Wedges and plate-bandes: mechanical theories after De la Hire

Plate-bandes are straight masonry arches (they are called, also, flat arches or lintel arches). Ideally they have the surfaces of extrados and intrados plane and horizontal. The stones or bricks have radial joints converging usually in one centre. The voussoirs have the form of wedges and in French they are called "claveaux". A plate-bande is, in fact, a lintel made of several stones and the proportions of lintels and plate-bandes are similar. Proportions of plate-bandes, that is the relationship between the thickness t and the span s (t/s)varies, typically between 1/4–1/3 in thick plate-bandes, and is less than 1/20 in the most slender ones. A ratio of circa 1/8 was usual in the 18th Century and follows a simple geometrical rule: the centre form with the intrados an equilateral triangle and the plate-bande should contain an arc of circle. The joints are usually plane, but in some cases present a «rebated» or «stepped» form. Plate-bandes exert an inclined thrust as any masonry arch. This thrust is usually very high and it requires either massive buttresses, or to be built in the middle of thick walls. Master builders and architects have tried since antiquity to calculate the abutment necessary for any arch. A modern architect or engineer will measure the arch thrust in units of force, kN or tons. Traditionally, the thrust has been measured as the size of the buttresses to resist it safely. Old structural rules, then, addressed the design problem establishing a relationship between the span and the depth of the buttress. These were empirical rules, particular for every type of arch or structure in every epoch. Thus, the typical gothic buttress is 1/4 of the vault span, but a Renaissance or baroque barrel vault will need more than 1/3 of the span. A plate-bande would require more than one half of the span; this is precisely the rule cited by the French engineer Gautier, who tried unsuccessfully to justify it by static reasons. They were used, typically, to form the lintels of windows or doors (1-2 m, typically); in Antiquity they were used, also, though rarely, at the gates of city walls or in niches (ca. 2 m, reaching 5.2 m). Plate-bandes may show particular problems: it is not unusual that some sliding of the voussoirs can be observed, particularly in thick plate-bandes. The stepped joints on Fig. 1, left, were used to avoid this problem. There are other «hidden» methods, like iron cramps or the use of stone wedges, etc. In seismic zones these devices were usual. Another problem relates to the deformation; a slight yielding of the abutments, or even the compression of the mortar joints, may lead to some cracking and the descent of the central keystone. Even a tiny descent will convert the original straight line of the intrados in a broken line with a visible «kink» or angle in the middle. Of course, both problems should be avoided. Finally, the wedge form of the voussoirs lead to acute angles in the stones and this can produce partial fractures; this occurs usually at the inferior border of the springers at the abutments. It follows, that to build a successful plate-bande is not an easy matter. Also, the structural study of plate-bandes is far from simple, and mechanics and geometry are related in a particular way. In the present paper we will concentrate on the structural aspects and their constructive consequences, with a historical approach. We will outline the development of structural analysis of plate-bandes from ca. 1700 until today. This brief history has a more than purely academic interest. Different approaches and theories pointed to particular problem, and though the solution given may have been incorrect, the question posed was often pertinent. The paper ends with the application of modern Limit Analysis of Masonry Structures, developed mainly by professor Heyman in the last fifty years. The work aims, also, to give some clues for the actual architect and engineer involved in the analysis or restoration of masonry buildings.

Low-dimensional modeling of streaks in a wedge flow boundary layer

This paper is concerned with the low dimensional structure of optimal streaks in a wedge flow boundary layer, which have been recently shown to consist of a unique (up to a constant factor) three-dimensional streamwise evolving mode, known as the most unstable streaky mode. Optimal streaks exhibit a still unexplored/unexploited approximate self-similarity (not associated with the boundary layer self-similarity), namely the streamwise velocity re-scaled with their maximum remains almost independent of both the spanwise wavenumber and the streamwise coordinate; the remaining two velocity components instead do not satisfy this property. The approximate self-similar behavior is analyzed here and exploited to further simplify the description of optimal streaks. In particular, it is shown that streaks can be approximately described in terms of the streamwise evolution of the scalar amplitudes of just three one-dimensional modes, providing the wall normal profiles of the streamwise velocity and two combinations of the cross flow velocity components; the scalar amplitudes obey a singular system of three ordinary differential equations (involving only two degrees of freedom), which approximates well the streamwise evolution of the general streaks.

Electrooptical behavior of twisted-wedge nematic structures

A new type of domain in twisted-wedge negative nematics is reported. The wedge shape allows the separation of ordinary and extraordinary beams that can be studied separately. In these conditions, at least five different regions can be detected, depending on applied voltages and frequencies. Both rays are shown to yield different diffraction patterns. The relative light intensity of several spots of these paterns is also studied.

Electrohydrodynamic behavior in twisted-wedge nematic structures

A new type of domain for nematic liquid crystals with a twisted-wedge structure is presented. This new type of domain appears from the low frequency range to 10 kHz. This behavior was observed for square and pulsed excitations. The liquid crystal was N-(p-methoxybenzylidene)-p'-butylaniline) (MBBA) used at room temperature. These domains offer a higher degree of complexity than conventional Williams domains. The corresponding stability chart is presented.

Numerical and experimental study of the parametric rolling of a fisihing vessel in regular head waves

The dynamic behaviour of a fishing vessel in waves is studied in order to reveal its parametric rolling characteristics. This paper presents experimental and numerical results in longitudinal regular waves. The experimental results are compared against the results of a time-domain non-linear strip theory model of ship motions in six degrees-of-freedom. These results contribute to the validation of the parametric rolling prediction method, so that it can be used as an assessment tool to evaluate both the susceptibility and severity of occurrence of parametric rolling at the early design stage of these types of vessels.

Urban trees and atmospheric pollutants in big cities: Effects in Madrid

Una investigación sobre la mejora de la contaminación del aire (CA) por medio de arbolado urbano se realizó en Madrid, una ciudad con casi 4 M de habitantes, 2,8 M de vehículos y casi 3 M de árboles de mantenimiento público. La mayoría de los árboles estaban en dos bosques periurbanos. Los 650.000 restantes era pies de alineación y parques. Los taxones estudiados fueron Platanus orientalis (97.205 árboles), Ulmus sp. (70.557), Pinus pinea (49.038), Aesculus hippocastanum (22.266), Cedrus sp. (13.678) y Quercus ilex (1.650), de calles y parques. Muestras foliares se analizaron en diferentes épocas del año, así como datos de contaminación por PM10 de 28 estaciones de medición de la contaminación durante 30 años, y también la intensidad del tráfico (IMD) en 2.660 calles. La acumulación de metales pesados (MP) sobre hojas y dentro de estas se estimó en relación con la CA y del suelo y la IMD del tráfico. La concentración media de Ba, Cd, Cr, Cu, Mn, Ni, Pb y Zn en suelo (materia seca) alcanzó: 489,5, 0,7, 49,4, 60,9, 460,9, 12,8, 155,9 y 190,3 mg kg-1 respectivamente. Los árboles urbanos, particularmente coníferas (debido a la mayor CA en invierno) contribuyen significativamente a mejorar la CA sobre todo en calles con alta IMD. La capacidad de las seis sp. para capturar partículas de polvo en su superficies foliares está relacionada con la IMD del tráfico y se estimó en 16,8 kg/año de MP tóxicos. Pb y Zn resultaron ser buenos marcadores antrópicos en la ciudad en relación con el tráfico, que fue la principal fuente de contaminación en los árboles y suelos de Madrid. Las especies de árboles variaron en función de su capacidad para capturar partículas (dependiendo de las propiedades de sus superficies foliares) y acumular los MP absorbidos de los suelos. Las concentraciones foliares de Pb y Zn estuvieron por encima de los límites establecidos en diferentes sitios de la ciudad. La microlocalización de Zn mediante microscópico mostró la translocación al xilema y floema. Se detectaron puntos de contaminación puntual de Cu and Cr en antiguos polígonos industriales y la distribución espacial de los MP en los suelos de Madrid mostró que en incluso en zonas interiores del El Retiro había ciertos niveles elevados de [Pb] en suelo, tal vez por el emplazamiento la Real Fábrica de Porcelana en la misma zona hace 200 años. Distintas áreas del centro de la ciudad también alcanzaron niveles altos de [Pb] en suelo. Según los resultados, el empleo de una combinación de Pinus pinea con un estrato intermedio de Ulmus sp. y Cedrus sp. puede ser la mejor recomendación como filtro verde eficiente. El efecto del ozono (O3) sobre el arbolado en Madrid fue también objeto de este estudio. A pesar de la reducción de precursores aplicada en muchos países industrializados, O3 sigue siendo la principal causa de CA en el hemisferio norte, con el aumento de [O3] de fondo. Las mayores [O3] se alcanzaron en regiones mediterráneas, donde el efecto sobre la vegetación natural es compensado por el xeromorfismo y la baja conductancia estomática en respuesta los episodios de sequía estival característicos de este clima. Durante una campaña de monitoreo, se identificaron daños abióticos en hojas de encina parecidos a los de O3 que estaban plantadas en una franja de césped con riego del centro de Madrid. Dada la poca evidencia disponible de los síntomas de O3 en frondosas perennifolias, se hizo un estudio que trató de 1) confirman el diagnóstico de daño de O3, 2) investigar el grado de los síntomas en encinas y 3) analizar los factores ambientales que contribuyeron a los daños por O3, en particular en lo relacionado con el riego. Se analizaron los marcadores macro y micromorfológicos de estrés por O3, utilizando las mencionadas encinas a modo de parcela experimental. Los síntomas consistieron en punteado intercostal del haz, que aumentó con la edad. Además de un punteado subyacente, donde las células superiores del mesófilo mostraron reacciones características de daños por O3. Las células próximas a las zonas dañadas, presentaron marcadores adicionales de estrés oxidativo. Estos marcadores morfológicos y micromorfológicos de estrés por O3 fueron similares a otras frondosas caducifolias con daños por O3. Sin embargo, en nuestro caso el punteado fue evidente con AOT40 de 21 ppm•h, asociada a riego. Análisis posteriores mostraron que los árboles con riego aumentaron su conductancia estomática, con aumento de senescencia, manteniéndose sin cambios sus características xeromórficas foliares. Estos hallazgos ponen de relieve el papel primordial de la disponibilidad de agua frente a las características xeromórficas a la hora de manifestarse los síntomas en las células por daños de O3 en encina. ABSTRACT Research about air pollution mitigation by urban trees was conducted in Madrid (Spain), a southern European city with almost 4 M inhabitants, 2.8 M daily vehicles and 3 M trees under public maintenance. Most trees were located in two urban forests, while 650'000 trees along urban streets and in parks. The urban taxa included Platanus orientalis (97'205 trees), Ulmus sp. (70’557), Pinus pinea (49'038), Aesculus hippocastanum (22’266), Cedrus sp. (13'678 and Quercus ilex (1'650) along streets and parks. Leave samples were analysed sequentially in different seasons, PM10 data from 28 air monitoring stations during 30 years and traffic density estimated from 2’660 streets. Heavy metal (HM) accumulation on the leaf surface and within leaves was estimated per tree related to air and soil pollution, and traffic intensity. Mean concentration of Ba, Cd, Cr, Cu, Mn, Ni, Pb and Zn in topsoil samples (dry mass) amounted in Madrid: 489.5, 0.7, 49.4, 60.9, 460.9, 12.8, 155.9 and 190.3 mg kg-1 respectively. Urban trees, particularly conifers (due to higher pollution in winter) contributed significantly to alleviate air pollution especially near to high ADT roads. The capacity of the six urban street trees species to capture air-born dust on the foliage surface as related to traffic intensity was estimated to 16.8 kg of noxious metals from exhausts per year. Pb and Zn pointed to be tracers of anthropic activity in the city with vehicle traffic as the main source of diffuse pollution on trees and soils. Tree species differed by their capacity to capture air-borne dust (by different leaf surface properties) and to allocate HM from soils. Pb and Zn concentrations in the foliage were above limits in different urban sites and microscopic Zn revelation showed translocation in xylem and phloem tissue. Punctual contamination in soils by Cu and Cr was identified in former industrial areas and spatial trace element mapping showed for central Retiro Park certain high values of [Pb] in soils even related to a Royal pottery 200 years ago. Different areas in the city centre also reached high levels [Pb] in soils. According to the results, a combination of Pinus pinea with understorey Ulmus sp. and Cedrus sp. layers can be recommended for the best air filter efficiency. The effects of ozone (O3) on trees in different areas of Madrid were also part of this study. Despite abatement programs of precursors implemented in many industrialized countries, ozone remained the main air pollutant throughout the northern hemisphere with background [O3] increasing. Some of the highest ozone concentrations were measured in regions with a Mediterranean climate but the effect on the natural vegetation is alleviated by low stomatal uptake and frequent leaf xeromorphy in response to summer drought episodes characteristic of this climate. During a bioindication survey, abiotic O3-like injury was identified in foliage. Trees were growing on an irrigated lawn strip in the centre of Madrid. Given the little structural evidence available for O3 symptoms in broadleaved evergreen species, a study was undertaken in 2007 with the following objectives 1) confirm the diagnosis, 2) investigate the extent of symptoms in holm oaks growing in Madrid and 3) analyse the environmental factors contributing to O3 injury, particularly, the site water supply. Therefore, macro- and micromorphological markers of O3 stress were analysed, using the aforementioned lawn strip as an intensive study site. Symptoms consisted of adaxial and intercostal stippling increasing with leaf age. Underlying stippling, cells in the upper mesophyll showed HR-like reactions typical of ozone stress. The surrounding cells showed further oxidative stress markers. These morphological and micromorphological markers of ozone stress were similar to those recorded in deciduous broadleaved species. However, stippling became obvious already at an AOT40 of 21 ppm•h and was primarily found at irrigated sites. Subsequent analyses showed that irrigated trees had their stomatal conductance increased and leaf life-span reduced whereas their leaf xeromorphy remained unchanged. These findings suggest a central role of water availability versus leaf xeromorphy for ozone symptom expression by cell injury in holm oak.