Abstract This paper presents a procedure for estimating the minimum thickness of vaults and catenary arches to resist seismic in-plane horizontal loading. In order to define the shapes of the thrust lines that result from combining the self-weight of the arches and horizontal in-plane acceleration, an inverted hanging chain was used. The chain inclines the line by connecting its ends horizontally. Subsequently, iterative calculation help find the minimum thickness of the catenary arches with constant section that inscribe the defined thrust lines. This procedure was applied to different arches by considering ten values of horizontal acceleration. Consequently, graphs were drawn that relate the horizontal acceleration to the minimum thickness, contingent on the rise/span ratio of each arch, as well as the position of the hinges created in the collapse mechanism. Finally, the application is shown through the intersections of several catenary arches. In order to define the shapes of the thrust lines that result from combining the self-weight of the arches and horizontal in-plane acceleration, an inverted hanging chain was used. The chain inclines the line by connecting its ends horizontally. Subsequently, iterative calculation help find the minimum thickness of the catenary arches with constant section that inscribe the defined thrust lines. This procedure was applied to different arches by considering ten values of horizontal acceleration. Barock Elasticity Structural Mechanics Structural Geology Structural Materials Light-weight Construction, Steel and Timber Construction Seismic Performance of Masonry Structures Introduction The principle of using an inverted hanging chain to define the ideal shape of an arch or vault under self-weight loads (Hooke 1676 ) is a well-established concept which has been widely used since the eighteenth century (Graefe 2020 ). The Catalan modernist architect Antoni Gaudí took the literal interpretation of this principle to an extreme. Gaudí used hanging models made of ropes and small sandbags to define arches and vaults with highly intricate shapes, as evidenced by his work on the Sagrada Familia temple in Barcelona and the crypt of the Colonia Güell (Huerta 2006 ). More than half a Currently, the use of hanging chain physical models remains a simple and precise method for obtaining funicular shapes such as those that describe the thrust lines of arches and vaults (Sunguroglu and Baraut 2013 ; Miccoli et al. 2023 ; Afonso and Fialho 2024 ; Borhani and Kalanta 2024 ; Fallacara et al. 2024 ). The catenary is the curved shape that a chain of uniform weight assumes when suspended between two fixed points. The equation was formulated several years after the discovery of the principle of the inverted hanging chain (Bernoulli 1691 ; Huygens 1691 ; Leibniz 1691 ; Gregory 1697 ). Subsequently, the catenary has been regarded as the best form of an arch of constant section when the sole load acting is its self-weight. In a manner analogous to the simple tensioning of a chain as a result of inversion, the catenary arch should be simply compressed, without bending moments. The subject of catenary arches remains a focus of research, with the objective of developing new methods of structural analysis and design (Gohnert and Bradley 2020 ; Li, et al. 2024 ; Wang and Zhang 2024 ). The use of hanging chain models could present two difficulties: the equivalence of the load distribution of the chain model and the real model, and the capture of the funicular shape obtained. Nevertheless, these difficulties are minimal in the presented method. Firstly, because the load is uniformly distributed in a catenary arch and, for this reason, only a chain of uniform weight is required. Secondly, because, in this case, the hanging chain model used is two-dimensional and the capture of the funicular shape obtained can be done through a simple photograph. Method Outline of the Procedure The procedure is comprised of two parts. The first part involves the use of hanging chain models. tilting the analysed object) to obtain the thrust lines of a catenary arch subjected to horizontal seismic acceleration. Next, a rigid rectangular panel 70 × 50 cm was taken, and placed it in a vertical plane with a suspended chain composed of spheres ø 3.5 mm. As a result, the ends were anchored at a fixed distance s  = 60 cm along one of the edges. When this edge coincides with the horizontal direction, the chain takes the form of a catenary whose axis of symmetry is vertical (Fig. 1 a). It is known that this inverted catenary coincides with the thrust line of a one-dimensional arch with evenly distributed weight. 12 Full size image Seismic acceleration and the angles φ 1 and φ 2 corresponding to the position of the hinges created in the collapse mechanism of a catenary arch of minimum thickness considering three different r/s ratios Table 2 Parameters m and n of the equations for the straight lines which relate the horizontal acceleration to the angles φ 1 and φ 2 for each ratio r/s Full size table Table 3 shows the relationship between the slenderness of the arch, i.e., the ratio between the length L of the catenary curve and the thickness t , and the slenderness of the hanging chain, i.e., the ratio between its length L and the spheres’ diameter ø. Here, the straight line joining the ends of the chain is tilted at an angle α from the horizontal direction, such that the tangent of α is equal to the ratio between the horizontal acceleration due to the earthquake and the acceleration of gravity. Consequently, three distinct thrust lines can be derived for a given arch. The first corresponds to the arch in a self-weight scenario, which is the catenary. The second corresponds to the arch in a self-weight scenario with the addition of a horizontal acceleration. The third is the symmetrical version of the second, accounting for the same horizontal acceleration but in an opposing direction.

suspended from two points will form a catenary, the curve with the lowest possible center of gravity available to any chain hung between two fixed points. He Physics is a branch of science in which the primary objects of study are matter and energy. These topics were discussed across many cultures in ancient times by philosophers, but they had no means to distinguish causes of natural phenomena from superstitions. The Scientific Revolution of the 17th century, especially the discovery of the law of gravity, began a process of knowledge accumulation and In… Hooke first described this discovery in an anagram "ceiiinosssttuv", whose solution he published in 1678 as Ut tensio, sic vis ("As the extension, so the force").His work on elasticity culminated in his development of the balance spring or hairspring, which for the first time enabled a portable timepiece – a watch – to keep time with reasonable accuracy.Hooke's announcement of his law of elasticity using an anagram was a method scientists, such as Hooke, Huygens, and Galileo, sometimes used to establish priority for a discovery without revealing details.Hooke used mechanical analogues to understand fundamental processes such as the motion of a spherical pendulum and of a ball in a hollow cone, to demonstrate central force due to gravity, and a hanging chain net with point loads to provide the optimum shape for a dome with heavy cross on top. To arrive at his results, Newton invented one form of an entirely new branch of mathematics: calculus (also invented independently by Gottfried Leibniz), which was to become an essential tool in much of the later development in most branches of physics. Using his three laws of motion and law of universal gravitation, Newton removed the idea that objects followed paths determined by natural shapes and instead demonstrated that all the future motions of any body could be deduced mathematically based on knowledge of their existing motion, their mass, and the forces acting upon them. However, In 1690, James Bernoulli showed that the cycloid is the solution to the tautochrone problem; and the following year, in 1691, Johann Bernoulli showed that a chain freely suspended from two points will form a catenary, the curve with the lowest possible center of gravity available to any chain hung between two fixed points. He then showed, in 1696, that the cycloid is the solution to the brachistochrone problem. ==== Early thermodynamics ==== A precursor of the engine was designed by the German scientist Otto von Guericke who, in 1650, designed and built the world's first vacuum pump to create a vacuum as demonstrated in the Magdeburg hemispheres experiment. In 1733, Daniel Bernoulli derived the fundamental frequency and harmonics of a hanging chain by solving a differential equation. In 1734, Bernoulli solved the differential equation for the vibrations of an elastic bar clamped at one end. Bernoulli's treatment of fluid dynamics and his examination of fluid flow was introduced in his 1738 work Hydrodynamica. Rational mechanics dealt primarily with the development of elaborate mathematical treatments of observed motions, using Newtonian principles as a basis, and emphasized improving the tractability of complex calculations and developing of legitimate means of analytical approximation. The accepted theory of heat in the 18th century viewed it as a kind of fluid, called caloric; although this theory was later shown to be erroneous, a number of scientists adhering to it nevertheless made important discoveries useful in developing the modern theory, including Joseph Black (1728–1799) and Henry Cavendish (1731–1810). Opposed to this caloric theory, which had been developed mainly by the chemists, was the less accepted theory dating from Newton's time that heat is due to the motions of the particles of a substance. In 1849, Joule published results from his series of experiments (including the paddlewheel experiment) which show that heat is a form of energy, a fact that was accepted in the 1850s. The relation between heat and energy was important for the development of steam engines, and in 1824 the experimental and theoretical work of Sadi Carnot was published. Carnot captured some of the ideas of thermodynamics in his discussion of the efficiency of an idealized engine. These shortcomings of classical physics were never to be resolved and new ideas were required. At the beginning of the 20th century, a major revolution shook the world of physics, which led to a new era, generally referred to as modern physics. === Radiation experiments === In the 19th century, experimenters began to detect unexpected forms of radiation: Wilhelm Röntgen caused a sensation with his discovery of X-rays in 1895; in 1896, Henri Becquerel discovered that certain kinds of matter emit radiation on their own accord. In 1897, J. J. In 1905, Einstein used the quantum theory to explain the photoelectric effect, and in 1913 the Danish physicist Niels Bohr used the same constant to explain the stability of Rutherford's atom as well as the frequencies of light emitted by hydrogen gas. The quantized theory of the atom gave way to a full-scale quantum mechanics in the 1920s. New principles of a "quantum" rather than a "classical" mechanics, formulated in matrix-form by Werner Heisenberg, Max Born, and Pascual Jordan in 1925, were based on the probabilistic relationship between discrete "states" and denied the possibility of causality. The Standard Model, based on the Yang–Mills theory groups the electroweak interaction theory and quantum chromodynamics into a structure denoted by the gauge group SU(3)×SU(2)×U(1). The formulation of the unification of the electromagnetic and weak interactions in the standard model is due to Abdus Salam, Steven Weinberg and, subsequently, Sheldon Glashow. Electroweak theory was later confirmed experimentally (by observation of neutral weak currents), and distinguished by the 1979 Nobel Prize in Physics.

alongside a ship. See also CAMEL. catenary . , n. The curve formed by a uniform cable supported only at its ends. Navigators are concerned with the catenary of

A chain assumes the well-known shape known as a catenary when it hangs loosely from two points in a gravitational field. The correct solution of the catenary was one of the early triumphs of the newly invented calculus of variations at the end of the 17th century. Here we revisit the catenary and show that, for a chain hanging from a horizontal rod, three new and distinct configurations are possible if a soap film covers the area bounded by the chain and the rod. We first review the general problem and discuss the conditions under which the chain assumes a concave, triangular or convex configu