Designing by Geometry. Rankine's Theorems of Transformation of Structures.

William John Macquorn Rankine (1820-1872) was one of the main figures in establishing engineering science in the second half of the 19th. Century. His Manual of Applied Mechanics (1858) gathers most of his contributions to strength of materials and structural theory. A few additions are to be found in his Manual of Civil Engineering (1862). The book is based in his Lectures on Engineering delivered in the Glasgow University, and formed part of his intention of converting engineering science in a university degree (Channell 1982, Buchanan 1985). Both in plan and in content the book shows and enormous rigour and originality. It is difficult to read. As remarked by Timoshenko (1953, 198): "In his work Rankine prefers to treat each problem first in its most general form and only later does he consider various particular cases which may be of some practical interest. Rankine's adoption of this method of writing makes his books difficult to read, and they demand considerable concentration of the reader." Besides, Rankine does not repeat any demonstration or formula, and sometimes the reader must trace back the complete development through four or five previous paragraphs. The method is that of a mathematician. However, the Manual had 21 editions (the last in 1921) an exerted a considerable influence both in England and America. In this article we will concentrate only in one of the more originals contributions of Rankine in the field of structural theory, his Theorems of Transformation of Structures. These theorems have deserved no attention either to his contemporaries or to modern historians of structural theory. It appears that the only exception is Timoshenko (1953,198-200) who cited the general statement and described briefly its applications to arches. The present author has studied the application of the Theorems to masonry structures (Huerta and Aroca 1989; Huerta 1990, 2004, 2007). Rankine discovered the Theorems during the preparation of his Lectures for his Chair of Engineering in the University of Glasgow . He considered it very important, as he published it in a short note communicated to the Royal Society in 1856 (Rankine 1856). He included it, also, in his article "Mechanics (applied)" for the 8th edition of the Encyclopaedia Britannica (Rankine 1857). Eventually, the Theorems were incoroporated in the Manual of applied mechanics and applied to frames, cables, rib arches and masonry structures. The theorems were also included in his Manual of civil engineering (1862), generally in a shortened way, but with some additions

Constraint-bounded design search

The design process requires continual checking of the consistency of design choices against given sets of goals that have been fulfilled. Such a check is generally performed by comparing abstract representations of design goals with these of the sought real building objects (RBO) resulting from complex intellectual activities closely related to the designer's culture and to the environment in which he operates. In this chapter we define a possible formalization of such representations concerning the goals and the RBO that are usually considered in the architectural design process by our culture in our environment. The representation of design goals is performed by expressing their objective aspects (requirements) and by defining their allowable values (performance specifications). The resulting system of requirements defines the set of allowable solutions and infers an abstract representation of the sought building objects (BO) that consists of the set of characteristics (attributes and relations) which are considered relevant to represent the particular kind of RBO with respect to the consistency check with design goals. The values related to such characteristics define the performances of the RBO while their set establishes its behaviour. Generally speaking, there is no single real object corresponding to an abstract representation but the whole class of the RBO that are equivalent with respect to the values assumed by the considered characteristics. The more we increase the number of these, as well as their specifications, the smaller the class becomes until it coincides with a single real object - given that the assessed specifications be fully consistent. On the other hand, the corresponding representation evolves to the total prefiguration of the RBO. It is not therefore possible to completely define a BO representation in advance since this is inferred by the considered goals and is itself a result of the design process. What can only be established in advance is that any set of characteristics assumed to represent any RBO consists of hierarchic, topological, geometrical and functional relations among the parts of the object at any level of aggregation (from components to space units, to building units, to the whole building) that we define representation structure (RS). Consequently the RS may be thought as the elementary structures that, by superposition and interaction, set up the abstract representation that best fit with design goals

Recent advances in approximation concepts for optimum structural design

The basic approximation concepts used in structural optimization are reviewed. Some of the most recent developments in that area since the introduction of the concept in the mid-seventies are discussed. The paper distinguishes between local, medium-range, and global approximations; it covers functions approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It also appears that some new methodologies are emerging which could greatly benefit from the introduction of new computer architecture

AFLOW-SYM: Platform for the complete, automatic and self-consistent symmetry analysis of crystals

Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. Here, we present a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point-, factor-, space groups, site symmetries, and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space group information provided for over 54,000 entries in the Inorganic Crystal Structure Database. Subsequently, a complete symmetry analysis is applied to all 1.7$+$ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated, $\textit{ab-initio}$ framework AFLOW.Comment: 24 pages, 6 figure