improve the performance of pseudo-spherical shells under external pressure. It has been shown that the buckling pressures of the shell designs obtained from the optimizations are much higher than the spherical shells and not sensitive to imperfections.

Avaliação da degradação de elásticos ortodônticos intraorais de látex

Avaliou-se a taxa de degradação de elásticos ortodônticos de látex de diferentes fabricantes e diâmetros em diversos intervalos de tempo. Grupos de 15 elásticos de força média dos fabricantes AmericanOrthodontics (Sheboygan, Wis, EUA), Tp (La Porte, IN, EUA), Morelli (Sorocaba, SP, Brasil) e Uniden (Sorocaba, SP, Brasil); de diâmetros 3/16, 1/4 e 5/16 foram analisados nos intervalos de 0,1,3,6,12 e 24 horas, totalizando-se 1080 espécimes. Os elásticos foram estirados individualmente à distância de 30mm, respeitando-se o intervalo de um minuto entre cada estiramento. Os materiais foram acondicionados imersos em água deionizada à 37C. Realizou-se leitura das forças na máquina de ensaios Emic DL 500 (Emic Co, Sao Paulo, Brasil) à velocidade de 30 mm/min, com uma célula de 2Kg (Emic Co, Sao Paulo, Brasil). A leitura de cada elástico consumiu aproximadamente um minuto. Teste Kruskal-wallis com correções por Dunns aferiu significância estatística dos resultados. Foram observadas diferenças entre os grupos analisados, exceto às marcas Morelli e Tp. Foram significativas as inferências das variáveis tempo e marca comercial. Em 0 hora, a relação entre as forças geradas foi Morelli>AO>Uniden>Tp para elásticos 3/16 (p=0,0016) e 1/4 (p=0,0016) e, de Morelli>AO>Tp>Uniden para elásticos 5/16 (p=0,0087). Após 24 horas, as porcentagens de degradação dos elásticos foram AO>Morelli>Uniden>Tp no diâmetro 3/16; AO>Tp>Morelli>Uniden no diâmetro 1/4 e Tp>AO>Uniden>Morelli para elásticos de diâmetro 5/16. O comportamento no intervalo de 0-24 horas demonstrou uma queda acentuada no período de 0-3 horas, um ligeiro aumento em 3-6 horas, seguido de uma queda progressiva no intervalo de 6-24 horas.

Análise da dissipação das tensões em dentes humanos restaurados com facetas laminadas de cerâmica, com três tipos de preparos, através do método dos elementos finitos

O objetivo deste trabalho é analisar in vitro a dissipação de tensões em incisivos centrais superiores humanos restaurados com facetas de cerâmica feldspática, através da análise do método dos elementos finitos, considerando cargas funcionais de mastigação e corte dos alimentos, em função de três tipos de preparos utilizados: sem proteção incisal; com proteção incisal em ângulo e com proteção incisal em degrau palatino. Foram utilizadas modelagens bidimensionais de um incisivo central superior e suas estruturas de suporte, simulando três situações: (Primeira modelagem) incisivo central superior com desgaste vestibular (em forma de janela); (Segunda modelagem) incisivo central superior com desgaste vestibular e proteção incisal em plano inclinado; (Terceira modelagem) incisivo central superior com desgaste vestibular, e proteção incisal com degrau palatino. Foi considerada uma carga (P=100N) com uma inclinação de 45 concentrada, simulando a região de contato do incisivo central inferior com o superior durante a mastigação e uma na região de contato topo a topo dos incisivos superior e inferior, simulando o corte dos alimentos. Após a análise dos dados obtidos pela distribuição de tensões, pode-se concluir que quanto à dissipação das tensões em todo o sistema proposto, com a aplicação de carga em 45, não foram observadas mudanças no estado tensional nos três diferentes preparos. Quando foi aplicada carga vertical, simulando o contato de topo, houve variação no estado tensional no sistema do dente com preparo em janela. Nas facetas, com a aplicação de carga em 45, nos preparos em janela e com proteção incisal em plano inclinado o resultado foi semelhante nos valores tensionais enquanto, nas facetas em dentes preparados com proteção incisal com degrau palatino, a distribuição foi mais homogênea tendo valores superiores, mostrando que o abraçamento do dente diminuiu a flexão.

Estratégia para validação de processos de esterilização por vapor saturado em autoclaves nas indústrias farmacêuticas

O objetivo deste trabalho foi criar uma metodologia de validação e revalidação dos processos de esterilização por calor úmido em autoclaves horizontais, destacando os pontos críticos do processo e concentrando esforços onde são realmente necessários. Foram realizados estudos de distribuição térmica, de penetração de calor e de desafio microbiológico na validação da autoclave STERIS FINNAQUA 6912. Com o objetivo de avaliar o impacto de uma mudança e compreender a relação entre os fatores e suas interações para o processo de esterilização, foi utilizado o planejamento fatorial 23 dos fatores densidade da carga (quantidade de itens), embalagem do produto e localização na câmara interna. Os estudos de distribuição térmica confirmaram a distribuição homogênea de calor na câmara interna durante o tempo de exposição a 121C. As temperaturas variaram entre 120,35C e 120,92C com desvio padrão máximo de 0,12C. Os estudos de penetração de calor confirmaram exposições equivalentes a 121C por 24 minutos em todos os itens da carga (F0 > 24 minutos). Em todos os estudos para cargas secas, os índices de capacidade do processo (Cpk) foram maiores que 1,33. Os ensaios de desafio microbiológico garantiram níveis de esterilidade (S.A.L.) maiores que 12 reduções logarítmicas em relação aos indicadores biológicos Geobacillus stearothermophilus. Não foi detectada a presença de endosporos sobreviventes nos 132 indicadores biológicos utilizados nos quatro ciclos desafiados. Com base no planejamento experimental verificou-se que, para o nível de significância de 95% , as mudanças nos fatores posição, embalagem e quantidade da carga não são significativas para o processo de esterilização, em autoclave com remoção forçada de ar. Já para o nível de significância de 90%, a interação Posição x Embalagem apresentou significância estatística no processo de esterilização com valor P de 0,080

Fabrication, Characterization, And Deformation of 3D Structural Meta-Materials

Current technological advances in fabrication methods have provided pathways to creating architected structural meta-materials similar to those found in natural organisms that are structurally robust and lightweight, such as diatoms. Structural meta-materials are materials with mechanical properties that are determined by material properties at various length scales, which range from the material microstructure (nm) to the macro-scale architecture (μm – mm). It is now possible to exploit material size effect, which emerge at the nanometer length scale, as well as structural effects to tune the material properties and failure mechanisms of small-scale cellular solids, such as nanolattices. This work demonstrates the fabrication and mechanical properties of 3-dimensional hollow nanolattices in both tension and compression. Hollow gold nanolattices loaded in uniaxial compression demonstrate that strength and stiffness vary as a function of geometry and tube wall thickness. Structural effects were explored by increasing the unit cell angle from 30° to 60° while keeping all other parameters constant; material size effects were probed by varying the tube wall thickness, t, from 200nm to 635nm, at a constant relative density and grain size. In-situ uniaxial compression experiments reveal an order-of-magnitude increase in yield stress and modulus in nanolattices with greater lattice angles, and a 150% increase in the yield strength without a concomitant change in modulus in thicker-walled nanolattices for fixed lattice angles. These results imply that independent control of structural and material size effects enables tunability of mechanical properties of 3-dimensional architected meta-materials and highlight the importance of material, geometric, and microstructural effects in small-scale mechanics. This work also explores the flaw tolerance of 3D hollow-tube alumina kagome nanolattices with and without pre-fabricated notches, both in experiment and simulation. Experiments demonstrate that the hollow kagome nanolattices in uniaxial tension always fail at the same load when the ratio of notch length (a) to sample width (w) is no greater than 1/3, with no correlation between failure occurring at or away from the notch. For notches with (a/w) > 1/3, the samples fail at lower peak loads and this is attributed to the increased compliance as fewer unit cells span the un-notched region. Finite element simulations of the kagome tension samples show that the failure is governed by tensile loading for (a/w) < 1/3 but as (a/w) increases, bending begins to play a significant role in the failure. This work explores the flaw sensitivity of hollow alumina kagome nanolattices in tension, using experiments and simulations, and demonstrates that the discrete-continuum duality of architected structural meta-materials gives rise to their flaw insensitivity even when made entirely of intrinsically brittle materials.

Strength of thin-walled elliptical cylinders supported at the minor axis

In this investigation it was found that the instability failure of curved sheet is nearly independent of the type of loading and is primarily a function of the maximum stress, radius-thickness ration and modulus of elasticity. A method of correlating the critical stress of thin sheet under several different types of loading is given. An explanation for the experimental critical stress of thin walled cylinders under bending being greater than that for pure compression is given. The strength of unstiffened thin walled circular nose sections under pure bending was found to be controlled by local instability of the section, rather than a large scale instability. The equation of local instability of curved sheet gives values which are in fair agreement with those found experimentally.

The strength of elliptical cylinders supported at the minor axis under bending plus shear loads is governed primarily by the bending strength, and is little effected by the sheer force unless the amount of shear is quite large with respect to the moment. The effect of increasing the amount of elliptically greatly reduces the bending and shear strength of nose sections. Under torsional loads the stress at buckling falls off as the ration of the major to minor axis increases but the failure stress decreases at a slower rate than the buckling stress. The length effect of semi-circular sections under torsion is similar to that of a circular tube, and can be obtained by Donnell's theoretical equation.