GYANMANJARI INSTITUTE OF TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING HYPERBOLOID STRUCTURE YOUR NAME/s HYPERBOLOID STRUCTURE 1 TOPIC TITLE HYPERBOLOID STRUCTURE By DAVE MAITRY [151290106010] [[email protected]] SOMPURA HEETARTH [151290106024] [[email protected]] GUIDED BY VIJAY PARMAR ASSISTANT PROFESSOR, CIVIL DEPARTMENT, GMIT. DEPARTMENT OF CIVIL ENGINEERING GYANMANJRI INSTITUTE OF TECHNOLOGY BHAVNAGAR GUJARAT TECHNOLOGICAL UNIVERSITY HYPERBOLOID STRUCTURE 2 TABLE OF CONTENTS 1.INTRODUCTION ................................................................................................................................................ 4 1.1 Concept ..................................................................................................................................................... 4 2. HYPERBOLOID STRUCTURE ............................................................................................................................. 5 2.1 Parametric representations ...................................................................................................................... 5 2.2 Properties of a hyperboloid of one sheet Lines on the surface ................................................................ 5 Plane sections .............................................................................................................................................. 5 Properties of a hyperboloid of two sheets .................................................................................................. 6 Common parametric representation .......................................................................................................... 6 Symmetries of a hyperboloid ...................................................................................................................... 6 2.3 Benefits: .................................................................................................................................................... 6 2.4 Disadvantages: ...................................................................................................................................... 7 2.5 Parameters: ............................................................................................................................................... 7 2.6 Design:....................................................................................................................................................... 7 Design in GSA Software: .............................................................................................................................. 7 2.6 Construction Techniques: ..................................................................................................................... 9 3. SURVEY ..........................................................................................................................................................10 3.1 List of existing hyperboloid structures ....................................................................................................10 4. MODEL DESIGN .............................................................................................................................................12 4.1 Materials & Quantity ..............................................................................................................................12 4.2 Costing of Model .....................................................................................................................................12 4.3 Estimation of Full Scale model ................................................................................................................12 4.4 Design Drawings ......................................................................................................................................12 5. MODEL MAKING ............................................................................................... Error! 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CREDIT .............................................................................................................. Error! Bookmark not defined. Created by: ....................................................................................................... Error! Bookmark not defined. Guided by: ........................................................................................................ Error! Bookmark not defined. HYPERBOLOID STRUCTURE 3 1.INTRODUCTION 1.1 Concept In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. A hyperboloid is a surface that may be obtained from a paraboloid of revolution by deforming it by means of directional scaling, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a centre of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has also three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Hyperboloid of two sheets Hyperboloid of one sheet conical surface in between Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are axes of symmetry of the hyperboloid, and origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations: or Both of these surfaces are asymptotic to the cone of equation One has a hyperboloid of revolution if and only ifa2=b2. Otherwise, the axes are uniquely defined (up to the exchange of the x-axis and the y-axis. There are two kinds of hyperboloids. In the first case (+1 in the right-hand side of the equation), one has a one-sheet hyperboloid, also called hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies that the tangent plane at any point intersect the hyperboloid into two lines, and thus that the one-sheet hyperboloid is a doubly ruled surface. In the second case (−1 in the right-hand side of the equation), one has a two-sheet hyperboloid, also called elliptic hyperboloid. The surface has two connected components, and a positive Gaussian curvature at every point. Thus the surface is convex in the sense that the tangent plane at every point intersects the surface only in this point. HYPERBOLOID STRUCTURE 4 2. HYPERBOLOID STRUCTURE 2.1 Parametric representations Cartesian coordinates for the hyperboloids can be defined, similar to spherical coordinates, keeping the azimuth angle θ ∈ [0, 2π), but changing inclination v into hyperbolic trigonometric functions: One-surface hyperboloid: v ∈ (−∞, ∞) Two-surface hyperboloid: v ∈ [0, ∞) Hyperboloid of one sheet: generation by a rotating 2.2 Properties of a hyperboloid of one sheetLines on the surface hyperbola (top) and line (bottom: red or blue) A hyperboloid of one sheet contains two pencils of lines. It is a doubly ruled surface. If the hyperboloid has the equation then the lines. are contained in the surface.In case a=b he hyperboloid is a surface of revolution and can be generated by rotating one of the two lines + - g0 or g0 which are skew to the rotation axis (see picture). The more common generation of a hyperboloid of revolution is rotating a hyperbola around its semi-minor axis (see picture). Remark: A hyperboloid of two sheets is protectively equivalent to a hyperbolic paraboloid. Hyperboloid of one sheet: plane sections Plane sections For simplicity the plane sections of the unit hyperboloid with 2 2 2 equation H1: x +y -z =1 are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too. A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects H1 in an ellipse, A plane with a slope equal to 1 containing the origin intersects H1 in a pair of parallel lines, A plane with a slope equal 1 not containing the origin intersects H1in a parabola, A tangential plane intersects H1 in a pair of intersecting lines, A non-tangential plane with a slope greater than 1 intersects H1 in a hyperbola. Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case. HYPERBOLOID STRUCTURE 5 Properties of a hyperboloid of two sheets The hyperboloid of two sheets does not contain