PSI - Issue 10

C.B. Demakos et al. / Procedia Structural Integrity 10 (2018) 148–154

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C.B. Ddemakos et al. / Structural Integrity Procedia 00 (2018) 000 – 000

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formed in axial forces. An important question, which is automatically generated, is what is the appropriate equation describing this curved geometry? Galileo (1638) postulated that this shape is a parabola and Euler (1744) stated that “Since all effects of nature obey some maximum or minimum law, we cannot deny that the trajectories described by projectiles under the influence of central forces, will follow a property of maximum or minimum ” and the integral of the velocity multiplied by curve element is always a maximum or a minimum. In this way, Euler postulated an expression for the least action principle and proved in this work that the catenary is the curve which, when rotated about the x -axis, gives the surface of minimum-surface area (the catenoid) for the given bounding circles. Finally, Nicolas Fuss (1796) provided equations describing the equilibrium of a chain under any force in 1796. He has revealed that the geometry of the catenary curve is not a parabola but a cosines hyperbola. This was applied by Antonio Gaudi (1915), who found an optimal geometry and built the famous and beautiful Basílica i Temple Expiatori de la Sagrada Família church. Since past, engineers developed empirical methods for structures mainly in stone bridges, which are always in full compression under external loadings. Catenary arches are often used in the construction of kilns and bridges (Sigmund (2003)). The arch is encountered apart of civil engineering constructions in human beings (https://en. wikipedia.org/wiki/Arches_of_the_foot#Medial_arch). The method of arch constructing was extensively used from past with masonry, concrete and steel construction to bridge large spans. The arch analysis for these constructions has been applied to geotechnical projects (http://www. obvis.com, 21/08/2017) as well as to bridge engineering. The fixed arch is most often used in reinforced concrete bridges and tunnel constructions, where the spans are short. Because it is subject to additional internal stress caused by thermal expansion and contraction, this type of arch is considered to be statically indeterminate (Ambrose (2012)), which is a soft compression form. It can span a large area by resolving forces into compressive stresses and, in turn eliminating tensile stresses. This is sometimes referred to as arch action (Vaidyanathan (2004)). As the forces in the arch are carried to the ground, the arch will push outward at the base, called thrust. As the rise, or height of the arch decreases, the outward thrust increases (Ambrose (2012)). In order to maintain arch action and prevent the arch from collapsing, the thrust needs to be restrained, either with internal ties or external bracing, such as abutments (Ambrose (2012)). Arches have many forms, but all fall into three basic categories: circular, pointed, and parabolic and can also be configured to produce vaults and arcades (Ambrose (2012)). The parabolic arch employs the principle that when weight is uniformly applied to an arch, the internal compression resulting from that weight will follow a parabolic profile. Of all arch types, the parabolic arch produces the most thrust at the base, but can span the largest areas. It is commonly used in bridge design, where long spans are needed (Ambrose (2012)). During 2017 in the Reinforced Concrete Laboratory of Civil Engineering Department at University of West Attica, an experimental MSc thesis was prepared (Liveris (2017)), part of which is described in this paper. In these experiments three curved arches were constructed and tested in bending. The geometry of the specimens was formed such that L=70 cm in span and h= 30 cm high following the parabola shape x x y h L L                       2 4 (1) The optimal height of the arch, i.e. the height at which the arch’s volume is minimized, is given by the ratio of height to span expressed by: h L  3 4 (2) The fact that this is the optimal ratio of span over height for the funicular parabolic curve, is widely known in literature (Block (2009); Kilian (2005)) and this confirms that load path optimization is indeed a valid approach to 2. Experiment setup