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Wind design of fabric structures: Determination of gust factors for fabric structures
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Wind design of fabric structures: Determination of gust factors for fabric structures
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Content
WIND DESIGN OF FABRIC STRUCTURES
DETERMINATION OF GUST FACTORS FOR FABRIC STRUCTURES.
by
Neha Sivaprasad
A Thesis Presented to the
FACULTY OF THE SCHOOL OF ARCHITECTURE
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF BUILDING SCIENCE
May 2006
Copyright 2006 Neha Sivaprasad
UMI Number: 1437584
1437584
2006
UMI Microform
Copyright
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, MI 48106-1346
by ProQuest Information and Learning Company.
ii
Acknowledgements
This thesis is dedicated to my very loving family. It is quite unbelievable that I
have reached almost the end of my master’s thesis. This end to me, is not just
the end of my thesis but is a gratifying and fulfilling end of a very distinct
phase of my life, my two year long stay in Los Angeles as a student the masters
program at University of Southern California; a phase in my life, which taught
me a lot academically, professionally and personally.
I would like to thank my mother, father, grandmother, grandfather, and my
sister, Niyatee who were a constant support even though they were very far
away from me. A huge thanks to my uncle Umesh and my aunt Jagi who were
a great support through out these two years of my stay here. I would like to
extend a warm thanks to my friend Anish for his support.
This thesis would have not been possible with out the constant guidance and
sincere support of my thesis chair Professor Goetz Schierle. I am very grateful
to Professor Douglas Noble for all his contribution to my thesis and I am
thankful for all his help during the course of the whole masters program at
iii
USC. I thank Professor Dimitry Vergun for his extremely valuable technical
advice, and Professor Jim Tyler for all his time and help.
I would like to take this chance to thank the entire MBS faculty who have
always been very willing to share their knowledge and have been very
encouraging. And finally, I thank my roommates and my friends for standing
by my through this thesis.
iv
Table of Contents
Acknowledgements
List of Tables
List of Figures
Abstract
Chapter 1: Introduction
IBC / ASCE 7 wind design has no gust factor for fabric
structures
Currently wind tunnel tests have to be conducted for
determining the gust factor.
The code does has not enough design information
This discourages the design of fabric structures
Chapter 2: Wind
The effects of wind on structures
Interaction of wind and structures
ASCE 7 methods for wind load depending on structure
type
ASCE 7 formulas for gust factor analysis
The formula for the gust factor of flexible structures
requires natural frequency of vibration of the structures
Chapter 3: Tensile Structures
Classification of tensile structures
Fabric membrane structures, a type of tension structure,
can also be classified into various types.
Classified by boundary conditions
Classified by shape
Considering the classifications a designer has many
options to design fabric structure for a given space
ii
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1
7
29
v
Chapter 4: Previous Research Related to Gust Factors and Wind
Scaling
Related research relevant to this thesis
Research on gust factors
Research on wind scaling
Chapter 5: Proposed Method To Determine Gust Factor
Natural frequency method to determine gust factor
Proposed method to find the natural frequency for
various fabric structures shapes
Chapter 6: Model Building and Wind Tunnels
Fabric testing
Model building procedures
Wind tunnel types
Wind tunnel used for testing
Chapter 7: The Natural Frequency Test
Wind tunnel tests
Dynamic responses of models
Chapter 8: The Dynamic Static Load Test
Failure of natural frequency test
Dynamic loading test
Static loading test
Gust factor calculation
Chapter 9: Results and Conclusions
Results and conclusions of dynamic lateral test
Relationship between prestress and lateral load for
saddle model
Relationship between prestress and gust factor
Pulley test, results and conclusions
56
66
71
95
102
125
vi
Chapter 10: Future Work
Relationship between lateral load and openness
Relationship between openness and gust factor
Aero elasticity
Bibliography
138
143
vii
List of Tables
Table 2A: Definitions of Importance factors as per ASCE‐7
Table 2B: Building Exposures as per IBC
Table 2C: Table from ASCE‐7 Minimum Standards For Building
Loads; Section 6: Wind Loads
Table 4A: Exposure types, according to ASCE‐95
Table 6A: Weights in pounds attached to the fabric and the
respective elongation
Table 6B: Weights in pounds attached to the fabric and the respective
elongation (Second test)
Table 8A: Results of Stress strain test for rubber strip.
Table 9A: Amplitude of vibration of membranes at wind speed of
17mph
Table 9B: Lateral loads exerted by wind on membrane surfaces
Table 9C: Prestress and lateral load for Saddle shape model
Table 9D: Static loads that cause deformation equal to that caused by
the wind
Table 9E: Gust factors for tested models
Table 9F: Gust factors for saddle shape model at different prestresses.
Table 9G: Results of the Pulley test.
22
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viii
List of Figures
Figure 2.1: Velocity profiles over terrain with 3 different roughness
characteristics
Figure 2.2: Formation of Hurricanes
Figure 2.3: Tornadoes
Figure 2.4: Positive Pressure and Negative Pressure on a Structure in
the path of the wind
Figure 2.5: Vortex Shedding; plan view
Figure 2.6: Relation of wind velocity to wind pressure exerted on a
stationary object
Figure 2.7: Average wind speeds (IBC)
Figure 3.1: Prestress
Figure 3.2: Air supported structure
Figure 3.3: Air inflated structure
Figure 3.4: Cable suspended roof Dulles International Airport
Figure 3.5: Cable force for different sag to span ratio
Figure 3.6: Cable Stayed Roof
Figure 3.7: Cable force for different sag to span ratio
Figure 3.8: Cable truss configurations
Figure 3.9: Hyperbolic Paraboliod and Minimal Surface for Square and
Rhomboid Plan
9
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16
24
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36
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ix
Figure3.10: Cable net configurations: parallel to boarder, diagonal to
boarder, triangular mesh
Figure 3.11: First Cable net Structure; The German Pavilion Expo 67
Figure 3.12: Cable (tension) edge, Rigid Arch (compression) edge,
Straight Beam (bending) Edge
Figure 3.13: Saddle Shaped membrane
Figure 3.14: Wave shaped membrane
Figure 3.15: Arch supported membrane
Figure 3.16: Point shaped membrane
Figure 3.17: Fabric‐to‐Fabric connection with cable
Figure 3.18: Connection at mast (plan view)
Figure 3.19: Fabric edge detail, Cable to mast connection (plan view)
Figure 4.1: Average wind speeds based on ASCE‐95 3 second gust
speed data
Figure 5.1: Saddle shape membrane structure
Figure 5.2: Wave shape membrane structure – Denver International
Airport
Figure 5.3: Arch shape membrane structure
Figure 5.4: Point shape membrane structure
43
43
45
45
46
47
48
53
54
55
59
67
67
68
68
x
Figure 5.5: Saddle, wave, arch and point shaped models
Figure 6.1: Fabric stress strain test
Figure 6.2: Calibration of weighing instrument
Figure 6.3: Stress to strain graph of the model fabric.
Figure 6.4: Stress over Strain graph of model fabric.
Figure 6.5: Step 1: Plywood base of saddle shape model
Figure 6.6: Step 2: Saddle shape model with fabric
Figure 6.7: Step 3: Finished saddle shape model
Figure 6.8: Saddle shape model with three different sag to span ratios,
as shape S1, S2 and S3
Figure 6.9: Wave shape model
Figure 6.10: Wave shape model
Figure 6.11: Step 1 of arch shape model
Figure 6.12: Step 2 of arch shape model
Figure 6.13: Step 3 of arch shape model
Figure 6.14: Step 3 of arch shape model
Figure 6.15: Step 1 of point shape model
Figure 6.16: Step 2 of point shape model
Figure 6.17: Step 3 of point shape model
Figure 6.18: Point shape model
Figure 6.19: Open Circuit Wind Tunnel
69
73
73
74
76
79
80
81
82
83
83
84
84
85
85
86
87
87
87
89
xi
Figure 6.20: Closed Circuit Wind Tunnel
Figure 6.21: Turbulent boundary layer generated in a wind tunnel
section
Figure 6.22: Wind Tunnel; USC School of Engineering, Mechanical
and Aerospace Engineering Department
Figure 6.23: Wind Tunnel; USC School of Architecture
Figure 6.24: Wind Meter
Figure 7.1: Saddle Shape Model in the Wind Tunnel
Figure 7.2: Dynamic response of Saddle shaped membrane
Figure 7.3: Dynamic response of wave shape membrane
Figure 7.4: Dynamic response of arch shape membrane
Figure 7.5: Dynamic response of arch shape membrane
Figure 8.1: Stress vs. Strain graph of rubber strip
Figure 8.2: Lateral load testing assembly
Figure 8.3: Elongation in rubber strip due to movement of model.
Figure 8.4: Test stand
Figure 8.5: Pieces of paper used to simulate uniform loading
Figure 8.6: Pieces of paper used to simulate uniform loading
Figure 8.7: Static Loading assembly for saddle shape model
Figure 8.8: Static Loading assembly for wave shape model
Figure 8.9: Static loading of saddle shape model – procedure 2
90
91
92
93
94
96
99
100
101
101
106
108
108
111
113
113
115
116
118
xii
Figure 8.10: Static loading of wave shape model – procedure 2
Figure 8.11: Static loading of arch shape model – procedure 2
Figure 8.12: Static loading of point shape model – procedure 2
Figure 8.13: Pulley test
Figure 8.14: Pulley test
Figure 9.1: Stress vs. Strain graph of rubber strip used to calculate
lateral/dynamic load
Figure 9.2: Lateral load vs. prestress graph for saddle shaped
membrane
Figure 9.3: Graph showing decrease in the gust factor with increase in
prestress for saddle shape model.
Figure 10.1: Uplift force
119
120
121
123
123
129
130
133
141
xiii
Abstract
Fabric structures are a classification of tensile structures in which the spanning
material is fabric. Prestress and anticlastic shape stabilizes these structures and
facilitates spanning large distances. The design of such lightweight structures is
usually governed by wind.
The American Society of Civil Engineer’s publication, the ASCE 7‐ 02 Minimum
Design Loads for Buildings and Other Structures is the design standard
adopted by the International Building Code. This standard does not concise
gust factors for the design of fabric structures. A gust factor takes into account,
the dynamic amplification of structural response due to wind‐structure
interaction and is crucial to design of fabric structures.
This thesis describes a method used to determine gust factors for fabric
structures and uses the proposed method to calculate gust factors for certain
basic shapes of structures. An optimum gust factor of about 1.4 is suggested for
all types of fabric structures in general.
1
Chapter 1: Introduction
1.0 Introduction
This chapter gives an over view of the question and importance of gust factors
in fabric membrane structures. The chapter also briefly reviews the proposed
method for seeking a resolution.
1.1 Problem
Fabric Structures are dynamically sensitive. The American Society of Civil
Engineers (ASCE) Standard, Minimum Design Loads for Buildings and Other
Structures (which is the recommendation of the IBC) does not adequately
define a gust factor for the wind design of dynamically sensitive structures.
1.1.2 Tensile structures
Tensile structures primarily resist loads in tension and attain stability due to
curvature and shape. Tensile structures are becoming increasingly popular
throughout the world. These structures not only have the advantage of being
beautiful but also are fast to erect and most importantly, can span very large
distances with a minimum amount of material. Also, fabric structures provide
natural lighting to save energy. Tensile structures can be classified in different
2
ways into different categories. In an architectural perspective, they may be
classified as tensile membrane or spanning structures and tensile supporting
structures. For example, suspension bridges and cable‐stayed roofs have a
structural supporting system that primarily resists loads in tension. (They do
include some compression members too.)
Tensile structures include:
• Stayed structures
• Suspension structures
• Anticlastic membranes and cable nets
• Cable trusses
• Pneumatic structures
This thesis deals with tensile membranes, and specifically with anticlastic fabric
membranes. Recently a few airport designs have incorporated large span
tensile membrane roofs to get unobstructed column‐free spaces. Small
temporary tensile membranes have been used for over 40 years. Even though
the popularity of such structures is increasing, given their economy and
elegance, only specialists primarily do the manufacturing and designing. This
may be because there is not enough transfer of technical information between
architects, engineers, the construction industry and the research organizations.
3
Also, there are no uniform and well‐established guidelines or codes for design
of such structures. This lack of information and expertise, limits the designing
of these extraordinary structures to relatively few designers, engineers and
architects.
Tensile membranes are very light and flexible and thus have their structural
design governed by wind load. Wind forces can induce large deformations and
vibrations in the membrane. The large variety of shape, size and curvature
possibilities makes every membrane structure almost a unique design,
requiring considerable analysis and research during the design stages. Also the
shapes being aerodynamic, pose challenges in predicting their responses to
wind forces. The aerodynamic behavior of these structures makes it imperative
to conduct wind tunnel tests; however, tests are too costly.
1.2 The Building Code
The IBC (International Building Code), which is the code used in the United
States of America, recommends the ASCE (American Society of Civil Engineers)
standard design loads as a standard to be used for wind design. In 1972 the
American National Standards Institute published a standard for design loads. It
was called the ANSI A58.1‐1972. This standard was revised 10 years later to
4
include an innovative approach to wind loads for components and cladding.
These wind loads were based on aerodynamic responses of building corners,
ridges and eaves to lateral wind loading. In the mid 1980s ASCE took over the
responsibility to lay down the minimum load standards. ASCE published the
ASCE 7‐88, which contained design load criteria for snow loads, wind loads,
earthquake loads, and other environmental loads. In 2000 they published
ASCE 7‐98. The International Building Code adopted the load criteria of this
publication by reference. The latest publication by the ASCE is the ASCE 7‐02
Minimum Design Loads for Buildings and Other Structures.
1.3 Gust factor
As most codes do, ASCE 7 calculates wind loads using the ‘equivalent static
load’ method. The wind load is calculated as the total static force exerted by the
wind on a structure. Wind, however is dynamic and consists of turbulences and
gusts. In addition, flexible structures like tall buildings and tensile structures
tend to increase this dynamic effect. To account for the dynamic response of
buildings to wind, a gust factor is included in wind load calculations.
The gust factor is defined as a dynamic response factor, which accounts for
dynamic amplification of loading due to interaction between wind turbulence
and the structure. ( Taranath, 2004, p.38 ). Gust factors are crucial to wind
5
design of lightweight fabric structures, which are dynamically sensitive and
responsive to wind.
The ASCE standards include gust factors, based on very long and tedious
formulas for dynamically sensitive structures, which include tall structures. The
ASCE standard recommends determination of the gust factor by wind tunnel
experiments and tests, for sensitive and flexible structures like tensile
structures. These dynamic response factors can be studied for a specific
structure with the help of wind tunnel tests, simulating the wind speed, the
wind direction and the neighboring surface conditions, thus assisting in
accurate designing. Wind tunnel tests can be very expensive and tedious and
may be economical only for relatively big fabric structures. However, for
smaller structures it will be useful to have a range of dynamic response factors
for wind design.
This thesis thus proposes to find a range of gust factors and also to study the
dynamic effects of wind on fabric structures. The thesis proposes to classify the
structures into types and test each type to determine a gust factor. The intention
is also to study the impacts, influences and relationships of variables, such as
wind speed, shape, prestress, and sag to span ratio on each other.
6
The thesis proposes to study the dynamic effects on fabric membrane structures
by testing four different study models at a scale of 1:50. This scale is chosen,
taking into consideration the size of test area of the wind tunnel and also the
ease of model building. The thesis also includes background research on tensile
structures and wind design in general.
7
Chapter 2: Wind
2.0 Introduction
This chapter discusses the background study carried out with brief definitions
and explanations of all the important terms to be understood for the purpose of
this thesis. The chapter covers terms and important concepts related to
1. Wind
2. Wind structure interaction
3. Wind resistant design for buildings
4. The building code and wind design
5. Introduction to tensile structures
2.1 Wind
Wind is a natural phenomenon, which generally means movement of air
parallel to the earth’s surface. Wind can be defined as movement of air caused
by a force, which is usually a difference of air pressure. Wind is generated
because air acts like a fluid; temperature variations change its density and
obstacles in its path change its speed. Wind can either flow in a direction
parallel to isobars, or in a direction perpendicular to isobars. An isobar is a line
joining points of equal air pressure. If the wind is gentle, it is usually traveling
8
parallel to isobars. This is because every object moving across the earthʹs
surface deflects to the right in the northern hemisphere (to the left in the
southern) because of rotation of earth (Pattric, Blas, 2000). This force exerted on
the object due to the rotation of the earth is called Coriolis force.
Wind, which is developed by pressure variations usually, starts of in a direction
perpendicular to isobars, but gradually, due to the deflection to the right (or
left, depending on the hemisphere) its direction becomes nearly parallel to the
isobars.
2.1.1 Turbulence
The irregularities on the earth’s surface cause a drag in the path of the wind
and convert a part of it into turbulence. Turbulence includes both, vertical and
horizontal movement of air. Wind turbulence decreases with increase in height
above the surface of the earth. With more surface roughness, like trees,
buildings etc, there is more turbulence and a decrease in wind speed with
height. Figure 2.1 shows mean velocity profiles over terrains with 3 different
roughness characteristics for gradient wind of 100mph.
9
Figure 2.1: Velocity profiles over terrain with three different roughness
characteristics (Pattric, Blas, 2000)
2.1.2 Gust
Gusts are random changes of wind speed above and below the mean speed
level. Fast moving parcels of air, from high‐speed moving air to low speed
strata, cause sudden gusts. This is caused due to obstacles in the path of the air
that induce movements that are perpendicular to the flow of air. Thermal
instabilities induce considerable increase in turbulence.
10
2.1.3 Hurricanes
According to the National Hurricane Center, ʺhurricaneʺ is a name for a
tropical cyclone that occurs in the Atlantic Ocean. ʺTropical cycloneʺ is the
generic term used for low‐pressure systems that develop in the tropics. Tropics
are areas around the equator near the ocean. Hurricanes swirl in circles around
an “eye”, and are counterclockwise in the northern hemisphere and clockwise
in the southern hemisphere. The eye or center of the cyclone has the lowest
pressure. A hurricane can sustain storm winds for several days. This thesis
however, does not deal with hurricanes.
Figure 2.2: Formation of Hurricanes
(www.howstuffworks.com. 21st September 2005)
11
2.1.4 Tornados
Tornados, like hurricanes, are cyclic. Unlike hurricanes however, they have
maximum pressure in the center. Tornados occur when a vortex of air is formed
in a thunderstorm cloud. The tornado moves downward in the thunderstorm
forming a rope of air. This rope of air moves along a path, governed by the
movement of the thunderstorm. If the tornado touches the ground, then it can
cause tremendous damage. This thesis does not deal with tornados.
Figure 2.3: Tornadoes (www.nssl.noaa.gov/NWSTornado. 21st September 2005)
2.2 Wind effects
When wind hits a freestanding structure, the streamlines touch the structure
and then diverge. On the other side of the structure these streamlines don’t
meet immediately. This causes a drop of pressure on the leeward side of the
structure.
12
Wind effects on bodies can be generalized and grouped into the following:
Positive Pressure: Surfaces on the windward side perpendicular to the flow of
the wind experience a direct force, which is called direct positive pressure.
Negative Pressure: Surfaces on the leeward side of the wind generally
experience a suction force, pulling it outward, in the direction of the wind. On
comparison to direct positive pressure on the windward side of the object, this
suction pressure is called negative pressure.
Drag Effect: Surfaces that are parallel to the direction of wind flow experience a
drag effect, as a result of the liquid like behavior of wind. Wind flows around
an object like a liquid.
Figure 2.4: Positive Pressure and Negative Pressure on a Structure in the path
of the wind
Apart from the above effects, any structure or body in the path of airflow is
affected by forces, which can cause effects including vibration, vortex shedding,
turbulent buffeting, galloping and flutter.
13
Consider an object in the path of airflow. As the flow of air passes over the
object, it loses pressure. The pressure around the sides of the object decreases.
This causes suction and formation of eddies or vortices on the opposite side
(leeward side). This phenomenon is called vertex shedding. The pressure drop
at the sides causes the body to move in the direction of the drop. The pressure
drop usually happens alternately on the two sides, thus pushing and pulling
the body in these two directions, causing it to vibrate.
Figure 2.5: Vortex Shedding; plan view (www.galleryoffluidmechanics.com.
21st September 2005)
Turbulent buffeting is a phenomenon similar to vortex shedding. It occurs
when the object is in the path of gentle flow of air causing the object to oscillate
with low amplitude.
14
2.2.1 Wind structure interaction
All building structures have a natural period of vibration and a natural
frequency. The frequency or frequencies at which an object or body tends to
vibrate with when hit, struck, or somehow disturbed is known as the natural
frequency of the object. If the period of the wind‐induced oscillations or
vibrations on a structure match the natural period of vibration of the structure,
then a resonance is created, which amplifies the frequency of vibration to a very
high extent. This high frequency vibration can cause damage to structures.
Thus while designing structures one should be aware of the range of frequency
of the general wind in that location, the natural period of the designed structure
and the damping capacity of the structure. Generally, as a rule of thumb for
most structures, the natural period of vibration is 1/10 second times the number
of stories. Thus a ten‐story building will have a period of about one second.
Therefore, if the wind induces a vibration of a one second period (or a
frequency of 0.1 Hz) it may cause the structure to resonate and may cause
severe vibrations of increased frequencies, causing damages. Also, periodic
gusts in the wind may find resonance with the natural frequency of vibration of
the building and the building may resonate, causing dangerous building
oscillation. Such situations can cause damage even though the equivalent static
15
load of the wind on the building is much less than that for which the building
was designed.
If the entire energy in the wind is converted into pressure exerted on the
structure, then this pressure is given by the following formula.
q = ½ pV
2
Where,
q = wind pressure
p = mass density of the air
V = velocity of the wind
16
.
Figure 2.6: Relation of wind velocity to wind pressure exerted on a stationary
object (Ambrose, Vergun 1995, p.9)
The formula thus shows that wind pressure on a surface perpendicular to the
wind direction is equal to a constant multiplied by the square of the wind
velocity, i.e., wind pressure rises exponentially with an increase in wind
velocity.
17
2.3 Wind resistant design of structures
All structures have to be designed to resist gravity and lateral loads. Lateral
loads chiefly include wind loads and earthquake loads. IBC, the International
Building Code refers to the “ASCE‐7 Standard, Minimum Design Loads for
Buildings and Other Structures” as a model design method. The ASCE‐7
Standard provides a basis for determining wind loads for designing structures.
Presently most structures are designed to resist a lateral force equal to the static
equivalent of the wind pressure. For wind design of structures, it is important
to find out the pressure being exerted by the wind. The remainder of section 2.3
describes the effects of wind on structures.
2.3.1 Inward pressure and outward suction
Surfaces on windward and leeward sides of the structure are designed for full
base pressure. However, the actual windward or leeward force accounts for
only 60% of the total force on the building. Designing for only the 60% of the
total forces is compensated by the fact that base pressures do not take in to
account gust effects as well. Gust effects do not affect the building structure as a
whole but do impact building cladding. (Ambtose, Vergun. 1995, p.13)
18
2.3.2 Pressure on roofs
The roofs if not horizontal may experience inward positive or suction negative
pressure, depending on their form.
2.3.3 Overall force on buildings
The overall force on the building is taken to be the horizontal pressure and
suction on the building surface, and the lateral force resistive system is
designed to resist this force.
2.3.4 Other effects
Dynamically sensitive buildings have to be designed for effects such as
vibration, flutter, whipping etc. These effects, however, need to be dynamically
analyzed for accurate results and cannot be taken care of by the equivalent
static force method. Stiffening of buildings with bracing, shear walls help
reduce these effects. In case of structures of high dynamic sensitivity, like, cable
net structures, suspension bridges, membrane roofs etc, very careful, analysis,
like computational fluid dynamics, or wind tunnel tests should be carried out.
The orientation of lateral resistive systems is also important in buildings. If the
lateral resistive system is not symmetrical in a building, the building might
experience torsion. This is primarily if the center of gravity of the building does
19
not coincide with the centroid of the lateral resistive system. Very large
openings or voids on the surface of the building can cause effects like cupping
of the wind. Such irregularities in building shape and layout, and irregularities
on the building surface can be analyzed for wind only through wind tunnel
tests. Effects such as uplift could also occur.
2.4 Building Code
2.4.1 Introduction
The American Society of Civil engineer’s publication, ASCE Standard,
Minimum Loads for Buildings and Other Structures is a consensus standard.
The standard was first introduced in 1972. Ten years later, it was revised, with
inclusions of an approach to wind loads for components and cladding of
buildings. Later in the 1980s the ASCE took the responsibility of the Minimum
Design Loads of Buildings and Other Structures Standards Committee, which
establishes design loads. The document ASCE‐7 published later included
design‐loading criteria for earthquake loads, wind loads and snow loads.
The International Building Code adopted the ASCE‐7 98 as a recommended
wind loading and design criteria. The latest publication however, is ASCE 7‐02
published in 2003. Model codes like the International Building Code are not
legally binding unless they are adopted by state or city or county ordinances.
20
Section 6.0 of ASCE 7 talks about wind Loads. (Mehta, Delahay. 2004).
Following are brief descriptions of some important points from the ASCE 7‐02,
which are relevant to this thesis.
2.4.2 Design Methods
The ASCE 7‐02 gives three methods to design for wind.
Method 1 is the simplified procedure and can be used if the building is a simple
diaphragm building, regular in shape, has a mean roof height less than or equal
to 60ft, when the mean roof height does not exceed the least horizontal
dimension and if it meets the other requirements given in section 6.4 of the
code. In this method wind pressures are not calculated, they are just selected
directly from a table.
Method 2 is the analytical procedure, which can be used for almost any type of
building, which is regularly shaped. It is, however, not recommended for
buildings that are sensitive to wind effects like vortex shedding, galloping
flutter etc.
Method 3 is for dynamically sensitive buildings. A rational analysis is allowed,
in the case of such buildings.
21
This category includes buildings that have the following characteristics:
1. Non‐uniform in shape.
2. Dynamically sensitive and flexible structures with a natural frequency of less
than 1 Hz.
The fist two methods provide specific formulas and steps to follow to calculate
wind loads on main wind force resisting systems (MWFRS) and also on
Components and Cladding (C&C). Wind design for components and cladding
is not included in this thesis.
2.4.2.1 Abbreviations and symbols
Following are the abbreviations and symbols used in “ASCE 7‐02 Minimum
Design Loads for Buildings and Other Structures”.
Cp = external pressure coefficient to be used in determination of wind loads on
buildings
G = Gust Factor
Gf = Gust Factor for main wind force resisting system of flexible buildings on
other structures.
h = mean roof height of a building (mid‐height between eave and ridge)
22
Kzt = topography factor {Kzt = (1+k1 k2 k3)
2
}
Kd = wind directionality factor
Kh = velocity pressure coefficient evaluated at height z = h
Kz = velocity pressure coefficient evaluated at height z
p = design pressure in pounds/ft
2
q = velocity pressure in pounds/ft
2
qh = velocity pressure evaluated at height h pounds/ft
2
I = Importance factor
This factor accounts for the importance of the building. The following table 2A
gives the importance factor classification of buildings. (IBC table 1604.5
Importance classification for wind loading excerpt)
Category
Nature of Occupancy
Importance
Factor
I
Low hazard structures: Agriculture
temporary, minor storage.
0.87
II
Structures not in categories I, III, and IV 1
III
Structures such as Occupancy > 300 people
per area, Elementary Schools >250 students,
Colleges > 500 students, Occupancy > 5000
1.15
IV Essential facilities such as: Hospitals, police
and fire stations
1.15
Table 2A: Definitions of Importance factors as per ASCE‐7
23
Building Exposures
Buildings are also categorized by surface area exposed to the wind, with
respect to surface roughness around. The following table lists exposures
defined by American National Standards Institute. Exposures B, C and D are
used by the IBC.
Exposure A Large City Centers
Exposure B Urban and Suburban areas, wooded areas
Exposure C Flat open country with minimal obstructions
Exposure D
Flat unobstructed coastal areas
Table 2B: Building Exposures as per IBC
The first step to wind design is to determine the basic wind speed of the region
or location. The following map (Figure 2.7) shows average wind speed to be
considered while designing according to the IBC.
24
Figure 2.7: Average wind speeds (IBC)
25
2.4.2.2 Method 1; Simplified procedure
In this method, the design wind pressures (ps) represent net internal and
external pressure on the main force resisting systems. Ps is to be applied on all
horizontal and vertical projections of building surface.
Ps is given by
ps = λ I psz
λ = Adjustment factor for building height and exposure from figure 6‐2 in
ASCE7‐02 standards.
I = Importance factor
ph = Simplified design wind pressure at height z from figure 6‐2 in ASCE 7‐02
standards.” For exposure B and for Importance factor I = 1.0
2.4.2.3 Method 2; Analytical procedure
This method is to be used for regular shaped buildings that do not have
response characteristics that create vortex shedding, or instability due to
galloping or flutter and likewise, and that confirm to the definition in section
6.2 of the standard. In this procedure the following two formulas are used to
find out the wind pressure and the velocity pressure respectively.
26
p = qGCp ‐ qiGCp
q = 0.00256 IKZKZtKdV
2
(Variables defined at the beginning of this section)
2.4.3 Gust factor calculations
Gust factor, or gust effect factor, is a dynamic response factor which accounts
for an additional dynamic amplification of loading in the along wind direction
due interaction between wind turbulence and the structure. (TARANATH, S
Wind and Earthquake Resistant Buildings, Structural Analysis and Design, p38)
The Gust effect factor also accounts for along wind loading effects due to
dynamic amplification of flexible structures. (Taranath, 1997, p.189)
2.4.3.1 Gust factor for rigid structures
The gust factor “G” for rigid structures is defined for two categories; rigid
structures simplified method, rigid structures analytical method.
The gust factor for wind load simplified (method 1) calculations of a rigid
structure is to be taken as 0.85, as per ASCE standards.
The Gust effect factor for calculation of wind loading on rigid structures by the
analytical method is given by
G = 0.9 {(1+7IiQ) / 1+7Ii) }
27
Ii = C (33/z)
1/8
Where,
Ii = the intensity of turbulence at height z
z = The equivalent height of structure =0.6h
Exp α b c L (ft) ∈
A 1/3.0 0.30 0.45 180 1/2.0
B 1/4.0 0.45 0.30 320 1/3.0
C 1/6.5 0.65 0.20 500 1/5.0
D 1/9.0 0.80 0.15 650 1/8.0
Table 2C: Table from ASCE‐7 Minimum Standards For Building Loads; Section
6: Wind Loads
c = Value given in the above table 2C
Q = Background response given by the formula
Q
2
= 1 / {1+0.63 [(b +h)/Li]
0.63
}
Where, b = Building width parallel to wind
H = Building height
Li = Integral length of scale of turbulence at the equivalent height given by
Lz = λ(z/33)
∈
Where,
λ and ∈ are as given in table 2C
28
2.4.3.2 Gust factor for flexible and dynamically sensitive structures
G = 0.925 {1+1.7Iz (gq
2
Q
2 +
gR
2
R
2
)
1/2
} / {1+1.7 gvIz }
Where gq and gv shall be taken as 3.4 and gR is given by
gR = [{ 2ln (3600n1) }
1/2
] +[ 0.577 /{ 2ln (3600n1) }
1/2
]
Where R is the resonant response factor
R= {1/β Rn Rh RB (0.3 + 0.47 RL)}
1/2
Rn = 7.47 N1 / (1+10.3 N1)
5/3
N1= n1 LZ / VZ
Rl = 1/η ‐ 1/2η
2
(1‐e
‐2
η
)
for η > 0
Rl =1 for η = 0
Where n1 = building natural frequency.
The natural frequency of the building is thus crucial to using this formula. The
natural frequency of vibration of a simple building can be estimated
mathematically. Physical model tests are recommended for buildings with large
heights or complex layouts.
29
Chapter 3: Tensile Structures
3.0 Introduction
This chapter describes tensile structures with respect to history, types,
categories and materials. Towards the end it focuses more on fabric membrane
structures, due to their relevance to this thesis.
3.1 Histories and Introduction
The history of tensile architecture goes as early as the first few forms of man‐
made shelters. Tents were probably one of the first tensile structures ever made.
The evolution of modern tensile architecture seems to have begun with the
construction of suspension bridges; although in many parts of the world, the
rope bridge was already a widely used method to get across rivers and other
water bodies. The use of the principle of suspension of cables over large spans
for bridges impacted building design as well. Today we see a growing
popularity of tensile structures, due to their lightweight visual appearance, free
flowing and sleek look. Tensile structures are defined as those in which the
main support members resist loads by tensile stresses. Tension structures being
usually made up of flexible materials are tensioned and thus are stabilized by
geometric form and shape. The geometry of these flexible structures thus
follows certain rules, governed by force transfer and stiffness. Surface curvature
30
in two mutually opposite curvatures stabilizes anticlastic membranes and cable
nets.
3.1.1 Pretension
Pretension or prestress allows flexible members to absorb compressive stress
without slack, which would cause instability. The following figure explains the
concept. Consider a vertical string fastened on its top and bottom. Apply a load
in the mid height of the string, the lower portion of the string will slack, and
will be unstable. Now, consider the same string stretched or prestressed. The
same load applied at mid height will be taken by the top link by stretching
(increase of prestress) and by the lower link through decrease in prestress. The
compression force, in this case instead of causing instability was converted in to
reduction of tension, thus not causing instability.
31
Figure 3.1: Prestress
Conventional buildings work primarily on the principle of compression,
tension and bending. Gravity and rigidity are their most important
characteristics. The load resisting system needs these structures to be bulky,
because a slender structure would buckle under compression loading. Tensile
structures on the other hand are stabilized by curvature and prestress. On
application of load, there is no compression that occurs. Load application
instead reduces the pre‐stress or the tension in the structural member.
Anticlastic curvature and prestress provide stability and allow these structures
to be slender and lightweight.
32
3.2 Classification of tensile structures
• Pneumatic Structures
• Suspension Structures
• Stayed Structures
• Cable Trusses
• Anticlastic Membranes and Cable Nets
All the mentioned types are worth a brief mention, however only anticlastic
fabric membranes are included in the research of this thesis.
3.2.1 Pneumatic structures
Pneumatic structures are those that are held up by air pressure. The material
that is held up is usually fabric panels that may be attached to a cable mesh.
There are two types of pneumatic structures. Pneumatic Structures can be air
supported or air inflated.
3.2.1.1 Air supported structures
These structures are usually a single membrane enclosing a large space held up
by air pressure, which is slightly more than the outside normal pressure. The
enclosed usable space is thus under high pressure, making it not very
comfortable for the user. Moreover, the pressure not being very high, the
33
membranes are not stiff enough and are susceptible to flutter under high wind
loads.
Figure 3.2: Air Supported Structure (http://www.yeadondomes.com. 30th
September 2005)
3.2.1.2 Air inflated structures
These structures are typically enclosed volumes that are inflated under high
pressure to provide stability. They usually consist of two layers with high‐
pressure air filled in between them, making the enclosed space usable under
normal pressure. Air supported structures can be categorized again into two
types, depending on their height to span ration.
• High profile
• Low profile
34
High profile structures are generally freestanding structures, used for storage
facilities etc. Low profile structures are those, which have lesser height as
compared to their large spans. They are used mostly to span large areas like
sports facilities etc, where the primary purpose of the air supported structure is
roofing.
Figure 3.3: Air Inflated Structure (http://itek‐usa.com/emergency. 30th
September 2005)
3.2.2 Suspended structures
Suspension structures are those in which horizontal planes (road decks, roofs)
are supported by the sag of large, high‐strength steel cables suspended from
supports on either side.
35
Suspended cables are used to support large span roofs. The suspended cable
takes the load of the structure and transfers it to the support elements on the
two sides. In some case, the structure is hung on hangers or cables, which are
hanging on the suspended cable. Suspended cables resist gravity loads;
however, their flexibility makes them unstable under wind loads. They can be
stabilized by various means for example; adding dead load or adding a
stabilizing cable. Under a uniformly distributed load the funicular of a cable is
parabolic.
An important design consideration for suspended structures would be the span
to dept ratio. The following figure 3.5 shows a vector diagram of three spans to
depth ratios. Greater the sag the lesser the tension in the cable, but the greater is
the force on the vertical supports. Thus a greater cable force would give a lesser
force on the supports and lesser cable force will mean a larger force on the
support. Optimum sag to span ratio would approximate 1/10, however it
would vary from project to project depending on other design constraints.
36
Figure 3.4: Cable Suspended Roof Dulles International Airport
(www.fairfaxcountyeda.org. 15th October 2005)
Figure 3.5: Cable force for different sag to span ratio
37
3.2.3 Stayed structures
Cable‐stayed structures are structures in which horizontal planes (like bridge
decks, roofs, beams, trusses or floors) are supported with inclined cables that
are attached to, or run over towers. Stays usually support structures by pulling
from above, however, in certain cases they may be used below the structure, as
compression struts. An important design consideration for stayed structures
would be the span to depth ratio. The figure 3.7 shows a vector diagram of
three span to depth ratios. A greater depth results in reduced force in the stay,
but a large vertical force in the tower, and a shallow sloped stay results in
greater stay force, but a lower vertical force on the tower. Optimum sag to span
ratio would depend on structural as well as architectural concerns. An
optimum slope for the stays would be about 30 degrees.
Figure 3.6: Cable Stayed Roof (http://www.andrew.cmu.edu/course/48‐
300/CableStayed.pdf. 4th September 2005 )
38
Figure 3.7: Cable force for different sag to span ratio
3.2.4 Cable Truss
Cable trusses evolved from the need to stabilize suspension structures against
wind and uplift forces, using a second set of cables with an opposing curvature
to that of the suspension cable. (Schierle, 2005, p14‐17). A cable truss consists of
usually 2 main cables with compression struts or diagonal truss cables in
between them.
Figure 3.8 Cable truss configurations
39
3.2.5 Anticlastic membranes and cable nets
An anticlastic curvature is used to stabilize flexible surfaces like cable nets and
fabric membranes. An anticlastic surface is a surface with two opposite
curvatures, in two opposite directions, with the centers of radii on either side of
the surface. A simple illustration of anticlastic stability can be observed with
string models. Two strings pulled in opposite directions stabilize a point. If the
two stings are in non‐parallel planes, the point is stable in three dimensions.
Now consider a series of such strings crossing in opposite directions, forming a
series of stabilized points. The plane formed by these points is stabilized by an
anticlastic curvature. A simple anticlastic curvature can be made by using a
square (or rectangular) stretch fabric, and pulling its opposite vertices in
upward and downward direction respectively. The surface is stable when their
tensioning is in two opposing directions. The tension in the opposing curves
balances and stabilizes the structure.
In addition to curvature, membranes require to be prestressed to be stable. In a
string model of two strings on opposite directions, application of external load
will cause tension in one string and compression in the other. The string under
compression will slack. However, if this string is prestressed, an application of
load would only cause a reduction in tension, rather than compression.
40
3.2.5.1 Minimal surface
There are innumerable possible surface shapes that can span a boundary. One
of the ways to obtain surface shape would be using the minimal surface
principle. These are shapes that have the minimal possible surface area to cover
a particular boundary. Minimal surfaces are could be anticlastic or flat. A
triangular shaped boundary would always have a flat minimal surface.
Three conditions determine the minimal surface:
1. Connection of any given boundary lines with the absolute minimal
amount of surface area.
2. At any point on the surface the sum of positive and negative radii
equals zero.
3. Equality of surface tension in all directions throughout the surface.
(Schierle, 1968 )
One very simple and reasonably accurate method for form finding on the
principle of minimal surface is using a soap film to connect between the set
boundary conditions. However, using minimal surface principle for structures
is not very feasible, because this shape is stable only when load applied on the
structures is equal in both (opposite) directions. For anticlastic structures,
minimum sag to span ratio, proportional to the applied load is usually used to
optimize shape and economy. The shape is also governed by factors such as the
volume to be enclosed and minimum height requirements. Anticlastic surfaces
are formed by a cable net or by fabric membranes.
41
3.2.5.2 Hyperbolic paraboloid surfaces
Hyperbolic paraboloid surfaces are anticlastic surfaces with opposite
curvatures. These surfaces are hyperbolic paraboloids because they are at mid
height between the low points and high points, regardless of boundary
condition. Following are diagrams comparing minimal surfaces and hyperbolic
paraboloid surfaces for a square and rhomboid boundary shape. An interesting
observation is that a minimal surface for a square boundary shape is a
hyperbolic paraboloid.
Figure 3.9: Hyperbolic Paraboloid and Minimal Surface for Square and
Rhomboid Plan
42
3.2.5.3 Cable nets
The most common use of cables for spanning a structure is cable nets. Pre‐
tensioned cables cross each other in opposite directions, and in different
curvatures forming a grid form planes. When load is applied on to the
configuration of a flat cable net it generates forces by redirection, which is of
very high magnitude. If the cable net has predetermined sag, the redirection
will be of lesser magnitude. (Schierle, 1968)
Cable net structures can further be categorized according to the grid shape.
1. Square grid net parallel to border: In this type the cables run parallel to the
edge cable. The mesh spaces are a perfect square before deformation. The
deformation of the square will be maximum, towards the highest and the
lowest points of the form. This type deforms six times more than other
types
2. Square grid net diagonal to border: This type of cable net is similar to the
previous type, with the difference that the cables run diagonal to the edge.
The cables thus run from high point to low point.
3. Triangular mesh net: This type of configuration has a mesh composed of
45°‐45°‐90° or 60°‐60°‐60° triangles. Triangular mesh cable nets have
shearing stability.
43
Figure 3.10: Cable net configurations: parallel to boarder, diagonal to boarder,
triangular mesh
Frei Otto built one of the first cable net structures, The German Pavilion Expo‐
67. (Figure 3.11) He built a network of intersecting cables and suspended a
fabric membrane from it. It was one of the first structures to introduce free
flowing, organic shapes of tensile architecture.
Figure 3.11: First cable net structure; The German Pavilion Expo 67
(http://naid.sppsr.ucla.edu/expo67/map‐docs/images/germany3.jpg. 11th
October 2005)
44
3.2.5.4 Membrane structures
These are structures in which, the spanning material consists of high strength
fabric. Similar to cables, fabric cannot take load in compression because it is
flexible. The fabric has to be pre‐tensioned, and it has to have curvature to be
made rigid, so that it can take loads in tension.
Depending on boundary or edge conditions, fabric membrane structures can be
classified into the following types:
Cable (tension) edge: In this case the membrane is supported on the edges by
pre‐tensioned cables. The membrane thus transfers load purely in tension.
Rigid Arch (compression) edge: In this case the membrane is supported on the
edge by an arch, which transfers the load from the membrane through
compression.
Straight Beam (bending) Edge: The beam at the boundary of the membrane
transfers the load from the membrane through bending. In this case however,
the size of the beam is usually very large, in order to resist large bending
moments.
45
Figure 3.12: Cable (tension) edge, rigid Arch (compression) edge, straight Beam
(bending) Edge
Depending on surface conditions fabric membrane structures can be classified
into the following types:
Saddle shape, Wave shape, Arch supported shape, Point supported shape.
2.3.5.4.1 Saddle shape
This shape has a single surface of anticlastic curvature, and could have square,
hexagon, octagon, or any symmetrical free form shape.
Figure 3.13: Saddle shape membrane
(http://www.fabricarchitecture.com/il_189.htm 5th February 2006)
46
3.3.5.4.2 Corrugated or wave shape
A wave shaped roof can have a linear or a curvilinear composition. Anticlastic
curvature is achieved by various means like alternate ridge and valley cables,
or alternately sloped beams. The edges may have cables or could have rigid
edge beams. Wave shapes can be used to span very large areas.
Figure 3.14: Wave shape membrane (Berger, 1996, p.66)
3.3.5.4.3 Arch supported shape
In an arch supported membrane, a vertical or tilted arch holds up the
membrane. The arch should be designed to bisect the angle made by the
membrane to the arch on both sides. The arch should also be on pin supports to
resist lateral forces of wind and earthquake.
47
Figure 3.15: Arch supported membrane (Berger, 1996, p.121)
3.3.5.4.3 Point supported shape
This type typically has a high point at the center. To prevent the membrane
puncturing in the center due to high concentration of loads, a tension ring, loop
cable, a ridge dish or an eye cable can be used. Point shapes can have single or
multiple high points. Typically the mast occurs in the center, however recently
many beautiful point supported fabric structures have been built with the mast
outside the structure.
48
Figure 3.16: Point shape membrane
(http://www.rainierindustries.com/comtents/tensile.htm. 5th February 2006)
3.4 Aero elastic behavior of membranes
Aero elasticity is the study of the deformations and displacements of an elastic
structure in the path of airflow and the aerodynamic forces produced. It was
defined by Collar in 1947 as the study of the mutual interaction that takes place
within the triangle of the inertial, elastic, and aerodynamic forces acting on
structural members exposed to an airstreams, and the influence of this study on
design
An elastic surface in the path of airflow will always be subjected to motion
because of aerodynamic effect. The flow of the air itself is changed due to the
boundary layer over the elastic body. The change in the airflow pattern changes
the aerodynamic forces on the structure, which changes its deformations and
deflections. These further cause changes in the boundary layer, forming a
49
continuous feedback loop. The structure thus deforms, to maintain equilibrium,
with the external forces. This deformation or movement could some times
cause oscillating movements, which could grow in amplitude and cause
damage.
For example, flutter is an aero elastic effect. It is a self‐induced vibration when a
flexible membrane deforms under a aerodynamic load. Reduction of the load
causes reduction in deformation. This causes the surface to gain back its
original shape increasing the load and then causing deformation. This cycle
continues and is called flutter. Reduction of flutter can be done by either
increasing the stiffness of the surface, or by increasing mass or curvature.
3.5 Components of fabric membrane structures
3.5.1 Structural fabric
Structural fabric is the primary spanning element in fabric membrane
structures. The fabric must be strong enough to span between the supporting
members and also strong enough to take self weight and imposed loads like
wind and snow load. Apart from these attributes the fabric should be water
tight, air tight, fire resistant, and weather proof. Other architectural
considerations would include light transmitting and reflecting properties,
50
sound insulating, reflecting or absorbing properties, and heat transmitting or
insulating properties. Structural or architectural fabrics are typically woven
fabrics in which bundles of fiber in orthogonal directions are interlaced
together to form a surface on which coatings or laminations are applied.
Fiberglass or polyester is the material usually used for the base fabric. The
common surface coatings used are Polyvinylchloride (PVC), silicon and
polytetrafluoroethylene (PTFE or Teflon). Teflon or PVC coated fiberglass
fabrics are generally used for long span structures. While, PVC coated polyester
is used for temporary structures.
The behavior of these fabrics primarily depends on the type of weave used. The
stiffness properties in both directions of the fabric may not be the same
depending on the type of weave.
One of the most significant issues for fabric structures are its life span. The
fabrics usually deteriorate with time owing to rain snow, sun and other
weather and environmental effects. Polyester based fabrics are more susceptible
to damage due to ultraviolet (UV) effects than, fiberglass‐based fabrics.
Teflon coated fiberglass is the most expensive of all structural fabrics, however,
it has stood the test of time, maintaining sufficient strength for over 30 years in
some roof structures.
51
3.5.2 Other considerations for fabric selection
3.5.2.1 Lighting
Membrane structures often allow light to penetrate through them. The range of
translucency depends on the material properties of the fabric. For example
PTFE glass cloth can have a transparency ranging from 5‐ 15% and glass coated
cloth can have the same ranging from 8 to 30%. Apart from this diffused light in
the membrane structure, direct daylight sources can also be created. A good
place to do so is places where the fabric membrane connects with the support
structure, mast truss, arch or any other structural component. Various design
options exist to direct daylight into membrane structures.
3.5.2.1 Acoustics
The factors that influence the acoustic performance of a space in a membrane
structure are as follows:
1. The sound absorptive or reflective characteristics of the material of the
membrane.
2. The shape and geometry of the space.
3. The size and volume of the space.
52
Designing a space for an acoustic performance with a membrane structure
should be done with great care. The fabric membranes tend to reflect low
frequency sound, and also tend to have a high sound transmissibility.
3.5.3 Connections and detailing
Connections between various parts and components are very crucial and have
to be designed with a lot of care. The connections are points where the stress
are very high, due to the forces exerted by the tensioned fabric. Fabric‐to‐fabric
connections are also a line of high stress in a membrane structure.
3.5.3.1 Fabric to fabric connections
During the design process after the determination of the shape of the fabric
structure, the fabric has to be patterned. This process involves determining the
shape of each piece of fabric, cutting it and then joining all the pieces to achieve
the desired shape. Thus every fabric structure involves many fabric‐to‐fabric
connections. Depending on the shape of the fabric there may or may not be a
cable inserted with the fabric to fabric joint. Usually large structures would
have this cable. The prestress in the cable helps in controlling the prestress of
the fabric membrane, and thus the shape of the membrane.
53
Figure 3.17: Fabric‐to‐fabric connection with cable (Scheuermann, Boxer, 1996,
p.125)
3.5.3.2 Fabric to edge cable connections
Edge cables used in membrane structures are usually wire ropes. Fabrics are
wrapped around the cable and welded to form a strong connection. Changing
the prestress on the edge cable changes the prestress in the fabric in the
perpendicular direction. Among the many methods used to increase or
decrease prestress of a cable, turnbuckles are most commonly used.
3.5.3.3 Fabric to supporting member connection
Various innovative connections have been designed over the years for
connections between fabric and mast, arch, truss and other support structures.
The following figures show few typical details for connections.
54
Figure 3.18: Connection at mast (plan view)
55
Figure 3.19: Fabric edge detail, Cable to mast connection ‐plan view (Scheuermann, Boxer, 1996, p.125)
56
Chapter 4: Previous Research Related to Gust Factors and Wind
Scaling
4.0 Introduction
This chapter presents brief descriptions of research done by other researchers,
and papers written by other authors, on topics that have relevance to this thesis.
4.1 Formation of gust effect factor in ASCE 7‐ 95
The wind load provisions of ASCE 7‐ 95 (American Society of Civil Engineers)
incorporated significant revisions with respect to gust factors. The changes
were caused mainly because of the adoption of wind speed maps based on 50
year peak 3 second gust speeds, which replaced the wind speed map that was
based on 50 year peak fast mile wind speed. The following is a brief
understanding of the paper “On Formulation of ASCE 7‐95 Gust Loading
Factor” by Giovanni Solari and Ahsan Kareem published in the Journal of
Wind Engineering and Industrial Aerodynamics 77 & 78 (1998), and the paper
“Wind Speeds in the ASCE 7 Standard peak‐Gust Map: An Assessment” by
Simiu, R. Wilcox, F Sadek and J. Filben in the 2001 NIST Building Science Series
178. Part of the paper relevant to this thesis is briefly explained in the following
sections.
57
4.2.1 Three‐second‐gust speed wind map
The new wind speed map is based on 50‐year peak 3‐second gust speed
because; 3‐second gust speed data were collected at a large number of stations
in the country. It provides a more consistent measure of wind speed to be used
and understood by design professionals, building code officials, the media and
the public. The ASCE 7‐ 93 wind load provisions included the effect of gusts by
taking into consideration the maximum expected mean wind effects of a given
probability. These gust factors were based on an average interval of the design
winds (1 minute interval for 60 mph fastest mile), which are less than typically
occurring short gusts.
The new wind speed map for ASCE 7‐95 was developed by Colorado State
University and Texas State University.
Based on research conducted at Texas Tech University in July 2002 for five
National Weather Service stations (Lubbock, TX; Amarillo, TX; Kansas City,
MO; Minneapolis, MN; and Syracuse, NY), a ratio between 3 s peak‐gust
speeds and the corresponding fastest‐mile wind speed of about 1.2 was
judged to be reasonable. If this ratio is used, 3 s speeds of 85 mph and 90
mph correspond approximately to 70 mph and 75 mph fastest‐mile speeds,
respectively. (Solari, Kareem, 1998.)
Apart from this, another significant difference between the two wind maps are
that, the new map is based on analysis of data from sets of stations rather than
that from individual stations. The advantage of the new map is that data from a
58
set of stations yield estimates based on a large amount of data, which reduces
possibility of sampling errors.
Except for hurricane prone areas, ASCE7 95 three second peak gust map
consists of three wind zones.
• Zone 1: Most of the United States at 90 mph
• Zone 2: California, Oregon, and Washington at 85 mph
• Zone 3: Along the gulf and east coasts at 100 to 150 mph
59
Figure 4.1: Average wind speeds based on ASCE 7‐95 three‐second gust speed data
60
The ASCE 7‐ 95 requires the gust factor to be 0.8 for exposures B and 0.85 for
exposure C and D for the main wind force resisting systems of buildings and
other structures and for components and cladding of open buildings and other
structures.
Exposure A Large City Centers
Exposure B Urban and Suburban areas, wooded areas
Exposure C Flat open country with minimal obstructions
Exposure D
Flat unobstructed coastal areas
Table 4A: Exposure types, according to ASCE7 ‐95
The Gust factor is presented for three major categories
1. Rigid Structures – simplified method
2. Rigid Structures – complete analysis
3. Flexible or dynamically sensitive structures of > 1Hz
E. Simiu, R. Wilcox, F Sadek and J. Filben in their analysis on the new gust
factors and the new map conclude the following:
The ASCE 95 peak‐gust map division of the United States into three adjacent
wind speed zones does not reflect the differentiated extreme climate of the
United States. The methodology used for the purpose of mapping tends to
61
average out differences among the stations for various reasons. The wind speed
information that was used to compute the ASCE 7‐95 peak‐gust map was taken
from data collected by super stations. There are some minor but still
noteworthy problems with the data. For example, it was observed that in 80%
of cases every component station belonged to more than one super station
which would result in double‐counting the data.
Also, off‐the‐shelf smoothing software was used, which did not account for
physical geography and meteorological differences, which played a significant
role in the development of the wind map. The authors of the paper thus
conclude that the ASCE 95 peak gust map can overestimate wind loads in some
cases and underestimate losses due to wind loads in other cases. The map thus
needs substantial improvement.
The latest publication of the ASCE is the ASCE 7‐02, which, for the purpose of
this thesis is taken to be the standard as recommended by the IBC. The wind
speed map used in the ASCE –02 is the same as that was introduced in the
ASCE 7‐95.
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4.2 Wind tunnel testing and wind speed scaling for aero elastic models
The following is the brief understanding of relevant information from the ASCE
publication, “Wind Tunnel studies of Buildings and Structures”. This
publication talks about types of wind tunnels, types of testing, methods of
scaling, methods of measurement of various parameters etc. Following this
section is information pertaining to aero elastic testing and modeling.
4.2.1 Aeroelastic simulations
Aeroelastic simulations provide information regarding wind induced responses
and dynamic loads of an aeroelastic structure. These structures include tall
buildings long span roofs, long span bridges and flexible roofs like fabric
membrane structure roofs, which undergo aeroelastic effects. To be able to
model such structures accurately for wind tunnel tests, it is necessary to
simulate stiffness, mass, damping and the wind itself. For proper study of the
dynamic responses the models should be tested at various wind speeds and all
critical directions. Scaling the mass of the model is done so that the inertial
forces of the structure and the wind flow are consistent. Scaling the critical
damping ratio of the structure in the model scales the damping. Stiffness
scaling is required by the fact that the forces that resist deformation should be
scaled consistently with the inertial forces.
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4.2.2 Aeroelastic Models
There are three types of aeroelastic models.
Replica Models: These are models in which the structure is simulated or
replicated and the geometric scaling of dimensions results in scaling of elastic
properties.
Equivalent models: These are models in which only certain desired aspects of
the dynamic character of the structure are simulated by some mechanical
analog. These models are thus used to simulate only certain particular
structural behavior.
Section Models: These models have only a certain part or section of the
structure to analyze wind forces. Models of long span bridges or tall chimneys
where the wind can be considered two‐dimensional are usually section models.
Such models can be made in much larger scales.
4.2.3 Modeling flexible roofs
According to the ASCE publication, for tensioned fabric membrane structures,
geometric stiffness dominates the responses and thus the elastic scaling can be
relaxed. Also, scaling structural damping may be less crucial than scaling the
aerodynamic damping. This is because the aeroelastic forces induced in the
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structure due to wind‐ structure interaction governs the dynamic response and
behavior of these structures. Aeroelastic simulations of these structures can be
done to evaluate the response to dynamic loading. It can also be done to study
the influence of surrounding surface conditions and environment on structural
response.
4.2.4 Scaling the wind
Aeroelastic simulations are typically carried out in lower wind speeds as
compared to commonly done tests, which are done for pressure measurements.
The ASCE manual recommends that, for structures like long span bridges,
suspension bridges and guyed towers, the wind speed scaling is done based on
Froude number. Based on this the wind speed is the square root of the length
scale. Length scales are in the range of 1:100 to 1:500, and so the typical velocity
scales range from 1:10 to 1:22. The dominance of elastic forces in such
aeroelastic models calls for such low wind speeds. This wind‐scaling concept
has been considered to a certain extent as a rational for scaling the wind foe the
purpose of aeroelastic tests conducted in this thesis. The model scale is 1:50,
and the tests (described in the chapters to follow) are done in a maximum wind
speed of 17mph, which is a wind speed scale of approximately 1:6, which is
close to the square root of the length scale. The wind speed thus simulated by
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the 17mph wind is close to 100mph. This wind scaling criteria was considered
for this thesis due to time limitations and limitations of the wind tunnel. It
would be an interesting research to find the best velocity scale for aeroelastic
tests for fabric membrane structures.
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Chapter 5: Proposed Method to Determine Gust Factor
5.0 Introduction
This chapter describes the relevance of natural frequency of vibration in the
determination of the gust factor. It describes the proposed testing method to
determine the gust factor.
5.1 Proposed method
The importance of natural frequency of vibration of a structure for wind design
was earlier described in chapter 2, section 2 .2 .1. The ASCE formula for
calculation of gust factor for flexible structures is also described in chapter 2
section 2 .4 .3. This formula to calculate the gust factor involves using the
natural frequency of vibration of the considered flexible structure. The
proposed method to find the gust factor is to find the natural frequency of
vibration of membrane structures and use it in the formula.
Membrane structures are built in different shapes. Each of these shapes is
distinctly different from each other. Their respective responses of the different
shapes may also be very different, with respect to factors such as aerodynamics
and lateral forces acting on the surfaces. Considering the variety of shapes of
membrane structures, the proposed method calls for finding the natural
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frequency for four basic shapes of membrane structures. The following basic
shapes are proposed for testing with scaled models, for the purpose of this
thesis:
Saddle shape, Wave shape, Arch shape, Point shape
Figure 5.1: Saddle shape membrane structure
(http://www.fabricarchitecture.com/il_189.htm. 5th February 2006)
Figure 5.2: Wave shape membrane structure – Denver International Airport
(Berger, 1996, p.66)
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Figure 5.3: Arch shape membrane structure (Berger, 1996, p.121)
Figure 5.4: Point shape membrane structure
(http://www.rainierindustries.com/comtents/tensile.htm. 5th February 2006)
69
Figure 5.5: Saddle, wave, arch and point shape models
The saddle shaped being the most basic anticlastic shape, the thesis proposes to
do tests on this model for various parameters such as prestress and sag to span
ratio. A minimum number of three different prestresses and shapes has been
chosen for testing the parameters, in order to determine if the effect is linear or
nonlinear. The saddle shaped model will thus be tested for natural frequency of
vibration, in a combination of the following conditions; three sag to span ratios
with three prestress levels each. This makes nine different cases for the saddle
shaped model. All the models are proposed to be tested in a wind tunnel at
three different wind speeds to see the changes in behavior with changing wind
speed.
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5.3 Proposed method to determine the natural frequency of the membrane
model
The initial method proposed to find out the natural frequency of vibration of
the membrane, with the help of a wind tunnel, was as follows:
Place the fabric membrane model in the wind tunnel. Video record 30 second
clips of the response of the models at three different wind speeds. Reduce the
speed of the video recordings, to make them one minute long. Closely observe
and count the number of oscillations in one minute. Calculate an average
number of oscillations per minute of the three wind speeds. Mathematically
calculate the natural period and the natural frequency of vibration.
Apart from finding the natural period of vibration of the structure, the dynamic
response of the structure to the wind force is also proposed to be studied. The
relationship of shape to dynamic response is very crucial and is to be observed
in detail. The relationship between prestress and deformation or amplitude is
also to be recorded.
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Chapter 6: Model Building and Wind Tunnels
6.0 Introduction
This chapter describes the tests conducted before building the model and the
method to build models of tensile membrane structures. Brief introductions to
various facilities and instruments used are also included in this chapter.
6.1 The test models
The membranes in the test models are made of stretch fabric. The advantage of
using stretch fabric is the elimination of the need to pattern the fabric to get the
desired shape of the model. The bases of the models are made of medium
density fiberboard. Fishing line or nylon string simulates the steel cables.
6.1.1 Selecting the model fabric
The model fabric has to satisfy the following criteria:
• The fabric has to be reasonably equally elastic in both perpendicular
directions.
• The fabric has to be relatively linearly elastic.
• The modulus of elasticity of the fabric has to be determined before
making the model.
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Modulus of elasticity, also called Young’s modulus, is a measure of the stiffness
of a given material. It is defined as the rate of change of strain with stress. This
can be experimentally determined from the slope of a stress‐strain curve
plotted with the results of tensile tests conducted on a sample of the material.
To determine the modulus of elasticity, the fabric has to be tested for stress and
strain. A stress to strain graph has to plotted. The determination of modulus of
elasticity is crucial to be able to determine the strain scale of the model as
compared to the real structure.
6.1.2 Fabric testing
The tensile test for the fabric chosen for the physical models was done in the
following manner:
A strip of fabric 10” in width and 8’ in length was cut, and hung vertically on a
wall, with dowels attached to its upper and lower end. The dowels help the
fabric to hang stably.
Weights were hung at the bottom of the fabric symmetrically, in increasing
increments and the elongation was recorded. Lead balls in cups were used as
weights. The small lead balls helped in being able to get accurately small
increments in the weights. The weighing instruments (figure 6.1, 6.2) were
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calibrated and checked for accuracy. The elongation in the fabric due to every
increasing increment in the weight was measured. The elongation was used to
mathematically calculate strain.
Figure 6.1: Fabric stress strain test
Figure 6.2: Calibration of weighing instrument
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Strain = elongation of fabric / original length (Stress = dL/L)
The stress was also calculated mathematically;
Stress = Load/ cross‐section area. (P/A) Cross Section area in this case will be
replaced by width because fabrics have negligible thickness. A stress over strain
graph was plotted out of the results of the fabric test
6.1.2.1 Results
Following is the stress strain graph that was plotted. The graph was reasonably
linear, which indicated that the fabric stretches fairly linearly.
stress/ strain
y = 1.1435x + 0.0038
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.020.040.060.08 0.1 0.12
Strain dL/L
Stress P/A
Figure 6.3: Stress to strain graph of the model fabric
The conclusion derived from the graph is that the e modulus of the fabric is
approximately 0.82
Modulus of elasticity = stress/ strain
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Applied load Elongation Strain Stress E modulus
Weight (pounds) dL (inches) dL/L P/A (pound/ sq inch)
0.125 1.75 0.018 0.0125 0.69
0.25 3 0.031 0.025 0.8
0.375 4.25 0.0442 0.0375 0.85
0.5 6 0.0625 0.05 0.8
0.625 7.5 0.078 0.0625 0.8
0.75 8.75 0.091 0.075 0.82
0.875 10 0.104 0.0875 0.84
1 11 0.114 0.1 0.87
1.1 12.5 0.130 0.11 0.85
Average E modulus 0.82
Table 6A: Weights in pounds attached to the fabric and the respective
elongation
6.1.2.2 Observation
A crucial observation made after the fabric test was carried out was regarding
the deformation of the fabric in the width direction. Due to elongation in the
length direction the fabric could have shortened in the width direction. This
depends upon the Poisson’s ratio of the fabric. When a sample of material is
stretched in one direction, it tends to get thinner in the other two directions (in
the case of fabric it tends to get thinner in the other one direction, which is the
width, since thickness is almost negligible). Poissonʹs ratio (ν), is a measure of
this tendency. It is defined as the ratio of the strain normal to the applied load
to the strain in the direction of the applied load.
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6.1.3 Fabric Testing
To avoid the above‐mentioned discrepancies the fabric test was carried out
again with a 3’ wide by 5’ long fabric, Following are the results of the second
test.
Stress Strain graph‐ Fabric
y = 0.6255x + 0.0089
0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Strain = DL/L
Stress = P/a
Figure 6.4: Stress over Strain graph of model fabric
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Applied
load Elongation Elongation average Strain Stress E modulus
on right side on left side elongation
P (pounds) Dl1 (inches) Dl2 (inches) Dl Dl/l
P/A
(pound/ sq
inch)
1 1.65 1.9 1.775 0.029 0.027 0.938
1.5 3.5 2.75 3.125 0.052 0.041 0.8
2 4.75 4.25 4.5 0.075 0.055 0.740
2.5 6 5.75 5.875 0.097 0.069 0.709
3 7.25 7.15 7.2 0.12 0.083 0.694
3.5 8.5 8.25 8.375 0.139 0.097 0.696
Average E modulus 0.763
Table 6B: Weights in pounds attached to the fabric and the respective
elongation. (Second test)
6.2 Model building procedure
An appropriate fabric is to be chosen to simulate the fabric. The stress strain
graph for the fabric should be relatively linear in order to define a linear elastic
modulus. A linear graph suggests that the fabric is linearly elastic and is fit for
simulating the membrane in the model. This graph is of prime importance in
this thesis. It will be used to determine the stress in the model fabrics by
recording the elongation (or strain) in the fabric, caused by wind loads. A
detailed procedure for determination of fabric prestress (induced stress) is
described in section 6.3 of this chapter.
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After choosing a suitable fabric for the model, and after determining the stress
strain graph for the fabric, the models can be built. As stated earlier, four basic
shapes of fabric structures were chosen to make test models. Following are the
detailed model making procedures used to make the models
6.2.2.1 Saddle shape model
This shape being the most basic anticlastic shape, the tests are to be conducted
for a number of variables including prestress and sag to span ratios. This helps
in the study of the dynamic behavior of the structures with respect to these
variables. The model was built for three sag to span ratios, and three prestress
levels.
Steps to make the saddle shaped model
Step 1: Make a 14” square base for the model medium density fiberboard, of ½”
thickness. Drill holes of 3/8” diameter at two inches form the corner on two
diagonally opposite corners. Cut two 3/8
”
dowels (equal to the diameter of the
hole drilled into the base) to height of 12”. Drill holes of a diameter of 1/12 “ at
one, two and three inches from the end of the dowel at one end. Insert and
tighten the dowel with some wood glue into the holes on the base, in such a
way that the holes on the dowel are at the upper end.
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Figure 6.5. Step 1: Plywood base of saddle shape model
Step 2: Cut a 14inch square piece of fabric, draw a one inch square grid on it.
The grid is drawn to assist in latter measuring the stress in the fabric. (The
method of measuring the stress in described in section 6.3 of this chapter.)
Stretch this piece of fabric on the base in such a way that two diagonally
opposite ends of the fabric are on the dowels, and the other two are on the
wooden base near its corners. Now stretch the fabric and pin it down to the
base and the top of the dowels, so that the fabric is tight. Draw with a pen on
the stretched fabric the profile of the curves joining the high points and the low
points forming a square with inwardly curved edges. Remove the fabric off the
base and cut it half an inch parallel (along the edge) to the drawn edge curves.
At the perimeters, roll this extra half‐inch fabric inward with a fishing line
string inside it (as edge cable) and stitch it along the edge curve. After this is
done for all four sides the fabric can be mounted on the base.
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Figure 6.6: Step 2: Saddle shape model with fabric
Step 3: Insert firmly pushpins on all four corners of the base. Attach two
diagonal corners of the fabric to two diagonal corners of the base (low points of
the saddle shape), by tying the corner nylon strings of the fabric to the pushpins
on the base. Now lift the nylon strings at the other two diagonal corners, to the
high points and pass them into the first holes on the dowels, and then pull them
downwards and anchor them to the pins on the base near the dowels. Adjust
the tightness of the nylon strings at all corners, at all pins to get good shape and
desired prestress.
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Figure 6.7. Step 3: Finished saddle shape model
Simply pulling and tightening the edge strings at the corners can adjust the
prestress. The sag to span ratio can be reduced by lowering the high points to
the next lower hole in the dowel. The three shape variations generated by
changing the sag to span ratios are named as S1, S2 and S3, with ratios of 0.05,
0.12 and 0.16 respectively. Tests were conducted at three prestress levels;
namely 0.28 pounds/inch, 0.19 pounds/inch and 0.09 pounds/inch for each of
these shapes.
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Figure 6.8 Saddle shape model with three different sag to span ratios, as shape
S1, S2 and S3
6.2.2.2 Wave shaped model
The procedure to make the wave shape model is similar to that of the saddle
shaped model. The only difference is that there are four high points and four
low points. Also the dowels have only one hole to fit in the nylon edge strings
at the high points. Another important addition to this model is the ridge and
valley cables. The cables that are simulated by the nylon strings are added to
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the model at the end after the membrane is in place. Increasing or decreasing
the prestress in the ridge and valley cables adjusts the shape of the membrane.
Figure 6.9: Finished Wave shape model
Figure 6.10: Wave shape model
6.2.2.3 Arch shape model
Steps to make the arch shaped model
Step1: Make a 14” square base for the model with medium density fiberboard
of 1/2ʺ thickness. Drill holes of a 1/4
ʺ
diameter, to fit the base of the 2 arches as
shown in figure 6.7. Make the arches out of bent 1/4ʺ wooden dowels. Fit in the
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arches in to the holes in such a way that the arches are slightly inclined
outwards as shown in the figure so that the angles made on both sides of the
arch, by the membrane are relatively equal.
Figure 6.11: Step 1 of arch shape model
Step 2: Draw a one‐inch grid on the fabric to be used. Stretch the fabric over the
arches and attach it to the base with pushpins. Draw with a pen the curved
edges on the stretched fabric. Remove the fabric off the base and cut it half an
inch parallel to the drawn edge curves. At the perimeters, roll this extra half‐
inch fabric inward with a fishing line string inside it and stitch it along the edge
curve.
Figure 6.12: Step 2 of arch shape model
85
Step3. After this is done for all sides the fabric can be mounted on the base, by
anchoring the nylon strings to the base with the help of pushpins.
Figure 6.13: Step 3 of arch shape model
Figure 6.14: Step 3 of arch shape model
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6.2.2.4 Point shape model
Steps to make the point shape model
Step1: Make a 14” square base for the model with medium density fiberboard
of 1/2“ thickness as shown in the figure. Drill a hole of a 1/4” diameter in the
center of the base frame to accommodate a dowel of 1/4” diameter and 10
inches height. After tightening the dowel into the hole, add a flat disk shaped
piece of 1” diameter on the top of the dowel.
Figure 6.15: Step 1 of point shape model
Step2: Stretch a square piece of fabric with a one‐inch grid drawn on it on the
base, over the central mast as shown in the figure. Attach the fabric to the base
at the four corners of the base with pushpins. Draw the edge curves on the
stretched fabric. Remove the fabric off the base and cut it half an inch parallel to
the drawn edge curves. At the perimeters, roll this extra half‐inch fabric inward
with a fishing line string inside it and stitch it along the edge curve.
87
Figure 6.16: Step 2 of point shape model
Step3. After this is done for all sides of the fabric can be mounted on the base,
by anchoring the nylon strings to the base with the help of push pins.
Figure 6.17: Step 3 of point shape model
Figure 6.18: Point shape model
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6.2.3 Method to measure stress in model fabric
The following procedure is used to determine the model fabric stress under
prestress, wind or gravity loading:
Draw 1” by 1” square grid on the unstretched model fabric before building the
model. Now to attain certain prestress in the membrane one can find the
corresponding strain needed in the fabric with the help of the stress strain
graph. This strain data is used to find the elongation required at every square
drawn on the fabric. The elongation can be visually measured by the increase in
the size of the squares when the fabric is stretched. Also, with the help of the
grid drawn on the fabric, an evaluation of uniform distribution of stress over
the membrane can be visually estimated.
This method can be reversed to measure the stress in the fabric due to wind or
gravity loading. This loading will result in stretching of the fabric, causing
elongation of the square drawn on the fabric. The corresponding stress is
computed with the help of the graph.
6.3 Wind Tunnel
Most of the tests done in this thesis have used a wind tunnel. This section gives
a brief description of the equipment, its various types and its working
89
procedure. Wind tunnels are used to simulate winds or other atmospheric
conditions for the purpose of testing. They can be used to simulate steady
laminar flows of air or turbulent gusty wind patterns. Numerical analysis or
computational methods may provide insufficient or inaccurate information on
responses of structures like long span structures, high‐rise buildings, long span
bridges, and most importantly flexible structures. Wind tunnel testing is
recommended in such cases. Wind tunnels can be broadly classified as open
circuit or closed circuit type. An open circuit type is straight and it has air
entering form one side and exiting from the other, facilitated by a suction fan at
the outlet end. (Figure 6.15). A closed circuit wind tunnel the air circulates in a
loop. (Figure 6.16)
Figure 6.19: Open Circuit Wind Tunnel (Liu, H. 1991)
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Figure 6.20: Closed Circuit Wind Tunnel (Liu, H. 1991)
Wind tunnels are of various types, depending on their use. For architectural
purposes, boundary layer wind tunnels are used. Research has shown that the
turbulent boundary layer generated in a wind tunnel due to the interaction of
the airflow with the physical boundary of the wind tunnel, is a very good
simulation of the atmospheric boundary layer.
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Figure 6.21: Turbulent boundary layer generated in a wind tunnel section
An atmospheric boundary layer wind tunnel may thus be defined as a wind
tunnel having a test section sufficiently large enough to produce a tall vertical
boundary layer, simulating the atmospheric boundary layer. Atmospheric
boundary layer wind tunnels thus prove to be good testing tool for
architectural models, in which flow pattern around the structure is an
important consideration. However, for the purpose of this thesis, due to the
non‐availability of the facility a boundary layer wind tunnel is not be used. The
wind tunnel used generates a fairly laminar flow of air though out its test
section. This however does not alter our results to a great extent because the
thesis focuses on the dynamic response of different kinds of membranes to
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wind. Thus a reasonably laminar flow of wind proves to be more helpful in this
thesis rather than a turbulent boundary layer, because a laminar flow has
fewer variables as compared to the turbulent flow of a boundary layer wind
tunnel.
Figure 6.18 shows the wind tunnel in the USC School of Engineering,
Mechanical and Aerospace Engineering Department. The following are the
specifications of the wind tunnel:
Air flow type: suction
Test area cross section: 1’6” X 1’6” ht
Other attached devices: pressure balance
Maximum wind speed: 100 mph
Figure 6.22: Wind Tunnel; USC School of Engineering, Mechanical and
Aerospace Engineering Department
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Figure 6.19 shows pictures of the wind tunnel owned by the USC school of
Architecture. This wind tunnel was built in the 1960‘s and is primarily made
out of wood. The following are the specifications of the wind tunnel:
Air flow type: suction
Test area cross sectional size: 1’6” X 11” ht
Maximum wind speed: 17mph.
Figure 6.23: Wind Tunnel; USC School of Architecture
94
Due to the non‐availability of the wind tunnel in the aerospace department, this
wind tunnel was used for the purpose of the experiments for this thesis. This
wind tunnel was checked for its wind speed, and calibrated using a wind
meter.
Figure 6.24: Wind Meter
The process of calibration of the wind tunnel involved measuring the wind
speed at different points in the test area, perpendicular to the wind speed. The
wind speeds measured at different points were same, thus confirming a
laminar flow at the test section area. The maximum wind speed measured was
17 mph. The wind speed can be varied between 10mph and 17mph.
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Chapter 7: The Natural Frequency Test
7.0 Introduction
This chapter describes the tests, the observations and preliminary results which
indicate a need for a modification of the procedure to compute the gust factor.
7.2 Natural frequency method
The following method was chosen to arrive at the gust factor for membrane
structures
Step 1: Find natural frequency of vibration of fabric membrane structures,
through physical model experiments
Step 2: Use the results (natural frequency of vibration) in the ASCE gust factor
formula for dynamically sensitive structures to calculate the gust factor.
7.2.1 Fabric structure models
Following four basic shapes of models were tested under lateral wind loading
in the wind tunnel, at 10mph, 13mph 17mph wind speeds.
Saddle shape; 3 prestress levels, 3 sag to span ratios, Wave shape, Arch shape,
Point shape
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7.2.2 Model testing to determine natural frequency
After determining the fabric modulus of elasticity and after confirming that the
fabric can be used for the elastic model, the models were built. The saddle
shape model was built first. The saddle shape being the mode basic anticlastic
shape, was chosen for testing with a number of variable parameters. The saddle
shaped model was tested under varying wind speeds. 30‐second samples of the
tests conducted at wind speeds of 10mph, 15mph and 17 mph were video
recorded. This model was also tested with other variations. Three different sag
to span ratios were tested. Each of these sag to span ratios were tested at three
prestress levels. 30‐second video clips of the model responses to the wind
tunnel were recorded for each of these nine cases. The speed of the recordings
was halved to be able to observe clearly the response of the structure and to be
able to count the number of oscillations.
Figure 7.1: Saddle Shape Model in the Wind Tunnel
97
7.2.3 Observations
It was observed that, the fabric membrane vibrated with different frequencies at
different wind speeds. The time period of the membrane could not be
computed by counting the number of oscillations in the video, the reason being,
a consistent number for the period for various wind speeds for a particular case
could not be estimated. Every case had a different number of oscillations for
different wind speeds.
7.2.4 Conclusions
It was concluded that the frequency of vibration of the membrane structure is
dependent on the wind speed. This could be because of the aero elastic
behavior of the membrane. The increase in wind speed causes an increase in
wind pressure, which causes an increase in the deformation. The increased
deformation causes a slowing down of fabric oscillation, which causes a
decrease in frequency. The membrane structure thus may not have a natural
period of vibration, due to its aero elastic behavior. It would be interesting
however, to plot the relationship between vibration frequency and wind
velocity for different shapes of structures.
98
After testing the saddle shape model with variations in sag to span ratio,
prestress and wind speed, the other models were tested. 30‐second videos of
their responses to different wind speeds were recorded. After slowing down
these videos and carefully observing them to count the number of oscillations,
conclusions similar to that for the saddle shaped model were arrived at. The
increase in wind speed caused an increase in frequency and decrease in
amplitude of vibration. Thus the natural frequency of vibration for all shapes
could not be determined by this method. The natural frequency method thus
did not help in determining the gust factor. Therefore, another method had to
be arrived at, to determine the gust factor. The next chapters’ focus on the
second method formulated.
7.2.5 Observations with respect to response characteristics
The previous section focused on observations with respect to vibration of the
membranes. This section focuses on observations regarding the response
behavior of the different shapes of membranes, with respect to shape,
aerodynamic character and forces acting on it.
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7.2.5.1 Saddle shape model
The following figure shows the shape of model when deformed by the lateral
wind force. The lateral wind exerts two kinds of forces on the saddle shaped
model. An uplift force acts upon half the surface area on the leeward side and a
down ward force acts upon the other half on the windward side. It was
observed that the magnitude of deformation due to the uplift was considerably
more than that due to the downward force. This was because the structure
under consideration was an open structure. A closed structure would have a
different response to the same forces. The upward deformation in the case of a
closed structure would probably be lesser in magnitude. It would be because of
a negative pressure above the structure on the outside rather than an upwardly
moving current inside the structure.
Figure 7.2: Dynamic response of saddle shape membrane
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7.2.5.2 Wave shape model
The figure below shows the shape of model when deformed by the lateral wind
force. The lateral wind exerts two kinds of forces on the wave shaped model.
An uplift force acts upon the leeward surface and a downward force acts upon
the windward surface. And increase of prestress in the ridge and valley cables
reduces the amplitude of vibration. However, the response of the model would
be different with an increase in stiffness of the ridge and valley cables.
Figure 7.3: Dynamic response of wave shape membrane
7.2.5.3 Arch shape model
The arch shaped structure being more stable due to the rigid arches deforms
much lesser than the other shapes. In this case also an uplift force and a
downward force act on the surface of the membrane.
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Figure 7.4: Dynamic response of arch shape membrane
7.2.5.4 Point shape model
The point shape model was relatively stiffer than the other forms due to the
compression mast in the center. A downward force acted upon the windward
side and an uplift force acted upon the leeward side. The form being more
stable, the deformations due to both forces were almost equal, in contrast to the
wave and saddle shape structures.
Figure 7.5: Dynamic response of arch shape membrane
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Chapter 8: The Dynamic Static Load Test
8.0 Introduction
The natural frequency method used to determine the gust factor of fabric
structures was described in chapter 6. It was observed from the tests carried out
in accordance with that method that the frequency of vibration varied with
wind speed and prestress to a very large extent. It was concluded the natural
frequency of vibration might vary greatly for such dynamically sensitive fabric
structures. This method did not yield valuable results as far as the gust factor is
concerned. To proceed with the determination of the gust factor, another
method was developed. This chapter describes this second method and the
tests conducted in accordance with this second method.
8.1 Introduction to method 2
The Gust factor is a dynamic response factor, which accounts for dynamic
amplification of loading due interaction between wind turbulence and the
structure. Wind design is usually done in the static equivalent method. The
structure is thus designed to resist wind loads equal to the static equivalent of
the total wind force. A gust factor when multiplied to dynamic load will give
the equivalent static load taking into account the dynamic amplification due to
the nature of the structure.
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The gust factor is thus a comparison or a ratio of dynamic loading to static
loading. It can be calculated by dividing dynamic wind load with the
equivalent static load. In this case, the equivalent static load can be defined as
the load required, to cause a deformation equal to the amplitude of vibration
caused by a dynamic load.
For example, consider a dynamic force or load caused by wind loading on a
structure to be five pounds. Consider this loading causes a maximum
deformation or amplitude of one inch on the fabric membrane. The equivalent
static load will be the static force (could be gravity force) required to cause a
deformation of one inch. Now, consider this equivalent static load is four
pounds.
The gust factor (G) will be
G= DYNAMIC LOAD/ STATIC LOAD = 5/4 = 1.25
The second method to determine the gust factor is developed based on the
above‐mentioned concept. Physical model testing was done to calculate the
dynamic and static loading. This method will be referred to as the Dynamic/
Static Loading Method in the rest of this thesis.
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8.2 The Dynamic/Static Loading Method (D/SLM)
Steps involved in this method are as follows:
Step 1: Find the total dynamic lateral load due to wind pressure on the model
at maximum wind speed of 17 mph
Step 2: Find maximum amplitude of the fabric displacement at 17mph wind
speed
Step 3: Find equivalent static load required to cause a deformation equal to that
caused by the 17mph wind.
Step 4: Define gust factor G = Static lateral load / dynamic lateral load
For the purpose of this thesis, a maximum wind speed of 17mph has been used
to conduct tests. The reason being the maximum available speed in the
available wind tunnel is 17mph. For the purpose of ease and convenience, it has
been assumed that tests conducted for computing the gust factor (with the
method proposed in this thesis) are not influenced to a great extent by the wind
speed. The reason being, the proposed method to determine the gust factor is
calculated as a ratio and thus the wind speed used does not influence the ratio.
This however holds true for a certain range of wind speeds, exceeding which
the responses could change. This range is again unknown, unless tests are
performed. However, due to limitations of time and facility, the thesis has been
105
done under a set of assumptions. Chapter 4 section 4.2.4 talks about wind
scaling.
8.2.1 Rubber strip test
A rubber strip was used in the process of computing the lateral load. To be able
to use the rubber strip, its E‐modulus was first determined. The rubber strip
was hung on a vertical surface. Weights were hung in the bottom of the strip
symmetrically, in increasing increments and the elongation was recorded. A
stress / strain graph was plotted based on results of the rubber strip test.
Define elastic E‐ modulus
E = stress / strain
Stress = Load/area (area = cross section area of rubber strip)
Strain = change in length / original length, (which is basically the fraction of
elongation)
Following is the stress strain graph that was plotted. The graph was reasonably
linear, which indicated that the rubber strip stretches linearly, i.e. the E‐
modulus is linear.
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Stress Strain Graph‐ Rubber Strip
y = 50.207x + 29.183
0
20
40
60
80
100
120
140
0 0.5 1 1.5 2
Strain dL/L (inch/ inch)
Stress P/A
(Pounds/sq inch)
Figure 8.1: Stress vs. Strain graph of rubber strip
Weight dL dL/L P/A E modulus
Pounds Inches Strain Stress
0.5 1 0.25 40.3 161.2
0.75 2.5 0.625 60.4 96.64
1 4 1 80.6 80.6
1.25 5.5 1.375 100.8 73.30
1.5 7.5 1.875 120.9 64.48
E modulus 424.64
Table 8A: Results of Stress strain test for rubber strip.
The E‐modulus of the rubber strip was found to be 424.64.
8.2.2 Lateral Load Testing
The proposed method to determine the lateral load exerted on the membrane
surface due to the wind was as below.
107
Attach wheels to the underside base of the model. Attach a rubber strip to the
model. Place the model in the wind tunnel and attach the other end of the strip
to a fixed point in the wind tunnel. Let the wind force move the model. Record
the elongation in the rubber strip, due to model displacement. Convert this
elongation mathematically into stain. Find out the corresponding stress in the
rubber strip with the help of the graph and convert the stress in to load or force.
This load will be lateral load on the model.
The following was carried out to calculate the lateral force. Wheels were
attached to the base of the models, so that they can move only in one direction.
This was done to minimize friction between the model base and the wind
tunnel. The model was placed in the test area of the wind tunnel. The base of
the model was attached with a rubber strip to a fixed point in the test area. (See
figure 8.2)
This assembly was loaded at the maximum available wind speed of 17mph.
Section 8.2 contains a description of the reason of the use of a 17mph wind
speed. The elongation in the rubber strip was recorded. The maximum
amplitude of membrane vibration was also measured.
108
Figure 8.2: Lateral load‐testing assembly
Figure 8.3: Elongation in rubber strip due to movement of model.
109
The elongation in the rubber strip was due to the model moving in the direction
of the wind. This was caused due to the force exerted by the wind on the
model, which includes the base of the model and the fabric membrane itself.
For the purpose of this thesis we need to find out the force exerted only on the
membrane. To calculate the force exerted only on the membrane, the test
carried out on the model was repeated with only the model base.
The net elongation of rubber strip due to wind load was calculated
Net elongation = Elongation with membrane less elongation with base only.
This net elongation thus was due to the force exerted by the wind only on the
fabric membrane.
The net elongation was mathematically converted into strain. With the help of
the stress strain graph of the rubber strip the load per unit inch (stress) in the
rubber strip for the net elongation was calculated.
Strain = Dl/l ( ratio of change of length (Dl) to original length(l) )
Stress = P/A (ratio of Load to cross section area
The total lateral force was calculated as
Lateral dynamic load = (load per square inch) (cross section area of rubber
strip)
i.e.; (P/A) A
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This load is the lateral load acting only on the fabric membrane.
For the saddle shape model this test was repeated for the following cases:
• 3 sag to span ratios; named as shapes S1, S2 and S3 having the sag to
span ratios as 0.05, 0.12 and 0.16 respectively.
• 3 prestress levels; 0.28 pounds/inch, 0.19 pounds/inch and 0.09
pounds/inch, for each sag to span ratio
• The tests were also conducted for the other models, namely;
• Wave shaped
• Arch shaped
• Point shaped
The test results were tabulated and the net lateral load for each was calculated.
The results and observations of the tests are presented in chapter 9.
8.2.3 Static load testing
Static load tests were conducted on all models including all the nine cases of the
saddle shape, wave shape, arch shape, and point shape models. The amount of
gravity load required to cause a deformation equal to the amplitude of
vibration due to the wind in the model (for the corresponding sag to span ratio
and prestress) was found. A distributed and relatively uniform gravity load
was applied to the membrane by means of a testing stand. The testing stand
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facilitates the application of hanging gravity loads on a surface simultaneously
at the same time. Static load was applied as gravity load, for the purpose of
simulating the effect of dynamic wind load on the structure. This is because
when the fabric membrane is subjected to wind load, two kinds of forces act on
the surface; an uplift force and a downward pushing force. It has been observed
that these forces acting on the surfaces are perpendicular to the direction of the
wind. To simulate this effect, the static loading on the model is applied as
gravity load acting downward.
Figure 8.4: Test stand
112
8.2.3.1 Load application procedure on membrane
The test stand consists of two horizontal frames, on which the models to be
tested can be kept. It also has a hanging horizontal plane called the loading
platform. This platform is usually a metal sheet or a plank of wood. The plank
or metal sheet hangs on steel cables, which are attached to pulleys and then to a
rotating handle. Rotating the handle lowers the platform down. A load is
simulated on the model by hanging cups filled with lead balls of the desired
weight and resting them on the hanging platform. Load is applied on the model
by lowering the platform. The test stand facilitates placement of a model at a
certain height on frames and application of a number of hanging gravity loads
simultaneously, to simulate a reasonably uniform loading.
To be able to create a relatively uniform loading on the membrane surface of
the model, small paper cups with lead balls, in them were hung on the
membrane in a uniform pattern with the help of a nylon string. A square or
circular piece of cardboard was placed at the point where the nylon string went
through the fabric, so that the weight of the cups is distributed more uniformly,
rather than them being a series of point weights. (Figure 8.4, 8.5)
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Figure 8. 5: Pieces of thick paper used to simulate uniform loading
Figure 8.6: Pieces of thick paper used to simulate uniform loading
8.2.3.2 Static loading of the model
All the models were attached with the hanging cups in the above‐mentioned
manner. A gravity load was to be applied to the membrane by filling the cups
with lead balls. The amount of load required to create a deformation on the
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model, which was equal to the deformation caused in the same model by the
lateral wind load, was to be discovered.
The model was thus kept on the loading table, and to start with, lead balls
measuring a certain weight were distributed uniformly in the cups. To begin
with, this load or weight was equal to almost 80% of the corresponding lateral
load. This load was applied on that surface of the membrane that was on the
windward side in the lateral loading test. The idea behind this was that, the
side was loaded by the lateral (dynamic) load should receive the gravity (static)
load. The following figure 8.6 shows the surfaces that were gravity loaded in
the saddle and wave shaped models. The tests were conducted on these models
first. The models were inverted during the test for ease of application of load;
the models were symmetrical along the horizontal plane. So, the models could
be inverted for loading. The models were loaded with the help of the cups, as
mentioned above. After loading the cups the loading platform was lowered to
apply the load on the membrane. The deformation in the membrane was
measured. Depending if the measured deformation was more or less than the
corresponding deformation in dynamic loading the weight was increased or
decreased. The whole procedure was repeated till the deformations due to the
gravity load and the corresponding dynamic loading matched.
115
Figure 8.7: Static Loading assembly for saddle shape model
116
Figure 8.8: Static Loading assembly for wave shape model
117
The Gust factor was then calculated as
Gust factor = dynamic load / gravity load for all the cases of the saddle shaped
model and the wave shaped model.
It was then realized that the deformation caused due to static loading did not
simulate that of the dynamic loading. The wind lateral load (or dynamic load)
had caused two effects on the surface. There was uplift force acting on the
leeward surface and a downward force on the windward surface. Thus the
static loading should also simulate this deformation to be able to get a correct
gust factor. The static load tests were thus repeated. This second loading
pattern is referred to as “procedure 2” of static load testing and the previous
one is referred to as ”procedure 1” of static load testing in the rest of the thesis
book.
8.2.3 Procedure 2 for static loading
Now in accordance with procedure 2, an upward pulling force would have to
be created to simulate uplift. The following method was used to do so:
The uplift was simulated by strings attached to the membrane with cups for
loading at the other end, similar to that of the downward force. The important
difference is that these strings were pulled in an upward direction and hung on
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dowels. The strings were long enough thus creating an uplift force
perpendicular to the surface.
The following figure shows the static load test assembly for all models. Similar
to procedure 1, the load required to cause a deformation similar to that caused
by the dynamic wind load was to be established.
Figure 8.9: Static loading of saddle shape model – procedure 2
119
Figure 8.10: Static loading of wave shape model – procedure 2
120
Figure 8.11: Static loading of arch shape model – procedure 2
121
Figure 8.12: Static loading of point shape model – procedure 2
122
The results for procedure 2 of the static load tests were different from those of
procedure 1. The gust factor calculated by dividing the dynamic load by the
static load was smaller for procedure 2 than procedure 1. Chapter 9 shows all
results in tabulated form.
8.3 The Pulley test
It was realized after conducting all testing that, there could have been some
friction between the dowel, which was used to simulate the uplift load, and the
nylon string in the static loading test. This could alter the results to a certain
extent. A better option to eliminate friction would have been the use of pulleys
in place of the dowels. A test was thus conducted to clarify this discrepancy.
The test consisted of the static loading assembly with only the uplift load on the
membrane. Two sets of three strings each were attached symmetrically to the
arch model in the upward direction (figure 8.12). One of the sets was lifted up
and hung over dowels and the other set of three strings were hung over
pulleys. Every string was attached with the same load at the end. The uplift
load was applied to the arch model by lowering the loading platform. The
deformation due to uplift force was measured for all six points. The difference
due to the use of the pulley as against the dowel was recorded and averaged
out.
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Figure 8.12: Pulley test
Figure 8.13: Pulley test assembly
124
The arch shape model was used for this test because this model had most of the
membrane almost parallel to the horizontal plane. Thus uplift caused due to the
hanging load would produce a deformation in an almost vertically upward
direction. Comparison of this deformation would be more accurate as against
comparing deformations of a non‐horizontal surface (like in the other three
models). This test helped in finding a correction factor, to eliminate the possible
friction caused by the use of the dowel. The results of this test are presented in
chapter 9.
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Chapter 9: Results and Conclusions
9.1 Introduction
This chapter presents all the results of all tests conducted. It also includes
conclusions and observations made.
9.2 Lateral/ dynamic loading test
The lateral loading test was carried out to calculate the lateral load exerted by
the wind on the surface of the membrane. This test was conducted in the wind
tunnel. A detailed description of the test is given in chapter 8 section 8.2.2.
Following is a table giving the amplitudes of vibration of all models against a
wind speed of 17mph in the wind tunnel.
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Model
Prestress (Pounds/inch) Shape Amplitude (inch)
SADDLE 0.28 S1 0.5
SADDLE 0.19 S1 0.75
SADDLE 0.09 S1 1.1
SADDLE 0.28 S2 0.85
SADDLE 0.19 S2 1
SADDLE 0.09 S2 1.35
SADDLE 0.28 S3 1.5
SADDLE 0.19 S3 1.65
SADDLE 0.09 S3 1.75
WAVE 0.28 ‐ 1.5
ARCH 0.28 ‐ 0.25
POINT 0.28 ‐ 0.2
Table 9A: Amplitude of vibration of membranes at wind speed of 17mph
9.2.1 Results: Lateral loading test
Following table shows the lateral loads calculated for various models. Twelve
models were tested, nine of them being shape and prestress variations of the
saddle shape model.
127
Model Prestress Shape ElongationNet Elongation Lateral force
Pounds/inch inches inches pounds
SADDLE 0.28 S1 0.65 0.3 4.08
SADDLE 0.19 S1 0.5 0.15 3.8
SADDLE 0.09 S1 0.35 0 3.6
SADDLE 0.28 S2 0.75 0.4 4.2
SADDLE 0.19 S2 0.65 0.3 4.08
SADDLE 0.09 S2 0.5 0.15 3.8
SADDLE 0.28 S3 1 0.65 4.6
SADDLE 0.19 S3 0.85 0.5 4.4
SADDLE 0.09 S3 0.75 0.4 4.2
WAVE 0.28 ‐ 1 0.65 4.6
ARCH 0.28 ‐ 0.55 0.25 4.07
POINT 0.28 ‐ 1 0.65 4.6
Table 9B: Lateral loads exerted by wind on membrane surfaces
Model = Model shape‐ saddle/ wave/ arch/ point
Prestress = Prestress in pounds per inch in membrane (method of measuring
the stress in the fabric is given in chapter 6 section 6.2.3.
Shape S1= Saddle model with sag to span ratio of 0.05.
Shape S1= Saddle model with sag to span ratio of 0.12.
Shape S1= Saddle model with sag to span ratio of 0.16.
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Amplitude = Maximum amplitude of vibration of the membrane when acted
upon by a 17mph wind in the wind tunnel.
Elongation = Elongation of the rubber strip attached on one side to the model
and a fixed point in the wind tunnel, due to model movement, when acted
upon by lateral forces of the 17mph wind in the wind tunnel.
Net Elongation = Elongation of the rubber strip with the model – Elongation of
the rubber strip with only base of model.
Lateral Force or Lateral load = Load acting on the model membrane due to the
17mph wind, calculated from the rubber stress strain graph. (Table 9B and
figure 9.1)
9.2.2 Rubber strip – stress strain test
Method used to calculate lateral force or load with the help of a rubber strip is
presented in chapter 8 section 8.2.1 A stress to strain graph of the rubber strip
was plotted to help calculate the lateral load exerted on the membrane .
Following is the method to calculate the lateral load
Net elongation/ original length = strain in the rubber strip
Thus stress in the rubber strip can be found out for the corresponding strain
with the help of the following graph
Force causing elongation in the rubber strip = Stress/ cross sectional area
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Force causing elongation in the rubber strip = Lateral force or lateral load acting
on the membrane due to the wind.
Stress Strain Graph‐ Rubber Strip
y = 50.207x + 29.183
0
20
40
60
80
100
120
140
00.5 11.5 2
Strain dL/L (inch/ inch)
Stress P/A
(Pounds/sq inch)
Figure 9.1: Stress vs. Strain graph of rubber strip used to calculate
lateral/dynamic load
9.2.3 Observations of the lateral load test
It was observed from testing the various models that the lateral load acting on
the surface of a membrane structure increases with increase in stiffness.
This observation was made on the basis of the following:
1. Saddle shape model: Three varying sag to span ratios of the saddle shape
model were tested. The lateral load test was carried out for 3 different prestress
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for each of the three variations of the sag to span ratios. In all three cases an
increase in lateral load was observed with an increase in the prestress.
The following graphs show the observation
Figure 9.2: Lateral load vs. prestress graph for saddle shaped membrane
Prestress Lateral Load in Pounds
pounds / sq inch shape 1: S1 Shape 2: S2 Shape 3: S3
0.09 psi 3.6 3.8 4.2
0.19 psi 3.8 4.08 4.4
0.28 psi 4.08 4.2 4.6
Table 9C: Prestress and lateral load for Saddle shape model
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9.3 Gravity/ static loading test
This test was done to calculate the static or gravity load required to cause a
deformation equal to that caused by the wind for the given model at the same
prestress. A detailed description of the testing method is given in chapter 8,
section8.2.3.2. Following are the results of the test. The static load was
conducted in two procedures. Procedure 1 was conducted on saddle and wave
shaped models. This procedure only took into account downward force.
Procedure 2 was conducted on all four models, which took into account
downward force and uplift force. Chapter 8 section8.2.3 gives a detailed
description of the procedures and reason they were conducted.
Model Shape Prestress Static Load Static Load
pounds/ inch pounds pounds
Procedure 1 Procedure 2
SADDLE S1 0.28 1.5 2.33
SADDLE S1 0.19 1.35 2.1
SADDLE S1 0.09 1.15 1.93
SADDLE S2 0.28 1.65 2.65
SADDLE S2 0.19 1.5 2.38
SADDLE S2 0.09 1.4 2.2
SADDLE S3 0.28 2 3.2
SADDLE S3 0.19 1.75 3
SADDLE S3 0.09 1.6 2.73
WAVE ‐ 0.28 2.2 2.9
ARCH ‐ 0.28 Not conducted 2.7
POINT ‐ 0.28 Not conducted 2.9
Table 9D: Static loads that cause deformation equal to that caused by the wind
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9.4 Gust factor calculation
The gust factor is mathematically calculated as the ratio of dynamic load to
static load or the ratio of lateral load to gravity load. The following table gives
the calculated gust factors for all the models.
Model Shape Prestress Lateral load Static Load Gust Factor
pounds/ inch pounds pounds
(Procedure 2)
SADDLE S1 0.28 4.08 2.33 1.75
SADDLE S1 0.19 3.8 2.1 1.8
SADDLE S1 0.09 3.6 1.93 1.86
SADDLE S2 0.28 4.2 2.65 1.58
SADDLE S2 0.19 4.08 2.38 1.7
SADDLE S2 0.09 3.8 2.2 1.72
SADDLE S3 0.28 4.6 3.2 1.4
SADDLE S3 0.19 4.4 3 1.46
SADDLE S3 0.09 4.2 2.73 1.5
WAVE ‐ 0.28 4.6 2.9 1.58
ARCH ‐ 0.28 4.07 2.7 1.5
POINT ‐ 0.28 4.6 2.9 1.58
Table 9E: Gust factors for tested models
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9.4.1 Conclusions
An interesting observation was made by comparing the gust factors of the three
variations of prestress for each shape of the saddle shape model. The gust factor
increased with a decrease in prestress for all three cases. The following graph
illustrates the same.
Prestress Vs Gust factor‐ Saddle Shape Model
0.09 0.19 0.28
Prestress
Gust factor
Saddel Shape S3
Saddle Shape s2
Saddle Shape S1
Figure 9.3: Graph showing decrease in the gust factor with increase in prestress
for saddle shape model.
134
Model Shape Prestress Gust Factor
pounds/ inch
SADDLE S1 0.28 1.75
SADDLE S1 0.19 1.8
SADDLE S1 0.09 1.86
SADDLE S2 0.28 1.58
SADDLE S2 0.19 1.7
SADDLE S2 0.09 1.72
SADDLE S3 0.28 1.4
SADDLE S3 0.19 1.46
SADDLE S3 0.09 1.5
Table 9F: Gust factors for saddle shape model at different prestresses.
Thus, the stiffer the membrane the lesser is the gust factor. On the other hand,
increase in membrane stiffness will increase lateral loads to a certain extent. The
aerodynamic property of the shape of the membrane however, also influences
the lateral loads exerted on it by the wind. An aerodynamic shape will pose
least resistance to wind thus resisting least lateral load.
9.5 The Pulley Test
As described in chapter 8 a test was conduct with an aim to reduce the effect of
friction between the dowel and the nylon string in the static load test. This test
compared the deformations caused by an upward load when hung over a
pulley and when hung over a dowel.
135
Following table shows results of the test.
Pulley Dowel
Weight Deformation Deformation Percentage difference
Pounds Inches Inches
0.375 0.15 0.2 25
0.375 0.25 0.4 37.5
0.375 0.5 0.6 16.66
0.562 0.3 0.45 33.33
0.562 0.4 0.5 20
0.562 0.55 0.65 15.38
0.75 0.45 0.5 10
0.75 0.6 0.7 14.28
0.75 0.7 0.8 12.5
Table 9G: Results of the Pulley test.
9.5.1 Results and conclusions
The results of the pulley test show that the deformations when using a pulley
are lesser than the deformations while using a dowel. This was in contrary to
the expected results, which were an increase in deformation due to reduction or
elimination of when the dowel was replaced by the pulley. It was thus
concluded that the friction within the pulley must have been more than that
between the dowel and the nylon string. The difference in the deformations
was recorded and the percentage difference was also calculated. The percentage
increase in deformations due to the pulley however cannot be taken as precise,
since the scales of the models were very small and actual differences were very
small in dimension (even though if scaled to full size, would be considerable).
136
The pulley test thus did not yield expected results however, it opened many
questions, which have not been taken further and handled in the scope of this
thesis.
9.6 Suggested gust factor
Considering that the gust factor for all the shapes of membrane structures
tested have been in a certain close range, it is suggested that an optimum
average number be found, which can be used for designing fabric membrane
structures in general. As explained in chapter 4 section 4.2 .1, the ASCE 7 –02
incorporated a three second gust wind speed map. The wind speeds prescribed
in this map take into consideration the gustiness of the wind. The gust factor
prescribed is 0.8 for exposures B and 0.85 for exposure C and D for the main
wind force resisting systems of buildings and other structures and for
components and cladding of open buildings and other structures. The reason
for the gust factors to be less than 1 is the fact that the wind speed used for
designing takes into consideration the gustiness of the wind (three second gust
speeds).
The gust factors based on tests performed for this thesis have been calculated
for laminar steady wind speed and thus do not take into consideration wind
137
gustiness. These factors thus need to be adjusted to be compatible with the
other gust factors, which have been numerically reduced due the new three‐
second‐gust wind speed map. The gust factors determined have to thus be
multiplied by 0.8 or 0.85, to be compatible with ASCE 7 ‐02.
To define a general gust factor, the average of the gust factors of the saddle
shape model is suggested. The saddle shaped model has been tested for various
prestress and sag to span ratio cases. Thus an average of the gust factors of
these cases would be more accurate than an average of all the model’s gust
factors. This average is multiplied with 0.85 to find a suggested gust factor. The
reason of choosing 0.85 is that it would yield a larger number than when
multiplied with 0.8. This larger gust factor could be used for exposures B. C and
D.
Suggested gust factor = average gust factor (saddle shape) X 0.85
i.e. Suggested gust factor = 1.642 X 0.85 = 1.394
This could be rounded to 1.4.
This thesis thus suggests a gust factor of 1.4 for wind design of fabric
membrane structures.
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Chapter 10: Future Work
10.0 Introduction
This chapter describes the prospective future research that can be done with
this thesis as foundation.
10.1 Determination of gust factor
This thesis proposed a method to determine the gust factor for fabric
membrane structures. The facilities and equipment used in the tests carried out
as per the method were subject to many limitations. An important area of
research would be to conduct the same tests as per the method used in this
thesis with all or at least few of the following variations:
1. Higher speed wind tunnel, with ability to have a large range of wind speeds.
2. Bigger wind tunnel with larger test section to accommodate larger models.
3. Ability to simulate and measure wind gusts in the wind tunnel.
4. Accurate measurement devices inside the wind tunnel.
5. Accurate measuring devices for the test stand.
6. Bigger test models that simulate more accurately the material connections of
the real structure.
139
7. More accurate and easy to handle system of weights used to load the models
for the static load tests.
8. Apart from the above‐mentioned points, tests could also be carried out using
bigger models in an atmospheric boundary layer wind tunnel. This would help
simulate the actual atmospheric conditions with surface roughness conditions
surrounding the structure.
9. Apart from the dynamic responses of the membranes studied in this thesis,
the airflow pattern around membranes can be studied with the help of smoke
generators in the wind tunnel.
10. An improved method could be devised to determine the gust factor on the
basis of the method proposed in this thesis.
11. Influence of factors like specific edge conditions (beam, arch and cable) on
the gust factor can be determined.
12. Gust factors for cable nets can be computed in the similar manner, with
considerations of shape, prestress and sag to span ratio.
10.2 Gust factors used in design
It would be interesting to find the actual gust factors used by designers for the
many fabric membrane structures that exist. It will also be interesting to
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document membrane structures that failed or collapsed due to wind load. Tthe
gust factors used in the design of these structures can be of great use.
10.3 Lateral load and openness:
All models studied in this thesis were open models. It was observed that the
lateral loads acting on surfaces of membrane structures were causing primarily
two kinds of deformations. One was in the downward direction due to a
downward force. The other is in the upward direction due to uplift. The
deformation due to uplift was much more than the downward deformation for
all models. This uplift was because of the following reasons:
Uplift force due to upward push of air inside the structure. (Figure 10.1)
Negative pressure built up on the top pf the structure. (Figure 10.1)
The first of the two reasons stated above was primarily due to the openness of
the structures. It would be interesting to find the relationship between
openness and lateral load exerted on the surface of a membrane structures in
wind load. A reduction in openness may reduce uplift, however it may not
necessarily reduce total lateral load exerted on the surface. A closed structure
may have more surface area against the wind; which exerts more lateral load.
Further the nature of the enclosure material may influence the lateral load on
the membrane. A rigid wall may cause an increase in the lateral load. However,
141
a complete membrane enclosure may be more flexible and aerodynamic to
reduce lateral load.
Considering all these variables that influence the lateral loads exerted by the
wind on a membrane structure, the research could define an optimum
openness ratio for minimal uplift for all shapes.
Figure 10.1: Uplift force
10.4 Openness and gust factor
Gust factors are a function of lateral loads. As mentioned above lateral loads
are influenced by openness. A relationship between openness and gust factor
will be interesting to know.
142
10.5 Aero elasticity
While performing any sort of wind test on scaled models of fabric membrane
structures it is very important to be aware of the fact that the surface responses
are due to aero elastic properties of the membrane. Aero elastic effects
(explained in chapter 3, section 2.4) may not be same in the real structure and
the model. To be able to match the aero elasticity, in the model and the
structure, the aeroelastic‐damping ratio of the structure and the model will
have to be matched. Damping is any effect, either intentionally produced or
inbuilt to a system, that tends to reduce the amplitude of oscillations or
vibrations. An interesting area of research would be to find out damping factors
or damping constants for fabric structures. This will help greatly in accurate
scaling of models for wind tunnel testing.
143
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Sivaprasad, Neha
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Wind design of fabric structures: Determination of gust factors for fabric structures
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