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.

Design and construction of the Barriga Dam spillway through an improved wedge-shaped block technology

The Barriga Dam (Burgos, Spain) is a unique case study because its trapezoid spillway is located on the dam body and is composed of wedge-shaped concrete blocks (WSB) that include certain relevant improvements. This note summarizes the main features of the studies, the key aspects of the final design of the WSB and their placement on the dam, and important details of the spillway design. The design team concluded the study by showing the suitability of this enhanced technology for application to small dams and ponds in the short term, even with unit flows above 5 m2/s.

Comparisons between analogue and numerical models of thrust wedge development

Analogue and finite element numerical models with frictional and viscous properties are used to model thrust wedge development. Comparison between model types yields valuable information about analogue model evolution, scaling laws and the relative strengths and limitations of the techniques. Both model types show a marked contrast in structural style between ‘frictional-viscous domains’ underlain by a thin viscous layer and purely ‘frictional domains’. Closely spaced thrusts form a narrow and highly asymmetric fold-and-thrust belt in the frictional domain, characterized by in-sequence propagation of forward thrusts. In contrast, the frictional-viscous domain shows a wide and low taper wedge and a thrust belt with a more symmetrical vergence, with both forward and back thrusts. The frictional-viscous domain numerical models show that the viscous layer initially simple shears as deformation propagates along it, while localized deformation resulting in the formation of a pop-up structure occurs in the overlying frictional layers. In both domains, thrust shear zones in the numerical model are generally steeper than the equivalent faults in the analogue model, because the finite element code uses a non-associated plasticity flow law. Nevertheless, the qualitative agreement between analogue and numerical models is encouraging. It shows that the continuum approximation used in numerical models can be used to model frictional materials, such as sand, provided caution is taken to properly scale the experiments, and some of the limitations are taken into account.