hyperboloid - gmit...hyperboloid structure 4 1.introduction 1.1 concept in geometry, a hyperboloid...
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HYPERBOLOID STRUCTURE 1
GYANMANJARI INSTITUTE OF TECHNOLOGY
DEPARTMENT OF CIVIL ENGINEERING
HYPERBOLOID
STRUCTURE
YOUR NAME/s
HYPERBOLOID STRUCTURE 2
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 3
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! Bookmark not defined.
5.1 Photographs of making / crafting model [Stage by Stage] ........................ Error! Bookmark not defined.
6. FINAL MODEL ................................................................................................... Error! Bookmark not defined.
6.1 Model Photographs ................................................................................... Error! Bookmark not defined.
7. REFERENCE ....................................................................................................... Error! Bookmark not defined.
Web Reference ................................................................................................ Error! Bookmark not defined.
Research Papers ............................................................................................... Error! Bookmark not defined.
Thesis / Books .................................................................................................. Error! Bookmark not defined.
8. CREDIT .............................................................................................................. Error! Bookmark not defined.
Created by: ....................................................................................................... Error! Bookmark not defined.
Guided by: ........................................................................................................ Error! Bookmark not defined.
HYPERBOLOID STRUCTURE 4
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 one sheet
conical surface in between
Hyperboloid of two sheets
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 5
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, ∞)
2.2 Properties of a hyperboloid of one sheetLines on the surface 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.
Plane sections For simplicity the plane sections of the unit hyperboloid with
equation H1: x2+y2-z2=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 of one sheet: generation by a rotating
hyperbola (top) and line (bottom: red or blue)
Hyperboloid of one sheet: plane sections
HYPERBOLOID STRUCTURE 6
Properties of a hyperboloid of two sheets The hyperboloid of two sheets does not contain lines. The discussion of plane sections can be performed for
the unit hyperboloid of two sheets with equation
H2: x2+y2-z2=-1
which can be generated by a rotating hyperbola
around one of its axes (the one that cuts the
hyperbola).
A plane with slope less than 1 (1 is the slope of
the asymptotes of the generating hyperbola)
intersects H2 either in an ellipse or in a point or not
at all,
A plane with slope equal to 1 containing the
origin (midpoint of the hyperboloid) does not
intersect H2,
A plane with slope equal to 1 not containing the origin intersects H2 in a parabola,
A plane with slope greater than 1 intersects H2 in a hyperbola.
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the
general case (see circular section).
Remark: A hyperboloid of two sheets is protectively equivalent to a sphere.
Common parametric representation The following parametric representation includes
hyperboloids of one sheet, two sheets, and their common
boundary cone, each with the z-axis as the axis of
symmetry:
For d>0 one obtains a hyperboloid of one sheet,
For d < 0 a hyperboloid of two sheets, and
For d=0 a double cone.
One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of
symmetry by shuffling the position of the cs term to the appropriate component in the equation above.
Symmetries of a hyperboloid The hyperboloids with equations are,
point symmetric to the origin,
symmetric to the coordinate planes and
rotational symmetric to the z-axis and symmetric to any plane containing the z-axis, in case of a=b
(hyperboloid of revolution).
2.3 Benefits: So why are hyperboloid structures of interest today? The two main reasons, apart from aesthetic
considerations, are strength and efficiency.
Hyperboloid of two sheets: generation by
rotating a hyperbola
Hyperboloid of two sheets: plane sections
HYPERBOLOID STRUCTURE 7
As hyperboloid structures are double curved, that is simultaneously curved in opposite directions, they are
very resistant to buckling. This means that you can get away with far less material than you would
otherwise need, making them very economical.
Single curved surfaces, for example cylinders, have strengths but also weaknesses. Take a drinks can
for example: these are made extremely thin, with sides only a fraction of a millimetre thick, yet contain the
pressurised beverage and if stood on end can support the weight of a grown adult even when empty. But,
once you have enjoyed the contents, you can push in the side with just a slight pressure from your finger.
Alternately, if you were to push with your finger from the inside of the can (being careful to avoid any sharp
edges of course) then you will find that you have to apply considerable effort to make any impression.
Double curved surfaces, like the hyperboloids in question, are curved in two directions and thus
avoid these weak directions. This means that you can get away with far less material to carry a load, which
makes them very economical.
The second reason, and this is the magical part, is that despite the surface being curved in two
directions, it is made entirely of straight lines. Apart from the cost savings of avoiding curved beams or
shuttering, they are far more resistant to buckling because the individual elements are straight.
This is an interesting paradox: you get the best local buckling resistance because the beams are
straight and the best overall buckling resistance because the surface is double curved. Hyperboloid
structures cunningly combine the contradictory requirements into one form.
Hyperboloid structure can be used as diagrid structure.
Gives a very pleasant look to structure.
Hyperboloid structure can also be used to make sculpture.
Can be used for making of roof.
To make high-rise towers.
To make Cooling Towers.
To make water tanks.
2.4Disadvantages:
Skilled labour required to construct hyperboloid structure.
Special equipment required for critical hyperbola curves.
2.5Parameters: Following parameters should be taken care of, for design of hyperboloid structure.
1. Span
2. Curvature
3. Area
4. Material
5. Loads
2.6 Design: Manual Design of hyperboloid
structure is lengthy task and
required very high accuracy. Now
a days computer aided software
are used to analyse and design of
Hyperboloid structure. STAAD,
GSA, Ansys software are
generally used to design
hyperboloid structures.
Methodology of designing
structure in GSA is given below.
Design in GSA Software:
Link for software: http://www.oasys-software.com/products/engineering/gsa-suite.html
Methodology:
HYPERBOLOID STRUCTURE 8
Draw a square or rectangular array of crossed beams, but make sure that all the elements or members are full length: do not
split them at the intersection points at this time.
To make the twist select all the nodes on two adjacent sides, then right click on the middle corner and invoke the Flex
command.
Use a linear flex to move that node up by the appropriate amount and note that all the other nodes, and hence beams, have
followed.
Repeat for the other two sides and you have created your hyperbolic paraboloid.
To finish simple, select all the elements and connect them using the sculpt menu.
Rectangular hyper shells are even easier:
Take a Quad4 element to cover the whole roof, adjust the corners to the appropriate elevation, then split the warped quad
into suitably sized pieces.
Finish by splitting the Quads into Triangle elements as the Quads will likely be too warped to analyse.
GSA Hyper
The key to hyperboloid towers is the use of cylindrical axes. If you set the current grid (Ctrl+Alt+w) to the Global Cylindrical Axis
(or your own as appropriate) you will note that the nodal coordinates are now given not as X, Y, & Z but Radius, Theta (angle) & Z (height).
Define a node on the top and bottom rings and join with a
beam.
Copy this round to form a circular array (note that the copy
command defaults to the current axis set).
Mirror all the resulting beams through the radius/theta plane
(you will see why in a moment).
Select all the nodes in the top ring and move them through a
suitable theta angle (hint: make it a multiple of the beam
spacing).
Do the same for the twin ring at the bottom but make the angle the negative
of that you used at the top.
Select all the lower beams and Move them (not copy this time) back through the radius/theta plane to make them overlap
with the original set.
Select all the beams and Connect them to form the surface.
To complete the surface, select all the nodes and extrude them by the angle you used to create the beams, including beam
elements along the extrusion to create hoops.
Alternately, trace Quads onto the beam grillage, split them into triangles, and copy them around. Finish by deleting the beam formwork.
Another way to form a hyperboloid shell tower is to create two nodes, one on the top ring and one on the bottom, with one
set at an angle. Join the nodes up with a beam, split the beam into sufficient pieces, delete the beams and extrude the resulting nodes
around, creating Quads in the process. Finish by splitting the Quads into Triangles. Note that while the surface will be the same as the previous method, the mesh will be in a different pattern.
HYPERBOLOID STRUCTURE 9
Beam vs Shell meshing
2.6 Construction Techniques: Construction techniques mostly depend of type of material to be used in hyperbolic structure as well as size
of the structure.
Following images shows stage by stage construction of cooling tower (hyperboloid structure):
HYPERBOLOID STRUCTURE 10
3. SURVEY
3.1 List of existing hyperboloid structures
Sr.No. Description Photograph
1. The world's first hyperboloid
structure, Nizhny Novgorod, 1896
2. Shukhov Tower in
Moscow.
3. Hyperboloid
Adziogol Lighthouse near
Kherson, Ukraine, 1911
4. Shukhov towers
on the Oka River in the suburb of
Nizhny Novgorod, 1989
5. Shukhov tower in
Krasnodar, 2005
HYPERBOLOID STRUCTURE 11
6. Mae West in Munich, 2011
7. Kobe port tower
8. Catedral de
Brasília by Oscar Niemeyer
9. Sydney Tower
10. Tower by Jerzy
Michał Bogusławski in
Poland
11. Cooling tower of
Padwa power plant, Bhavnagar
HYPERBOLOID STRUCTURE 12
4. MODEL DESIGN
4.1 Materials & Quantity
Material-1 wooden sticks
Material-2 Photo + Quantity
4.2 Costing of Model
1. Material 500/-
2. Labour 300/-
3. Man Hour required 30 hours
4.3 Estimation of Full Scale model
1. Material 500/-
2. Labour 300/-
3. Man Hour required 30 hours
4.4Design Drawings
Technical Drawing-1
Technical Drawing -2
5.1 Photographs of making / crafting model[Stage by Stage]
HYPERBOLOID STRUCTURE 13
7. REFERENCE
Web Reference 1. https://en.wikipedia.org/wiki/Hyperboloid_structure 2. https://commons.wikimedia.org/wiki/Hyperboloid_structure 3. https://en.wikiversity.org/wiki/Building_construction_techniques 4. http://www.oasys-software.com/blog/2012/03/hyperboloid-structures-in-gsa/
Research Papers 1. A hyperboloid structure as a mechanical model of the carbon bond by, I.E. Berinskii, A.M.
Photo-1
Photo-2
Photo-3
6. FINAL PHOTOGRAPHS OF MODEL
HYPERBOLOID STRUCTURE 14
Krivtsov, 2016.
2. Wind-induced loads and integrity assessment of hyperboloid reflector of solar power plants by M. Eswaran, R.K. Verma, G.R. Reddy, 2016.
3. Ultraspinning limits and rotating hyperboloid membranes by, Robie A. Hennigar, David Kubizňák, 2016.
4. On the dynamics of a particle on a hyperboloid by, K. Kowalski, J. Rembieliński, 2013
Thesis / Books 1. Hyperboloid Structures by General Books LLC, 2010, ISBN - 1155627067, 9781155627069 2. Tensile Architecture by Philip drew.
3. An Introduction to Shell Structures: The Art and Science of Vaulting By Michele Melaragno,
4. Structural Steel Design By Russell Jesse
8. CREDIT
Created by:
DAVE MAITRY B.E.Civil, 5th sem
[email protected] , Mo.9428188903
SOMPURA HEETARTH B.E.Civil, 5th sem [email protected] MO.8460818190
HYPERBOLOID STRUCTURE 15
Guided by:
VIJAY PARMAR B.E.Civil, M.E.Structure,
