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Curves with Lines
Published in Joseph Choma, Études for Architects, 2018
A ruled surface is a curved surface generated with a series of straight lines. Prior to the computer, ruled surfaces were a popular way to construct and calculate curved surfaces. It is a simple, elegant and efficient geometric logic. Imagine trying to create a custom formwork for a concrete shell structure. If the formwork can be built out of a series of straight linear wood elements, the construction becomes feasible. In addition, similar to the folded plate structure, ruled surfaces can have inherent structural principles embedded within them. It is possible to increase the structural depth of a surface while maintaining a thin material thickness. Many architects, historical and contemporary, choose to design with ruled surfaces. Roofs, canopies, cantilevers, walls, ramps and spiral stairs are some of the architectural elements that have been designed with ruled surfaces. Wood, concrete, steel, cables and bricks are some of the materials that have been used to construct these surfaces in architectural applications. Outside of architecture, many artists have made drawings, sculptures and installations with the same basic logic. Simply put, it is a seductive system that makes economic, construction and structural sense.
Geometry of the Middle Surface
Published in Eduard Ventsel, Theodor Krauthammer, Thin Plates and Shells, 2001
Eduard Ventsel, Theodor Krauthammer
Ruled surfaces are obtained by the translation of straight lines over two end curves (Fig. 11.11). The straight lines are not necessarily at right angles to the planes containing the end curves. The frustum of a cone can thus be considered as a ruled surface, since it can be generated by translation of a straight line (the generator) over two curves at its ends. It is also, of course, a shell of revolution. The hyperboloid of revolution of one sheet, shown in Fig. 11.11a, represents another example of ruled surfaces. It can be generated also by the translation of a straight line over two circles at its ends. Figure 11.11b shows a surface generated by a translation of a straight line on a circular curve at one end and on a straight line at the other end. Such surfaces are referred to as conoids. Both surfaces shown in Fig. 11.11 have negative Gaussian curvatures.
Extraction of vertical cylinder contacting area for motorcycle safety verification
Published in Computer-Aided Design and Applications, 2018
For the purpose of maintaining confidentiality, only the computation result for two CAD demonstration models is shown here. Fig. 11 illustrates sample model A with 437,040 polygons (Fig. 11(a)) and sample model B with 2,700,707 polygons (Fig. 11(b)). The result of the offsetting and shrinking the projection of the model is also shown in blue in the same figure. A ruled surface is generated by moving a vertical line along the boundary of the shrunken figure. Fig. 12 illustrates the ruled surface in blue. By using the contact analysis between the ruled surface and surface polygons of the motorcycle model, polygons contacting the cylindrical column of diameter 300 mm in the forward motion are extracted as green polygons in Fig. 13. In these polygons, polygons whose radius of curvature are less than 4mm are illustrated in red in Fig. 14.
Terrestrial laser scanning for structural inspection with Kriging interpolation
Published in Structure and Infrastructure Engineering, 2022
Thomas Sanchez, David Conciatori, Mahdi Ben-Ftima, Bruno Massicotte
Other more complex surfaces, such as ruled surfaces or revolution surfaces can also be processed. A ruled surface is a surface swept by a range of lines, where two 3D-curve f(u1) and g(u1) are joined by line segments (Figure 4b):
Bending analysis of functionally graded CNT reinforced doubly curved singly ruled truncated rhombic cone
Published in Mechanics Based Design of Structures and Machines, 2019
Since casting and fabrication of doubly curved singly ruled truncated cone (also known as conoidal shell) possessing singly ruled surface is quite easy, it is favored in the construction industry. The advancements in composite technology have made it possible to design a strong and stiff composite has blossomed a new inspiration for researchers to study about conoidal shells. Conoidal shells are esthetically appealing, structurally stiff and are used to cover the column-free large area in aircraft hangars, industrial structures, assembly hall, and exhibition hall. A comprehensive study of bending response under various types of mechanical load is essential to access the optimal use of rhombic conoids. A combined variational approach was used by Hadid (1964) for the bending response of both simply supported and clamped elastic conoids. In his formulation, he modifies the shell equations (expressed in terms of displacement) in the ordinary differential equation using Kantorovich method. Ghosh and Bandyopadhyay (1990) and Dey et al. (1997) used finite element technique for the bending analysis of doubly curved isotropic shell and analysis of laminated composite conoidal shell structure, respectively. Bakshi and Chakravorty (2014) presented the first ply failure analysis of thin laminated composite conoidal shell using finite element method under uniformly distributed loading. The free vibration analysis of rotating CNTRC truncated conical shell are studied by Heydarpour et al. (2014) using FSDT. Mehri et al. (2016a, 2016b) investigated the dynamic instability and vibration analysis of FG-CNT reinforced truncated conical shell subjected to an external pressure and axial compression simultaneously. Nejati et al. (2017) explored the static behavior of FG truncated conical shells reinforced by carbon nanotubes using generalized differential quadrature method which is associated with the TSDT. Mehri et al. (2017) presented the aeroelastic response of FG-CNTRC truncated conical panel under combined action of axial load and aerodynamic load. Chaubey et al. (2018) presented a higher order mathematical model for the static analysis of laminated composite conoidal shell.