UNIVERSITY OF CALGARY
Low-cost Triangular Lattice Towers for Small Wind Turbines
by
Ram Chandra Adhikari
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
CALGARY, ALBERTA
September, 2013
© Ram Chandra Adhikari 2013
Abstract
This thesis focuses on the study of low-cost steel and bamboo triangular lattice towers for small
wind turbines. The core objective is to determine the material properties of bamboo and assess
the feasibility of bamboo towers. Using the experimentally determined buckling resistance,
elastic modulus, and Poisson’s ratio, a 12 m high triangular lattice tower for a 500W wind
turbine has been modeled as a tripod to formulate the analytical solutions for the stresses and
tower deflections, which enables design of the tower based on buckling strength of tower legs.
The tripod formulation combines the imposed loads, the base distance between the legs and
tower height, and cross-sectional dimensions of the tower legs. The tripod model was used as a
reference for the initial design of the bamboo tower and extended to finite element analysis. A 12
m high steel lattice tower was also designed for the same turbine to serve as a comparison to the
bamboo tower. The primary result of this work indicates that bamboo is a valid structural
material.
The commercial software package ANSYS APDL was used to carry out the tower analysis,
evaluate the validity of the tripod model, and extend the analysis for the tower design. For this
purpose, a 12 m high steel lattice tower for a 500 W wind turbine was examined. Comparison of
finite element analysis and analytical solution has shown that tripod model can be accurately
used in the design of lattice towers. The tower designs were based on the loads and safety
requirements of international standard for small wind turbine safety, IEC 61400-2. For
connecting the bamboo sections in the lattice tower, a steel-bamboo adhesive joint combined
with conventional lashing has been proposed. Also, considering the low durability of bamboo,
periodic replacement of tower members has been proposed. The result of this study has
established that bamboo could be used to construct cost-effective and lightweight lattice towers
for wind turbines of 500 Watt capacity or smaller. This study concludes that further work on
joining of bamboo sections and weathering is required to fully utilize bamboo in practice. In
comparison to steel towers, bamboo towers are economically feasible and easy to build. The
tower is extremely lightweight, which justifies its application in remote areas, where the
transportation is difficult.
ii
Acknowledgements
First of all, I would like to express my gratitude to my supervisor, Professor David Wood, for his
constant support and guidance throughout my master’s. Without David’s strong support, this
thesis would not have been possible. David has been more than an academic supervisor to me.
Thank you so much for everything you have done to support me. I would also like to thank my
co-supervisor, Professor Les Sudak, for his invaluable advice from the beginning of this
research.
I would like to thank all my friends in the EES specialization office space for their comradery
and exchanging ideas on interdisciplinary research. I would also like to thank ISEEE, SSAF,
SAF at the U of C and the NSERC/ENMAX Industrial Research Chair in Renewable Energy for
providing the financial support in my experimental work in Nepal.
On a more personal level, I am deeply indebted to my whole family for their endless support and
encouragement. I specially thank my wife Sushma, for her unwavering love and support
throughout the years of my master’s.
Last but not least, I am very grateful to Rajendra Pant, Lab Incharge at Pulchowk Campus, TU,
Nepal and Donald F. Anson at the University of Calgary for their support in the experimental
tests on bamboo. Particular thanks go to Pramod Ghimire and Kimon Silwal at KAPEG, Nepal
for their support during my experimental work in Nepal.
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Dedication
This piece of research is dedicated to the loving memory of my mother, Keshar Devi Adhikari.
iv
Table of Contents
Abstract ............................................................................................................................... ii
Acknowledgements.............................................................................................................iii
Dedication .......................................................................................................................... iv
Table of Contents .................................................................................................................v
List of Tables ................................................................................................................... viii
List of Figures .................................................................................................................... ix
List of Symbols and Abbreviations.................................................................................. xiv
CHAPTER 1: INTRODUCTION ........................................................................................1
1.1 Context of the Thesis .................................................................................................1
1.2 Small Wind Power Systems .......................................................................................2
1.3 Motivation for the Thesis ...........................................................................................3
1.3.1 Lattice Tower for Small Wind Turbines ...........................................................4
1.3.2 Bamboo for Wind turbine Towers .....................................................................4
1.4 Thesis Objectives and Approach ...............................................................................5
1.5 Organization of the Thesis .........................................................................................6
CHAPTER 2: LITERATURE REVIEW .............................................................................7
2.1 Chapter Overview ......................................................................................................7
2.2 Wind Turbine Towers ................................................................................................7
2.3 Types of Wind Turbine Towers .................................................................................8
2.3.1 Monopole or Tubular Tower .............................................................................9
2.3.2 Lattice Tower...................................................................................................10
2.3.3 Hybrid Tower ..................................................................................................10
2.4 Towers for Small Wind Turbines ............................................................................10
2.5 Costs of Small Towers .............................................................................................11
2.6 Materials for Wind Turbine Towers ........................................................................12
2.7 Bamboo ....................................................................................................................14
2.8 Physical Structure of Bamboo .................................................................................15
2.9 Micro-structure of Bamboo .....................................................................................16
2.10 Mechanical Properties of Bamboo .........................................................................19
2.11 Joining Methods for Bamboo ................................................................................22
2.12 Durability of Bamboo ............................................................................................27
2.13 Further Comments .................................................................................................28
2.14 Adhesives Joints ....................................................................................................29
CHAPTER 3: EXPERIMENTAL TESTS ON MECHANICAL PROPERTIES OF
BAMBOO .................................................................................................................32
3.1Chapter Overview .....................................................................................................32
3.2 Related Works..........................................................................................................32
3.3 Testing Protocol .......................................................................................................34
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3.3.1 Test Specimens for the Buckling Experiment .................................................34
3.3.2 Buckling Test Procedure .................................................................................35
3.3.3 Compression Test Procedures .........................................................................37
3.4 Results and Analysis ................................................................................................39
3.4.1 Buckling Strength ............................................................................................39
3.4.2 Compression Strength .....................................................................................43
3.4.3 Modulus of Elasticity and Poisson Ratio.........................................................46
3.5 Joint Testing.............................................................................................................49
CHAPTER 4: LOADS AND DESIGN REQUIREMENTS FOR WIND TURBINE
TOWERS ..................................................................................................................53
4.1 Chapter Overview ....................................................................................................53
4.2 Design Standards and Requirements .......................................................................53
4.3 Loads on Wind Turbine Tower ................................................................................55
4.3.1 Gravity Loads ..................................................................................................55
4.3.2 Aerodynamic Thrust on Rotor Blades .............................................................56
4.3.3 Drag on the Tower ...........................................................................................56
4.4 Load Safety Factors .................................................................................................57
4.5 Tower Design Methods ............................................................................................57
4.5.1 Allowable Strength Design..............................................................................57
4.5.2 Allowable Buckling Strength ..........................................................................58
4.5.3 Allowable Tower Deflection and Natural Frequency .....................................58
CHAPTER 5: DESIGN AND OPTIMIZATION OF LATTICE TOWERS FOR SMALL
WIND TURBINES ...................................................................................................59
5.1Chapter Overview .....................................................................................................59
5.2 Overview of Design Optimization and Objectives ..................................................59
5.3 The Triangular Lattice Tower ..................................................................................60
5.4 Design Procedure .....................................................................................................63
5.5 Structural Analysis of the Lattice Tower .................................................................64
5.5.1 Analysis of the Tripod Model..........................................................................65
5.5.2 Failure Criteria.................................................................................................71
5.5.3 Tower Deflection .............................................................................................75
5.6 Optimization of the Tripod Model ...........................................................................78
5.7 Finite Element Analysis ...........................................................................................78
5.7.1 The Methods of FEA .......................................................................................78
5.7.2 The FEA of the Lattice Tower.........................................................................78
5.7.3 FE Model of the Tower ...................................................................................79
CHAPTER 6: DESIGN OF STEEL LATTICE TOWER..................................................82
6.1 Chapter Overview ....................................................................................................82
6.2 The Steel Lattice Tower ...........................................................................................82
6.3 Design Optimization Procedure ...............................................................................82
6.4 Optimization of the Tripod Model ...........................................................................83
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6.4.1 Tower Loading ................................................................................................83
6.4.2 Optimization of Tower Legs ...........................................................................84
6.4.3 Results of Tripod Analysis ..............................................................................86
6.5 Finite Element Analysis of Tower ...........................................................................88
6. 6 Results and Discussion ...........................................................................................90
6.6.1 Design Examples with Horizontal Bracings....................................................96
6.6.2 Design Example including Cross-bracings .....................................................98
6.7 Design Loads for Foundation ................................................................................100
6.8 Tower Manufacture................................................................................................102
CHAPTER 7: OPTIMAL DESIGN OF BAMBOO TOWER .........................................104
7.1 Chapter Overview ..................................................................................................104
7.2 The Bamboo Tower ...............................................................................................105
7.3 Design Requirements for Bamboo Tower .............................................................107
7.4 The Proposed Joint.................................................................................................108
7.5 Design Procedure for the Bamboo Lattice Tower .................................................109
7.5.1 Structural Analysis of the Tripod Model .......................................................110
5.5.2 Results of Tripod Analysis ............................................................................111
5.5.3 Finite Element Analysis of the Tower ...........................................................114
7.5.3.1 FE Model of the Bamboo Tower .........................................................115
7.5.3.2 Results of Finite Element Analysis ......................................................116
7.5.4 Results and Assumptions of the Analysis .....................................................123
7.5.5 The Optimal Tower Design ...........................................................................123
CHAPTER 8: SUMMARY, CONCLUSIONS AND FUTURE WORK ........................127
8.1 Summary of Thesis ................................................................................................127
8.2 Conclusions ............................................................................................................129
8.3 Future Work ...........................................................................................................131
REFERENCES ................................................................................................................133
APPENDICES .................................................................................................................139
vii
List of Tables
Table 2.1 Mechanical properties of bamboo................................................................................. 20
Table 3.1 Test Results: Buckling Experiment (TU, Nepal) .......................................................... 39
Table 3.2 Test Results: Buckling Experiment (University of Calgary) ........................................ 41
Table 3.3 Test Results: Compression Experiment ........................................................................ 44
Table 3.4 Results of Pull-out Tests on Steel-bamboo Adhesive Joints ........................................ 52
Table 4.1 Extreme Wind Speeds for Different Classes of Wind Turbines [6]………………......53
Table 4.2 Load safety factors for the design loads [6]…………………………………………...57
Table 5.1 Values of Q [63]…………………………………………………………………....…73
Table 6.1 Optimum D for tower legs (t = 3 mm)………………………………………………...87
Table 6.2 Comparison of FEA, numerical and analytical results (Tripod tower)…………….…92
Table 6.3 Comparison of FEA and numerical results (Tower with horizontal bracings)……….93
Table 6.4 Results of FEA with horizontal bracings (b =1.2 m)………………………………….97
Table 6.5 Recommended tower design (tower with horizontal bracings)……………………….98
Table 6.6 Recommended tower design with cross-bracings……………………………………100
Table 7.1 Buckling strengths of 1.5 m long columns (t = 6 mm) (equation 3.6)………………112
Table 7.2 Tower configurations for the FEA (t=6 mm)………………………………………..114
Table 7.3 Comparison of the results of FEA and tripod analysis……………………………....119
Table 7.4 Optimized design of the bamboo tower……………………………………………...123
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List of Figures and Illustrations
Figure 1.1 A hybrid wind-photovoltaic power system in a remote village in Nepal (Photo: by
the Author) .............................................................................................................................. 3
Figure 2.1 Types of wind turbine towers ........................................................................................ 8
Figure 2.2 A small turbine tower with guy-wires [17] ................................................................... 9
Figure 2.3 Relative costs of lattice and other tower designs for a 10 kW wind turbine [7] ......... 12
Figure 2.4 World’s first 100 m high proto-type timber tower (left) made with timber
composite panels (right) [11]. ............................................................................................... 13
Figure 2.5 A bamboo plantation in Nepal (Photo: by the Author) ............................................... 15
Figure 2.6 A bamboo culm (left) and longitudinal cross-section of the culm (right) showing
its physical structure (Photos: by the Author)....................................................................... 16
Figure 2.7 Density of vascular bundles in the wall [26] ............................................................... 17
Figure 2.8 Cross-section of bamboo culm perpendicular to the longitudinal axis, showing
wall and diaphragm (Photo: by the Author).......................................................................... 17
Figure 2.9 Fraction of fibre-density with respect to distance from outer to the inner wall [24]... 18
Figure 2.10 Micro-structure of bamboo wall showing the vascular bundles [26] ........................ 18
Figure 2.11 Arrangements of fibres in the nodes [29] .................................................................. 19
Figure 2.12 Strength and stiffness comparison of bamboo with different materials [39] ............ 21
Figure 2.13 Methods of connecting two or more bamboos culms, the friction tight method
[42] ........................................................................................................................................ 23
Figure 2.14 Connection with ropes [42] ....................................................................................... 23
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Figure 2.15 Bamboo scaffoldings in construction (photos taken from [35])................................ 23
Figure 2.16 Space frame (left) and connection of bamboo columns by metallic joint (right)
[44] ........................................................................................................................................ 24
Figure 2.17 Connection with steel wire to a steel plate [46] ........................................................ 24
Figure 2.18 Interlocking with metal anchors [42] ........................................................................ 24
Figure 2.19 Bamboo-wood glued joint [39] ................................................................................. 25
Figure 2.20 Double layer grid (DLG) with PVC-bamboo joints .................................................. 26
Figure 2.21 Pull-out test of the joint ............................................................................................. 26
Figure 2.22 Load-deflection curve of the joint ............................................................................. 26
Figure 2.23 Components of an adhesive joint .............................................................................. 30
Figure 2.24 Single lap tubular joint .............................................................................................. 31
Figure 3.1 Freshly cut bamboo culms (left) and dried bamboo specimens (right)……………... 35
Figure 3.2 Experimental set-up for the buckling test in the MTS-100 test machine ……………37
Figure 3.3 Buckling mode of the bamboo column during the buckling test…………………….37
Figure 3.4 Experimental set-up for the compression test (left) and split bamboo………………38
Figure 3.5 Relationship between buckling strength and slenderness ratio of bamboo columns...42
Figure 3.6 Load-deflection behaviour of bamboo columns in buckling tests……………………43
Figure 3.7 Load-deformation of bamboo in compression……………………………………….45
Figure 3.8 Load-deformation behaviour of bamboo in compression ………………………….. 45
Figure 3.9 Stress-strain of bamboo in compression……………………………………………..47
x
Figure 3.10 Specifications of the bamboo (left) and cylindrical steel caps (right)……………...50
Figure 3.11 Specimen setup for the pull-out test on bamboo joints………………………….….51
Figure 3.12 Failure of the joint by slippage of bamboo culm from the steel cap…………….…51
Figure 5.1 Structural model of the Triangular Lattice Tower ………………………………..….61
Figure 5.2 Design optimization procedures for the lattice tower………………………….….....64
Figure 5.3 Free Body Diagram (FBD) of the tripod tower. Bracings are not included. The legs
are denoted by AD, BD, and CD. The turbine is mounted at point D. The arrows indicate the
direction of forces and moments in the tower ………………………………………………...…66
Figure 5.4 Lattice tower as a cantilever beam……………………………………………….…..67
Figure 5.5 Base cross-section of the tower as a composite beam of legs and bracings………....67
Figure 5.6 2-node 188 beam element [13]………………………………………………………80
Figure 5.7 3-node 189 element [13]……………………………………………………………..81
Figure 6.1 Loads on the Tower…………………………………………………………………..84
Figure 6.2 Optimum diameters of legs (D) for various base distances (b) and wall thickness (t) of
3 mm using (ASCE standard) ……………………………………………………………………87
Figure 6.3 Bottom section of the FE model of the lattice tower showing drag forces and
boundary conditions……………………………………………………………………………...90
Figure 6.4 Convergence test for stress with different lengths of beam elements (Tower with b =
1m, D =64 mm and 64 mm horizontal bracings)………………………………………………...91
Figure 6.5 Convergence test for deflection with different lengths of beam elements (Tower with
b = 1m, D =64 mm and 64 mm horizontal bracings) ……………………………………………91
Figure 6.6 Maximum stress and deflection of the tower at b =1m and D =64 mm……………...94
xi
Figure 6.7 Maximum stress and deflection at b =1.2 m and D =45 mm………………………...94
Figure 6.8 Maximum stress and tower-top deflection at b =1.4 m and D =35 mm……………..95
Figure 6.9 Maximum stress and deflection at b = 1.6 m and D =29 mm………………………..95
Figure 6.10 Maximum stress and deflection at b = 1.2 m, D =40 mm and 20 mm diameter
bracings ………………………………………………………………………………………..97
Figure 6.11 Maximum stress and tower deflection for D =35 mm, 20 mm diameter horizontal
bracings, and 10 mm cross-bracings…………………………………………………………….99
Figure 6.12 Schematic of the loads on tower foundation………………………………………102
Figure 6.13 Jig to make tubular lattice tower used by Kijito Windpower, Kenya. Photo taken
from [7]…………………………………………………………………………………………103
Figure 7.1 Study approach for the bamboo lattice tower…………………………………….....104
Figure 7.2 Bottom section of the proposed bamboo lattice tower………..………………...…..106
Figure 7.3 Joining methods for the leg sections………………………………………………..106
Figure 7.4 Steel connector cap for the adhesive joints…………………………………………106
Figure 7.5 Proposed joining methods in the lattice tower……………………………………...109
Figure 7.6 Maximum compressive stresses in tower legs for various leg diameters and base
distances………………………………………………………………………………………...112
Figure 7.7 Diameters of 1.5 m long bamboo columns that are marginally safe against buckling
for various base distances……………………………………………………………………....113
Figure 7.8 Maximum tensile forces on tower legs for various b…………………………….…113
Figure 7.9 Finite element models of the tower with horizontal bracings (left), with horizontaland cross-bracings (centre), and bottom section of the tower showing wind loading on the tower
(right)…………………………………………………………………………………………...116
xii
Figure 7.10 Tower-top deflection and compressive stress (b=1.6 m and D=70 mm)………….117
Figure 7.11 Tower-top deflection and compressive stress (b=1.85 m and D = 65 mm for legs and
bracings)………………………………………………………………………………………...117
Figure 7.12 Tower-top deflection and compressive stress (b=2.15 m and D = 60 mm for legs and
bracings…………………………………………………………………………………………118
Figure 7.13 Tower-top deflection and compressive stress (b=2.6 m and D =55 mm for legs and
bracings)………………………………………………………………………………………...118
Figure 7.14 Tower-top deflection and compressive stress in the tower with horizontal- and crossbracings (b= 1.85 m, D=65 mm)………………………………………………………………..120
Figure 7.15 Tensile forces in the tower legs at b= 2.6 m, D = 65 mm and 30 mm diameter for
bracings…………………………………………………………………………………………121
Figure 7.16 Effect of bracing sizes on maximum tensile forces in legs at b= 2.6 m (obtained from
FEA)…………………………………………………………………………………………….121
Figure 7.17 Lattice tower of 2.6 m with 65 mm leg size and 30 mm for horizontal- and crossbracings…………………………………………………………………………………………122
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List of Symbols and Abbreviations
Symbols
Definition
A SC
cross-sectional area of tower legs
b
base distance between the tower legs
Cd
drag coefficient
CT
thrust coefficient of rotor blades
D
diameter of the tower legs
D1
external diameter of the bamboo culm at the larger end
d1
internal diameter of the bamboo culm at the larger end
D2
external diameter of the bamboo culm at the smaller end
d2
internal diameter of the bamboo culm at the smaller end
df
diameter of the foundation
E
elastic modulus of the material
F
thrust on rotor blades at extreme wind speed of 50 m/s
F cr
buckling strength of bamboo column
Fb
allowable bending stress
g
acceleration due to gravity (9.81 m/s2)
h
height of the tower
hf
depth of the foundation
I(y)
moment of inertia of the tower section at ‘y’ from the tower top
l
length of the test specimen, tower leg section
li
length of the ith tower member
MC
moisture content
M(y)
bending moment at section ‘y’ from the tower top
Mf
resultant moment at the foundation
Mo
overturning moment at the foundation
M RS
resisting moment of the foundation
N
number of nodes in the bamboo specimen
Nt
thickness of the node of the bamboo culm
xiv
P cr
critical buckling load
q
drag force per unit length of the tower leg
R1
internal radius of tower legs
R2
external radius of tower legs
R HF
resultant horizontal force on the foundation
t
thickness of tower legs and bracings
U
extreme wind speed
W
weight of turbine and nacelle
Wt
weight of tower
y
distance of a tower section from the tower top
ρ
density of the tower material
v
deflection of tower section at distance ‘y’ from the tower top
𝝈𝝈𝒄𝒄
compressive strength of bamboo
𝜺𝜺
longitudinal strain
ρ
density of air
Abbreviations
Definitions
AISC
American Institute of Steel Construction
ASCE
American Society of Civil Engineers
IEA
International Energy Agency
IEC
International Electrotechnical Commission
INBAR
International Network for Bamboo and Rattan
ISO
International Organization for Standardization
xv
“The central activity of engineering,
as distinguished from science,
is the design of new devices, processes
and systems''
- Myron Tribus (1969)
xvi
Chapter 1
INTRODUCTION
1.1 Context of the Thesis
Universal access to clean, secure and sustainable energy systems is one of the biggest
development challenges today. In a recent report released by International Energy Agency (IEA)
[1], more than 1.3 billion people in the world lack access to electricity; about 95% of them are
from Asian developing countries and sub-Saharan Africa. Similarly, more than 2.6 billion people
rely on traditional biomass sources 1 for cooking and heating and about 84% of people live in
rural areas [1]. This energy poverty has depressingly impacted many areas of human endeavour
(health, education, food, water etc...), particularly in the developing world.
To respond to energy poverty, the United Nations secretary general announced in “Rio+20” [2],
the United Nations conference on sustainable development, a major initiative, “Sustainable
energy for all (SE4all)”, to provide electricity to all people by 2030 [3].The “SE4all” has stated
three major goals: ensuring universal access to modern energy services, doubling the share of
renewable energy in the global energy mix, and doubling the global rate of improvement in
energy efficiency by 2030.To achieve the target of universal access to energy services,
significant cost reductions and rapid deployment of current energy technologies, along with
commercialization of renewable energy technologies, will be needed.
Continuous research and development is essential for the technological innovation to bring down
the costs of energy systems [4]. While a set of policy instruments enable transitioning the energy
sectors to a more sustainable one, “technological development will significantly enhance the
portfolio options available and will bring down the costs of energy technologies” [4].
1
Traditional biomass energy sources include firewood, animal dung, and agricultural residues [1]
1
Consequently, there is an urgent need to establish clean and affordable energy systems, with
particular emphasis on the exploitation of renewable energy sources.
Among many renewable energy technologies, wind energy offers an immense potential to extract
clean energy; and its rapidly growing installations worldwide have shown that wind energy could
play a significant role in the future energy supply systems. In developing countries, such as
Nepal, where the “SE4all” is targeted, wind energy technologies are not fully developed due to
many barriers. The key barriers include lack of profitable markets and well-adapted technologies
to end user needs, inability to manufacture and manage technologies, high capital and life-cycle
costs, technological limitations, financing risks etc [5].
1.2 Small Wind Power Systems
Small wind turbines, which have rotor swept area smaller than 200 m2 [6] or rated power less
than about 50 kW [7], are increasingly relevant in rural or off-grid areas for generating costeffective electricity [5,8,9]. They can be installed on their own or in combination with
photovoltaic (PV) modules, to supply electricity to a small village, a health clinic, or a small
industry, by using a local transmission and distribution network. Figure 1.1 shows an example of
a small wind power system installed with photovoltaic (hybrid mode) in a remote village in
Nepal.
Small wind turbines are often installed in remote locations, where the best wind resources exist
but grid extension requires significant investment in building electricity transmission
infrastructure. Along with the opportunities, there are also many challenges to development of
small wind power systems. These include: high capital costs, lack of capabilities on design and
manufacturing of wind energy components, and difficulty in transportation and installation of
towers. Today’s small wind turbines are mostly installed on monopole towers, similar to those
shown in Figure 1.1,which are expensive, manufactured from steel, and are often difficult to
transport to remote locations.
2
Figure 1.1 A hybrid wind-photovoltaic power system
in a remote village in Nepal (Photo: by the Author)
1.3 Motivation for the Thesis
After the blades, the support structure or tower is the most critical part of the wind power system.
It is the most material-consuming structural unit that bears significant portion of the total cost [8,
10]. Cliffton-Smith and Wood [8] reported that the cost of manufacturing the towers could be 30
to 40% of the installation cost in the case of small wind turbines. In remote locations, where
there is no access to road transportation, the cost of transporting the tower would be even higher
and may be physically impossible in some cases.
The materials currently used in wind turbine towers are steel and concrete. Very recently, timber
has been successfully used to build wind turbine towers for large wind turbines [11].
Investigations on new materials, such as ultra-high reinforced concrete (UHRC) [12] and
composites [13], have been undergoing for developing environmentally sustainable, economic,
3
and light wind turbine towers. As far as the author is aware, steel is the only material being used
in small wind turbine towers and timber and concrete have not been yet used.
Within this broader context of sustainable energy and materials, it is relevant to look at
possibilities for minimizing the costs of towers using existing materials as well as using more
sustainable and low-cost materials, such as bamboo. The current research is aimed at minimizing
the costs of small wind power systems through: 1) design of light-weight and low-cost lattice
tower and 2) investigation on the applicability of renewable material, bamboo to lattice towers.
1.3.1 Lattice Tower for Small Wind Turbines
The biggest challenges in developing wind power system in remote areas are the high capital
costs of towers and the difficulty in transporting the towers from manufacturing facilities, which
are often located in urban areas. Lattice towers are an alternative to the conventional monopoles
[7]. Lattice towers provide a lighter and stiffer tower design, which can be easily manufactured,
installed and maintained with minimum equipment and workmanship and mostly at lower costs
[7]. This work proposes a triangular lattice tower because of: 1) ease of design, manufacture,
transport, and install and 2) economic competitiveness.
1.3.2 Bamboo for Wind turbine Towers
Bamboo is a superior natural material that has been used widely in various engineering structures
throughout the human history. As a cheap and sustainable material with impressive tensile,
compressive, and buckling strengths, bamboo shows its suitability for lattice towers for small
wind turbines. Until now, there is no engineering investigation about its use in wind turbine
towers. If bamboo is capable of meeting the loads and safety requirements of International
Electrotechnical Commission (IEC) IEC61400-2 [6] and the problem of connecting bamboo
sections is solved, then there is a significant potential for utilizing this material to reduce the cost
of small wind power systems in the developing countries. This thesis investigates the mechanical
strengths of bamboo through material testing. To assess the suitability of bamboo for small
4
towers, a design example of 12 m high bamboo lattice tower for a 500 W wind turbine is
presented and compared to a steel lattice tower design. The advantages of bamboo from
engineering design point of view are summarized below:
•
High tensile and buckling strengths
•
High growth rate and sustainable material
•
Easily available and low-cost material
•
Simple technology for processing
1.4 Thesis Objectives and Approach
The core objectives of this thesis are 1) to establish the mechanical properties of bamboo
required for the design and analysis of a bamboo lattice tower, 2) to develop a design
optimization methodology for the lattice tower in accordance with IEC61400-2 safety standard
for small wind turbines, and 3) to determine whether bamboo towers are feasible for small wind
turbines.
The specific objectives are:
1. To determine experimentally the buckling and compression strengths and elastic properties
of bamboo to determine its suitability for application in small wind turbine towers.
2. To develop a design procedure for a cheap, lightweight, and easily-manufactured lattice
tower using analytical methods and finite element (FE) modelling tool, ANSYS [13].
3. To design and optimize a 12 m high steel lattice tower for a 500 W wind turbine using
analytical and FE modeling.
4. To carry out the design assessment of a 12 m high bamboo lattice tower for the same 500 W
wind turbine using analytical and FE modeling on the basis of experimental results on
bamboo’s mechanical properties and load requirements of IEC61400-2 and compare with
steel tower.
5
1.5 Organization of the Thesis
This thesis is organized into 8 chapters as follows:
Chapter 1 introduced the context, motivation, and objectives of the thesis. Chapter 2 presents two
literature reviews: the first on tower design types for small wind turbines and the second on
bamboo’s mechanical properties and its applicability to lattice towers for small wind turbines.
Chapter 3 describes the experimental work on bamboo’s mechanical properties and steel-bamboo
adhesive joint and summarizes the main results. Chapter 4 discusses the main loads acting on
wind turbine towers and IEC design requirements for small wind turbine towers. Chapter 5
introduces the approximate mathematical analysis for triangular lattice towers and describes the
design procedure through FE modeling. Chapter 6 presents the design optimization of a 12 m
high steel lattice tower for 500W wind turbine. Chapter 7 presents the design and feasibility
analysis of a 12 m high bamboo lattice tower for 500W wind turbine. Finally, Chapter 8
summarizes the main conclusions of the thesis work and provides a brief summary for future
works.
6
Chapter 2
LITERATURE REVIEW
2.1 Chapter Overview
The literature review is divided into two sections. The first section presents an overview of wind
turbine towers and their design types. The second section presents an overview of bamboo as an
engineering material, past experimental research on bamboo’s mechanical properties, and its
potential for use in wind turbine towers.
2.2 Wind Turbine Towers
A wind turbine tower has two primary functions. First, it supports the wind turbine and
accessories at a desired height [10]. Second, it transfers the loads acting on the wind turbine and
the tower to the foundation. In addition, wind turbine tower often houses the electrical
components and accessories and also provides access to wind turbine [10].
Wind turbine towers must meet the functional requirements, which are the specifications that
define the intended functions of the tower, throughout the life-span, typically 20 years. The
functional requirements include withstanding the turbine and wind loads and the self-weight, and
vibration. Structural performance is defined as an acceptable structural behaviour, such as
minimal tower top deflection under ultimate loads, with reference to the specified functional
requirements.
7
2.3 Types of Wind Turbine Towers
There exists a great variety of wind turbine towers, which can be broadly classified into three
types:
1. Monopole or tubular tower
2. Lattice tower
3. Hybrid tower
The first two types of towers are commonly used in both large and small wind turbines, while the
hybrid towers are recently conceptualized by National Renewable Energy Laboratory (NREL)
[14] for large off-shore wind turbines. The towers can be installed with or without guy-wires;
towers without guy-wires are called free-standing or self-supporting towers. The purpose of guywires (Figure 2.2) is to reduce the bending stress at the tower base, but it requires more ground
space and may be subject to vandalism [7]. Therefore, self-supporting towers are desired.
a) Hybrid tower [14]
b) Monopole tower [15]
Figure 2.1 Types of wind turbine towers
8
c) Lattice tower [16]
Figure 2.2 A small turbine tower with guy-wires [17]
2.3.1 Monopole or Tubular Tower
The monopole or tubular designs are the most popular types of tower for both large and small
wind turbines (Figure 2.1 (a)). These towers require less ground space to install and are
aesthetically pleasing; however, they require more steel to manufacture [7] than the lattice tower
in Figure 2.1 (b). They are usually made from circular hollow sections for ease to manufacture as
well as for transportation and installation at sites. Small tubular towers are either made in single
sections or multiple sections that are slip-fit together or have flanges that are bolted. However,
they tend to be expensive to transport to remote locations. For example, Wood [7] described the
design of a three-sectioned 18 m monopole tower for a 5 kW turbine that weighs 530 kg.
9
2.3.2 Lattice Tower
Lattice towers are manufactured as truss or frame structure with the sectional or tubular members
connected through welding or use of mechanical fasteners (Figure 2.1 (b)). Lattice towers are
cheaper and easier to manufacture than monopoles and can be easily assembled on site, but their
life-span is often shorter than monopole towers due to corrosion at joints [7]. Despite the fact
that lattice towers are aesthetically less pleasing than monopoles, low-cost, longer life-span, and
light and stiffer towers can be manufactured using circular tubular members [7]. Also the
foundation cost is lower than the monopoles. The factors that make the cost of lattice towers
lower than monopoles are summarized below.
•
Manufacturing cost is lower, as it can be easily manufactured with less equipment and
processes
•
The material cost of tower is lower as the material use is less
•
Transportation cost is lower as it’s easy to transport light and smaller tower sections and
can be assembled easily on site
2.3.3 Hybrid Tower
The hybrid tower, shown in Figure 2.1 (c), combines a truss, tube, guy-wires, and cables to build
tall towers while avoiding requirement of large base diameter for tubular towers [14]. It
incorporates the advantages of both tubular and lattice tower structures in respects of stiffness
and cost [14]. It is also called the “guyed design” because it consists of truss, tube, and guywires. The hybrid towers are specifically intended for large wind turbines.
2.4 Towers for Small Wind Turbines
Free-standing triangular or square lattice towers and monopoles are popular for both off-grid and
grid-connected small wind turbines. Lattice towers are more common in off-grid or remote
10
applications, while monopoles are in grid-connected applications because of the aesthetics. Cost
is always the determining factor in choosing the wind power system over its alternative,
photovoltaic technology for off-grid applications. In remote or off-grid locations, the cost of
tower is often significantly higher due to higher transportation costs than in urban locations
where the transportation is easier.
There are also another variety of towers similar to wind turbine towers that should be mentioned:
electricity transmission towers or poles and telecommunications or meteorological masts.
Transmission towers carry mainly the weight of transmission wires and horizontal tension loads
and so the design is dictated by these loads. Masts are slender structures that are designed with
guy-wires. They are very sensitive to dynamic wind loads and the structural response is often
non-linear that require dynamic analysis. As such, the functional requirements of wind turbine
towers are different than these tower types. Consequently, the design methodology is different.
2.5 Costs of Small Towers
The process of estimating the cost of a tower is challenging as it depends upon the specific site
and its socio-economic context. Therefore, economic aspects can be judged only on the basis of
relative merits of design aspects and capital costs.
Wood [7] presented a cost comparison between the lattice towers and monopole towers as shown
in Figure 2.3 for a typical 10 kW wind turbine with reference to various tower heights. It is
evident that self-supporting lattice towers are the cheapest option for small wind turbines. These
costs exclude the foundation costs which are too site-specific to be included. This thesis will also
exclude detailed consideration of foundation costs.
11
Figure 2.3 Relative costs of lattice and other tower
designs for a 10 kW wind turbine [7]
2.6 Materials for Wind Turbine Towers
The tower is the most material-consuming component of a wind turbine structure. Presently, the
dominant materials for manufacturing the towers are steel and concrete, the latter for large
towers only. With the issues of climate change and raising costs of tower materials along with
their environmental impacts, investigations into new and sustainable materials for low-cost
towers have become very pertinent.
Very recently, timber has been successfully used to manufacture a prototype100 m high tower,
shown in Figure 2.4, to support a 1.5 MW wind turbine, installed in Hannover, Germany. Its full
potential is yet to be assessed but current industry reports [11] have indicated that wood is an
economic and sustainable material alternative to steel and concrete. Another potential material
being investigated is ultra-high performance fibre reinforced concrete (UHPC) [12].
12
Figure 2.4 World’s first 100 m high proto-type timber tower (left) made
with timber composite panels (right) [11].
When considering new materials for wind turbine towers, the tower design must satisfy certain
loads and safety requirements that the tower may encounter during its life-span. The tower
design is governed by the loads imposed on the tower, the strength of the tower materials, and
the stiffness requirement for the tower [10]. The main loads acting on the tower are the turbine
thrust, wind and gravity loads. The tower design should ensure that the tower response is linear
elastic under the imposed loads [10]. This can be achieved by ensuring that the tower top
deflection is a small proportion of the height, the maximum stress is below the allowable stress,
no tower section or component will buckle, and the tower’s natural frequency is not excited by
the blade passing frequency of the turbine [7, 10]. Almost all towers are designed against
extreme static loads that are expected to occur during the life-span of the tower, which are often
specified by the standards, such as IEC. The basis of the tower design presented here is the
ultimate load analysis for extreme wind speeds.
13
2.7 Bamboo
Bamboo is a member of the grass family “gramineae” that has more than 1200 known species
around the world. A typical bamboo plant is shown in Figure 2.5. Bamboo grows mostly in
tropical and sub-tropical regions of the world, particularly in South Asia and Africa, but it can be
grown almost everywhere. Bamboo grows very quickly; it has a remarkable growth rate,
sometimes reaching 15 to 18 cm in a day and gains its full height in 4 to 6 months [18]; some
species are reported to grow at up to 5 cm per hour [19]. Bamboo’s strength is fully developed
between three to five years and can then be used as structural material.
Bamboo is nature’s superior product; it was the first plant that grew in Hiroshima after the
bombing [19]. It is also the first material used in the filament of the electric light bulb developed
by Thomas Edison [19]. Bamboo can grow at an altitude of 3800 m in all types of lands and soils
[19]. Sometimes bamboo is planted on sloping lands to prevent landslides. Bamboo is recognized
as one of the highest carbon dioxide (CO 2 ) sequesters amongst plants [20].
Bamboo is very similar to wood at a macro-scale, such as appearance and material properties.
However, at the micro-scale, the outer layer of bamboo is harder than the inner, whereas it is the
opposite for wood. Bamboo is a versatile material; its uses span from foods and clothes to
building structures. As a low-cost, strong, and easily available material, bamboo has been
extensively used throughout human civilization. With the growing interest on eco-friendly and
sustainable materials in recent years, significant research has been focused on characterizing the
mechanical properties of bamboo to utilize its potential in modern structures.
Modern structural applications include: footbridges [19], scaffolding [21], composites for wind
turbine blades [22], reinforcement of concrete [23], laminated beams and composites [20] etc.
Bamboo is particularly suitable where high tensile, bending, and compression strengths are
required. Bamboo presents advantages in comparison to other construction materials for its
lightness, high bending capacity and low cost because it requires simple and low-cost processing
techniques with minimal workmanship [24].
14
Figure 2.5 A bamboo plantation in Nepal
(Photo: by the Author)
2.8 Physical Structure of Bamboo
Bamboo grows as a cylindrical hollow structure with thin transversal diaphragms spaced along
its length (Figure 2.6), forming closed cylindrical cavities called “lacuna” [24]. The bamboo
plant body or stem is called a “culm”. The outer part of the culm where these diaphragms are
present is called the node and the inter-nodal portion is a hollow circular section. The diaphragm
closes the hollow structure. Bamboo naturally exhibits dimensional variability along the length
to counteract natural loads; i.e. the diameters and thicknesses of bamboo decrease towards the tip
during its growth, but the variation is not significant for a short section of bamboo. The thickness
of the wall remains constant in the inter-nodal region and increases slightly near the diaphragms.
15
Node
Diaphragm
Wall
Inter-nodal region
Figure 2.6 A bamboo culm (left) and longitudinal cross-section of the
culm (right) showing its physical structure (Photos: by the Author)
2.9 Micro-structure of Bamboo
Bamboo is a natural fibre-composite material, in which cellulosic fibres are reinforced
longitudinally into the lignin matrix [25, 26]. The cellulose fibres run in the longitudinal
direction and the walls possess a graded structure; fibre density increases from the inner to the
outer wall [26, 27, 28] as illustrated in Figures 2.7-2.10. It is important to note that the
distributions of fibres in the walls and nodal sections or diaphragms are different and so is their
strength. The reported tensile strengths of outer fibres, middle, and inner fibres are given in
Table 2.1. It is noted that outer layer is stronger than that of steel in tension. Liese [29] observed
that the bamboo fibres are essentially vascular bundles Figure 2.10, which are composed of veins
and cellulose micro-fibres reinforced with lignin. The fibres are arranged in both transverse and
longitudinal directions in the nodal region as shown in Figure 2.11.
16
Outer wall
Inner wall
Figure 2.7 Cross-section of bamboo
culm perpendicular to the longitudinal
axis, showing wall
(Photo: by the Author)
Inner wall
Outer wall
Figure 2.8 Density of vascular
bundles in the wall [26]
17
Figure 2.9 Fraction of fibre-density with respect to
distance from outer to the inner wall [24]
Figure 2.10 Micro-structure of bamboo wall
showing the vascular bundles [26]
18
Figure 2.11 Arrangements of fibres
in the nodes [29]
It is important to note that bamboo has acquired this graded composite structure through many
years’ of evolutionary processes to resist bending and compression loads, such as wind and selfweight [26, 30, 31] as well as various natural loads in its environment. Due to its graded structure
for the function of resisting different types of loads, bamboo is known as functionally graded
material (FGM) [26]. Silva et al. [28] used FE modelling to study the composite structure as a
homogenized material domain and concluded that bamboo can be modeled as a homogeneous
material to determine its effective mechanical properties.
2.10 Mechanical Properties of Bamboo
Bamboo exhibits excellent tensile, compressive, and buckling strengths and stiffness properties
in the longitudinal direction [25, 26, 31]. This is attributed to the longitudinal reinforcement of
fibres into the lignin matrix, which form a hollow tubular composite structure with the
diaphragms spaced along its length. Due to the graded composite structure, bamboo is an
anisotropic material having low mechanical strength in the transverse direction [26, 31, 32]. The
general mechanical properties of bamboo are summarized in Table 2.1. Table 2.1 shows
considerable variation in mechanical properties within and across the species of bamboo. This is
typical of natural materials.
19
Table 2.1 Reported Mechanical Properties of Bamboo
Material Properties
Values
Tensile strength (MPa)
[17]
- Inner layer
135-163
- middle layer
165-275
- outer layer
306-357
- average strength
162-275
Compression strength (MPa)
Source and Remarks
63-86
[34]
35-79
Kao Jue for wet and dry [21]
44 -117
Mao Jue for wet and dry[21]
45-65
Kao and Mao Jue [33]
46- 68
Bambusa Stenostachya (Tre
Gai) [32]
Buckling strength (MPa)
12-37
[21]
Bending strength (MPa)
50-75
Kao Jue [21]
Shear strength (MPa)
9.6
For nodal region [32]
8
For inter-nodal region [32]
16-23
[34]
9-14
[35]
Elastic Modulus (GPa)
12-22
[28, 34]
Poisson’s Ratio
0.30- 0.35
[28]
20
Bamboo is an extremely light material that has a dry density between 600-800 kg/m3 [36].
Bamboo’s specific modulus of elasticity (22889 k m2/s2) is comparable to mild steel (25316 k
m2/s2) and specific strength (214 k m2/s2) four times higher than mild steel (50 k m2/s2) [37]. A
comparison of strength and stiffness of bamboo with different materials is shown in Figure 2.12.
Figure 2.12 Relative strength and stiffness of bamboo with
different materials [39]
Although bamboo has excellent mechanical properties in the longitudinal direction, it is weak in
the transverse direction (about 10% of the tensile strength [40]) due to the predominantly axial
orientation of the fibres; that is why bamboo is susceptible to splitting or fracturing along the
longitudinal axis. Cracks appear in the hollow section along fibre directions as the nodal section
consists of diaphragms. When a crack or split occurs in the internodal region, it spreads to the
nodal sections but the nodal section prevents further spreading; hence it is safe from the fracture
of the whole culm [19]. Therefore, nodes increase the splitting or fracture strengths of bamboo
culms.
Mitch [32] studied the splitting characteristics of bamboo using a “split-pin” method. Tan et al.
[31] studied the micro-structure crack phenomena. Their results suggest that the low fracture
resistance of bamboo must be taken into account while designing bamboo structures.
21
Since bamboo is not an isotropic material, the elastic modulus is different in tension, shear, and
compression. As a graded composite material, it can be viewed as a stack of several concentric
layers with varying properties [19, 28]. The material properties vary between the nodes and
around nodes. The nodes increase the compression and buckling strength in the culm.
The material properties of bamboo depend on a number of parameters, such as diameter,
thickness, inter-nodal distance, straightness, moisture content, species, age, and preservation
techniques [39, 40].
2.11 Joining Methods for Bamboo
Several joining methods are available for bamboo columns; some of the relevant joints to the
lattice tower are discussed here. Different types of joints have been investigated by Arce [39] and
Janssen [40]. Laraque [19] divided joints into two groups: traditional and modern connectors.
The most common traditional method of joining is the lashing technique or wrapping the ropes
or fibres or plastic cords around two or more bamboo columns (Figures 2.13 and 2.14). These
joints are also called friction-tight joints. They are highly effective and provide high strength and
stiffness when two or more bamboo columns are connected at right angles. This is the cheapest
way to connect bamboo in structures and is the most commonly used in scaffoldings,
footbridges, houses, and other structures. The most successful application of lashings or ropes is
scaffoldings [21] of bamboo columns (Figure 2.15) and puja pandals [41]. It is important to note
that these structures or columns fail in buckling rather than in joints [21] under compressive
loads, validating that the lashing of joints are sufficiently strong and stiff. Although lattice towers
are self-supporting structures, unlike scaffolding which needs support, the lashing of joints could
be effectively utilized for bracings and legs in lattice towers. That means the load directions must
be carefully analyzed in the joints that utilize lashing.
22
Figure 2.13 Methods of connecting two or more
bamboos culms, the friction tight method [42]
Figure 2.14 Connection
with ropes [42]
Figure 2.15 Bamboo scaffoldings in construction (photos taken from [35])
23
Figure 2.16 Space frame (left) and connection of bamboo columns by
metallic joint (right) [44]
Figure 2.17 Connection with
Figure 2.18 Interlocking with metal
steel wire to a steel plate [46]
anchors [42]
There are some modern trends in bamboo joining with the aim to transfer loads in an efficient
and reliable way without compromising with the limitations of bamboo culms. These are
illustrated in Figures (2.16-2.18).
Arce [39] investigated the bamboo-wood glued joint, shown in Figure 2.19, and carried out
experimental tests to determine the strength of joints (Figure 2.19) under static loads and load
24
reversals. In this joint, a round piece of wood is glued inside the bamboo culm. Experimental
results showed that there is no impact on the joint strengths by static loads and load reversals;
however, it was not understood at what stress levels the joint would fail under cyclic-loads. The
parameters, bamboo density and bamboo thickness showed no significant influence on the
strength of joints, and bamboo was found to show the weakest bonding in the joint. Glued joints,
between wood and bamboo, were found among the best joints and Arce recommends that stress
design levels should be kept in the elastic range for the safe design of structures.
Figure 2.19 Bamboo-wood glued joint [39]
In a recent work by Albermani et al. [18], the pull-out resistance or strength of the Polyvinyl
Chloride (PVC) - bamboo glued joints, shown in Figures 2.20-2.22, was investigated . In this
joint, a grooved bamboo end is encased inside the PVC cylindrical connector using megapoxy
grouting material. The results of the pull-out tests with and without grooves show that the pullout or tensile resistance of the grooved joints is significantly higher than the non-grooved ones.
The reported maximum pull-out resistance was about 18 kN for the grooved specimens of 61
mm diameter bamboo.
25
Figure 2.20 Double layer grid (DLG) with
PVC-bamboo joints [18]
Figure 2.21 Pull-out test of the joint [18]
Figure 2.22 Load-deflection curve of the
joint [18]
26
2.12 Durability of Bamboo
As a natural biological material, bamboo has inherently low resistance to weathering (drying and
wetting processes) and biological decay (ageing and insects and fungi). Under open
environmental conditions without any protection, bamboo can last upto 3-5 years. However, in
protected and indoor applications, the durability is considerably higher depending upon the use.
The longevity of bamboo can be improved if certain protective measures are applied. However,
cost is always the determining factor in deciding whether the preservative techniques should be
used or not.
Since moisture content has influence on the mechanical strength of bamboo, it should be kept as
low as possible and it should not change due to weathering effects when it is used in wind
turbine tower. As a biological material, bamboo is attacked by insects or other micro-organisms
if not treated properly. Insects and fungi degrade the micro-structure of bamboo, which reduces
the mechanical strength over time [29]. There is a strong correlation between insects and fungi
attacks and harvesting time, humidity and starch content (nutrition for insects) of bamboo [46].
In order to reduce the nutrition and moisture contents, bamboo should be harvested in dry
seasons and dried properly [46]. Use of driers and smokers are effective and low-cost methods to
dry bamboo poles. There are many commercial preservative techniques available [46, 47]; but a
low-cost and effective method is to spray the bamboo with borax salt [46, 47]. Surface coatings
may be applied to reduce the effect of humidity or moisture absorption. The details of drying and
curing can be found in [46].
Lima et al. [48] investigated experimentally bamboo’s durability and changes with time in
material properties such as tensile strength and Young’s Modulus when it was used as concrete
reinforcement. Tensile strength and Young’s Modulus showed little difference over time. This
verified that the durability of bamboo does not change in concrete reinforcement. However, in
many applications, weathering often leads to splitting and degradation of bamboo.
27
2.13 Further Comments
Systematic engineering investigations that focused on the design of bamboo structures were
carried out by Arce [39] and Janssen [40], who determined material properties and examined
joint designs. Arce explored various material and design constraints, design objectives, and
developed general design approaches for bamboo structures along with the options for joints.
The conclusions of their studies relevant to this thesis are summarized below:
•
Moisture content has significant effect on the strength of bamboo [40]
•
The dimensional variation and modulus of elasticity along the length of bamboo does not
reduce bending and axial stiffness by more than 15% [39]
•
The critical value of buckling load on bamboo columns can be determined as a
conservative estimate by assuming bamboo is a hollow section [39]
•
Splitting is the dominant limit state or failure mode for most bamboo in structural
applications [39,40]
•
Tensile strength and density of bamboo are correlated in longitudinal direction but not in
the transverse direction. Tension modulus in the transverse direction is about 1/8th of the
longitudinal modulus [40]
•
Use of pins, screws, bolts, or drilling for joints would concentrate stress and induce
splitting of the culm [39,40]
•
For designing joints, insertions and gluing of wooden plugs inside the bamboo ends
would eliminate the splitting and weathering effects in the joints [39]
•
In glued joints, such as wood and bamboo, stress levels should be kept in the elastic
range for safe design of the structure [39]
•
In glued joints, the bamboo density, thickness, and initial diameter, and type of wood did
not influence the strength of joints. It was determined that bamboo was the weakest phase
in the joint and there is no impact on the joint by load reversals, but it was not understood
at what stress levels the joint would fail under cyclic-loads [39]
•
Use of “steel fittings as central elements in bamboo connections in bamboo structures” is
recommended [39]
28
Overall, bamboo is a versatile structural material that has been used throughout the human
history. It is a natural material that is cheap and readily available and grows quickly. It possesses
excellent mechanical properties. However, until recently, it is mostly used in temporary
structures (e.g. scaffoldings, footbridges, houses etc...). The main advantages of bamboo for
lattice towers are summarized below:
•
grows quickly and is easily available
•
low-cost structural material for support structures
•
natural composite with excellent mechanical properties along the longitudinal axis
•
high strength to weight ratio, superior than steel
•
sustainable material and excellent CO 2 sequester
There are also limitations that make challenging to use bamboo in lattice towers.
•
Lack of appropriate joining techniques due to circular hollow structure and dimensional
variability along the length
•
Variability in physical and mechanical properties
•
Low durability compared to the life-span of wind turbines
In order to address above challenges, steel-bamboo adhesive joint for leg sections and periodic
replacement of tower members are proposed.
2.14 Adhesives Joints
Adhesives are materials that join two similar or dissimilar structural materials (either metallic or
non-metallic) through surface attachment. A typical physical configuration of an adhesive joint is
illustrated in Figure 2.23 [53]. The materials being joined together are called adherends. The
advantages of adhesive joints are: minimum stress concentration in the joint (than in welded and
riveted joints), high resistance to moisture, light-weight, and easily producible [53]. Adhesive
29
joints are mostly used in aircraft and marine structures. A detailed theory on adhesive joints can
be found in [53].
Figure 2.23 Components of an adhesive joint
The joint design involves the calculation of adhesive thickness and the joint length or the overlap
length [53]. Most of the structural adhesives have shear strength in the range 13 MPa-38MPa
[53].
Most adhesive joints are loaded in tension. When an adhesive joint is loaded, shear stresses and
strains are developed within the adhesive and interface rather than in the adherends. Assuming
that a tensile load (F) is applied to the joint of width b and length l, the shear stress (𝜏𝜏𝑦𝑦 ) in the
joint is given by [53]:
𝜏𝜏𝑦𝑦 = 𝐹𝐹/𝑏𝑏𝑏𝑏
(2.1)
𝑙𝑙 = 𝐹𝐹/𝜏𝜏𝑦𝑦 𝑏𝑏
(2.2)
The minimum joint length is given by:
By knowing the shear strength of the adhesive (𝜏𝜏𝑦𝑦 ) and the load on the joint (F), the length and
width of the joint can be selected. From this simple formula, it is evident that the strength of the
30
joint can be improved by increasing the bonding area of the joint. The required thickness of the
adhesive can be found by using the Volkerson’s equation [54]:
𝐸𝐸 𝑡𝑡 𝑡𝑡𝑎𝑎
𝐹𝐹 = 2𝑏𝑏𝜏𝜏𝑦𝑦 �
2𝐺𝐺
𝐺𝐺𝑙𝑙2
tanh�𝐸𝐸𝐸𝐸𝑡𝑡
𝑎𝑎
(2.3)
Where, E is the elastic modulus of the adherend, t is the thickness of the adherend, l is the
overlap length, G is the elastic modulus of adhesive, and t a is the adhesive thickness. For the
joint with two dissimilar materials, t and E should be taken of the weaker adherend. It is evident
from equation (2.3) that strength is proportional to �𝑡𝑡𝑎𝑎 .Volkerson [54] assumed that adhesives
deform only in shear whereas adherends deform in tension.
Due to the polymeric layer in the joint, the adhesives have good damping property or fatigue
strength and good resistance to vibration. Also stress concentration is much smaller than in
welded and riveted joints [55]. Among the wide variety of joint designs, those commonly found
in engineering structures are: single lap, double lap, scarf, bevel, step, single butt, double butt,
and tubular lap [54]. However, only the tubular single lap joint, shown in Figure 2.24, was found
an appropriate option for connecting circular bamboo sections in the lattice tower, described in
Chapter 7. The fabrication and testing of the joint strength is described in Chapter 3.
Figure 2.24 Single lap tubular joint
31
Chapter 3
EXPERIMENTAL TESTS ON MECHANICAL PROPERTIES OF BAMBOO
3.1Chapter Overview
In order to assess the feasibility of bamboo for designing lattice towers for small wind turbines, it
is crucial to establish the mechanical properties of bamboo columns. The essential mechanical
properties for design and analysis of the bamboo lattice tower are the compression and buckling
strengths and elastic constants (elastic modulus and Poisson’s ratio). It is first necessary to
establish linear behaviour of bamboo. These material properties provide basic input data and
ultimate design limits for the analysis of the tower. Tensile strength was not determined as the
literature review in Chapter 2 established that bamboo is stronger in tension and due to the fact
that the main design criteria are expected to be the buckling strength and the tensile strength of
the proposed joint. This chapter describes the experimental work on bamboo’s mechanical
properties and strength of steel-bamboo adhesive joints.
3.2 Related Works
As a natural material, both similarity and variation on physical and mechanical properties of
bamboo are evident across different species. Moreover, as bamboo grows under various natural
loads, variation in physical and mechanical properties exists even within the same species.
Therefore, researchers have focused their studies on a specific species. Here, a brief review of
past experimental studies on compression and buckling strengths of different species is
presented.
Cylindrical columns subjected to compressive loading become structurally unstable well below
the yield strength of the material [49]. This phenomenon is called buckling. In the context of
industrial materials such as steel, material properties are well established through theory and
experimental investigations over a long period of time [50]. However, in the case of bamboo,
32
very little information is available and therefore it is required to determine their values through
material testing when a specific species of bamboo is considered.
Arce Villabos [39] evaluated the critical buckling strengths of bamboo columns by considering
variation of second moment of inertia and Young’s Moduli along their length and derived a
mathematical equation, using the “Southwell Plot” procedure, to account for the variation of
elastic and moment of inertia. However, this procedure is too complex to be applied in practice.
Yu et al. [21] investigated the axial buckling behavior of bamboo columns in scaffolding. They
conducted experimental tests on two bamboo species, Kao Jue and Mao Jue, and established the
buckling strengths by calibrating against the characteristic compressive strengths. The PerryRobertson interaction formula was used to take into account the modified slenderness ratio and
initial imperfections in the columns.
Richard and Harries [51] determined the buckling strengths of bamboo columns of the species
Bambusa Stenostachya. The experimental results were also compared with the theoretical values
predicted using Euler’s formula for column buckling taking into account the dimensional
variability on diameter, straightness, and material properties along the length. Normalized critical
stress (critical stress times slenderness ratio) and slenderness ratio were used for comparing the
theoretical and experimental results. The slenderness ratio is the ratio of the length and the radius
of gyration of the column. Their results showed that initial out of straightness and tapering
should be taken into account in the buckling analysis.
Yu et al. [21] determined the compressive strengths of two species of bamboo, Kao Jue and Mao
Jue and the mean values reported are 79 MPa and 117 MPa respectively. Chung and Chan [35]
showed that moisture content is important in determining the compressive strengths of culms;
properly dried bamboo ( < 5% moisture content) can have three times higher strengths than the
wet bamboo. Moreover, dried bamboo will have consistent properties. Mitch [32] reported
average compression strength of 56.7 MPa for the bamboo species Bambusa Stenostachya (Tre
Gai).
33
3.3 Testing Protocol
In view of the practical context of this study in developing countries such as in Nepal, the
experimental tests were carried out on the Nepalese bamboo species, bambusa arundinacea,
known as Tama bamboo in Nepal. This species is one of the strongest available and grows in
most parts of the country. The mechanical properties of this species of bamboo have not been
studied prior to this investigation.
3.3.1 Test Specimens for the Buckling Experiment
The bamboo specimens for this experiment were collected by the author in May, 2012 from a
village in Kavre district of Nepal. Straight bamboo culms were obtained from plantations of
about 3-4 years of age, as specified in International Organization for Standardization (ISO)
22156 (2004(b)) test protocol [52], for preparing the test specimens. In general, bamboo starts to
develop mechanical strengths by lignifying and silicating processes after reaching this age [34].
For the test specimens, straight sections of the culms were cut into lengths, ranging from 700 to
1500 mm with diameters from 40 mm to 64 mm. The specimens were cut from fairly straight
sections based on visual inspection (Figure 3.1).
Since bamboo does not possess perfect circular cross-sections and varies in diameter and
thickness along the length, the diameters were measured at four different sections and averaged.
Only specimens with less than 5 mm variation in external diameters between the two ends were
used in the buckling tests. The wall thickness was measured by splitting each specimen after the
buckling tests. Similarly the thickness of diaphragms was measured to get the net cross-sectional
areas of specimens. The ends of the test specimens have nodes at both ends, as shown in Figure
3.1.
34
Figure 3.1 Freshly cut bamboo culms (left) and dried bamboo specimens
for the buckling tests (right) (Photos: by the Author)
Since bamboo exhibits higher strengths as well as consistent material properties at moisture
levels below 20% [35], the specimens were air-dried in a natural drying chamber for one and half
month until the moisture levels reduced below 20% before conducting the tests. 20% moisture
level is taken as a reference for experimental tests. It is assumed that about 20% moisture level
would be maintained when bamboo is used in lattice towers.
3.3.2 Buckling Test Procedure
The buckling experiment was accomplished in two phases. The first phase was carried out at
Tribhuvan University (TU), Nepal using a Universal Testing Machine (UTM) having 100 tons
load testing capacity. In this machine, the maximum testing loads can be set at 20, 40, 60, 80,
and 100 tons limits. The machine has a data logging capability of 20 kg/division. The axial
deformations could not be measured during this test due to lack of measuring instruments in the
UTM and so only the critical buckling load could be measured. So, the author brought a few test
specimens to the University of Calgary to determine the load-deflection behaviour of columns,
35
compression strength, and elastic modulus and Poisson ratio. An MTS-100 test machine, shown
in Figure 3.2, rated at 10 tons was used for these tests.
In the experiment conducted at TU, the test specimen was vertically aligned in the UTM as
illustrated in Figure 3.2. Both ends of the test specimens were fixed on flat metal plates to avoid
rotation and translation when loaded. These types of end conditions were chosen to simulate the
rigidly fixed end connections of the tower members. An axial compressive load was then applied
at a rate of 500 N/sec until buckling failure was observed in the specimen. The critical load was
recorded when the specimens began to buckle. A typical indication of buckling failure was a
small indentation in the surface (see Figure 3.3) along the transverse axis. No longitudinal
splitting was observed in any sample at any time.
In the second phase of the experiment, the buckling tests were carried out at the University of
Calgary (Figure 3.2) in the same manner. An MTS-100 Test Machine was used with two test
specimens to determine the load-deflection behaviour of the columns. These measurements were
not possible in Nepal.
For the buckling test, the specimens were aligned axially in the machine as shown in Figure 3.2
and axial compressive load was gradually applied at a rate of 1mm/min until the buckling failure
was noticed in the specimen. The loads and deformations were recorded at every 15 seconds with
a FlexTest® SE Controller-MTS used to control the load. A typical buckling failure mode of the
bamboo column is shown in Figure 3.3.
36
Figure 3.2 Experimental set-up for the
Figure 3.3 Buckling mode of the bamboo
buckling test in the MTS-100 test machine
column during the buckling test
(Photo: by the Author)
(Photo: by the Author)
3.3.3 Compression Test Procedures
The compression tests were carried out at the University of Calgary. The test specimens were
obtained from the same bamboo culms that were used to prepare buckling test specimens in
Nepal. The test specimens were prepared according to the ISO 22157-1:2004 (b), which
prescribes that the length of the test specimens must be between D (diameter of the culm) and
2D. Ten test specimens without nodes were prepared and both ends were finely grinded.
According to [38, 39], there is no effect of nodes in the compressive strength. ISO 22157-1:2004
(b) also outlines that nodes are not required in the test specimens. Therefore, only the specimens
without nodes were tested.
37
In order to establish the deformation behaviour of bamboo, load and deformation were measured.
Two strain gauges were fixed at mid-height in each diagonally opposite side of the specimens to
measure the strains in the specimens as illustrated in Figure 3.4. Two strain gauges measured the
longitudinal strains (labelled 2 in Figure 3.4) and the other two measured lateral strains (labelled
1 in the Figure 3.4) due to compressive load. The outputs of the strain gauges were used to
determine Poisson’s ratio. Thin rubber pads were placed at each end of the specimens to ensure
uniform loading in the specimen during compressive loading. ISO 22157-1:2004(b) emphasized
that uniform compressive loads must be applied to test specimens. The experimental set-up is
illustrated in figure 3.4.The loads and deformations were recorded at every 15 seconds by
applying load at the rate of 1mm/min in the load head until the failure was noticed in the
specimen. The compressive failure was initiated by splitting along the longitudinal direction as
shown in Figure 3.4.
Figure 3.4 Experimental set-up for the compression test
(left) and split bamboo after the compression test (right)
(Photos: by the Author)
38
3.4 Results and Analysis
3.4.1 Buckling Strength
The physical dimensions of the test specimens and the corresponding test results of the buckling
experiment are presented in Table 3.1 and Table 3.2. The load-deformation curve is depicted in
Figure 3.5.
Table 3.1 Test Results: Buckling Experiment (TU, Nepal)
Specimen MC
L
D1
d1
D2
d2
N
Nt
P cr
Fc
(mm) (N)
(MPa)
3
6
41225
41
44
3
4
37345
48
57
44
3
10
32980
29
52
62
50
3
7
36375
33
43
29
42
28
3
3
28130
36
1300
49
39
48
38
3
5
19400
28
18
1160
59
44
58
44
4
14
31525
26
BT08
18
1220
60
50
58
48
3
5
37345
43
BT09
18
1280
54
44
52
43
3
4
30070
39
BT10
18
1170
59
45
57
43
4
5
44135
39
BT11
19
1240
58
44
56
43
3
6
58685
52
BT12
19
1320
47
38
46
36
3
4
16490
27
BT13
19
1050
64
50
63
48
3
6
69840
56
(%)
(mm) (mm) (mm) (mm) (mm)
BT01
20
880
60
47
56
46
BT02
17
1060
55
45
52
BT03
20
1000
59
45
BT04
19
1070
64
BT05
15
1060
BT06
16
BT07
39
BT14
18
1170
57
44
56
42
3
6
42195
41
BT15
20
1160
56
44
54
42
3
5
32980
35
BT16
18
1100
50
42
49
40
3
4
21340
37
BT17
17
1140
47
35
46
33
3
4
25220
33
BT18
19
1280
50
38
48
36
3
4
25220
30
BT19
18
1220
47
39
45
37
3
3
22795
42
BT20
17
1410
42
34
40
32
3
4
11155
23
BT21
19
1100
47
37
46
36
3
3
27160
41
BT22
17
1210
41
31
40
29
3
4
12610
22
BT23
16
1200
53
43
52
41
3
5
40827
54
BT24
17
1060
45
35
44
33
3
3
28130
45
BT25
14
670
62
49
60
48
3
2
52000
52
BT26
12
960
72
62
71
61
3
3
49400
48
BT27
15
1010
76
63
74
61
3
3
77600
57
BT28
16
1080
69
58
67
56
3
3
49800
45
BT29
15
1120
72
58
69
56
3
4
54000
43
BT30
12
1110
73
58
69
57
4
3
56000
46
BT31
13
1100
71
53
69
51
3
2
77000
45
BT32
13
1200
71
57
69
56
3
3
75000
52
BT33
15
1010
61
44
58
43
4
3
47000
58
BT34
12
1200
68
57
65
55
3
4
50600
55
BT35
11
1250
69
55
67
54
3
3
68000
54
40
BT36
11
1340
70
57
68
56
3
2
46000
37
Table 3.2 Test Results: Buckling Experiment (University of Calgary)
Specimen
MC
L
D1
d1
D2
d2
N
Nt
P cr (N)
(%)
(mm)
(mm)
(mm)
(mm)
(mm)
BT01
15
640
57
49
53
45
2
2.5
38730
60
BT02
17
655
52
42
47
39
2
3.0
27050
50
(mm)
Fc
(MPa)
To assess the buckling strengths of columns, Euler’s formula for column buckling was adopted
by incorporating the variation in second moment of inertia along the length (due to the variation
of diameter and thickness), as suggested by previous studies, eg. [21] and [51]. The crosssectional area and second moment of inertia were computed at the smallest cross-section of the
column. The cross-sectional area (A) and the second moment of inertia (I) of the bamboo column
are given by:
𝐴𝐴 = 𝜋𝜋�𝐷𝐷𝑒𝑒 2 − 𝐷𝐷𝑖𝑖 2 �⁄4 = 𝜋𝜋[𝐷𝐷2 − (𝐷𝐷 − 2𝑡𝑡)2 ]⁄4
(3.1)
𝐼𝐼 = 𝜋𝜋[𝐷𝐷4 − (𝐷𝐷 − 2𝑡𝑡)4 ]⁄64
(3.2)
The slenderness ratio (λ) was computed at the smallest section of the column, given by:
λ = 𝐿𝐿𝑒𝑒 ⁄𝑟𝑟
(3.3)
𝑟𝑟 = �𝐼𝐼 ⁄𝐴𝐴
(3.4)
Here, L e is the effective length of the column and r is the radius of gyration given by:
41
In the tests, the column was supported at both ends that simulate the pinned-pinned column
supports. So the effective length of column is taken as equal to the actual length of the bamboo
specimens. The Euler’s elastic buckling strength (F cr ) for the column is given by:
𝐹𝐹𝑐𝑐𝑐𝑐 = 𝜋𝜋 2 𝐸𝐸 ⁄λ2
(3.5)
As shown in Tables 3.1 and 3.2 above, various column sizes were tested in the buckling
experiment. Buckling strengths of bamboo columns as a function of slenderness ratio are plotted
in Figure 3.5. The least squares method was used to fit a quadratic equation to the data at 95%
confidence level as required by [6], so that buckling strength for any bamboo column with
known slenderness ratio, i.e. length, diameter and thickness, can be determined using the lower
bound in Figure 3.5.
Buckling Strength (MPa)
60
2rd order polynomial
Pred bounds (95% confidence interval)
BucklingStrength vs. SlendernessRatio
50
40
30
20
10
40
50
60
70
80
Slenderness Ratio
90
100
110
Figure 3.5 Relationship between buckling strength and slenderness
ratio of bamboo columns
The quadratic equation of the lower bound at 95% confidence level is:
𝐹𝐹𝑐𝑐𝑐𝑐 = −0.0061λ2 + 0.47λ + 24.77
42
(3.6)
Figure 3.6 Load-deflection behaviour of bamboo
columns in buckling tests
The results of buckling tests obtained above are comparable to the results obtained by Yu et al
[21] on similar bamboo columns of the Kao Jue and Mao species. It is also observed from Figure
3.6 that the load-deflection behaviour of bamboo columns is approximately linear, which is a
useful material property for designing lattice towers. It is also evident that there exists a
significant variation in the buckling strengths. This variation was taken into account by
considering the 95% confidence level values, as required by IEC.
3.4.2 Compression Strength
The test results obtained from the compression experiment are presented in Table 3.3. The loaddeflection behaviour is depicted in Figure 3.7. According to ISO22157-1:2004(b), the
compression strength (σ c ) of the culm is determined by using equation (3.7):
𝜎𝜎𝑐𝑐 = 𝐴𝐴
𝐹𝐹𝑐𝑐
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
(3.7)
43
where, F c is the maximum compressive load and A culm is the cross-sectional area of the culm,
given by equation (3.1).
Table 3.3 Test Results: Compression Experiment
Specimen
No
MC (%)
Outside
Diameter (mm)
Thickness
(mm)
Length
(mm)
Load
(kN)
CT01
12
56.62
8.03
56.30
73.41
Compression
Strength
(MPa)
59.86
CT02
13
56.66
8.19
57.00
86.57
69.36
CT03
11
61.11
5.77
67.17
74.83
74.53
CT04
14
61.72
5.10
61.30
70.81
77.98
CT05
11
62.02
5.19
60.50
70.06
75.59
CT06
9
61.85
5.46
59.80
72.00
74.43
CT07
10
56.57
10.36
62.61
82.35
54.74
CT08
11
63.32
6.05
65.66
56.47
51.87
CT09
12
56.77
8.68
59.40
71.60
54.58
CT10
10
56.50
9.02
55.28
74.05
55.00
From Table 3.3, the compressive strength of bamboo was found in the range, 51-77 MPa, which
gives compressive strength of 44 MPa at 95% confidence level. The results compare well with
the reported compressive strengths by various authors for different species of bamboo, e.g. Yu et
al. [21], Mitch [32], and Chung and Chan [35].
The load-deformation behaviour of bamboo in compression is shown in Figure 3.7. It is
important to note that a non-linear section of the curve is observed at the beginning. This should
not be treated as the material behaviour because it is due to the initial deformation of rubber pads
at both ends of the specimens before the test specimens take the compressive loads. After this
44
region, the overall load-deformation is approximately linear. Previous studies performed by
Chung and Chan [35] have also shown the linear load-deformation behaviour (Figure 3.8).
Linear region
Figure 3.7 Load-deformation of bamboo in compression
Figure 3.8 Load-deformation behaviour of bamboo
in compression (adapted from [35])
45
3.4.3 Modulus of Elasticity and Poisson Ratio
Stress-strain curves are an important measure of material’s mechanical properties. The
longitudinal strains were measured using strain gauges installed in four different test specimens.
It is important to note that a non-linear section of the curve is observed at the beginning;
otherwise the results can be misleading. This should not be treated as the material behaviour, but
due to the effect of test-setup. This is due to the initial deformation of rubber pads used at both
ends of the specimens until the test specimens started taking the actual loads. The elastic
modulus E c was determined by the slope of stress-strain graph in the linear region (Figure 3.9).
𝐸𝐸𝑐𝑐 =
𝜎𝜎𝑐𝑐
(3.8)
𝜀𝜀
From the stress-strain graph, elastic modulus was computed in the linear-elastic region as 16.2
GPa at 95% confidence level as required by [6] in tower design. This value is similar to the
reported values for other species of bamboo [28].
The Poisson ratio,ν, is the ratio of lateral to longitudinal strain measured in the linear region
during compression loading and is expressed as:
𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
ν = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
(3.9)
The lateral and longitudinal strains were measured by strain gauges installed in the test
specimens. Poisson’s ratio was computed in the linear region. The average Poisson’s ratio for
each specimen was 0.29, 0.33, 0.35, and 0.36. The average Poisson ratio is determined as 0.33.
For most engineering materials, such as steel and aluminum, the values lie between 0.25-0.35. A
Poisson’s ratio of 0.35 is reported in [28].
46
Linear region
Figure 3.9 Stress-strain of bamboo in compression
It is evident from the results of buckling (Table 3.2 and Figure 3.5) and compression tests (Table
3.4) that the buckling strengths of bamboo column for slenderness ratio below 60 are comparable
to the compressive strengths. For the longer columns the failure occurs due to bending across the
transverse axis rather than splitting along the longitudinal axis (due to shear stress), whereas for
the short columns, the failure occurs due to splitting along the longitudinal axis (due to shear
stress).
In order to determine the dominant failure criteria (i.e. shear or yield) in buckling of columns,
maximum shear stress criterion was used. The principal stresses in a biaxial loading are
expressed as:
𝜎𝜎1,2 =
𝜎𝜎𝑥𝑥 +𝜎𝜎𝑦𝑦
2
± ��
𝜎𝜎𝑥𝑥 +𝜎𝜎𝑦𝑦 2
2
47
� + 𝜏𝜏𝑥𝑥𝑥𝑥 2
(3.10)
Maximum shear stress is given by:
𝜎𝜎𝑥𝑥 +𝜎𝜎𝑦𝑦 2
𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = ��
� + 𝜏𝜏𝑥𝑥𝑥𝑥 2
2
Alternatively, maximum shear stress is expressed as: 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚
(3.11)
𝜎𝜎1 −𝜎𝜎2
⎡� 2 �⎤
⎢ 𝜎𝜎 −𝜎𝜎 ⎥
= max ⎢� 2 2 3 �⎥
⎢ 𝜎𝜎1 −𝜎𝜎3 ⎥
⎣� 2 �⎦
(3.12)
In a uniaxial compressive loading, the buckling failure is dominated by yield rather than shear
stress ( or splitting along the longitudinal axis) if the maximum shear stress is less than the
maximum shear stress of the test specimen in tension at yield.
𝜎𝜎𝑌𝑌
𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 < 𝜏𝜏𝑌𝑌 =
(3.13)
2
In the present experimental study, the tensile strength of the species, bambusa arundinacea, was
not measured. The tensile strengths of different bamboo species have been reported in the range
162-275 MPa (Table 2.1). For the purpose of this analysis, the minimum average tensile stress
(𝜎𝜎𝑌𝑌 ) of 162 MPa is used. Similarly, the minimum average shear stress (𝜏𝜏𝑥𝑥𝑥𝑥 ) is taken as 10.6 MPa
(Table 2.1). In the compression test, 𝜎𝜎𝑦𝑦 = 0.
From Table 3.4, the maximum compressive stress was experimentally determined as: 𝜎𝜎𝑥𝑥 = 77.98
MPa. Using equations (3.10-3.13):
𝜎𝜎𝑥𝑥 +𝜎𝜎𝑦𝑦 2
𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = ��
2
77.98+0 2
� + 𝜏𝜏𝑥𝑥𝑥𝑥 2 = ��
2
� + 10.62 = 40.40 MPa < 81 MPa
From this analysis, it is evident that the shear failure is not the dominant failure mode in bamboo
columns. In other words, the design of bamboo lattice towers should be based on buckling
strength, which is the dominant failure mode of bamboo columns.
48
3.5 Joint Testing
As far as the author is aware, no previous experimental studies on steel-bamboo adhesive joints
have been carried out. This experimental study on joints focused on measuring the tensile or
pull-out resistance of a particular size of steel-bamboo adhesive joint. The analysis and modeling
of the joints were not performed due to the requirement of extensive experimental works.
As an initial step to calculate the size of the test specimens, a 12 m high tripod model, described
in Chapter 7, was optimized against buckling strength of leg sections for various base distances.
The minimum possible diameter of tower legs, that is safe against buckling, was computed. The
dimensions used in the fabrication of steel-bamboo adhesive joints are presented in Table 3.4.
Five test specimens of bamboo-steel adhesive joints were fabricated using structural epoxy resin.
Each joint was fabricated by encasing the ends of 52 cm long and 65 mm diameter bamboo
sections into the cylindrical steel cap as shown in Figure 3.10. A 68.6 mm diameter steel cap was
chosen based on the available size of steel pipe in the market, whereas the length of the steel
caps was chosen as 46 mm by assuming the practicable size of the joint in the lattice tower.
As reported in [18], surface roughening and grooves could improve the strengths of the adhesive
joints by more than 50%. The surface of the bamboo ends was first roughened using sand paper
and 3 groves (4 mm wide by 2 mm depth) were made at each end of bamboo. Each groove was
made at 10 mm apart. Epoxy resin was mixed with the hardener and then applied around the
external surface of the bamboo ends and internal surface of the steel cap. Bamboo ends were
then encased inside the cylindrical cap by applying slight pressure until epoxy was properly
filled in the gap. The joint sections were clamped to ensure straightness and uniform gluing until
the joint was properly cured. The joints were cured for one and half day before the test was
carried out. The normal curing time of the used epoxy is about 8 hours.
49
Figure 3.10 Specifications of the bamboo (left) and cylindrical steel caps (right)
used in the fabrication of epoxy grouting joint specimens
The pull-out experiment was carried out in a UTM at Tribhuvan University, Nepal .The solid
rods welded at the ends of steel caps were clamped in the UTM (Figure 3.11) and axial tensile
load was gradually increased until failure was noticed in the joint. The load-deflection graph
could not be measured due to lack of measuring instrument in the UTM. Only the ultimate
failure loads were measured (Table 3.4). The failure occurred in the joints mostly by slippage of
the culm (Figure 3.12), and no crack and splitting were observed in the bamboo surface. It is
evident from Figure 3.12 that the bonding between epoxy-steel is stronger than that of between
the epoxy-bamboo.
50
Steel-bamboo
adhesive joint
Figure 3.11 Specimen setup for the pull-out
test on bamboo joints in a Universal Testing
Machine (Photo: by the Author)
Figure 3.12 Failure of the joint by
slippage of bamboo culm from the
steel cap (Photo: by the Author)
51
Table 3.4 Results of Pull-out Tests on Steel-bamboo Adhesive Joints
Test
specimens
Length of
bamboo(cm)
Dia. of
bamboo,
D 1 (mm)
Dia. of
bamboo,
D 2 (mm)
Joint length
(mm)
Pull-out
resistance
(kN)
64.4
Steel cap
diameter
and length
(mm)
68.6×46
TS1
52
64.7
46
21.46
TS2
52
64.1
64.3
68.6×46
46
22.45
TS3
52
62.8
62.6
68.6×46
46
23.23
TS4
52
63.4
63.3
68.6×46
46
22.91
TS5
52
63.4
63.3
68.6×46
46
22.58
Pull-out resistance at 95% confidence level
20.32
It is noted that the results of this study are in close agreement with the pull-out strength (18 kN)
of the PVC-bamboo adhesive joint reported in [18]. Further, it was experimentally determined
that the load-deformation of PVC-bamboo adhesive joint is linear-elastic (Figure 2.22). This
indicates that the load-deformation behavior of the bonding between bamboo-adhesive is linearelastic.
From this experimental work, it is concluded that the tubular single lap joints have good pull-out
strengths, which is about one half of the buckling strength reported above. Further, this is the
simplest type of adhesive joint that is easy to fabricate. The joints can be produced easily at low
cost by encasing the bamboo ends inside the steel tubular caps and using structural adhesives
(e.g. epoxy resins) between the outer surface of bamboo and inner surface of steel. Steel tubular
caps of desired diameter and thickness are easily available in the market.
52
Chapter 4
LOADS AND DESIGN REQUIREMENTS FOR WIND TURBINE TOWERS
4.1 Chapter Overview
This chapter outlines the design standards and requirements for small wind turbine towers and
different loads acting on wind turbine towers.
4.2 Design Standards and Requirements
The design of wind turbine towers is governed by the loads acting on the tower and the strength
of materials. The loads should consider the extreme load-cases that the wind turbine tower may
encounter during its life-span. Standards such as IEC 61400-2 and Germanischer Lloyd (GL)
certification guidelines [56] for wind turbines require that ultimate and fatigue strengths of
structural elements must ensure structural integrity of wind turbines and components during
extreme wind loads. IEC 61400-2 is the most commonly used safety standard for small wind
turbines.
IEC61400-2 has categorized wind turbines into four classes in terms of maximum wind speed
and turbulence parameters. Extreme wind speeds, which are 3-second gust wind speed with 50
years’ recurrence period, are given in Table 4.1 for different classes of wind turbines.
Table 4.1 Extreme Wind Speeds for Different Classes of Wind Turbines [6]
Class of wind turbines
50-years, 3-Second gust wind speed (m/s)
I
70
II
59.5
III
52.5
IV
42
53
The simple load model (SLM) of IEC 61400-2 ensures safety by setting different load cases for
designing small wind turbines. The different load cases of the SLM are:
1. Load case A: Normal operation – fatigue load due to rotating blades
2. Load case B: Yawing – gyroscopic forces and moments
3. Load case C: Yaw error – flap wise bending moment is caused due to yaw error
4. Load case D: Maximum thrust – maximum thrust on rotor blades
5. Load case E: Maximum rotational speed- centrifugal forces are created due to rotation of
blades
6. Load case F: Short at load connection – high moment is created due to the short circuit
torque of the generator
7. Load case G: Shutdown (braking) – maximum blade load during shutdown
8. Load case H: Parked wind loading – the loads on the parked blades are calculated using
the extreme wind speed, which is the 3-second 50 years recurrence wind speed.
9. Load case I: Parked wind loading, maximum exposure – in the case of failure of yaw
mechanism, the turbine may be exposed to wind loads from all directions, which must be
considered
According to the SLM, load case H gives the maximum bending stress on the tower. It is an
extreme load case due to the maximum thrust on turbine blades and the drag on the tower at the
extreme wind speed, taken here as 50 m/s. This is close to the 3-second 50 years recurrence wind
speed specified for class III wind turbines, 52.5 m/s, and is consistent with the limited
measurements of extreme wind speeds in Nepal. IEC 61400-2 also requires that the drag on
tower is calculated using the same extreme wind speed.
It is important to note that Load case A gives the fatigue load on tower; however, the stress on
tower due to fatigue load is relatively small compared to the load case H. Therefore, the design
of tower should be based on the extreme load case H. Other load cases are only useful in the
design of wind turbines.
54
4.3 Loads on Wind Turbine Tower
Wind turbine towers are subjected to a combination of three main types of loads throughout their
life-span, which define the final size of the tower [10]. These are:
•
Gravity or dead loads due to turbine, nacelle, and tower masses
•
Aerodynamic thrust on turbine blades
•
Aerodynamic drag forces on tower structure
Almost all wind turbine towers are designed to withstand these loads. In addition, there are also
loads generated by the steady and unsteady blade torque transmitted to the tower. Fatigue loads
are caused by the fluctuating thrust on turbine blades. Gyroscopic loads occur as a result of
yawing the rotating blades. Only fast yawing rates lead to significant gyroscopic moments in the
vertical plane at the top of the tower. However, as yawing rate is relatively slow in small
turbines, the effect is relatively small and is not usually considered in the design of tower [7].
For the SLM of IEC61400-2, the main load is the ultimate load due to the extreme wind speeds,
for which the deflection of the tower should be small. The descriptions of the main loads are
summarized below.
4.3.1 Gravity Loads
Gravity or dead loads are comprised of the lumped mass of the rotor and nacelle and the
distributed mass of the tower. These loads are static and do not change throughout the service
period of the tower. Gravity loads contribute to both the axial and bending stresses in the tower.
The centre of mass of a wind turbine is usually off-set from the tower axis. The clearance
between the rotational plane of the blades and the tower is kept as small as possible in order to
minimize the length of the nacelle and to avoid the bending moment induced due to turbine
overhang. As the design should ensure linear elastic deflection, the secondary effect, which is
known as load-deflection (P-δ) magnification, is not considered in this study.
55
4.3.2 Aerodynamic Thrust on Rotor Blades
The aerodynamic thrust is generated by the blades. It acts in the horizontal direction at the tower
top. It is determined by multiplying the turbine thrust coefficient and the swept area of the rotor:
𝐹𝐹 =
𝐶𝐶𝑇𝑇 𝜌𝜌𝜌𝜌𝑈𝑈 2
(4.1)
2
where, F is the thrust on turbine, C T is the thrust coefficient of blades, A is the projected area of
the blades in the plane of rotation, and U is the extreme wind speed (m/s).
To calculate the maximum thrust on rotor blades using equation (4.1), load case H of IEC 614002 is used [6, 7]. In this load case, the rotor is assumed stationary at extreme winds, in which case
the torque coefficient is equal to the drag coefficient of the blades [7]. For the stationary rotor
blades, the drag coefficient is 1.5 [6].
4.3.3 Drag on the Tower
Drag forces are caused by the flow of wind around the tower. The tower components can be
considered as bluff bodies. The drag, which is force per unit length of tower component, is
expressed as:
𝑞𝑞 =
𝐶𝐶𝑑𝑑 𝜌𝜌𝜌𝜌𝑈𝑈 2
2
(4.2)
IEC 61400-2 has recommended the drag coefficient of 1.3 for circular cylindrical members. This
value is used here. Despite the fact that wind speed varies along the tower height, it is easier and
conservative to assume the extreme wind speed to be constant throughout the tower height to
calculate the drag forces. The drag force in a tower member is computed by using equation (4.2).
The drag on each tower members is applied as uniformly distributed loads.
56
4.4 Load Safety Factors
In designing the wind turbine tower, load safety factors are applied in order to provide the safety
margins of the tower. IEC load safety factors are the most commonly used factors in the design
of wind turbine towers, which are given in Table 4.2. Here, we use these factors in the tower
design.
Table 4.2 Load safety factors for the design loads [6]
Sources of loading
Load safety factors
Wind loads
1.35
Gravity loads
1.10
4.5 Tower Design Methods
Once the loads acting on the tower are determined, resultant stresses and deflections within the
structure are determined. These values are then analyzed to determine the optimum dimensions
of the structure and to minimize the material cost of the structure. The commonly used tower
design methods are described below.
4.5.1 Allowable Strength Design
The allowable strength design (ASD) is the most commonly used criterion in the design of
structures and the only one to be used in the context of the SLM. In this method, the stresses
developed within the structural elements are compared with the allowable stress of the material
[7] using factor of safety (FS) or its inverse, the capacity factor (CF).The design stresses and
allowable stresses are expressed as:
𝜎𝜎𝑦𝑦
𝜎𝜎𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ≤ 𝐹𝐹𝐹𝐹 = 𝜎𝜎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
57
(4.4)
where, 𝜎𝜎𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 is the design tensile or compressive stress for the structural element, 𝜎𝜎𝑦𝑦 is the
yield strength of the material, 𝐹𝐹𝐹𝐹 is the factor of safety (=1/ CF), and 𝜎𝜎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 is the allowable
stress of the structural element.
An ultimate strength is the maximum strength beyond which the structure subjected to extreme
loads is assumed to fail to meet the design requriements or lose stability.
4.5.2 Allowable Buckling Strength
Since wind turbine towers are slender structures subjected to compressive loading, buckling is
often the governing design criterion. IEC 61400-2 does not give detailed specification for
buckling, but it requires the tower to satisfy relevant local codes as well and hence its buckling
analysis must be based on those codes. This study uses the standards, ASCE [57], the Eurocode 3
[58], and the AISC [59] for the steel tower. Small tower designs, e.g. [7] and [8], have used the
ASCE and Eurocode 3 to assess the buckling strengths.
4.5.3 Allowable Tower Deflection and Natural Frequency
Most of the tower designs are governed by allowable stress and buckling or stability criteiria of
the tower. In addition, tower top deflection or stiffness and natural frequency of tower vibration
are also considered.Although there are no standards that set the limit for tower-top deflection, it
is mentioned in [8] that 5% of the tower heigth is acceptable for small wind turbine towers to
ensure linear-elastic response for the worst case loads. It is also important to ensure that natural
frequency of the tower should not be excited by the blade passing frequency.
58
Chapter 5
DESIGN AND OPTIMIZATION OF LATTICE TOWERS FOR SMALL WIND TURBINES
5.1 Chapter Overview
This chapter introduces the proposed lattice tower, the analytical formulation of the tower, and
then describes the design optimization procedure using analytical and FE analysis techniques.
5.2 Overview of Design Optimization and Objectives
According to Dieter [60], to design a product is “to pull together something new or arrange
existing things in a new way to satisfy a recognized need of society that has never been done
before”. Designing a new thing is primarily driven by society’s needs to solve problems or
modify the current design due to design inadequacy. In [60], it has been emphasized that “design
is a multifaceted process and encompasses various considerations: design requirements, lifecycle issues, and regulatory and social issues”. In general, design is governed by the choice of
materials or their properties and design needs. In the context of this thesis, we investigate the
application of a new material bamboo and optimization procedure of wind turbine tower with an
objective of minimizing the cost of material and manufacture while satisfying the safety
requirements of IEC41400-2.
Multiple design objectives are combined in the design of wind turbine towers. These
fundamentally arise from different design requirements, such as: maximizing structural
performance and minimizing weight, costs, and maintenance. In the process of tower design, it is
critical to synthesize these objectives simultaneously for the design to be ultimately successful in
practical situations. However, simultaneous optimization is a complicated design process and
therefore trade-offs between design objectives are often required.
59
As outlined in Chapter 1, the underlying motivation of this study is to reduce the costs of
material, manufacture, transportation, and installation and maintenance of tower. In order to
achieve those objectives, triangular lattice tower is proposed, which can be manufactured using
steel tubular sections or bamboo. Clearly, it is entirely feasible to make lattice towers from steel
pipe, but the feasibility of bamboo lattice towers has not been established. A major aim of this
chapter is to establish this feasibility.
To design a low-cost and easily transported lattice tower, mass minimization as well as load and
safety are important for steel towers, whereas load and safety requirements are important for a
bamboo tower. Two tower design examples each having 12 m height considering the load cases
of same 500 W wind turbine are examined. These sizes of wind turbines are typical for remote
power systems in off-grid applications or in rural areas of developing countries, such as in Nepal.
The basic load information for this turbine is taken from [7].
5.3 The Triangular Lattice Tower
The structural model of the proposed triangular lattice tower is shown in Figure 5.1. The tower is
basically a tripod. The tower consists of three circular hollow columns (legs or pods) as the main
load-carrying structural elements, positioned at the corners of an equilateral triangle at the base
and fixed together into the turbine mounting flange at the tower top. The tower-top width is kept
as minimum as possible. The legs are braced to each other with horizontal and cross bracings at
intermediate heights in order to prevent buckling and to enhance stiffness of the tower.
Horizontal- and cross-bracings are joined to the legs either by welding or use of mechanical
fasteners that constitute the lattice structure.
The key structural feature of the lattice structure is that the legs carry axial and bending loads
and the horizontal and cross-bracings increase the lateral stability of the legs by resisting shear
forces and bending moments between the legs. Once the tower height is fixed, the load carrying
capability of the tower is determined by the base distance between tower legs (b) and the cross60
sectional dimensions, D and t, of legs and bracings. It is important to note that for any given
tower height (h) and tower-top width, there is no unique base distance between the legs or sizes
of legs and bracings. As the legs are columns, the governing design criteria are the buckling
strengths of legs and the tensile strengths of joints connecting the tower members.
Tower top width
A lattice section
Cross-bracing
Horizontal bracing
Leg section
Y
Base distance
b
Z
X
Figure 5.1 Structural model of the triangular
lattice tower
Lattice towers can be designed with different bracing configurations. The commonly used
configurations are the x- bracings with horizontal bracings as shown in Figure 5.1. Alternatively,
only x-bracings are used without horizontal bracings. The cross-section of the bracings may be
angular or circular hollow sections, which can be joined by welding or using mechanical
fasteners. Wood [7] mentioned that corrosion could occur at the joints of angular sections and
recommended galvanized tubular sections. The present study for steel tower considers the
61
galvanized circular hollow steel sections for legs and bracings, whereas round and solid steel
rods for cross-bracings.
The triangular lattice tower presents significant advantages over monopole towers. First, it is
structurally efficient in transferring the loads to the foundation due to its increasing cross-section
towards the base (lower stress and higher stiffness). So the tower can be designed lighter with
minimum use of material even if the monopole is manufactured in sections with varying
thickness. Second, it is easy to manufacture, transport, and install and repair lattice towers with
short and smaller tubular sections as tower components. Furthermore, the spacing between the
legs means lattice towers require small foundations. This would further reduce the cost of
building the towers. The main design features of the triangular lattice tower are summarized
below:
The design of the triangular lattice tower is governed by several variables and design constraints
which are summarized below:
•
Tower height (h)
•
Number of tower sections or lattices
•
Tower top width
•
Base distance between the tower legs (b)
•
Diameters (D) and thicknesses (t) of the legs and bracings
•
Loads and boundary constraints imposed on the tower
•
Strength of the materials used
Among these variables, h, the number of lattice sections, and the tower top width are assumed to
be specified at the initial stage of tower design. The remaining design parameters are b, D, and t,
which depend on the loads and boundary constraints and the strength of materials. The loads are
specified by the standards such as IEC61400-2. To assess the strength of tower, this study uses
the ASCE, Eurocode 3, and AISC codes for steel tower, whereas for the bamboo tower, material
properties determined from the experimental tests are used.
62
5.4 Design Procedure
Self-supporting lattice towers are indeterminate structures. Consequently, it is not possible to
obtain exact analytical solutions for stresses and deflections in the structure to apply directly to
the design of tower. However, the structural model of the tower can be simplified to obtain
approximate analytical solutions, which can then be extended to a more detailed analysis by FE
modeling.
In the first step, structural analysis is carried out to obtain approximate analytical solutions by
assuming a simplified structural model of the lattice tower. Equations are formulated to
determine the stresses on tower legs and tower-top deflection for various base distances. Using
these analytical solutions, a simple optimization is carried out to get the first approximations on
tower dimensions. In the second step, FEA using ANSYS APDL is used to extend the
approximate analytical solutions and to provide detailed tower analysis and design. The
methodology adopted in the tower design optimization is briefly summarized in Figure 5.2.
63
Calculation of ultimate
loads on tower
• Gravity loads
• Wind loads
Structural analysis of the
lattice tower
Approximation of the
tower dimensions using
analytical solutions
Design validation using
FE modeling
Optimum Tower Design
Figure 5.2 Design optimization procedures for the lattice tower
5.5 Structural Analysis of the Lattice Tower
Assuming that the tower top width is relatively small compared to the base distance between the
tower legs, the triangular lattice tower was modeled as a tripod structure consisting of three legs
as its main structural elements as shown in Figure 5.3. This simplifies the analysis for tower
design and optimization.
64
5.5.1 Analysis of the Tripod Model
Chapter 4 described the main loads acting on the tower along with the design requirements.
Using these loads, structural analysis is carried out to compute the internal forces, stresses, and
deflections in the tower.
A free-body diagram (FBD) of the lattice tower as a tripod consisting of three legs is shown in
Figure 5.3.The bracing elements are not included in the model. The tower loads are due to the
lumped mass of turbine and nacelle and horizontal thrust at the tower top, gravity load due to
tower mass, and uniform drag forces along the lengths of tower legs.
For any h, the design variables are D, t, and b. So the total mass of the tower depends on the base
distance and dimensions of legs. The FBD shown in Figure 5.3 is now used to find the analytical
solutions for the axial and bending stresses on tower legs.
Assumptions made in the analysis:
•
The drag coefficient of 1.3 was obtained from IEC61400-2 by assuming the tower
members to be circular cylinders. IEC61400-2 has specified the extreme wind speed of
52.5 m/s for class III wind turbines. As discussed in Chapter 4, we used the extreme wind
speed of 50 m/s to calculate the turbine thrust and drag on tower legs, which was
assumed to be constant throughout the tower height.
•
The bending moment due to drag on the tower was calculated by assuming the tower as a
cantilever beam.
•
The response of the tower to the imposed loads is linear elastic, i.e. secondary effects,
also known as moment amplification or P-δ effects are negligible for the small deflection
of the tower. In other words, it is assumed that “A static-linear-three dimensional
structural analysis is sufficient for almost all lattice tower structures” is valid as
mentioned in SCI (2003) [7].
65
•
Aerodynamic damping [57], which arises due to the relative motion between the tower
and the wind, is negligible as the tower deflection is linear elastic and static ultimate
analysis is sufficient for lattice tower structures as discussed above.
•
The maximum compressive stress occurs in the single rear leg, CD, when the wind is
normal to the line drawn between the two front legs, AD and BD, and the maximum
tensile stress occurs when the wind direction changes by 180° (Figure 5.3). These are the
two extreme load cases that are examined here. It is further assumed that the maximum
compressive and tensile stresses occur at the base of the tower and are therefore
determined by the reactions of the foundations.
y
h
Figure 5.3 Free Body Diagram (FBD) of the tripod tower.
Bracings are not included. The legs are denoted by AD, BD,
and CD. The turbine is mounted at point D. The arrows
indicate the direction of forces and moments in the tower
66
x
5.5 Base cross-section of the tower
Figure 5.4 Lattice tower as a
as a composite beam of legs and
cantilever beam
bracings
Choosing x - , y - , and z - axes as a set of mutually perpendicular directions in Figure 5.3, the
positions of the tower legs at the base and top are expressed as:
A (x, y, z) = 0, 0, 0
B (x, y, z) = b, 0, 0
C (x, y, z) = 𝑏𝑏⁄2 , ℎ, √3 𝑏𝑏⁄2
D (x, y, z) = 𝑏𝑏⁄2 , ℎ, √3 𝑏𝑏⁄2
The external and internal forces acting on the tower legs can be expressed as vectors:
𝑭𝑭𝑨𝑨𝑨𝑨 = 𝐹𝐹𝐴𝐴𝐴𝐴
𝑭𝑭𝑩𝑩𝑩𝑩 = 𝐹𝐹𝐵𝐵𝐵𝐵
𝑏𝑏 𝒊𝒊
𝑏𝑏 𝒌𝒌
+
ℎ
𝒋𝒋
+
2
2√3
�ℎ2 + 𝑏𝑏 2 ⁄3
𝑏𝑏
𝑏𝑏 𝒌𝒌
− 2 𝒊𝒊 + ℎ𝒋𝒋 +
2√3
2
2
�ℎ + 𝑏𝑏 ⁄3
67
𝑭𝑭𝑫𝑫𝑫𝑫 = 𝐹𝐹𝐷𝐷𝐷𝐷
𝑭𝑭 = 𝐹𝐹 𝒌𝒌
𝑏𝑏 𝒌𝒌
√3
�ℎ2 + 𝑏𝑏 2 ⁄3
−ℎ 𝒋𝒋 +
𝑾𝑾 = −𝑊𝑊𝒋𝒋
𝑾𝑾𝒕𝒕 = −𝑊𝑊𝑡𝑡 𝒋𝒋
𝒒𝒒 = 𝑞𝑞 𝒌𝒌
and the unit vectors (i, j, k) are in the x, y, and z directions respectively, as defined in Figure 5.3.
Assuming that the tower legs are vertical cantilever beams, the bending moment M(y) at a
section ‘y’ from the tower base due to turbine thrust (F) and drag forces (q) is computed by:
𝑀𝑀(𝑦𝑦) = 𝐹𝐹(ℎ − 𝑦𝑦) + 3𝑞𝑞(ℎ − 𝑦𝑦)2⁄2
(5.1)
The governing equations for the static equilibrium of the tower are:
Σ 𝑭𝑭𝒙𝒙 = 0 , Σ 𝑴𝑴𝒙𝒙 = 0
Σ 𝑭𝑭𝒚𝒚 = 0 , Σ 𝑴𝑴𝒚𝒚 = 0
(5.2)
Σ 𝑭𝑭𝒛𝒛 = 0 , Σ 𝑴𝑴𝒛𝒛 = 0
To find the bending stresses in tower legs, force and moment equations (5.2) are applied at the
equilibrium point or an element of the FBD.
From the FBD (Figure 5.3), summing the moments about the positive x–axis (assumed to lie
along AB) gives the value of the reaction force at point C. Writing the moment equilibrium
equation Σ M AB =0 and omitting the zero moment terms gives:
√3 𝑏𝑏
𝒌𝒌
2
𝑏𝑏
𝑏𝑏
× 𝑅𝑅𝐶𝐶𝐶𝐶 𝒋𝒋 + ℎ𝒋𝒋 × 𝐹𝐹𝒌𝒌 − 2√3 𝒌𝒌 × 𝑊𝑊𝒋𝒋 − 2√3 𝒌𝒌 × 𝑊𝑊𝑡𝑡 𝒋𝒋 +
68
3𝑞𝑞ℎ2
2
𝒊𝒊 = 0
(5.3)
Equation (5.3) contains a single unknown, the reaction force R CY , which can be computed for
any given base distance and tower loads.
The compressive force in the tower leg DC (F DC ) due to bending loads can be computed by
assuming static equilibrium of forces at point C.
(5.4)
𝐹𝐹𝐷𝐷𝐷𝐷 = 𝑅𝑅𝐶𝐶𝐶𝐶 �ℎ2 + 𝑏𝑏 2 ⁄3⁄ℎ = 𝑅𝑅𝐶𝐶𝐶𝐶 �1 + 𝑏𝑏 2 ⁄3ℎ2
Similarly, the tensile forces 𝐹𝐹𝐵𝐵𝐵𝐵 and 𝐹𝐹𝐴𝐴𝐴𝐴 in legs AD and BD can be computed by summing the
moments about the point C. These forces are required to determine the loads on the foundation.
Writing the moment equilibrium equation (Σ M C =0) at point C and omitting the zero moment
terms gives:
𝑏𝑏
𝑏𝑏
𝒓𝒓𝐴𝐴𝐴𝐴 × 𝑭𝑭𝐴𝐴𝐴𝐴 + 𝒓𝒓𝐵𝐵𝐵𝐵 × 𝑭𝑭𝐵𝐵𝐵𝐵 + ℎ𝒋𝒋 × 𝐹𝐹𝒌𝒌 − 2√3 𝒌𝒌 × 𝑊𝑊𝒋𝒋 − 2√3 𝒌𝒌 × 𝑊𝑊𝑡𝑡 𝒋𝒋 +
𝑏𝑏
where, 𝒓𝒓𝐴𝐴𝐴𝐴 = 2 𝒊𝒊 +
𝑏𝑏√3
2
𝑏𝑏
𝒌𝒌 and 𝒓𝒓𝐵𝐵𝐵𝐵 = − 2 𝒊𝒊 +
𝑏𝑏√3
2
𝒌𝒌
3𝑞𝑞ℎ2
2
𝒊𝒊 = 0
(5.5)
It is important to note that the front legs AD and BD share equal tensile or compressive forces
due to symmetry of the load cases shown in Figure 5.3. Forces in the members can be computed
by solving the equation (5.3) and (5.5) for any b and tower loads. It is important to note that the
wind and self-weight loads would change when the size of tower members and b are changed.
The maximum compressive stress due to bending loads (thrust, drag, and gravity) can be
determined from equations (5.3) and (5.4) and cross-sectional area of each leg, A CS :
σ𝑏𝑏 = 𝐹𝐹𝐷𝐷𝐷𝐷 /𝐴𝐴𝐶𝐶𝐶𝐶
(5.6)
Alternative to the above analysis, the maximum bending stress in the most compressed tower leg
can be determined by assuming the tower as a composite cantilever beam (Figure 5.4 and 5.5),
69
which gives the same result for the bending stress as obtained above. It is noted that the
maximum bending stress occurs in the back-leg at the base section of the tower.
σ𝑏𝑏 = 𝑀𝑀𝑀𝑀⁄𝐼𝐼
(5.7)
where, M is given by equation (5.1), z is the distance of the back leg from the centroidal axis,
𝑧𝑧 = 𝑏𝑏⁄√3 , and I is the moment of inertia of the section. Referring to the figure (5.5), the
moment of inertia of the composite beam is computed by assuming the tower legs as an
equivalent beam [61]. The moment of inertia of the beam about the centroidal axis of tower is
obtained by:
𝐼𝐼 = 2𝐴𝐴𝐶𝐶𝐶𝐶 (𝑏𝑏⁄2√3 )2 + 𝐴𝐴𝐶𝐶𝐶𝐶 (𝑏𝑏⁄√3 )2 + 3𝐴𝐴𝐶𝐶𝐶𝐶 �𝑅𝑅2 2 + 𝑅𝑅1 2 �⁄4
(5.8)
𝐼𝐼 = 2𝐴𝐴𝐶𝐶𝐶𝐶 𝑏𝑏 2 /2 + 3𝐴𝐴𝐶𝐶𝐶𝐶 �𝑅𝑅2 2 + 𝑅𝑅1 2 �⁄4
(5.9)
where, A CS , R 1 and R 2 are the cross-sectional area, inner and outer radius of the tower legs
respectively. The last term on the right of equation (5.8) is the moment of the three cylinders
about their axes, and the first two come from the parallel axis theorem.
Assuming that the three legs share equally the gravity loads (turbine and tower mass), despite the
fact that most turbines have their centre of mass offset from the tower apex, the axial stress in
each leg can be computed by:
𝜎𝜎𝑎𝑎 = (𝑊𝑊 + 𝑊𝑊𝑡𝑡 )⁄3𝐴𝐴𝐶𝐶𝐶𝐶 = (𝑤𝑤 + 𝜌𝜌𝜌𝜌 ∑ 𝐴𝐴𝐶𝐶𝐶𝐶 𝑖𝑖 𝑙𝑙𝑖𝑖 )⁄3𝐴𝐴𝐶𝐶𝐶𝐶
70
(5.10)
5.5.2 Failure Criteria
A lattice tower subjected to axial compression and bending loads often fails by buckling rather
than yielding. Consequently, the design of the tower is controlled by their buckling strengths.
Therefore, the key design problem in lattice towers is to prevent the buckling of tower members,
particularly the legs, which otherwise may lead to overall collapse of the tower.
Some manufacturing defects always occur in real towers [7]. However, linear buckling analysis,
which is based on theoretical buckling strength, does not take into account these factors. Also,
buckling is not explicitly covered by IEC 61400-2, but it requires meeting the local standards and
codes, such as ASCE, Eurocode3, and AISC. These standards and codes have incorporated the
practical aspects of buckling, e.g. manufacturing imperfections. For the 18 m high monopole
tower design described in [7] considering on ASCE and Eurocode 3, the manufacturing defects
reduced the buckling strength nearly by one half of the ideal structure, which is very important to
take into account while designing towers.
To assess the buckling strength of the steel lattice tower, described in Chapter 6, ASCE (1990)
guidelines, Eurocode 3, and AISC 360-05 are used. Assuming the tower legs as circular hollow
pipes, the limiting stresses are determined by combined axial and bending stresses. The
combined axial and bending stresses in the tower legs must satisfy the interaction equation
(5.11):
σ𝑎𝑎 ⁄𝐹𝐹𝑎𝑎 + σ𝑏𝑏 ⁄𝐹𝐹𝑏𝑏 ≤ 1
(5.11)
where, σ a is the axial compressive stress due to turbine and tower weight, F a is the allowable
axial stress or buckling stress, σ b is the bending (compressive or tensile) stress on tower legs due
to turbine thrust and drag forces, and F b is the allowable bending stress of tower legs. It is noted
that equation (5.11) should include required safety factors or capacity factors.
71
ASCE (1990) gives the limiting axial and bending stresses (MPa) for steel circular tubes in terms
of outer diameter (D) and thickness (t) are given by equations (5.12) and (5.13):
𝐹𝐹𝑎𝑎 = �
𝐹𝐹𝑦𝑦
0.75𝐹𝐹𝑦𝑦 + 6550𝑡𝑡/𝐷𝐷
for 𝐷𝐷/𝑡𝑡 ≤ 26203/𝐹𝐹𝑦𝑦
for 26203/𝐹𝐹𝑦𝑦 < 𝐷𝐷/𝑡𝑡 ≤ 82745/𝐹𝐹𝑦𝑦
𝐹𝐹𝑦𝑦
0.7𝐹𝐹𝑦𝑦 + 12411𝑡𝑡/𝐷𝐷
for 𝐷𝐷/𝑡𝑡 ≤ 41372/𝐹𝐹𝑦𝑦
for 41372/𝐹𝐹𝑦𝑦 < 𝐷𝐷/𝑡𝑡 ≤ 82745/𝐹𝐹𝑦𝑦
𝐹𝐹𝑏𝑏 = �
(5.12)
(5.13)
where, 𝐹𝐹𝑦𝑦 is the yield strength.
Using the equations (5.12 - 5.13), an optimum size (diameter and thickness) of the most stressed
tower leg can be computed for minimum possible tower mass. It is noted that the axial stress has
no correlation with the slenderness ratio.
Annex (D) of Eurocode 3 provides the equations for determining the critical linear meridional
buckling stress for cylindrical shells of constant wall thickness. The symbols used in the code are
also used here.
The critical meridional buckling stress is given by:
(5.14)
𝜎𝜎𝑥𝑥𝑥𝑥𝑥𝑥 = 0.605𝐸𝐸𝐶𝐶𝑥𝑥𝑥𝑥 𝑡𝑡⁄𝑟𝑟
where, r is the mid radius of the cylinder.
Here, the unknown C xb is calculated as follows:
Non-dimensional length parameter is given by:
(5.15)
𝑤𝑤 = 𝑙𝑙 ⁄�(𝐷𝐷 − 𝑡𝑡)𝑡𝑡⁄2
72
where, l is length of the column
𝐶𝐶𝑥𝑥 = max(0.6,1 + 0.2 �1 −
2𝑤𝑤𝑤𝑤
𝑟𝑟
(5.16)
��𝐶𝐶𝑥𝑥𝑥𝑥 )
For clamped-clamped end conditions in lattice towers, 𝐶𝐶𝑥𝑥𝑥𝑥 = 6
After the meridional buckling stress is calculated, it should be multiplied with the “meridional
imperfection reduction factor”, α x :
𝛼𝛼𝑥𝑥 = 0.62/[1 + 1.91(𝑤𝑤𝑘𝑘 ⁄𝑡𝑡)1.44 ]
(5.17)
𝑤𝑤𝑘𝑘 = √𝑟𝑟𝑟𝑟⁄𝑄𝑄
(5.18)
where, Q is the fabrication quality factor given in Tablem5.1.
Table 5.1 Values of Q [63]
Fabrication quality class
Description
Q
A
Excellent
40
B
High
25
C
Normal
16
The critical meridional buckling stress is multiplied with the imperfection reduction factor to get
the critical buckling stress.
If the relative slenderness ratio, λ = �𝐹𝐹𝑦𝑦 ⁄𝛼𝛼𝑥𝑥 𝜎𝜎𝑥𝑥𝑥𝑥𝑥𝑥 ≤ 0.2, the characteristic buckling strength is
equal to the yield strength, F y , of the material.
73
For round hollow structural sections (HSS), AISC 360-05 equations can also be used to assess
the buckling strength. In compression loading, AISC classifies a structural section as compact or
slender based on D/t and a limiting value as follows:
The structural sections are categorized as compact if 𝐷𝐷/𝑡𝑡 ≤ 0.11𝐸𝐸/𝐹𝐹𝑦𝑦 .
The structural sections are categorized as slender if 𝐷𝐷/𝑡𝑡 > 0.11𝐸𝐸/𝐹𝐹𝑦𝑦 .
The sizes of lattice tower members fall within the compact sections. So, only the governing
design equations for compact sections are presented below:
Critical buckling strength (𝐹𝐹𝑐𝑐𝑐𝑐 ) is computed as follows.
Elastic buckling stress = 𝐹𝐹𝑒𝑒 = 𝜋𝜋 2 𝐸𝐸/(𝑘𝑘𝑘𝑘 ⁄𝑟𝑟)2
(5.19)
Inelastic buckling occurs if 𝐹𝐹𝑒𝑒 ≥ 0.44𝐹𝐹𝑦𝑦 . Then the critical buckling stress is given by:
𝐹𝐹𝑐𝑐𝑐𝑐 = (0.658 𝐹𝐹𝑦𝑦 ⁄𝐹𝐹𝑒𝑒 )𝐹𝐹𝑦𝑦
(5.20)
Elastic buckling occurs if 𝐹𝐹𝑒𝑒 < 0.44𝐹𝐹𝑦𝑦 . Then the critical buckling stress is given by:
(5.21)
𝐹𝐹𝑐𝑐𝑐𝑐 = 0.877𝐹𝐹𝑒𝑒
It is noted that any one of the above standards or codes can be used to assess the buckling
strength of tower legs. In this study, the standard that gives the lowest value of the critical
buckling stress is used in the design of the steel tower. For the bamboo tower, buckling of tower
legs is assessed by using equation (5.11) with the experimentally determined buckling strengths
of bamboo columns.
74
5.5.3 Tower Deflection
As discussed in section 5.5.1, a linear static three-dimensional structural analysis is sufficient for
tower analysis. Consequently, the tower deflection must be kept as small as possible to ensure
the adequacy of the linear static analysis using linearized material properties. A small tower
deflection ensures that structural shape of the tower is maintained and does not amplify the
effects of load [7]. In tower designs, the key design criterion is the requirement of tower
strengths. Limiting criteria for tower top deflections are not found in standards (e.g. IEC614002).
Clifton-Smith and Wood [8] optimized an octagonal tower for a 5 kW wind turbine based on
buckling stability and concluded that tower top deflection might not be the “critical factor” in
tower design. It is mentioned that tower top deflection of 5% of tower height is adequate for the
design of small towers. In the present tower design, this limiting value is used.
From the tripod analysis, the tower top deflection could be determined by assuming the tripod
tower as a composite cantilever beam of three legs (Figures 5.4 and 5.5). It is assumed that the
lateral stability of the legs is maintained, i.e. the legs do not distort significantly from the
centroidal axis of the tower. The tower is loaded as described in the previous section. It is further
assumed that Euler-Bernoulli beam theory is valid for the tripod model. According to this theory,
the plane sections of the beam remain plane during deformation and perpendicular the axis of the
beam. Also the shear deformation is negligible and beam deflections are small. The best check
on the validity of the theory is to determine that the tower top deflection is a small fraction of h.
The tower deflection (v) is computed from the moment-curvature relationship given by:
𝑑𝑑2 𝑣𝑣
𝑑𝑑𝑑𝑑 2
𝑀𝑀(𝑦𝑦)
(5.22)
= 𝐸𝐸 𝐼𝐼(𝑦𝑦)
Equation (5.22) can be solved for the deflection ‘v’ by applying the boundary conditions of zero
slope and deflection at the base (y=0) of the tower.
75
The bending moment, M(y), is given by equation (5.1).
Referring to Figure (5.5) and equations (5.8) and (5.9), the second moment of inertia of the tower
section about the centroidal axis at a distance ‘y’ from the tower base is expressed as:
𝐼𝐼(𝑦𝑦) =
𝐴𝐴𝐶𝐶𝐶𝐶 [ 3�𝑅𝑅2 2 +𝑅𝑅1 2 �/4+[(𝑏𝑏−𝐷𝐷)(ℎ−𝑦𝑦)⁄ℎ+𝐷𝐷]2 ]
(5.23)
2
Integration of the equation (5.22) gives the derivative of the deflection as:
𝑑𝑑𝑑𝑑⁄𝑑𝑑𝑑𝑑 = ∫ 𝐸𝐸𝐸𝐸
2(𝐹𝐹(ℎ−𝑦𝑦)+3𝑞𝑞(ℎ−𝑦𝑦)2 ⁄2)
𝐶𝐶𝐶𝐶 [1.5�𝑅𝑅2
2 +𝑅𝑅 2 �+[(𝑏𝑏−𝐷𝐷)(ℎ−𝑦𝑦)⁄ℎ+𝐷𝐷]2
1
]
𝑑𝑑𝑑𝑑
(5.24)
The boundary conditions are:
At y =0,
(5.25)
𝑑𝑑𝑑𝑑⁄𝑑𝑑𝑑𝑑 = 0
To integrate the equation (5.25) second time to get the deflection requires:
At y =0,
(5.26)
𝑣𝑣(𝑦𝑦) = 0
To reduce the complexity of integration of the equation (5.24), the second term in the
denominator of (5.24) is replaced by b (h-y)/h. Integrating the equation (5.24) twice using
Mathematica and applying the boundary conditions, given by equations (5.25) and (5.26), the
equation for the tower deflection, 𝑣𝑣(𝑦𝑦), is obtained as:
76
𝑣𝑣(𝑦𝑦) =
ℎ2
2𝐸𝐸𝐴𝐴𝐶𝐶𝐶𝐶 𝑏𝑏 4 �𝑅𝑅2 2 +𝑅𝑅1 2
�−2√6𝑏𝑏ℎ𝑞𝑞 �𝑅𝑅2 2 + 𝑅𝑅1 2 (2𝐹𝐹 − 3𝑞𝑞𝑞𝑞) tan−1 �
2√6 𝑏𝑏ℎ�𝑅𝑅2 2 + 𝑅𝑅1 2 (2𝐹𝐹 − 3𝑞𝑞(ℎ − 𝑦𝑦)) tan−1 �
𝑏𝑏(ℎ−𝑦𝑦)�2⁄3
ℎ�𝑅𝑅2 2 +𝑅𝑅1 2
𝑏𝑏�2⁄3
ℎ�𝑅𝑅2 2 +𝑅𝑅1 2
� +
� + �𝑅𝑅2 2 + 𝑅𝑅1 2 �2𝑏𝑏 2 𝑦𝑦(4𝐹𝐹 + 3ℎ𝑞𝑞 −
3𝑞𝑞𝑞𝑞) + 2𝐹𝐹(ℎ − 𝑦𝑦) �−𝑙𝑙𝑙𝑙𝑙𝑙�ℎ2 �2𝑏𝑏 2 + 3(𝑅𝑅2 2 + 𝑅𝑅1 2 �� + 𝑙𝑙𝑙𝑙𝑙𝑙�3ℎ2 �𝑅𝑅2 2 + 𝑅𝑅1 2 � + 2(ℎ − 𝑦𝑦)2 𝑏𝑏 2 �� +
9ℎ2 𝑞𝑞(𝑅𝑅2 2 + 𝑅𝑅1 2 ) �−𝑙𝑙𝑙𝑙𝑙𝑙 �1 + 3(𝑅𝑅
2𝑏𝑏 2
2
2 +𝑅𝑅
1
2𝑏𝑏 2 (ℎ−𝑦𝑦)2
2 )� + 𝑙𝑙𝑙𝑙𝑙𝑙 �1 + 3ℎ 2 (𝑅𝑅
2
����
2 +𝑅𝑅 2 )
1
(5.27)
Tower-top deflection occurs when y=h in equation (5.27). The exact equation for the tower-top
deflection is:
𝑣𝑣(𝑦𝑦 = ℎ) =
ℎ3
12𝐸𝐸𝐴𝐴𝐶𝐶𝐶𝐶 (𝑏𝑏−𝐷𝐷)4 �𝑅𝑅2 2 +𝑅𝑅1 2
�−2√6�2𝑏𝑏𝑏𝑏(2𝐷𝐷2 − 3(𝑅𝑅2 2 + 𝑅𝑅1 2 )� + 𝐷𝐷[−2𝐷𝐷2 (2𝐹𝐹 +
3ℎ𝑞𝑞)] + 3�2𝐹𝐹 + 9ℎ𝑞𝑞(𝑅𝑅2 2 + 𝑅𝑅1 2 )� tan−1 �
𝑏𝑏�2⁄3
ℎ�𝑅𝑅2 2 +𝑅𝑅1 2
� + 3(𝑅𝑅2 2 + 𝑅𝑅1 2 )�−2(𝑏𝑏 −
𝐷𝐷)[4(𝑏𝑏 − 𝐷𝐷)𝐹𝐹 + 3(𝑏𝑏 − 5𝐷𝐷)ℎ𝑞𝑞]� + �8𝑏𝑏𝑏𝑏𝑏𝑏 − 2𝐷𝐷2 (4𝐹𝐹 + 9ℎ𝑞𝑞) + 9ℎ𝑞𝑞�𝑅𝑅2 2 + 𝑅𝑅1 2 �� �𝑙𝑙𝑙𝑙𝑙𝑙�2𝑏𝑏 2 +
3(𝑅𝑅2 2 + 𝑅𝑅1 2 )� − 𝑙𝑙𝑙𝑙𝑙𝑙�2𝐷𝐷2 + (𝑅𝑅2 2 + 𝑅𝑅1 2 )���
(5.28)
Considering the complexity of the above equations, a MATLAB program TowerDef.m
(Appendix B) was written to calculate the tower deflection from the equation (5.24).
77
5.6 Optimization of the Tripod Model
The structural analysis of the tripod model in the previous section forms the basis for the design
of lattice tower. The objective of the design optimization is to minimize the D, t, and b and to
minimize the mass while minimally satisfying the material strengths with appropriate safety
factors.
The tripod optimization gives the first approximation to the optimal dimensions of tower legs for
various b for the given loads. As this model does not take into account the bracings in the lattice
tower, a more accurate analysis, FEA, is used to finalize the design of the tower considering the
bracing elements.
5.7 Finite Element Analysis
5.7.1 The Methods of FEA
FEA is one of the most powerful computational methods for solving structural problems. In this
method, a structure is subdivided into geometrically smaller units, which are called finite
elements [62]. In each finite element, unknown quantities (e.g. stress, deflections, forces etc) are
approximated by linear combinations of algebraic equations and unknown parameters [62].
Algebraic equations among those parameters are obtained from the governing equations of the
problem. The unknown parameters represent the values at nodes of the elements. Then all
algebraic equations are assembled using the principles of continuity and equilibrium to get the
solutions of the problem. One of the widely used FEA tool utilized in modeling wind turbine
towers is ANSYS. In this study, the FEA of the lattice tower used ANSYS APDL.
5.7.2 The FEA of the Lattice Tower
In the tripod model of the lattice tower, bracing elements were ignored in the structural analysis.
Since exact analytical solutions are not possible to obtain for the structural analysis, FEA is an
78
appropriate approach to determine the internal stresses in members and tower deflection and to
simulate the structural behaviour of the tower considering bracings.
In order to carry out the FEA of the lattice tower, following steps were applied in ANSYS
APDL.
1. Create the FE model of the tower geometry with realistic assumptions for geometry,
loads, and boundary conditions
2. Discretize the tower structure into finite elements using appropriate beam elements and
meshing
3. Apply the material properties, loads, and boundary conditions to the finite element model
of the tower
4. Solve the problem and verify the results
5.7.3 FE Model of the Tower
The lattice tower is a three-dimensional structure consisting of hollow structural elements. One
of the important structural characteristics of such elements is that longitudinal dimension is
larger than the cross-sectional dimension, and hence they can be modeled as one-dimensional (1D) beams [49]. In 1-D beams subjected to axial and bending loads, longitudinal mechanical
properties such as tensile, compressive, and bending strengths, determine the structural
behaviour of the beams. In other words, longitudinal stresses and lateral deflections are always
critical. To simplify the FEA, some modeling assumptions were made, which are summarized
below:
1. Three-dimensional linear static analysis is sufficient for the lattice towers. The tower
design is governed by the extreme static loads.
2. Turbine weight and thrust on blades act as point loads at the tower top.
3. Drag forces and weight of the tower act as uniformly distributed loads.
4. The joints connecting the tower members are perfectly rigid.
79
5. Tower legs are rigidly fixed to the foundation.
A three-dimensional geometry of the tower was created in ANSYS APDL using beam elements.
There are two types of beam elements available in ANSYS: 2-node with 188 and 3-node with
189 elements. The usefulness of each element is discussed here.
2-node with188 beam element is a linear, quadratic or cubic two node element based on
Timoshenko beam theory and can “accurately model slender and moderately thick beam
structures” [13]. The beam element has six or seven degrees of freedom at each node, which
includes translation in x, y, and z directions and rotation about x, y and z directions, having an
option for the seventh degree of freedom. This element has the capability to model linear and
large rotations as well as large strain nonlinear problems.
Figure 5.6 2-node 188 beam element [13]
3-node 189 beam element has 3 nodes and is a quadratic element in three-dimensions. This
element is based on Timoshenko beam theory having capability to model large shear
deformations [13] and is suitable for non-linear analysis.
80
Figure 5.7 3-node 189 element [13]
For modeling the legs and bracings in steel and bamboo lattice towers, 2-node with188 elements
have been used because the analysis is limited to static linear and the tower members are slender
beam elements which are, effectively, one dimensional. No other element in ANSYS is suitable.
ANSYS has a wide range of material models depending upon their properties and nature of the
response of the structure under imposed loads. As the tower should function in the linear elastic
region, the beam is modeled as linear elastic. The material is modeled as an isotropic material for
both steel and bamboo. The validity of the model for bamboo is discussed in Chapter 7. The
material properties required for this model are the Modulus of Elasticity (E) and Poisson’s ratio
(ν ). The density of the material is also required to account for the effect of tower weight on
stress and deflection.
The turbine thrust and weight were applied as point loads at the tower top as shown in Figure
5.3. Drag forces due to wind act in the horizontal direction on beam elements. They were applied
as uniformly distributed loads on beam elements. It is assumed that the joints are perfectly rigid
and tower legs are rigidly fixed to the foundation.
81
Chapter 6
DESIGN OF STEEL LATTICE TOWER
6.1 Chapter Overview
The motivation behind the design of the 12 m high steel lattice tower has been described in
chapter 1. Chapter 5 presented a design optimization procedure for lattice towers. In this
Chapter, these procedures are implemented to design a 12 m high steel tubular lattice tower for a
500W wind turbine.
6.2 The Steel Lattice Tower
As discussed in Chapter 5, the tower is composed of steel circular pipes for legs and bracings.
Circular tubular sections have been chosen because they are easily available, are galvanized to
resist corrosion, and are easy to manufacture and transport to remote locations. The tower is built
with six sections, i.e. each lattice section has 2 m height. Steel sections are used for legs and
horizontal bracings, whereas circular steel rods are used for cross-bracings as shown in Figure
5.1. The leg sections and horizontal bracings are connected by welding and mechanical fasteners,
which will be described in later sections because their details are unimportant for FEA. Small
cross-section is sufficient for cross-bracings, which reduces the wind loads. They can be easily
joined to tower legs by mechanical fastening.
6.3 Design Optimization Procedure
As discussed in Chapter 5, the design optimization of lattice tower involves determining the
optimum dimensions, D and t, of the tower legs and bracings to minimize tower mass and b. This
is accomplished in two steps: 1) determine the approximate values of b, D, and t from the tripod
model (equation 5.11) and 2) conduct FEA using those parameters to obtain the intended design.
82
6.4 Optimization of the Tripod Model
As the first step to tower design, all the loads imposed on the tower, as shown in Figure 6.1, are
computed according to the requirements of IEC 61400-2 using the turbine load cases
documented in [7] and as described in Chapter 5. The analytical solutions presented in Chapter 5
are used to compute the stresses in tower legs and the tower-top deflection to determine the
optimum D, t, and b.
6.4.1 Tower Loading
Using the simple load model (SLM) of IEC61400-2, extreme static wind loads acting on the
tower were calculated. The loads considered in the analysis are summarized here:
•
Tower top weight (W): mass of turbine and nacelle is 30 kg and the mass of turbine
mounting flange and accessories has been assumed to be 20 kg. Considering the IEC load
factor 1.10 for gravity loads, the total weight at the tower-top is 550 N.
•
Rotor thrust (F): 1592 N [7] considering the IEC load factor 1.35 for wind loads, the rotor
thrust is 2150 N.
•
From equation (4.4) with drag coefficient, C d =1.3, density of air, ρ =1.225kg/m3, and
extreme wind speed, U = 50 m/s, uniformly distributed wind load per unit length, q
(N/m) = 1990D. Considering the IEC load factor 1.35 for wind loads, q is 2687D. It is
noted that drag force is dependent upon the diameter of tower legs.
•
Tower weight, Wt (N) = 3ρ s g A CS�ℎ2 + 𝑏𝑏 2 ⁄3 ; where, density of steel = ρ s =7800
kg/m3, g = 9.81m/s2, cross-sectional area of the leg =A CS = π (D t - t2 ), and h = 12 m.
IEC load factor 1.10 is multiplied to this load to get the effective tower weight in the
analysis.
83
√3𝑏𝑏/2
Figure 6.1 Loads on the Tower
6.4.2 Optimization of Tower Legs
The cross-sectional dimensions, D and t, of tower legs and b are determined as the main
optimization variables that define the tripod geometry.
In the analytical solution, the governing design criterion is the buckling strength of tower legs.
The allowable buckling strength of leg sections is determined by the combined compressive and
bending equation (5.11). D and t of tower legs are determined from equation (5.11). Axial and
bending stresses along with their allowable values are computed below.
•
Axial stress: σ𝑎𝑎 = (mass of turbine and tower legs)/3𝐴𝐴𝐶𝐶𝐶𝐶
=
�550+3×12×0.0078×10𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )�ℎ
3𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )�ℎ2 +𝑏𝑏2 /3
84
=
�550+8.82(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )�ℎ
3𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )�ℎ2 +𝑏𝑏2 /3
(6.1)
•
Allowable buckling strength of tower legs, 𝐹𝐹𝑎𝑎 : equation (5.10) of the ASCE guidelines,
equations (5.14) - (5.18) of the Eurocode 3 and AISC equations (5.19) - (5.21) are used
to determine the allowable axial stress. Then the smallest value of the buckling strength
is used in the tower analysis.
•
•
Bending stress due to thrust and drag: σ𝑏𝑏 =
2𝑀𝑀
√3 𝑏𝑏𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )
Allowable bending stress (equation 5.13): 𝐹𝐹𝑏𝑏 = F y
=
2(25800+584𝐷𝐷)
(6.2)
√3 𝑏𝑏𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )
(6.3)
Following design constants have been used in the analysis:
•
Modulus of elasticity for steel: E = 200 GPa
•
Allowable bending strength: F y =255 MPa
•
Linear buckling factor: BF: 2
By inserting the equations (6.1 - 6.3) and design constants into equation (5.11), then equation
(5.11) can be written as:
�550+8.82�𝐷𝐷𝐷𝐷−𝑡𝑡 2 ��ℎ
3𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )𝐹𝐹𝑎𝑎 �ℎ2 +𝑏𝑏2 /3
+
2(25800+584𝐷𝐷)
√3 𝑏𝑏𝜋𝜋(𝐷𝐷𝐷𝐷−𝑡𝑡 2 )𝐹𝐹𝑏𝑏
(6.4)
≤1
R
Solving the equation (6.4), an equally optimum set (or Pareto front) of D for any desired value of
t and b can be obtained. These values of D are the minimum to make the tower safe from
buckling. It is noted that the compressive stress due to axial forces is relatively small compared
to the bending stress. The calculation for the optimum values of D using ASCE, Eurocode 3, and
AISC equations are shown in Appendix A.
Since thickness is the main input variable in the optimization, it is critical to consider the
practicable size of steel pipes based on their availability in the market or that can be easily
manufactured. For example, the author inquired of several manufacturers in Nepal in order to
find out what sizes of steel pipes are manufactured and available in the market. It was found that
the most commonly used galvanized steel pipes of different outside diameters have 2 to 6 mm
85
wall thickness. Here, the author arbitrarily assumed the wall thickness of 3 mm, which is
commonly available in the market.
6.4.3 Results of Tripod Analysis
Figure 6.2 shows the resulting optimum values of D for various b obtained from equation (6.4)
using t =3 mm. For the allowable axial and bending stresses, ASCE equations (5.10) and (5.11)
were used. The Eurocode 3 and AISC calculations for the optimum D are shown in Appendix A.
It is observed from Figure 6.2 that the slope of the graphs is decreasing with increasing b. This
indicates that D or mass of the tower can be minimized more effectively if higher values of base
distance are selected. Consequently, it requires more ground space but less mass for the
foundation.
It is noted that the tripod analysis has given the optimal solution for D for various b, but not a
unique minimum. To achieve the best design, it is intended to minimize the D by increasing b.
However, it requires more ground space, which is linked with the cost of land; and moreover, it
is very site specific. So these factors are not considered in this study.
86
Figure 6.2 Optimum diameters of legs (D) for various base distances
(b) and wall thickness (t) of 3 mm (ASCE standard)
In order to carry out the FEA of the lattice tower, values of D were computed at some specific b
using ASCE, Eurocode 3 and AISC equations, which are shown in Table 6.1. The basic
calculations are shown in Appendix A. It is noted that the ASCE and Eurocode 3 have given the
same values for D, whereas AISC has given slightly higher values.
Table 6.1 Optimum D for tower legs (t = 3 mm)
b (m)
1
1.2
1.4
1.6
ASCE: D (mm)
64
45
35
29
Eurocode 3: D (mm)
64
45
35
29
AISC: D (mm)
65
47
37
31
87
It is noted that increasing t in equation (6.4) decreases both D and tower mass because this
increases the strength without increasing the wind load. Therefore, the best design strategy to
minimize the tower mass is to increase t considering the availability of those sizes of steel pipes
in the market.
6.5 Finite Element Analysis of Tower
The analytical solutions of the tripod model has given a set of optimum results for D at t = 3mm
(Figure 6.2). Using these results, the FEA described in Chapter 5 is implemented to check the
validity of the tripod optimization procedure and use the results for the design and analysis of the
lattice tower using FEA. Following design examples were considered:
•
Lattice tower with legs and horizontal bracings: In this configuration, the tower was
modeled with legs and horizontal bracings using the values of D and b obtained from the
analytical solution. Initially, the legs and horizontal bracings were modeled with same D
and t. Then some of the minimum possible sizes for legs and horizontal bracings were
investigated for minimum mass of the tower. The results of FEA are compared to the
results of tripod analysis.
•
Lattice tower with horizontal- and cross-bracings: In this configuration, the tower was
modeled with horizontal- and cross-bracings. Some of the possible sizes for legs and
horizontal bracings were investigated for minimum mass of the tower, buckling strength,
and stiffness.
Each tower configuration was analyzed for various b as given in Table 6.1.
In order to carry out the FEA of the towers in ANSYS APDL, three-dimensional FE models
were created using key points and line elements (Figure 6.3) for the tower geometry. The tower
base distance and leg cross-sections (D and t) were obtained from Table 6.1 (Figure 6.2). Excel
spreadsheet was used to determine the coordinates of the key points for modeling the tower. 2
nodes with 288 beam elements were used to model the tower members. As the tower top width
should be kept as minimum as possible to avoid the turbine blades hitting the tower, the width
88
was set to 0.15 m for fixing the turbine mounting flange. The turbine mounting flange, flanges in
the leg sections, and the base plates are not included in the FE model.
The material model for steel was chosen as linear elastic and isotropic with the elastic modulus
and Poisson ratio of 200 GPa and 0.3 respectively. The density of steel was taken as 7800 kg/m3.
The tower legs were assumed to be rigidly fixed to the steel base plates at the foundation; no
rotation and displacement were assumed during extreme loads. The turbine thrust (2.15k N) and
weight (550 N) were applied at the tower-top in the horizontal and vertical directions
respectively. The drag forces per unit length were computed in legs, horizontal- and crossbracings and then applied at each nodal point of the beam elements. This simulates the uniformly
distributed loads as illustrated in Figure 6.3. The gravity load due to tower mass was applied as
“inertia in global coordinates” by using the value of acceleration due to gravity as 9.81 m/s2.
Since buckling is the main failure criterion in steel lattice tower design, the analysis was carried
out in the maximum compression mode of the tower, which is explained in Chapter 5. Maximum
tensile mode was not considered.
In order to investigate the convergence of the FEA results, the beam elements were meshed with
different lengths. Then the resulting values of both the stresses and deflections were compared
for consistency. Convergence on values of stress and tower-top deflection was obtained when the
tower was meshed with element lengths of 10 mm or less. In the analysis, element length of 5
mm was used. To verify the results, the results of stress and deflections were also compared with
the results of tripod analysis.
89
Figure 6.3 Bottom section of the FE model
of the lattice tower showing drag forces and
boundary conditions
6. 6 Results and Discussion
In order to compare the results of the analytical solutions (tripod analysis) and the FEA, the
stresses on tower legs and tower-top deflections were determined from the tripod analysis, as
shown in Table 6.2.
To achieve convergence and consistency in the FEA, the convergence test for stress and
deflection for the tower with horizontal bracings was carried out with different lengths of beam
elements as shown in Figures 6.4 and 6.5. It was found that deflections converged more quickly
than the stresses.
90
Figure 6.4 Convergence test for stress with different
lengths of beam elements (Tower with b = 1m, D =64
mm and 64 mm horizontal bracings)
Figure 6.5 Convergence test for deflection with
different lengths of beam elements (Tower with b = 1m,
D =64 mm and 64 mm horizontal bracings)
91
The numerical solution of the equation 5.25 for the tower top deflection was obtained by using a
MATLAB program, TowerDef.m, given in Appendix B. Also, the calculation was done with the
analytical solution for the tower-top deflection, given by the equation (5.28). The results were
compared with the results of FEA of the tripod model and the tower with horizontal bracings.
For the later, the tower legs and horizontal bracings were modeled with the same D and t. The
results of ANSYS simulations at b = 1, 1.2, 1.4 and 1.6 m for the tower with horizontal bracings
are shown in Figures 6.6- and 6.9. The results of FEA and numerical and analytical solutions for
the two tower models are summarized in Table 6.2 and Table 6.3 respectively.
Table 6.2 Comparison of FEA, numerical and analytical results (Tripod tower)
Base distance, b (m)
1
1.2
1.4
1.6
Leg diameter, D (mm)
64
45
35
29
Maximum compressive stress on tower legs
113
111.05
110.86
107.3
127.5
127.5
127.5
127.5
Maximum tower-top deflection (mm): FEA
118.08
109.22
105.46
93.83
Maximum tower-top deflection (mm): numerical
91.15
85.12
78.45
71.83
91.17
85.13
78.46
71.83
(MPa): FEA
Maximum compressive stress (MPa) on tower
legs: analytical solution
solution
Maximum tower-top deflection (mm): analytical
solution (equation 5.28)
92
Table 6.3 Comparison of FEA and numerical results (Tower with horizontal bracings)
Base distance, b (m)
1
1.2
1.4
1.6
Leg diameter, D (mm)
64
45
35
29
Maximum compressive stress (MPa): FEA
131.11
128.15
127.44
125.77
Maximum compressive stress (MPa): analytical
127.5
127.5
127.5
127.5
Maximum tower-top deflection (mm): FEA
109.28
103.31
96.22
88.42
Maximum tower-top deflection (mm): numerical
91.15
85.12
78.45
71.83
solution
solution
From the tripod model, the results of FEA for the stresses on legs were obtained about 13 %
lower than the predicted buckling strength, 127.5 MPa. However, the tower-top deflections from
FEA were obtained about 22% higher than the numerical solutions. From the FEA of the tower
with horizontal bracings, the results of FEA for the maximum compressive stresses on tower legs
were found in good agreement to the results of numerical solutions. It is noted that the leg
stresses have decreased slightly with increase in b. However, the results for the tower-top
deflections showed about 16-18 % difference in values. In an 18m high monopole tower
analyzed in [7], about 10% variation in analytical and ANSYS results was obtained. As buckling
is the governing design criteria and the tower-top deflections are only less than 1% of the tower
height, it indicates that linear static assumption is valid and therefore errors in FEA and
numerical solutions would not make a significant difference in the design of tower legs (D).
93
Figure 6.6 Maximum stress and deflection of the tower at b =1m and
D =64 mm
Figure 6.7 Maximum stress and deflection at b =1.2 m and
D = 45 mm
94
Figure 6.8 Maximum stress and tower-top deflection at b =1.4 m
and D =35 mm
Figure 6.9 Maximum stress and deflection at b = 1.6 m and
D =29 mm
95
6.6.1 Design Examples with Horizontal Bracings
A few design examples of 12 m high steel lattice towers with horizontal bracings were shown in
the previous section. Here, a design example of the lattice tower, with different diameters for
legs and bracings are considered. As the change in D with b is rapid around b = 1.2 m (Figure
6.2), the author has chosen the base distance of tower as 1.2 m. At b =1.2 m, the tripod analysis
gives an optimum D as 45 mm (Table 6.1); and assuming that the tower-top width is 15 cm, the
length of each leg section was computed as 2003.3 mm for each 2 m high lattice section of the
tower.
Figure 6.7 shows that the maximum stress of 128 MPa and tower-top deflection of 103 mm for
the tower with b =1.2 m, D = 45 mm, and t = 3mm. The corresponding mass of the tower was
computed as 144 kg. Since allowable tower-top deflection is 600 mm, both the diameters of legs
and bracings could be reduced to minimize the tower mass provided that the leg stress does not
exceed the allowable buckling stress of 127.5 MPa.
To examine a tower with different D and bracing diameters, the tower was modeled with D =40
mm and 20 mm diameter horizontal bracings. The maximum compressive stress on legs and
tower-top deflection were obtained as 127.75 MPa and 129 mm respectively as shown in Figure
6.10. The tower is marginally safe from buckling. For this tower model, the mass is computed
as112 kg, which is only slightly lower than 144 kg for the tower with D =45 mm. It is noted that
the reduction of diameters of legs and bracings has resulted in slight increase in tower-top
deflection. Further optimization of legs and bracings may result in lighter towers, but it is
requires an extensive work with FEA. However, the reduction in mass will be only slightly.
When stiff tower designs are desired, same size of legs and horizontal bracings, as obtained from
the tripod analysis, are recommended. It was shown in the above example that tower mass could
be minimized by reducing both the diameters of legs and horizontal bracings; however, it results
into significant increase in tower-top deflection. It is concluded that same size of legs and
bracings should be used if stiffer tower designs are desired. If tower mass is an important factor
96
in the design, the diameters of legs and bracings can be reduced from the values, as obtained
from the analytical solution, but the design requires FEA for the design optimization process.
Also, empirical equations of ASCE, Eurocode 3 and AISC should be used to verify the final
design. A comparison of two tower models analyzed here is given in Table 6.4. From this
analysis, both tower designs can be used. Here, the light weight tower design is recommended
despite its larger tower-top deflection (Table 6.5).
Figure 6.10 Maximum stress and deflection at b = 1.2 m, D =40 mm
and 20 mm diameter bracings
Table 6.4 Results of FEA with horizontal bracings (b =1.2 m)
D (mm)
Stress
(MPa)
128
Tower-top deflection
(mm)
103
Tower mass (kg)
45
Diameter of
bracings (mm)
45
40
20
127.7
129
112
97
144
Table 6.5 Recommended tower design (tower with horizontal bracings)
Description
Base distance between the tower legs (mm)
1.2
Length of leg sections (mm)
2003
Outer diameter of leg (mm)
40
Outer diameter of horizontal bracing (mm)
20
Pipe wall thickness (mm)
3
Maximum tower top deflection (mm)
129
Maximum stress (MPa)
128
Total tower mass (kg)
112
Assumed tower mass including accessories (kg)
142
6.6.2 Design Example including Cross-bracings
A design example of the tower with horizontal- and cross-bracings is presented here. As the
simultaneous selection of D and bracing diameters using FEA is a time consuming task, the
diameter of cross-bracing, which is a round steel rod, was assumed to be 10 mm for all tower
configurations. This reference is taken from the 12 mm diameter steel rods used as crossbracings in the design of an 18 m high triangular lattice tower in [7]. Consequently, the
optimization of D and bracings is required using FEA.
The design example considers the case of the tower model presented in Table 6.4. The tower has
b =1.2m, D = 40 mm and bracing diameter of 20 mm. The corresponding tower top deflection
and mass are 129 mm and 112 kg respectively. Further reduction of leg diameter will increase
the tower-top deflection, but reduces the tower mass. To design this tower for minimum towertop deflection, cross-bracings are used. As the cross-bracings increase tower mass, the tower was
designed with smaller values of D = 35 mm, 20 mm diameter horizontal bracings, and 10 mm
diameter round solid steel rods. Here 20 mm diameter is assumed as the minimum size of hollow
steel pipe for horizontal bracings. The result of FEA for the tower with D=35 mm, 20 mm
98
diameter horizontal bracings, and 10 mm diameter cross-bracings is shown in Figures 6.11. The
result showed that by including the cross-bracings, the tower deflection reduced significantly
from 129mm to 81 mm. The corresponding stress on tower leg is 128 MPa, which gives the
linear buckling factor of about 2 as assumed in the analytical solution. The total tower mass is
computed as 124 kg, which is slightly greater than the previous model of the tower. In
comparison with the tower having equal leg and bracing diameters of 45 mm (Table 6.4), this
tower is slightly lighter and stiffer. However, the maximum stress on legs has increased in
comparison to the previous tower, but less than the tower having equal diameters for legs and
bracings. This is due to the increased drag on cross-bracings.
It is concluded that the tower with D = 35 mm, 20 mm diameter horizontal bracings, and 10 mm
diameter cross-bracings (Figure 6.11) gives a good design in terms of weight and stiffness.
Further reduction in tower mass is possible, but it requires an extensive finite element modeling.
The proposed tower design with cross-bracings are compiled Table 6.6.
Figure 6.11 Maximum stress and tower deflection for D =35 mm, 20 mm
diameter horizontal bracings, and 10 mm cross-bracings
99
Table 6.6 Recommended tower design with cross-bracings
Description
Base distance between legs, b (m)
1.2
Length of leg sections (mm)
2003
Diameter of legs, D (mm)
35
Diameter of horizontal bracing (mm)
20
Thickness of legs and bracings, t (mm)
3
Diameter of cross-bracing (mm)
10
Total tower mass of tower members (kg)
136
Buckling capacity factor
0.5
Maximum tower top deflection (mm)
104
From the above analysis of different tower models, it is concluded that cross-bracings are not
required to design light-weight and stiff towers. Simple tower designs with only horizontal
bracings are recommended. Such tower designs can be easily designed with the results of tripod
analysis and do not require extensive work on FEA.
6.7 Design Loads for Foundation
After the tower dimensions of the lattice tower are defined, foundation analysis is carried out.
Steel reinforced concrete is the most commonly used material for tower foundations. Foundation
for lattice tower can be built in two ways. It can be built as either single spread footing or
individual footing for each tower leg. Rectangular and cylindrical foundations are commonly
used in lattice towers. However, the cost of a specific type of foundation depends upon the base
distance between the tower legs. For the tower with small base distance, single foundation may
be an economic option, whereas for the tower with large base distance, foundations at each leg
may be economic.
100
Foundation design is site specific due to the local soil conditions and their bearing capacity. The
dimensions of the foundation are determined by the reaction forces and the moments imposed on
the foundation. The forces and moments consist of total vertical load and overturning and
resisting moments on the foundation (Figure 6.12). The resisting capacity of the foundation
depends upon its own weight and bearing capacity of the soil. The vertical load (V) is the total
gravity load of the turbine (W), the tower (Wt), and the foundation (Wf). It is expressed as:
(6.5)
𝑉𝑉 = 𝑊𝑊 + 𝑊𝑊𝑡𝑡 + 𝑊𝑊𝑓𝑓
For a cylindrical foundation, the total resisting moment of the foundation is the total vertical load
multiplied by the radius of the foundation. The total overturning moment is the sum of the
resultant moment and the resultant horizontal force multiplied by the depth of the foundation.
Σ 𝑀𝑀𝑅𝑅𝑅𝑅 = 𝑉𝑉𝑑𝑑𝑓𝑓 /2
(6.6)
Σ 𝑀𝑀 = 𝑀𝑀𝑓𝑓 + 𝐻𝐻 ℎ𝑓𝑓
(6.7)
By solving equations 6.5, 6.6 and 6.7, appropriate diameter and depth of the cylindrical
foundation can be determined. For the tower example presented in Table 6.5, the total vertical
load due to turbine and tower mass is 1420 N and the base overturning moment was determined
as 81,385 Nm by using equation (5.1).
101
Figure 6.12 Schematic of the loads on tower foundation
6.8 Tower Manufacture
Galvanized circular hollow steel sections of required diameters and thicknesses are readily
available in the market or can be produced easily.
An important design consideration in lattice towers is the joining technique for the circular
HSSs. As described in [7], the lattice tower can be manufactured very simply and accurately with
the help of a jig, like that shown in Figure 6.13. For the designed tower (Table 6.4), the leg
sections have 42 mm diameter steel pipes, whereas horizontal bracings have diameter 20 mm.
Both sections have the same wall thickness. The bracings are welded to the leg sections as shown
in the jig. The top section of a tower is fixed in the jig and the white arrow indicates one of the
three base plates for the bottom section of the tower.
102
Figure 6.13 Jig to make tubular lattice tower used by
Kijito Windpower, Kenya. Photo taken from [7]
103
Chapter 7
OPTIMAL DESIGN OF BAMBOO TOWER
7.1 Chapter Overview
The basic design optimization procedures for lattice towers have been described in Chapter 5.
This chapter describes the main aspects of design and analysis of bamboo lattice tower.
This study aims to investigate bamboo’s suitability for small wind turbine towers using the
mechanical properties of bamboo experimentally established in Chapter 3. As a design example,
the design and analysis of a 12 m high bamboo tower for the 500 W wind turbine used in
Chapter 6 was carried out to compare to the steel lattice tower. On the basis of mechanical
properties of bamboo and IEC61400-2 safety requirements, the design of the tower was assessed
using analytical and FEA techniques. The study was focused on safety requirements, rather than
detailed economic analysis. The methodology adopted in the study is illustrated in Figure 7.1.
Experimental tests on
mechanical properties of
bamboo
Design of 12 m high
lattice tower for a 500 W
wind turbine
Design optimization and
finite element analysis
Joint design and testing
Figure 7.1 Study approach for the bamboo lattice tower
104
7.2 The Bamboo Tower
As discussed in Chapter 5, buckling strength of tower members is the design criterion for lattice
towers. Consequently, the most desired property for the lattice tower members is their ability to
withstand compressive loads without buckling. One of the remarkable mechanical properties of
bamboo is its high buckling and tensile strengths in the longitudinal direction, which should
make bamboo a suitable material for lattice towers. In addition, it’s a natural material that is
cheap, easily available, and sustainable.
The proposed bamboo tower is composed of bamboo columns, connected together as beam
elements constituting the lattice structure as shown in Figure 7.2. The tower is built with 8 lattice
sections, each of 1.5 m height. The reasons for using short leg sections are that the dimensional
variability along the length should be minimized and the fact that shorter leg sections would have
better buckling strengths, as required in lattice towers. For the tower legs, approximately1.5 m
long bamboo columns (depending upon the base distance) are joined co-axially in the
longitudinal axis by using steel-bamboo adhesive joints. In steel-bamboo adhesive joints,
lashings are applied to add strength and stiffness in the joint. Horizontal- and cross-bracings are
connected to the tower legs by using lashings to add stiffness to the tower. The legs are
connected together into the turbine mounting flange (using both bolts and lashings) at the top of
the tower, whereas the steel caps in the bottom of leg sections would connect to the base plates
bolted at the foundation. The bottom sections of the legs are also tied up with ropes to the steel
caps, which are bolted to the foundation.
105
Bamboo-steel adhesive joint for
legs and lashings for both legs
and bracings
Figure 7.2 Bottom section of the proposed
bamboo lattice tower
Figure 7.3 Joining methods for the leg sections
Figure 7.4 Steel connector cap for the adhesive
joints
106
7.3 Design Requirements for Bamboo Tower
The purpose of the bamboo tower is to offer an economic and technological alternative to steel
towers for small wind turbines in off-grid remote regions of the developing countries. To verify
bamboo’s validity as a low-cost material, the tower design should meet certain requirements,
which are summarized below:
1) The bamboo tower will meet the load and safety requirements of the “SLM” of
IEC61400-2 for small wind turbines
2) The compressively loaded tower members will not buckle during extreme wind loads
determined by IEC61400-2
3) The tensile strengths of the joints connecting the bamboo columns will withstand the
tensile loads induced in the tower during extreme wind loads determined by IEC61400-2
4) The response of the structure will be linear elastic during extreme wind loads
5) The mechanical properties of the bamboo columns and joints will not change over the
designed life due to the effects of weathering or loadings
6) The design of joints for connecting bamboo columns will take into account the
dimensional variability of bamboo columns at the two ends, will protect splitting of
bamboo in the transverse axis, and will provide barrier to moisture ingression into the
joints
7) The joint design should ensure that the forces and moments acting in the joints would be
transmitted along the longitudinal axis of the columns so that excessive shear stress is not
developed in the joints leading to the splitting of bamboo columns
8) Due to the low-durability of bamboo, the joint design will allow flexibility for periodic
replacement of tower members or when required to meet the turbine service life-span
9) The tower will be cost-effective over the life-span of the project
With regard to the above requirements, bamboo can meet several design requirements. As
discussed in Chapter 2 and Chapter 3, bamboo is strong in tension, compression and bending that
the strengths are adequate to lattice tower. Moreover, its structural response is linear-elastic
107
when subjected to those loads. Since the failure criterion of lattice towers is mainly the buckling
of tower members, bamboo should be a suitable structural material as it exhibits excellent
buckling strength. From the above requirements, the issues of joints and durability become
apparent when bamboo is considered for lattice towers.
For bamboo towers, the main challenge is the need to connect the leg sections co-axially. It is not
possible to join the bamboo sections by welding or machining to a desired shape. Also, drilling
of holes for mechanical fasteners would induce splitting and conventional lashing alone cannot
effectively join two bamboo sections co-axially. The proposed solution to this challenge is to use
steel-bamboo adhesive joints, which would connect the bamboo co-axially in the longitudinal
direction. Then lashings will be used in the joint to strengthen the steel-bamboo adhesive joints.
However, the bracings are connected to legs by lashings only as they do not need to be connected
co-axially.
Bamboo has a low durability, e.g. 3-5 years, in open conditions due to weathering, whereas the
typical design life of a wind turbine is 20 years. In addition, due to weathering effects,
mechanical properties might change over time, if not protected properly. Longevity can be
improved if appropriate coatings or paints are applied, but it adds more costs to the material. To
address these issues, the proposed solution is to periodically replace the tower members after few
years of service. Consequently, this requires close monitoring of the tower members.
7.4 The Proposed Joint
As mentioned in earlier sections, steel-bamboo adhesive joints combined with lashings are
proposed to connect leg sections co-axially as required in lattice tower (Figures 7.3 and 7.5). As
the bamboo sections are connected co-axially, the compressive strength of the joint is equal to
that of the bamboo culm, but the tensile strength of the adhesive joint alone is less than the
tensile strength of the bamboo. To increase the tensile strength and stiffness of the joint in
tension, it is proposed to strengthen the joints by using lashings. Steel is chosen for the joint
because it is readily available as circular pipes and it would provide rigidity at the joints to
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prevent splitting of bamboo columns. In addition, steel protects the vulnerable ends of the culms
from external damage and splitting at the joints. It is noted that adhesive joint does not require
drilling the bamboo columns for connecting different members. Moreover, it prevents moisture
ingression into the joint. More importantly, it will preserve the mechanical properties of the
culms as well as accommodate the dimensional variability along the length of bamboo column.
This study has focused on fabrication and testing of steel-bamboo epoxy joint. Detailed joint
design, which involves characterization of adhesive thickness and the joint length or overlap
length [53], was not carried out in this study. The experimental program on joint testing has been
described in Chapter 3.
Figure 7.5 Proposed joining methods in the lattice tower
7.5 Design Procedure for the Bamboo Lattice Tower
In the design of bamboo tower, the objective is to determine the minimum possible D of bamboo
columns, which are safe against buckling, for the minimum b. As discussed in section 3.4.3,
buckling is the only failure mode for the bamboo columns. Using the equations (5.9- 5.11), the
buckling stress on tower legs can be determined for any b. By knowing the buckling stress on
tower legs, l , and t, it is possible to determine the minimum D by using the equation (3.6). It is
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assumed that the variation of buckling strength of bamboo columns with D, t, and l is given by
the equation (3.6).
7.5.1 Structural Analysis of the Tripod Model
The basic equations for determining the forces and stresses in the tower legs have been
formulated in Chapter 5. Optimum values of D, which are the values satisfying the maximum
compressive loads on tower legs, can be obtained from the experimentally determined buckling
strength of bamboo. In other words, the tower legs under compression loads should be
marginally safe from buckling.
The loads considered in the tower analysis are summarized below.
•
Maximum thrust on turbine, F: 2150 N
•
Weight of turbine + turbine mounting flange, W: 550 N
•
Weight of tower, Wt: ∑ 𝜌𝜌𝜌𝜌𝜌𝜌 �ℎ2 + 𝑏𝑏 2 ⁄3 where, h =12 m; density of bamboo= 𝜌𝜌 = 800
kg/m3 [36]; and g =9.81 m/s2. The IEC load factor of 1.1 is applied in the analysis.
•
Uniformly distributed wind load: 1.35q (N/m) (equation 4.4)
In scaffoldings described in Chapter 2, lashing joints are sufficiently strong that the main failure
criterion is the buckling of bamboo columns. Similarly, in lattice towers, if the adhesive joints
combined with lashings are assumed to be sufficiently strong to resist maximum tensile loads in
tower legs, the design of the tower is governed by the buckling strengths of bamboo columns.
The bamboo tower is designed with 1.5 m high lattice sections. Therefore, the length of tower
legs is about 1.5 m depending upon the choice of b. The buckling strengths of legs for a
particular diameter can be determined from the experimental results by computing the
slenderness ratio of the leg sections. With a known slenderness ratio, the buckling strength is
computed by using equation (3.6).
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To prevent buckling of tower legs under combined axial and bending loads, the strength of tower
leg must satisfy the equation (5.11). The axial and bending stresses are computed as follows.
•
•
•
•
Axial stress, σ𝑎𝑎 , is given by equation (5.10)
Allowable buckling strength of tower legs, 𝑓𝑓𝑎𝑎 , is given by equation (3.6)
Bending stress, σ𝑏𝑏 , is given by equation 5.5 and 5.8
Yield strength of bamboo in compression is 𝑓𝑓𝑏𝑏 = 44 MPa (Table 3.4)
It is noted that the axial stress in tower legs is much smaller than the bending stress. Using
equation (5.11), an equally optimum set of D is obtained, which are safe from buckling.
In another case, if lashing is not considered in the joints of leg sections, the tower will fail if the
maximum tensile load in leg sections exceeds the ultimate strength of the steel-bamboo adhesive
joints. Consequently, the optimum D and b should be determined based on the maximum pullout resistance of the joint. In other words, the adhesive joints will have lower pull-out strengths
than the buckling strengths of tower members. To examine the tower design with tensile strength
criterion, first the optimum values of D were obtained using the buckling strength criteria. Then
a particular size of bamboo was chosen for the fabrication of steel-bamboo adhesive joint. This
was necessary because characteristic values of pull-out strengths of steel-bamboo are not
available and the study is based on the experimental result. As a reference, the pull-out strength
of 19 kN for the PVC-bamboo adhesive joint [18] was taken. The size of the bamboo used was
61.18 mm. In this study, 65 mm diameter bamboo was chosen for the fabrication of the adhesive
joint.
7.5.2 Results of Tripod Analysis
In order to assess the buckling of tower, the combined maximum compressive stresses in the
tower legs were determined for different bamboo diameters as shown in Figures 7.6. The average
thickness of 6 mm was assumed for all sizes of bamboo columns, which was obtained from the
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average thickness of tests specimens, given in Table 3.1. Table 7.1 shows the buckling strengths
of 1.5 m long bamboo columns for different diameters obtained from the results of buckling tests
(Figure 3.5).
Figure 7.6 Maximum compressive stresses in tower
legs for various leg diameters and base distances
Table 7.1 Buckling strengths of 1.5 m long columns (t = 6 mm) (equation 3.6)
Diameters (mm)
50
55
60
65
70
75
Slenderness Ratio
95
86
78
71
66
61
Buckling Strength (MPa)
15.5
20.5
24
27.5
30
31.5
Figure 7.7 shows the minimum possible diameters of bamboo, obtained from Figure 7.6 and
Table 7.2, for the leg sections that are safe against buckling. It is observed that there is an equally
optimum set of D for various b, which can be chosen to design the tower. However, the design
goal is to design a tower with minimum possible D at minimum b, appropriate b should be
selected based on buckling strength of bamboo.
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Figure 7.7 Diameters of 1.5 m long bamboo columns that
are marginally safe against buckling for various base
distances
Figure 7.8 Maximum tensile forces on tower legs for various b
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It is noted that the experimental tests on pull-out strength of steel-bamboo adhesive joints were
carried out only for D =65 mm. The pull-out strength for this size of joint was 20.32 kN (Table
3.4). So the design of tower based on tensile strength criterion was examined only for this size of
bamboo. The effective tensile forces (combined axial and bending) on tower legs in the
maximum tensile mode are shown in Figure 7.8 for a 65 mm diameter bamboo. It is observed
that the minimum b = 2.7 m to withstand the tensile loads if only the adhesive joints are used in
the tower. However, such a large base distance would increase the tower top width and may not
be feasible for mounting the turbine unless special turbine mounting arrangements are made.
7.5.3 Finite Element Analysis of the Tower
FEA was carried out to determine the maximum stresses and forces in the tower legs and the
tower-top deflection. Using the results of tripod analysis, FEA was carried out in the maximum
compression and tensile modes of the tower as discussed in section 5.5.1. Two examples of
tower designs were examined as discussed above.
In the first example, the design of tower is based on the buckling strengths of tower legs. Here,
the joints are assumed sufficiently strong and stiff to withstand maximum tensile loads on the
tower legs and buckling is the failure criterion. The tower configurations considered for the FEA
are shown in Table 7.2.
Table 7.2 Tower configurations for the FEA (t=6 mm)
b (m)
1.6
1.85
2.15
2.6
D (mm)
70
65
60
55
In the second example, the design of tower was based on the ultimate tensile strength of 20.32
kN of steel-bamboo adhesive joints. As the strength of steel-bamboo joint is already determined
for D = 65 mm, the tower design requires appropriate selection of b. For the 65 mm legs, the
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base distance of the tower should be at least 2.7 m as shown in Figure 7.8 to prevent the failure
of joints in tension. To verify the result, FEA of the tower was carried out.
7.5.3.1 FE Model of the Bamboo Tower
Although bamboo possesses a graded composite structure across the wall or in the transverse
axis, compressive and tensile strengths and elastic properties do not vary noticeably along the
longitudinal axis. Silva et al. [28] applied the FE methods to determine the effective mechanical
properties and structural behaviour of bamboo culms by assuming the homogenized material
structure. The results showed that effective material properties could be determined by assuming
a homogeneous material. In this study, bamboo was assumed as homogeneous beam, i.e. it
possesses same material properties in the longitudinal direction. As a conservative approach,
bamboo was modeled as a linear elastic isotropic material because longitudinal properties of the
beam are important in lattice towers. In FE modeling, the required material properties are the
modulus of elasticity and Poisson’s ratio, which have been determined from experimental tests.
The modulus of elasticity and Poisson’s ratio used in the analysis are 16.32 GPa and 0.33
respectively. The tower was modeled in ANSYS APDL using 2-node with 288 beam elements
for bamboo columns (Figure 7.11). The properties of the beam elements are described in Chapter
5. As the analytical and FEA results for stress and deflection were obtained very similar, the
above material model was assumed to be valid.
The tower legs were assumed to be rigidly fixed to the foundation; no rotation and displacement
were assumed during the extreme loads. In addition, the tower joints were assumed to be rigid
and no rotation and transportation are allowed. This could be achieved by using lashing around
the joint, which would increase strength and stiffness of the joints. The turbine thrust and
weights considering the IEC load factors were applied at the tower top in horizontal and vertical
directions respectively. The drag forces per unit length, with IEC load factor, were computed in
legs, horizontal- and cross-bracings and then applied at each nodal point in the beam elements
(Figure 7.10). The bamboo density was taken as 800 kg/m3. The gravity load due to tower mass
was applied as “inertia in global coordinates” by using acceleration due to gravity as 9.81 m/s2.
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Figure 7. 9 Finite element models of the tower with horizontal bracings (left),
with horizontal- and cross-bracings (centre), and bottom section of the tower
showing wind loading on the tower (right)
7.5.3. 2 Results of Finite Element Analysis
Using the tower configurations given in Table 7.2, FEA was carried out in order to compare the
stress and deflection results with the tripod analysis. The results of FEA for the maximum
compressive stress and tower deflection considering the horizontal bracings are shown in Figure
7.10-7.13. The numerical results for the tower deflections were calculated by using the
MATLAB program, TowerDef.m, given in Appendix B. The comparison of the results obtained
from the tripod analysis and FEA are shown in Table 7.3.
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Figure 7.10 Tower-top deflection and compressive stress
(b=1.6 m and D=70 mm)
Figure 7.11 Tower-top deflection and compressive stress
(b=1.85 m and D = 65 mm for legs and bracings)
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Figure 7.12 Tower-top deflection and compressive stress
(b=2.15 m and D = 60 mm for legs and bracings
Figure 7.13 Tower-top deflection and compressive stress
(b=2.6 m and D =55 mm for legs and bracings)
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Table 7.3 Comparison of the results of FEA and tripod analysis
Diameter of legs and bracings (mm)
70
65
60
55
Base distance between the tower legs (m)
1.6
1.85
2.15
2.6
Allowable compressive stress (MPa)
29.55
26.51
23.78
20.63
Maximum compressive stress (MPa) (FEA)
30.30
24.17
21.17
18.67
Tower-top deflection (mm) (FEA)
222
193.50
169.25
133.62
Tower-top deflection (mm)
232
186.27
148.46
110.17
(Analytical)
(numerical)
From Table 7.3, it was found that the tripod analysis determines the maximum stresses on legs
with reasonable accuracy. Also, the tower-top deflections are comparable. It is noted that as the
base distance increases, the results of the FEA and tripod models are diverging. This indicates
that the FEA and tripod analysis give similar results for b up to 2 m. Therefore, tripod analysis
can be used to determine optimum values of D for this range of b with sufficient accuracy. For
better stiffness, same diameters of bamboo should be used for both legs and horizontal bracings.
From Table 7.3, it is evident that only the tower designs with 65, 60, and 55 mm bamboo are
feasible.
In order to determine the effect of cross-bracings, a tower design with b =1.85 m and D = 65 mm
was considered. Both the horizontal- and cross -bracings were modeled as 30 mm bamboo,
which is assumed as the minimum possible size of bamboo that can be obtained in practice. The
result of ANSYS simulation is shown in Figure 7.14.
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Figure 7.14 Tower-top deflection and compressive stress in the
tower with horizontal- and cross- bracings (b= 1.85 m, D=65 mm)
It is observed from Figure 7.14 that compressive stress on tower legs has increased when crossbracings were used. This is because of the increased drag on the tower. However, the tower
stiffness has increased considerably, with the tower-top deflection of 94 mm. The increased
compressive strength means that buckling could be a critical factor. For the 65 mm bamboo, the
allowable compressive stress is 26.51 MPa. Therefore, the tower may buckle. It is concluded that
cross-bracings increase the drag on tower considerably, which would increase the compressive
stress in tower legs.
In the second design example, FEA was carried out to calculate the maximum tensile force on
tower legs for b =2.6 m and D =65 mm. Figures 7.15 shows the distribution of effective tensile
forces in the tower legs. The influence of bracing sizes on effective tensile forces is shown in
Figure 7.16. The results show that smaller size of horizontal bracings should be used to reduce
the drag on tower.
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Figure 7.15 Tensile forces in the tower legs at b= 2.6 m, D = 65
mm and 30 mm diameter for bracings
Figure 7.16 Effect of bracing sizes on maximum tensile
forces in legs at b= 2.6 m (obtained from FEA).
121
Figure 7.17 Lattice tower of 2.6 m with 65 mm leg size and 30
mm for horizontal- and cross-bracings
In the tower model having horizontal- and cross-bracings of minimum possible diameter (Figure
7.17), the effective tensile forces increased significantly to 25 kN from 19 kN (tower without
cross-bracings). This shows that cross-bracings, which increase the drag on the tower, should not
be used in bamboo towers. Horizontal bracings are sufficient to maintain the stiffness of the
tower.
In conclusion, it was found that drag forces on cross-bracings considerably increase the
compressive stress and tensile forces in tower legs. Although, full sections of bamboo are not
recommended for cross-bracings, split bamboo sections of smaller cross-sectional areas may be
used to enhance stiffness. However, the analysis was not carried out for these sections.
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7.5.4 Results and Assumptions of the Analysis
From the analysis of different bamboo towers, it has been shown that tower deflections are small.
The maximum tower top deflection was found as 1.6 % of the tower height for the feasible tower
designs (Table 7.3). Although elastic behaviour of bamboo was not experimentally established in
this study, the bamboo tower is assumed to maintain linear-elastic behaviour for small tower-top
deflections during extreme wind loads.
7.5.5 The Optimal Tower Design
The above analysis has shown that bamboo towers can meet the IEC load and safety
requirements. The analysis has shown that lashings should be combined with the steel-bamboo
adhesive joints to design an optimum tower. It is concluded that bamboo tower is technically
feasible, although there are some inherent limitations of bamboo’s use, such as durability.
However, this limitation can be overcome by replacing the tower members periodically during
the life-span of the turbine. In summary, the specifications for the design of an optimum bamboo
tower are given table 7.4.
Table 7.4 Optimized design of the bamboo tower
Base distance between the tower legs, b
1.85 m
Diameter of tower legs, D (mm)
65
Diameter of horizontal bracings (mm)
65
Wall thickness of bamboo (mm)
6
Tower-top width (m)
0.15
Length of leg sections (mm)
1505
Tower mass (kg)
37
Tower-top deflection (mm)
193
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7.5.6 Tower Manufacture and Installation
Bamboo towers can be easily built with simple tools and minimum workmanship. It is
recommended that straight bamboo sections should be dried properly, such as below 20%
moisture content, to achieve good mechanical strengths. To improve durability and minimize the
effects of weathering, bamboo sections should be painted. Fabrication of adhesive joints is the
major task in the design and manufacture of bamboo tower. The procedure for fabricating the
steel-bamboo adhesive joints is a reasonably simple task as described in Chapter 3, which can be
carried out with simple tools. After the joints are fabricated, assembly and erection of tower
involves joining of the bamboo sections using lashings. The design of foundation can be done as
described in Chapter 6.
7.5.7 Comparison with the Steel Lattice Tower
Both the steel and bamboo towers were designed to satisfy the loads and safety requirements of
IEC. The basic differences in design between the bamboo and steel lattice towers, besides its
economic merits, are summarized below.
•
The bamboo tower is relatively light (37 kg) when it is compared to the equivalent steel
tower (112 kg).
•
In the design of bamboo tower, the available size of bamboo columns is an important
design factor for the selection of base distance between the tower legs. To minimize the
loads on legs, base distance should be increased.
•
The minimum base distance of the bamboo tower is determined by the buckling strengths
of bamboo sections to satisfy the load requirements of the tower, whereas for steel tower,
minimum base distance can be chosen because any size of steel pipes can be obtained to
satisfy the load requirements.
•
In bamboo towers, joining of bamboo sections is a major challenge, whereas for steel
tower, there are many options for joining the tower members.
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•
In bamboo tower, the joints should be sufficiently strong to ensure that buckling is a
major design criterion.
•
In steel tower, cross-bracings of smaller size may be used to increase stiffness with
minimum increase of drag on tower, whereas in bamboo tower, smaller bamboo sections
for cross-bracings could not be obtained in practice; so there is significant drag on the
tower if full bamboo sections are used.
•
In steel tower, durability of material is not a major challenge, whereas in bamboo towers,
it is a major challenge and requires periodic replacement of tower members to meet the
turbine service life of 20 years.
•
For the same loads, the bamboo tower requires larger base distance than the steel lattice
tower. Also, the tower-top deflection is higher.
•
The construction of bamboo tower is very simple than the steel lattice tower.
7.6 Economic Feasibility
Bamboo is an extremely cheap structural material. On a market survey conducted by the author
in order to determine the current material price of bamboo in Nepal, a typical freshly harvested
bamboo pole, which is about 8-12 m long, costs about $1.5 - $3 in urban areas. The whole
bamboo pole could not be utilized due to the dimensional variability along the length.
Consequently, several bamboo poles may be required to make the tower components. On a rough
calculation, about 8-12 bamboo poles would be required for building the whole lattice tower. In
average, the material costs of the bamboo would be about $20-$30. In addition, there are also
other material costs, such as adhesives, steel connectors, and ropes etc that drive the capital costs
of the tower. The same steel connectors can be used for the whole life-span of the wind turbine.
The only materials needed during replacement of tower members are the bamboo and adhesives.
The cost for adhesives is estimated about $30-$40 and that for steel connectors is about $15-$20.
Altogether, estimated material cost of bamboo tower is about $100. Ideally, there are no
manufacturing costs besides assembly of tower at the site. For a 20 year life-span the cost of
bamboo tower would be around $400-$ 500 assuming that the bamboo is replaced 4-5 times.
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In the context of tower design, material cost is one of the several cost components, such as labor,
transportation, erection, repair and maintenance etc. It is crucial to evaluate the consequences
and every aspects of how the structure is built and maintained over the desired life-span in
practical contexts. Such systems level costs can be examined mainly in terms of design costs,
material and foundation costs, build time and labor costs, and repair and maintenance costs.
As discussed above, the purchasing cost of bamboo is very low when it is compared to steel.
Currently, steel costs about $2.7-$3/kg in Nepal [63]. So the material cost of an equivalent steel
lattice tower weighing 150 kg, described in Chapter 6, is about $405-$450. Moreover, the
production of lattice tower (e.g. welding) adds more cost in the total cost of the tower. The
designed steel lattice tower can be produced approximately at $700-$800. However, the
production and transportation costs of steel towers depend upon the contexts where it is
designed.
The manufacturing sequence for bamboo towers is very short and simple. The towers can be
built and assembled quickly with minimum use of workmanships, from design to installation.
Among others, the main drawback of bamboo tower is the low durability, which can be
addressed only by periodic replacement of tower members and use of protective coatings over
the designed life-span (generally 20 years) of the wind turbine. As bamboo is a very cheap
material, replacement of the whole tower every three to five years is not likely to reverse the
costs of steel and bamboo towers.
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Chapter 8
SUMMARY, CONCLUSIONS AND FUTURE WORK
8.1 Summary of Thesis
The core objectives of this thesis were: 1) to investigate the feasibility of bamboo tower for small
wind turbines and 2) to develop an easy design procedure for the triangular lattice towers.The
context of this thesis is the developing countries, such as Nepal, where small wind turbines are
recognized as appropriate technologies to produce electricity, particularly in off-grid remote
areas where transportation and cost of towers are the main challenges.
Chapter 2 presented a brief overview of main design types of towers for small wind turbines and
indicated the economic competitiveness of the lattice towers, examined the mechanical
properties of bamboo and various joining techniques, and introduced the type of adhesive joint
intended for the design of bamboo lattice towers.
Chapter 3 described the experimental work on mechanical properties of the bamboo and steelbamboo adhesive joint and summarized the main results. The buckling and compression
strengths and elastic properties of bamboo were experimentally determined. The buckling
strength of bamboo columns was characterized in terms of buckling strength and slenderness
ratio. The results of the experiment showed a considerable variation in buckling and compression
strengths. To account for the variation of properties, all the values were computed at 95%
confidence level, as required by the International Electrotechnical Commission (IEC) for the
design of wind turbine components. It was shown that a considerable variation on buckling and
compression strengths was observed when compared to the measured data and the 95%
confidence level values. The buckling strength of bamboo was found in the range of 23 MPa -60
MPa for different sizes of bamboo columns. The compressive strength was found in the range of
51 MPa-78 MPa. The elastic modulus in compression and Poisson ratio were determined as
16.32 and 0.33 respectively. In addition, the characteristic values of the pull-out strengths of a
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specific size of steel-bamboo epoxy joint, intended for connecting the leg sections in bamboo
tower, was experimentally determined.
Chapter 4 discussed the load and safety requirements of IEC for small wind turbines,
summarized the main loads acting on the tower, their computation methods and presented a brief
overview of design methodology and standards for assessing material strengths. It was concluded
that extreme load case H of the simple load model of International Electrotechnical Commission
is an important design criteria for the tower design. Fatigue load gives relatively low bending
stress on the tower than the extreme wind load. So fatigue was not considered in the design and
analysis of the tower.
Chapter 5 introduced the design aspects of triangular tower as a low-cost alternative to the
monopole towers. To simplify the design process of the triangular tower, it was modeled as a
tripod consisting of three legs only, which allowed formulation of analytical solutions for
stresses on tower legs and tower deflection. In the tripod model of the lattice tower, the overall
dimension of the tower is governed by the base distance, tower height, imposed loads, crosssectional dimensions of the legs, and buckling strength of the leg sections. ASCE, Eurocode, and
AISC equations were used in the analytical solutions to determine the minimum dimensions of
the tower legs that are safe against buckling. This defines the basic geometry of the tower in
terms of base distance, tower height, imposed loads, and diameter of tower legs. Using the
results of tripod analysis, the design of lattice tower is possible. To extend this analysis to a more
accurate analysis, finite element modeling procedure for the lattice tower using the software
package ANSYS APDL has been described.
In Chapter 6, the results of tripod analysis were checked with finite element analysis. It was
shown that the tripod model gives approximately the stresses on legs and tower-top deflections.
The results of tripod analysis for stresses and tower-top deflections were found more accurate
when compared to the lattice tower with horizontal bracings. A design example of a 12 m high
steel lattice tower for a 500W wind turbine was also presented, which was intended for
comparison with the bamboo lattice. The tower consists 6 sections, to be designed with circular
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steel hollow sections. The design procedure established in Chapter 5 was implemented to
illustrate the design process. The design of tower was based on the load assumptions of a 500 W
wind turbine at an extreme wind speed of 50 m/s. First, the tripod analysis was carried out to
obtain optimum leg diameters for various base distances using ASCE and Eurocode 3 guidelines.
Then, the finite element analysis of different tower models using ANSYS APDL was carried for
the tripod and the lattice towers. Two design examples of lattice tower models, towers with only
horizontal bracings and towers with both horizontal- and cross-bracings, were examined to
minimize the tower mass. The effect of cross-bracing was investigated by comparing the two
tower models in terms of stiffness and mass of the tower.
In Chapter 7, the design of a 12 m high triangular bamboo lattice tower for a 500 W wind turbine
was presented. The aim of the design was to examine the validity of bamboo as a potential
structural material for small wind turbine towers. The basic design of the bamboo tower consists
of 8 lattice sections, to be designed with bamboo columns. The design procedure established in
Chapter 5 was implemented to illustrate the design process. The tower design was based on the
load cases of a 500W wind turbine in accordance with the load and safety requirements of
IEC61400-2 and the experimentally determined mechanical properties of the bamboo and the
strength of steel-bamboo adhesive joint.
8.2 Conclusions
Proposing a new and alternative material for wind turbine towers is a multidisciplinary design
task, requiring a lot of work on basic design process and assessment of different material
properties; so this thesis has only considered the fundamental design and safety requirements of
the tower. To justify the use of bamboo in wind turbine towers, this thesis has proposed the
triangular type of lattice tower design.
The 12 m high triangular lattice tower, proposed in this thesis, has been modeled as a tripod to
formulate the analytical solutions for the stresses and tower deflections. Analytical equations for
determining the forces and stresses in tower legs were formulated. The analytical equation for
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the tower deflection was derived by assuming the tripod as a cantilevered equivalent beam of
three legs. The tripod model combines the imposed loads on the tower, the tower height, the base
distance between tower legs, and the cross-sectional dimensions of tower legs. The stresses on
legs are used to assess the buckling strength of the towers using appropriate standards provided
by ASCE, Eurocode, and AISC. The ASCE and Eurocode 3 equations showed consistent results.
AISC equation gave slightly higher values of leg diameters. The AISC equations is safer than the
ASCE and Eurocode 3.As is clear from the tripod model, the analytical solutions served as a
reference for the initial tower design, which could be extended to finite element analysis.
ANSYS APDL was used as a finite element analysis tool to check the validity of the tripod
model, which is intended for basic tower design. The comparison of the results of analytical,
numerical, and finite element analysis has demonstrated that the tripod analysis can accurately
give the basic dimensions for the lattice tower with and without horizontal bracings. It was also
shown that the drag force on bracings increase the stresses on legs, however, cross-bracings
significantly increase the tower stiffness.
To assess the feasibility of tower or the structural integrity under extreme wind loads, the tower
design was based on experimentally determined mechanical properties of bamboo and the loads
and safety requirements of IEC 61400-2. For this purpose, a 12 m high bamboo tower for a 500
W wind turbine was designed using the preliminary results of tripod analysis. The results of
material testing showed that bamboo possesses good buckling resistance that meets the load
requirements of small wind turbines. During the experimental work, it was found that the desired
thickness of bamboo could not be found in practice. Therefore, the design of bamboo tower
should be based on the selection of minimum external diameter to reduce the drag on tower
while meeting the buckling resistance of the tower legs. In addition, steel-bamboo adhesive joint,
combined with conventional lashing, has been proposed for connecting the bamboo sections in
the lattice tower. To address the challenge of low durability of bamboo, periodic replacement of
tower members has been proposed.
130
The results of tripod analysis were used to design the bamboo tower. The tripod model gives the
direct relation of buckling stress and base distance which is very useful in the selection of
bamboo diameters. It was found from the tripod analysis that for reducing the compressive stress
on tower legs and pull-out load on joints, the base distance should be increased. It was shown
that bamboo tower requires larger base distance to withstand the tower loads than the steel tower.
As the tensile load at the base of tower is significant, the tower design required 2.7 m base
distance if only adhesive joints were considered. For the same tower with combined lashing in
the joints, the required base distance is 1.85 m. It was concluded that the geometry of the
bamboo tower is governed by the diameter of the bamboo, joint strength, and base distance.
Subsequent finite element analysis was carried out for the same tower to evaluate the buckling
strength of the tower legs. The results of finite element analysis for a 12 m high bamboo tower
were compared to the results of tripod analysis and it was found that buckling stresses on legs
and tower deflections could be approximately determined using the analytical equations, which
further validates that analytical equations can be used for the basic design of the bamboo tower.
Furthermore, a comparison made with the equivalent steel tower indicates that bamboo tower is
an extremely economical option. The results of this study shows that steel towers are about 4
times heavier. Bamboo towers can be constructed easily than any known towers for small wind
turbines. The implications of these results show that bamboo towers are relevant in remote
regions, where low-cost towers could be easily build. The results of this study justifies that the
design of bamboo towers is technically feasible.
8.3 Future Work
As an extension of this study, the author proposes the following:
The short-term and long-term effects of dynamic loads and weathering on the mechanical
properties of bamboo, the adhesive joint, and the structure are not fully known, which must be
investigated to further build confidence on designing bamboo tower.
131
A detailed design and analysis of the joint requires an extensive experimental work on
characterization of the steel-bamboo adhesive joint. Parametric studies on the relationship
between strength, adhesive thickness, and overlap length of the steel-bamboo adhesive joints are
recommended.
132
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138
APPENDICES
APPENDIX A: CALCULATION OF LEG DIAMETER TO AVOID BUCKLING
A.1. Calculation of D using ASCE (1990) guidelines
For the calculation of optimum D, following constants have been used:
Table A.1 Constants used in the Calculation
Description
Value
Tower height, h (m)
12
Wall thickness of steel, t (mm)
3
Capacity factor, CF
0.5
Axial yield stress: Fy (MPa)
255
Bending yield stress: Fy (MPa)
255
Assuming that the possible sizes of steel pipes have typical diameters ranging from 20 -100 mm
with wall thickness, t =3 mm, using ASCE equations (5.10) and (5.11) for D/t, the allowable
axial stress (Fa) was determined as 255 MPa.
Using equation (6.1) and appropriate constants, optimum D is calculated from:
[550 + 8.82(𝐷𝐷𝐷𝐷 − 𝑡𝑡 2 )]ℎ 𝐶𝐶𝐶𝐶
3𝜋𝜋(𝐷𝐷𝐷𝐷 − 𝑡𝑡 2 )𝐹𝐹𝑎𝑎 �ℎ2 + 𝑏𝑏 2 /3
+
2(25800 + 584𝐷𝐷)𝐶𝐶𝐶𝐶
√3 𝑏𝑏𝜋𝜋(𝐷𝐷𝐷𝐷 − 𝑡𝑡 2 )𝐹𝐹𝑏𝑏
The resulting values for D are given in Table A.2.
139
=1
Table A.2 Optimum values of D (t =3 mm)
base distance, b (m)
Leg diameter, D (mm)
1
64.12
1.2
45.23
1.4
35.27
1.6
29.06
A.2. Calculation of D using Eurocode 3
To calculate the optimum leg diameter, it is first necessary to calculate the allowable critical
buckling strength. Eurocode 3 equations (5.14) – (5.18) were used to determine the critical
buckling strength. It was assumed that the diameters of steel pipes for tower legs fall in the range
10 mm -100 mm with wall thickness of t = 3mm.
For a 10 mm diameter pipe, the critical meridional buckling stress is given by equation (5.14):
𝜎𝜎𝑥𝑥𝑥𝑥𝑥𝑥 = 0.605𝐸𝐸𝐶𝐶𝑥𝑥 𝑡𝑡⁄𝑟𝑟
= 0.605𝐸𝐸𝐶𝐶𝑥𝑥
The unknown Cxb is calculated as follows:
Non-dimensional length parameter for a 2003 mm long pipe is calculated from equation (5.15):
𝑤𝑤 = 𝑙𝑙 ⁄�(𝐷𝐷 − 𝑡𝑡)𝑡𝑡⁄2
= 617.18
Equation (5.16) gives:
𝐶𝐶𝑥𝑥 = max(0.6,1 + 0.2 �1 −
2𝑤𝑤𝑤𝑤
𝑟𝑟
��𝐶𝐶𝑥𝑥𝑏𝑏 )
For clamped-clamped end conditions in lattice towers, 𝐶𝐶𝑥𝑥𝑥𝑥 = 6
140
𝐶𝐶𝑥𝑥 = 0.6
The “meridional imperfection reduction factor”, αx , is given by equation (5.17):
𝛼𝛼𝑥𝑥 = 0.62/[1 + 1.91(𝑤𝑤𝑘𝑘 ⁄𝑡𝑡)1.44 ]
where, 𝑤𝑤𝑘𝑘 = √𝑟𝑟𝑟𝑟⁄𝑄𝑄 and Q is the fabrication quality factor given in Table 5.1. Using the normal
fabrication quality class, Q is 16. Now, 𝑤𝑤𝑘𝑘 = √𝑟𝑟𝑟𝑟⁄𝑄𝑄 = 0.205. The value of imperfection factor,
αx, was determined as 0.6.
Now,
𝜎𝜎𝑥𝑥𝑥𝑥𝑥𝑥 = 0.605𝐸𝐸𝐶𝐶𝑥𝑥 𝑡𝑡⁄𝑟𝑟 =62.22GPa
The value 62.22 GPa is multiplied with αx to get the critical buckling stress, 𝜎𝜎𝑐𝑐𝑐𝑐 = 37.33 GPa.
According to Eurocode 3, if the relative slenderness ratio, λ = �𝐹𝐹𝑦𝑦 ⁄𝛼𝛼𝑥𝑥 𝜎𝜎𝑥𝑥𝑥𝑥𝑥𝑥 ≤ 0.2 the
characteristic buckling strength is equal to the yield strength, Fy. Here, λ = 0.082 ≤ 0.2, so the
characteristic buckling strength of the assumed pipe is equal to the yield strength, 255 MPa.
Similarly, it was found that 2003 mm long steel pipe with D =100 mm and t =3 mm has the
characteristic buckling strength of 255 MPa. It is concluded that Eurocode 3 and ASCE give the
same optimum values of D for the range of assumed leg dimensions.
A.3 Calculation of D using AISC equations
For round hollow structural sections (HSS), AISC 360-05 equations (5.19-5.21) give the axial
buckling strength. For the assumed range of leg dimensions (e.g. 20 mm-100 mm), the sizes of
leg sections fall under the category of compact sections for which𝐷𝐷/𝑡𝑡 ≤ 0.11𝐸𝐸/𝐹𝐹𝑦𝑦 .
The critical buckling strength (𝐹𝐹𝑐𝑐𝑐𝑐 ) of compact sections is computed from equation (5.20):
141
Where, 𝐹𝐹𝑒𝑒 = 𝜋𝜋 2 𝐸𝐸/(𝑘𝑘𝑘𝑘 ⁄𝑟𝑟)2 and k =1 for the tower legs.
Now, 𝐹𝐹𝑒𝑒 = 𝜋𝜋 2 𝐸𝐸/(4𝑙𝑙 ⁄�𝐷𝐷2 + (𝐷𝐷 − 2𝑡𝑡)2 )2
For the range of leg dimensions considered, 𝐹𝐹𝑒𝑒 < 0.44𝐹𝐹𝑦𝑦 .The elastic buckling occurs and the
critical buckling stress is given by equation (5.21) as:
𝐹𝐹𝑐𝑐𝑐𝑐 = 𝐹𝐹𝑎𝑎 = 0.877𝐹𝐹𝑒𝑒
Now using this equation in equation (6.1), with BF of 2, the resulting equation is:
[550 + 8.82(𝐷𝐷𝐷𝐷 − 𝑡𝑡 2 )]ℎ 𝐶𝐶𝐶𝐶
3𝜋𝜋(𝐷𝐷𝐷𝐷 − 𝑡𝑡 2 )𝐹𝐹𝑐𝑐𝑐𝑐 �ℎ2 + 𝑏𝑏 2 /3
+
2(25800 + 584𝐷𝐷)𝐶𝐶𝐶𝐶
√3 𝑏𝑏𝜋𝜋(𝐷𝐷𝐷𝐷 − 𝑡𝑡 2 )𝐹𝐹𝑏𝑏
=1
Solving the above equation for D using MATLAB, following optimum values of D were
obtained:
Table A.3 Optimum values of D (t=3 mm)
base distance, b (m)
Leg diameter, D (mm)
1
65.34
1.2
47.23
1.4
37.12
1.6
31.08
The summary of the results for the optimum values of D from the three standards are presented
in Table A.4.
142
Table A.4 Comparison of optimum values of D (t = 3 mm)
base distance, b (m)
ASCE and Eurocode 3
AISC
D (mm)
D (mm)
1
64.12
65.34
1.2
45.23
47.23
1.4
35.27
37.12
1.6
31.08
31.08
143
APPENDIX B: NUMERICAL SOLUTIONS FOR DEFLECTIONS OF STEEL TOWER
B.1 MATLAB program for the tower-top deflection
The following program is the modified version of the monopole tower deflection program
documented in [7].
function TowerDef (b, D, t)
% d2x/dy2=M/(EI) is rewritten as two 1st order equations to use
% Matlab's Runge-Kutta routine ode45 for the deflection
%Function argument
%b = base distance between legs (m)
%D = leg outer diameter (mm)
%t = leg thickness (mm)
D=D/1000; %Diameter of tower leg (m)
t=t/1000; %thickness in m
R1=(D-2*t)/2; R2=D/2; % inner and outer leg radius respectively
h=12;%height of tower (m)
R1R22=(R1^2+R2^2);%R1^2 +R2^2 for tower leg (m^2)
q=0.5*1.3*1.35*1.225*D*50^2; %drag per unit length (N/m)
E=200e09; % Elastic modulus (Pa)
EA=E*(R2^2-R1^2)*pi; % E x cross-sectional area of leg (m^2)
F=2150; % Turbine thrust (N)
[Y, DEFL] = ode45(@(y,x) defderiv(y,x,F,q,R1R22,b,D,h,EA),[0 h],[0 0]);
[Y 1000*DEFL] % Output deflection in mm
end
function dx = defderiv(y,x,F,q,R1R22,b,D,h,EA) % Function for integration
dx=zeros(2,1);
dx(1)=x(2); % deflection
dx(2)=1/EA*(2*F+3*q*(h-y)).*(h-y)./(3/2*R1R22+((b-D)/h*(h-y)+D).^2); % equation (5.25)
end
144
B.2 Results of tower-top deflections
Table B.1 Constants used in the program
Description
Values
E (GPa)
200
F (N)
2150
t (mm)
3
Table B.2 Results of tower deflections
b (m)
D (mm)
Tower-top deflection (mm)
1
64
90
1.2
45
85
1.4
35
78
1.6
29
72
145
Figure B.2 Tower deflection (m) for b =1.2 m and D =45 mm
146
APPENDIX C: NUMERICAL SOLUTIONS FOR DEFLECTIONS OF BAMBOO TOWER
C.1 MATLAB program for the tower-top deflection presented in B1 was used with the following
constants.
Table C.1 Constants used in the program
Description
Values
E (GPa)
16
F (N)
2150
t (mm)
6
C.2 Results of tower-top deflections
Table C.2 Results of numerical solutions
b (mm)
D (mm)
Tower-top deflection (mm)
1.6
70
232.32
1.85
65
186.28
2.15
60
148.47
2.6
55
110.17
147
Figure C.2 Tower deflection for b =1.85 m, D =65 mm
148