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Computer Aided Engineering:
Midterm Design Project
Team: RotorStorm©
Enea Dushaj
Louis Lane
Chris Mignano
Yuval Philipson
The Cooper Union for the Advancement of Science and Art
ME 408: Intro to Computer Aided Engineering
Professor Scott Bondi
Table of Contents
Assignment Overview 3 Design Assumptions 3 Design Constraints and Parameters 4 Solution Concept Overview 5 Material Properties 5
Final Design Specifications 6 Design Sketches 8
Tower Operating Conditions & Load Cases 11 Overview of Loads 11 Operating Conditions 13 Load Case Configurations 14 Normal Operating Conditions Detailed Summary 15 Normal Operation Conditions Manual Calculations 16 Extreme Operating Conditions Detailed Summary 18 Extreme Operating Conditions Manual Calculations 19
Modeling 21 Preliminary Finite Element Model of Tower 22 Cable Tension Hand Verification 23 Finite Element Model of Tower 24 Buckling Modes 37 Fatigue Considerations 40 Modal Finite Element Model of Tower 40
Thermal Finite Element Model of Tower 44 Thermal Effects 44 Thermal Expansion of Tower 45
Base Flange 47 Structural Finite Element Model 49 Thermal Model 56 Mounting Hardware 59
Design Time Estimate & Conclusion 59
Appendices 60 Appendix A: Variables & Parameters 60 Appendix B: Formulae Used 61 Appendix C: Initial Modeling 62 Appendix D: MATLAB Scripts for Preliminary Calculations 64
Works Cited 68
1
Assignment Overview
Enercon has tasked RotorStorm© with designing the tower and support structure for their new on
shore 9.25MW wind turbine. The tower is expected to withstand a variety of loads and conditions as
defined by the contractor. Along with the expected loads, the design must account for potential failure
modes of the wind turbine. A successful design has been defined as one which endures all specified loads
and conditions with a factor of safety of 1.65 for the stresses sustained in the entire tower, factor of safety
of 4 for all connection hardware, and a buckling factor of safety of 4 for the tower under any loading
condition.
Design Assumptions
In order to ensure that the specified design will withstand all loading scenarios, it is assumed that
the air is at its highest possible density, which occurs at -10 °F and at a pressure of 101,325 Pa. It is also
assumed that the turbine base is installed at sea level to allow for the highest pressure air possible. These
assumptions were made to ensure that even under unusual weather phenomena, the structure will meet all
design criteria.
The unusual weather phenomenon considered for the design is one in which the structure is
initially at 120 °F at which point the air temperature rapidly drops to -10o F as may occur in a desert, or
grassland climate. In this scenario, the structure would maintain its steady state temperature of 120 °F due
to its massive size and material properties, while the air around the structure has cooled dramatically. In
this case, the structure would be at its maximum temperature, while experiencing wind loads from high
density air at -10 °F, until the wind turbine cools.
Additionally, the following assumptions apply to the final design: the air density is constant over
the height of the tower, the material properties remain constant over repeated loading cycles and over
time, the final design is reasonably manufacturable, and all load cases are taken to be at steady state.
2
Design Constraints and Parameters
Table 1: Provided Design Constraints and Parameters
Description Value
Height of Nacelle 500 ft
Weight of Nacelle 185,000 lbf
Weight of Each Blade 12,500 lbf
Blade Length 216 ft
Average Blade Width 18 ft
Cd of Blade 0.82
Cd of Tower (Modeled as Long Cylinder) 1.2
Number of Blades 3
Distance of Plane of Blades from Tower Central Axis 21 ft
Distance of Blade CG from Center of Rotation 81 ft
Maximum Operating Wind Speed 60 mph
Maximum Site Wind Speed 105 mph
Operating Range 6-12 rpm
Power Generation 9.25 MW
Ambient Temperature Range -10℉ : 120℉
Maximum Operating Deflection 1 ft
Maximum Critical Deflection 1.35 ft
Stress Factor of Safety on Tower and Flange 1.65
Stress Factor of Safety on Fasteners 4
Buckling Factor of Safety 4
The tower is required to be 500 ft. tall and must have a diameter of less than 42 ft. above an
elevation of 250 feet in order to allow for clearance with the swept area of the blades while spinning. In
order to meet the specified criteria, minimize weight, and manufacturing effort while satisfying standards
of safety, a tapered structure with supporting cables was designed with the dimensions summarized in
Table 2.
3
Solution Concept Overview
The wind turbine structure designed by team RotorStorm© provides a safe and cost efficient
solution to the design constraints by keeping the overall weight of the structure to a minimum. The weight
of the tower is kept to a minimum by utilizing a tapered tower design and external support cables in order
to mitigate deflection from wind loads.
The tower also features a tapered base plate structure to increase the tower’s stability. The base
plate structure is welded to the tower and bolted to the concrete foundation provided by Enercon. The
base plate structure is bolted to the foundation using eighty, 3 in. diameter bolts positioned along lines
spaced 9 degrees apart radially, with 2 bolts on each line. (see Figure 40,41,42 on page 47-48). This bolt
pattern was chosen as a means of optimizing spacing between bolt locations while also minimizing the
number of bolts required to support the structure. This design also features steel cables that act as
supports, reducing deflections to meet design requirements in all loading scenarios and conditions.
The wind turbine’s base structure covers an area of 1195 sqft., while the total footprint, including
the steel support cables, covers an area of 125,000 square feet, or 2.87 acres. Based on data collected by
the National Renewable Energy Laboratory, on average, a wind turbine in the U.S. will require 0.75 acres
per MW of power generated. Based on this average, RotorStorm© 's wind turbine requires 59% less area
to operate than the national average for an equivalently sized wind turbine, minimizing the environmental
impact.
Material Properties
The proposed tower design utilizes A514 steel which is a commonly used structural steel alloy. It
is used for its high ultimate strength and high modulus of elasticity. While it is more expensive than many
structural steels, its superior strength to weight ratio allows for lighter designs that are cheaper to
manufacture than other steel alloys of lower quality.
4
Final Design Specifications Table 2: Tower Geometry and Properties
Description Value
Material A514 Steel
Tower Height 500 ft
Average Cross Sectional Area 19.3 sq ft
Top Tower Diameter 35 ft
Bottom Tower Diameter 39 ft
Tower Wall Thickness 2 in
Tower Weight 2360 Tons
Tower Footprint 1195 sq ft.
Fully Assembled Wind Turbine Footprint
2.87 acres
Table 3 - Base Specifications
Description Value
Material A514 Steel
Outer Diameter 39.85 ft
Inner Diameter 37.96 ft
Max. Height 4 in
Table 4 - Cable Specifications
Description Value
Cable Specification ASTM A586-5 3/4
Cable Construction 1x19
The gross metallic area of the cable was accounted for in all calculations. ASTM A586 cable in
commonly used for bracing, suspension bridges, and suspended roofs.
5
Table 5 - Mounting Hardware Specifications
Description Value
Material A354 BD Steel
Number of Bolts 80
Diameter of Bolts 3 in
Tensile Strength 150,000 psi
Maximum Load per Bolt 256,000 lbf
Washer Diameter 5.5 in
Table 6 - Material Properties of A514 Steel
Property Value
Modulus of Elasticity psf.97x103 9
Poisson’s Ratio 0.29
Density 490 lbf t3
Yield Strength psf.61x101 7
Coefficient of Thermal Expansion .3x101 −5 1K
Table 7 - Material Properties of A354 BD Steel (Used in bolts)
Property Value
Modulus of Elasticity psf.97x103 9
Poisson’s Ratio 0.29
Density 490 lbf t3
Tensile Strength psf.16x102 7
Coefficient of Thermal Expansion .3x101 −5 1K
6
Design Sketches
Figure 1: Tower Drawing
7
Tower Cross Section
Through iterations of manual calculations, the cross section data were compiled in ANSYS
APDL while setting up Finite Element Analysis (FEA) simulations shown in following sections. Cross
sectional area, inner and outer diameters, and area moments of inertia for hand calculations were taken
from the resulting geometries. The main focus of the design was to optimize the area moment of inertia to
prevent the wind turbine from deflecting 1.35 feet in any direction.
Figure 2: Tower Base Cross Section [ft]
A bottom cross sectional diameter of 39 ft with thickness of 2 in. was determined to be the best
choice for satisfying all requirements, keeping the tower’s footprint to a minimum while providing an
area moment of inertia high enough to prevent large scale deflections. Having a large area moment of
inertia at the base allows for reduction of stresses at the base of the tower, and the 39 ft diameter avoids
having the tower below the blades at any point in the design.
8
Figure 3: Tower Top Cross Section [ft]
A top cross sectional diameter of 35 ft with thickness of 2 in. was chosen for the upper bound of
the tapered tower to reduce overall weight while maintaining the structural stability necessary to keep
deflections of the nacelle smaller than 1.35 ft while using cables of reasonable size.
Support Cables
In order to meet the deflection criteria as specified by the client, RotorStorm© suggests utilizing
steel cables to provide additional support to the turbine tower. Cables directly increase the tower’s ability
to resist bending as opposed to additional internal structures within the wind turbine tower. The design
incorporates a total of eight, 5.75 in. diameter, steel cables in sets of 2, anchored 90° apart at a distance of
250 feet from the centerline of the tower in order to provide external support. In the simulation, the cables
were modeled as having the equivalent cross sectional area of a 5 inch diameter solid steel rod. This is
equivalent to the gross metallic cross sectional area of 5.75 in. diameter steel cables specified to be used
by RotorStorm© Each cable weighs approximately 15 tons - across 8 cables this sums to ~125 tons. This
is much lighter than a 750 ton steel plate that could provide equivalent support.
9
Tower Operating Conditions & Load Cases
Overview of Loads
The tower experiences a torque about the central axis of the nacelle as a result of the power being
generated. It is assumed that maximum power generation (9.25 MW) occurs when the blades are spinning
at the highest operating rotational velocity (12 RPM).
The tower also experiences wind loads, the first of which is the sum of wind loads on the faces of
the blades and acts perpendicular to the top of the tower. An additional wind pressure acts on the frontal
area of the tower and is modeled as a transverse, perpendicular line load along the length of the tower.
The weight of the blades and nacelle acts downwards through the central axis of the tower, and
the blades being offset from the tower’s central axis creates a moment about the top of the tower, opposite
the moment generated by wind loads. The weight of the tower also contributes another substantial load to
be considered. The wind load on the cables is assumed to be negligible.
Finally, the tower is modeled as a cantilever beam with a fixed and free end. Therefore, a reaction
force in the x and y direction (Rx, Ry) act at the base of the tower along with a reaction moment (Mz). The
tower experiences the above loading conditions under both normal operating and extreme weather
conditions, however, no moment from power generation is acting on the tower under extreme weather
conditions since the blades do not spin at wind speeds past 60 mph. Additionally, loading in each case
differs in magnitude depending on respective weather conditions. A free body diagram depicting these
loads is shown in Figure 4.
10
Figure 4: Tower Loading Conditions
11
Operating Conditions
Normal Operating Conditions
Under normal operating conditions, the wind turbine is expected to be operating in steady state
between 6-12 rpm and operating at the previously specified temperature conditions. Thus, the normal
operating conditions were analyzed with the air at 1 atm of pressure, at -10°F with a wind speed of 60
mph (the maximum operating wind speed of the turbine), and the tower at 120° F. These conditions were
used to ensure that in the worst case scenario for deflection (under normal operating conditions), the
structure would be able to withstand wind loads based on the earlier assumption that a rapid air
temperature drop could occur.
It is assumed that the maximum power generation given (9.25 MW) occurs when the blades are
spinning at their fastest rotational speed of 12 RPM. Additionally, load cases when the wind and structure
are both at extreme cold (-10o F) and extreme heat (120o F) were analyzed as well to ensure all boundary
cases were considered.
Extreme Operating Conditions
Under extreme conditions, the tower is expected to experience wind speeds significantly higher
than operating speeds, up to 105 mph. Due to safety concerns, the wind turbine will not be allowed to
operate if wind speeds exceed 60 mph. Despite this, the wind turbine must be capable of withstanding the
forces exerted on the structure by wind speeds of 105 mph. Under extreme weather conditions, the nacelle
“weather vanes”, and turns such that it is always facing into the general direction of the wind. This is to
prevent any excess perpendicular loading of the blades and to allow for predictable behavior from all load
cases and wind blowing in all directions. This simplifies the analysis of the tower and allows for power
generation when the wind is blowing in any direction.
It was determined early on that the limiting factor in the design of the tower was not stress or
buckling failure. The limiting factor is the deflection of the tower in extreme weather conditions.
Therefore, what is defined as the “worst case scenario” in this report refers to the conditions that would
result in the highest deflection of the tower to ensure that this design criterion is met for any load case.
12
Load Case Configurations
The models were simulated with 2 different support cable configurations: one configuration
where only one set of the cables is in tension and lies in the plane in which the forces are acting, and
another case where the plane on which all the loads are acting on is rotated 90o to simulate two sets of
cables providing limited support to the tower. The configurations are summarized in Figure 5 below:
Figure 5: Diagram of load case configurations
13
Normal Operating Conditions Detailed Summary
The loads experienced by the tower under normal operating conditions are tabulated below.
Table 8 - Normal Operating Conditions Forces Applied
Normal Operating Conditions Loads
Weight of Nacelle 185,000 lbf
Weight of Blades (All 3 of them) 37,500 lbf
Weight of Tower 2360 tons
Moment from Weight of blades being offset 787500 lbf-ft
Wind Force on Blades 1.01 E+05 lbf (-10o F) 7.9 E+04 lbf (120o F)
Wind load line pressure on tower 500 lbf/ft (-10o F) 385 lbf/ft (120o F)
Power generation torque 4,004,000 lbf-ft
The results for the maximum deflections and stresses obtained from simulation in ANSYS APDL
experienced by the tower in all extreme temperature cases and in both load direction configurations under
normal operating conditions are tabulated below.
Table 9 - Normal Operating Conditions Deflections
Deflections Configuration 1 Configuration 2 Deflection (Worst Case)
Extreme Cold 0.42 0.39 L/1191
Extreme Heat 0.47 0.51 L/981
Worst Case Scenario 0.55 0.59 L/848
Table 10 - Normal Operating Conditions Stresses
Stresses Configuration 1 Configuration 2 FOS
Extreme Cold 1.5 E+06 1.2 E+06 10.7
Extreme Heat 6.4 E+05 6.9 E+05 23.3
Worst Case Scenario 7.1 E+05 7.7 E+05 20.9
The final tower design has a maximum deflection of L/848 and factor of safety of 10.7 under
normal operating conditions.
14
Normal Operation Conditions Manual Calculations
Load Calculation
To calculate the wind load, the density of air at the environmental temperature is used. Using this
constant density, the wind load could then be calculated. The drag coefficient of the blades is given to
0.82. The drag coefficient of the tower is assumed to be 1.2, the same as the drag coefficient of a long
cylinder. The air temperature is 249.8K in extreme cold and 322 K in extreme heat.
(1) .0027 ρ = PR Tair
= 101.325 kP a287.05 249.8KkJ
kg K* * * 1515.379 kg
m3
f t3slug
= 0f t3
slug
(2) (Wind Load)ρ Vw = 21 * d * Cd
2
(3) ρ 3 ) VF W ind = 21 * ( * ABlade * Cd
2
(4) (Torque from power generation)M P ower = ωP ower
(5) σ = IMy
(6) σ = AP
(7) τ = ItV Q
(8) otal StressT = IMy + A
P + ItV Q
(9) σAllowable = F OSσyield
(10) Elastic stretch of cable, where G=0.00000779 for a 1x19 cableE =D2F G
Maximum Deflection
The Maximum deflection is calculated by superimposing the solutions for the maximum
deflections of cantilever beams under different transverse loadings (distributed load along its length, point
load at its free end, moment about its free end).
(11) δMax = 3EI
F hW ind Blades 3
+ 8EIw hW ind T ower
4
+ 2EIM hBlades
2
+ 2EIM hP ower
2
Maximum Stress
The Maximum stress in the tower is determined by taking an imaginary cut at the base of the
tower and determining the internal axial and shear forces acting on the tower as well as the maximum
bending moment in the tower (which occurs at the base of the tower):
(12) WP = T ower + W Nacelle + 3 * W Blades
(13) V = wW ind T ower * h + F W ind Blades
(14) 21 f t) F M = − 3 * W Blades * ( + W ind Blades * h + 2w hW ind T ower*
2
+ M P ower
15
These internal loads were then used to find the maximum shear and axial stress acting in the
tower and superimposed to find the maximum stress in the tower:
(15) otal StressT = IMy + A
P + ItV Q
It is determined that the maximum stress in a straight 39 ft diameter annular beam with 2 in. thick
walls would be approximately 9 E+05 psf.
16
Extreme Operating Conditions Detailed Summary
Table 11 - Extreme Operating Conditions Forces Applied
Extreme Operating Conditions Loads
Weight of Nacelle 185,000 lbf
Weight of Blades (All 3 of them) 37,500 lbf
Weight of Tower 2360 tons
Moment from Weight of blades being offset 787500 lbf-ft
Wind Force on Blades 3.1 E+05 lbs (-10o F ) 1.01 E+05 lbs (120o F )
Wind load line pressure on tower 1443 lbf/ft (-10o F ) 1120 lbf/ft (120o F )
Power generation torque 0 lbf-ft
The results for the maximum deflections and stresses obtained from simulation in ANSYS APDL
experienced by the tower in all extreme temperature cases and in both load direction configurations under
normal operating conditions are tabulated below.
Table 12 - Extreme Weather Conditions Deflections
Deflections Configuration 1 Configuration 2 Deflection (Worst Case)
Extreme Cold 1.15 1.09 L/435
Extreme Heat 1.06 1.09 L/459
Worst Case Scenario 1.33 1.35 L/370
Table 13 - Extreme Weather Conditions Stresses
Stresses Configuration 1 Configuration 2 FOS
Extreme Cold 3.4 E+06 2.6 E+06 4.7
Extreme Heat 1.7 E+06 1.1 E+06 9.5
Worst Case Scenario 2.4 E+06 1.6 E+06 6.7
Under extreme weather conditions, the tower experiences the same types of loads in the same
directions, however, there is no torque from power generation and wind loads are larger than under
normal operating conditions. Therefore, the tower has a maximum deflection of L/370 and factor of safety
of 4.7 under extreme operating conditions.
17
Extreme Operating Conditions Manual Calculations
Load Calculation
(1) .0027 ρ = PR Tair
= 101.325 kP a287.05 249.8KkJ
kg K* * * 1515.379 kg
m3
f t3slug
= 0f t3
slug
(2) (Wind Load)ρ Vw = 21 * d * Cd
2
(3) ρ 3 ) VF W ind = 21 * ( * ABlade * Cd
2
(4) (Torque from power generation)M P ower = ωP ower
(5) σ = IMy
(6) σ = AP
(7) τ = ItV Q
(8) otal StressT = IMy + A
P + ItV Q
(9) σAllowable = F OSσyield
(10) Elastic stretch of cable, where G=0.00000779 for a 1x19 cableE =D2F G
Maximum Deflection
The maximum deflection is determined by superimposing the solutions for the maximum
deflections of cantilever beams under different transverse loadings (distributed load along its length, point
load at its free end, moment about its free end).
(11) δMax = 3EI
F hW ind Blades 3
+ 8EIw hW ind T ower
4
+ 2EIM hBlades
2
+ 2EIM hP ower
2
Maximum Stress
18
The Maximum stress in the tower is determined by taking an imaginary cut at the base of the
tower and determining the internal axial and shear forces acting on the tower and the maximum bending
moment in the tower (which occurs at the base of the tower): (12) WP = T ower + W Nacelle + 3 * W Blades
(13) V = wW ind T ower * h + F W ind Blades
(16) 21 f t) F M = − 3 * W Blades * ( + W ind Blades * h + 2w hW ind T ower*
2
These internal loads were then used to find the maximum shear and axial stress acting in the
tower and superimposed to find the maximum stress in the tower:
(15) otal StressT = IMy + A
P + ItV Q
Therefore, this tower has a maximum deflection of L/848 and factor of safety of 10.7 under
normal operating conditions, and a maximum deflection of L/370 and factor of safety of 4.7 under
extreme weather conditions. The limiting factor in this design is meeting the maximum deflection
requirements. Had more deflection been allowed, a much simpler, lighter, and cheaper design could have
been used, however to meet deflection requirements. The current design with a tapered tower and
supporting cables has been used to reduce weight as much as possible and meet the requests of the
customer.
19
Modeling The tower is modeled as a Beam 188 element type, with the support cables modeled as tension
only Link 180 elements. The tower is modeled as a cantilever beam with one fixed end and one free end.
The force from the wind on the blades is modeled as a single transverse load acting at the very top of the
tower. The wind force on the tower is modeled as a line load along the length of the tower. The weight of
the nacelle is modeled as a force acting downwards on the top of the tower. In manual calculations, the
weight of the tower is acting downward along the central axis of the tower. Gravity is modeled in ANSYS
APDL simulations as a constant 32.2 ft/s2. The weight of the blades is modeled as a downwards force
along the central axis of the tower along with a moment at the top of the tower caused by the offset of the
blades from the nacelle. The tower is fixed to the ground. The cables are not able to displace in x, y, and z
directions, but they are allowed to rotate about their hinged supports on the ground and on the tower.
A few different temperature cases and cable configurations are utilized to ensure all load cases are
covered. One model assumes the air providing the wind load is at -10o F and at 1 atm of pressure, while
the tower is at 120o F. This an absolute worst case scenario, in which deflections are amplified by the
increase in height of the tower, and the increased wind load by higher density air. Additionally,
recognizing that a tower at 120o F being loaded by air at -10o F is highly unlikely, two more models were
simulated: one where the air and tower are at 120o F, and one where the air and tower are at -10o F. It is
assumed that the tower is exactly 500 ft at 60o F, so all temperatures in each case are applied in APDL
with a reference temperature of 60o F (288 K).
Table 14 - Temperature Model Cases
Temperature Case Air Temperature (o F) Tower Temperature (o F)
Worst Case Scenario (Unlikely) -10 120
Extreme Heat 120 120
Extreme Cold -10 -10
20
Preliminary Finite Element Model of Tower
The first simulation in ANSYS APDL was completed with a 39 ft. constant diameter tower to
establish a gauge on the accuracy of the model as compared to the manual calculations.
Figure 6: Normal Operating Boundary Conditions
Figure 7: Normal Operating Condition Loads on 2” Thick 39’ diameter Tower [psf]
Simulating the structure in ANSYS APDL resulted in a simulated maximum stress of 8.2E+05
psf which is of the same order of magnitude as the manually calculated stress of 9.0 E+05 psf. This
establishes confidence in the FEA model. Since confidence in the model was established, the tapered
beam model with steel supporting cables was simulated in ANSYS.
21
Cable Tension Hand Verification
To verify that the model with supporting cables is valid, equilibrium equations under extreme
weather conditions were constructed. Manual calculations were done to determine the total reaction forces
in the vertical and horizontal directions, as well as the total moment assuming that the cables are pulling
on the tower with a tension of 100,000 lbf. Using the same assumption of 100,000 lbf as previously noted,
a simulation was performed modeling the tension as a force halfway up the tower, acting at 45o with
respect to the tower’s central axis. ANSYS APDL suggested reaction forces that are nearly identical to the
manual calculation, building confidence in the ANSYS model. A summary of the manually calculated
reaction forces and moments can be found in the table below. It is assumed that the cables would be
taught during installation, and that their structural stretch was already accounted for in the cable length.
Table 15 - Hand calculation checked with APDL model with assumed Tension = 100,000 lbf
Manual Calculation ANSYS APDL
Rx 8.9 e+05 lbf 9.6 e+05 lbf
Ry 5.1 e+06 lbf 5.0 e+06 lbf
Mz 3.0 e+08 ft*lbf 3.1 e+08 ft*lbf
A secondary simulation was run to determine the actual reaction forces and tensions in the cables
that the wind turbine would experience. The values predicted by ANSYS were then used as inputs for the
same manual calculation process previously noted to see if they matched. The ANSYS model suggested
an average tension in the two modeled cables of approximately 300,000 lbf. Using this value for manual
calculations resulted in the forces noted below. A summary of the reaction forces and moments based on
ANSYS input values and manual calculations can be seen below. Since the problem is statically
indeterminate, a verification of the model was required, and confidence in the model was established by
deriving the same reaction forces by hand as determined by simulation.
Table 16 - APDL Model Reaction Forces Verified with Manual Calculation Using Resultant
Tension Determined by APDL (T ~ 300,000 lbf)
ANSYS APDL Manual Calculation
Rx 6.0e+05 lbf 6.1e+05 lbf
Ry 5.5e+06 lbf 5.4 e+06 lbf
Mz 2.2e+08 ft*lbf 2.2e+08 lbf*ft
22
Finite Element Model of Tower
Extreme Weather Conditions
Loads and Boundary Conditions Under Extreme Weather Loads
Figure 8: Boundary Conditions of tower under extreme weather conditions
There are numerous ways to apply given forces as boundary conditions to a finite element model.
The boundary conditions for the tower were a pin support on the cable, and fixed support at the base of
the tower. It is assumed that the base that connects the tower to the ground acts as a fixed support for the
tower. The wind load of the blades was applied as a force centered on the nacelle at the top of the tower.
The wind pressure on the tower was applied as a beam pressure along the height of the tower.
Again, two different load configurations with respect to cable geometry were analyzed.
Configuration 1 looks at forces applied coplanar with the cables. Configuration 2 looks at forces applied
in a plane rotated 45 degrees (about the central axis of the tower) from two different cable supports.
23
Following are the plotted deflections and stress distributions for each temperature case and load plane
configuration:
Displacement Under Extreme Weather Conditions (Worst Case Scenario, Configuration 1)
Figure 9: Max deflection = 1.33 [ft]
Displacement Under Extreme Weather Conditions (Worst Case Scenario, Configuration 2)
Figure 10: Max deflection = 1.35 [ft]
24
Stress Under Extreme Weather Loads (Worst Case Scenario, Configuration 1)
Figure 11: Max stress = 0.24 E+07 [psf]
Figure 12: Von Mises stress at the base of the tower [psf]
25
Figure 13: Max stress occurs at the cables in this case (.24 E+07 psf)
26
Stress Under Extreme Weather Loads (Worst Case Scenario, Configuration 2)
Figure 14: Max stress = 0.16 E+07 [psf]
Figure 15: Von Mises stress at the base of the tower [psf]
27
Displacement Under Extreme Weather Loads (Extreme Heat, Configuration 1)
Figure 16: Max deflection = 1.06 [ft]
Displacement Under Extreme Weather Loads (Extreme Heat, Configuration 2)
Figure 17: Max deflection = 1.09 [ft]
28
Stresses Under Extreme Weather Loads (Extreme Heat, Configuration 1)
Figure 18: Max stress = 0.17 E+07 [psf]
Figure 19: Von Mises stress in the cables. Max stress occurs at in the cables [psf]
29
Figure 20: Von Mises stress at the base of the tower [psf]
Stresses Under Extreme Weather Loads (Extreme Heat, Configuration 2)
Figure 21: Max stress = 0.11 E+07 [psf]
30
Displacement Under Extreme Weather Loads (Extreme Cold, Configuration 1)
Figure 22: Max deflection = 1.15 [ft]
Displacement Under Extreme Weather Loads (Extreme Cold, Configuration 2)
Figure 23: Max deflection = 1.09 [ft]
31
Stresses Under Extreme Weather Loads (Extreme Cold, Configuration 1)
Figure 24: Max stress = 0.34 E+07 [psf]
Figure 25: Stress with interest at the base of the tower and in the cables. Max occurs in the cables
Stresses Under Extreme Weather Loads (Extreme Cold, Configuration 2)
32
Figure 26: Max stress = 0.26 E+07 [psf]
33
Normal Operating Conditions
Since the loads experienced by the tower are greater during extreme weather conditions, and extreme
weather conditions are when the tower reaches the limits of the design specifications, only the worst case
scenario under normal operating conditions is shown below. This is also done to maintain brevity of the
report. Results from extreme heat and extreme cold are displayed in table 9 and 10.
Displacement Under Normal Loads (Worst Case Scenario, Cable Configuration 1)
Figure 27: Max deflection = 0.55 [ft]
Displacement Under Normal Loads (Worst Case Scenario, Cable Configuration 2)
Figure 28: Max deflection = 0.59 [ft]
34
Stress Under Normal Loads (Worst Case Scenario, Cable Configuration 1)
Figure 29: Max stress = 0.71 E+06 [psf]
Figure 30: Von Mises stress at the base of the tower
35
Stress Under Normal Loads (Worst Case Scenario, Cable Configuration 2)
Figure 31: Max stress = 0.77 E+06 [psf]
Buckling Modes
Using Euler’s Column formula, the maximum allowable buckling load and factor of safety
against buckling can be calculated using the following formula:
F =L2
n π EI2
Where is a factor accounting for the supporting boundary conditions. defines the n .25n = 0
supporting end of the beam to be fixed, and the applied force to be at a free end: This load case is the
most similar to the boundary conditions of the wind turbine tower. Using only the axial loads on the
beam, it was found that the tower has a factor of safety against buckling of approximately 28.2.
36
Factor of Safety Against Buckling Under Normal Operating Conditions
Figure 32: Buckling factor of safety = 28
A buckling factor of safety of 28. was determined for the tower under normal loading
conditions. This matches closely with the hand calculated value of 28.2, and exceeds the required
buckling factor of safety set by the customer.
37
Factor of Safety Against Buckling Under Extreme Weather Conditions
Figure 33: Buckling factor of safety = 27.6
A buckling factor of safety of 27.6 was determined for the tower under extreme weather
loading conditions. This matches closely with the hand calculated value of 28.2, and exceeds the required
buckling factor of safety set by the customer.
38
Fatigue Considerations
To ensure the tower and base do not experience failure from fatigue, the endurance limit stress
must not be exceeded in order for the structure to sustain an unlimited amount of loading cycles. The
endurance limit for steel alloys are generally half their ultimate tensile strength. For A514 Steel, the
endurance limit is at a minimum, 9.81 E+06 psf, which is higher than the maximum stress the structure
can experience to have a factor of safety of 1.65 (9.75 E+06 psf). The max stress the tower should ever
experience is below the endurance limit of the steel, and so the structure should theoretically be able to
sustain an unlimited amount of cycles with an amplitude equal to the max. stress that the tower is allowed
to experience and never fail. Since the tower is designed to be under this limit by the customer’s factor of
safety requirements, fatigue need not be considered any further as a possible source of failure.
Modal Finite Element Model of Tower
A simplified model of the tower can be made by assuming the tower to be a beam with one fixed
end and one free end. The beam is also supporting a mass on its free end. The following definition of
natural frequency of this beam loading case was taken from Mark’s Standard Handbook for Mechanical
Engineers:
Figure 34: Diagram of natural frequency model
(17) ωn = √ 3EI(0.23 M +(M +3 M )) h* T ower Nacelle * Blade *
3
(18) /gM T ower = W T ower
/gM Nacelle = W Nacelle
/gM Blade = W Blade
It was determined through manual calculation that the natural frequency of the structure (Tower,
Nacelle, and Blades) is approximately 0.46 Hz. The natural frequency of the tower alone is approximately
39
0.50 Hz under the assumption that the tower is 500 feet tall, 39 ft in diameter and 2 in. thick. The natural
frequency of the tapered tower alone was determined to be about 0.52 Hz using the method mentioned
above.
While performing modal analysis on various tower configurations in APDL, the mass of the
nacelle and mass of the blades were not considered. This would suggest that the natural frequency
including the mass of the nacelle and blades would be slightly lower than predicted by ANSYS APDL.
The following table summarizes the hand calculated and simulated values for the natural
frequency of a 39 ft diameter tower with 2 in. wall thickness that is 500 ft tall:
Table 17 - Summary of Natural Frequency Results of Cylindrical Tower
39’ dia x 2” th x 500’ Tower Frequency (Hz)
Hand Calculated Natural Frequency 0.50
Simulated Natural Frequency to Verify Hand Calculations
0.49
Since the simulated natural frequency of the tower is similar to the manually calculated value, this
increases confidence in the simulation’s accuracy. The following table summarizes the manually
calculated and simulated values for the natural frequency of the final design:
Table 18 - Summary of Natural Frequency Results of Tapered Tower
Final Tower Design
(Taper from 39-35 feet in dia.)
Frequency (Hz)
Hand Calculated Natural Frequency 0.52
Simulated Natural Frequency to Verify Hand Calculations
0.49
The wind turbine’s operating speeds between 6 and 12 RPM translate to an operating frequency
of 0.1 to 0.2 Hz. To avoid achieving resonance of the structure if blade failure were to occur, the
operating frequency should be as far away as possible from the natural frequency of the structure. In other
words, the ratio of the driving frequency to the natural frequency should never approach a value ) (ω/ωn
of 1. Since the driving frequency at 12 RPM is closest to the natural frequency of the tower, the ratio of
the maximum operating frequency to the tower’s natural frequency was taken as the ratio to be analyzed.
40
Below is a table summarizing the range of operating frequencies, the natural frequency of the final tower
design, and the ratio of the driving frequency to the natural frequency of the tower:
Table 19 - Summary of Operating Frequencies
Item Frequency (Hz)
Minimum Operating Frequency 0.1
Maximum Operating Frequency 0.2
Tower Natural Frequency 0.49
ω/ω )( n Max 0.4
Modal Analysis Summary
Figure 35: Modal Analysis Analysis
The operating frequency closest to the natural frequency of the tower creates a value for
equal to 0.4. When the tower is under uneven loading, the transmissibility is in the highlightedω/ω )( n Max
region of the graph. This is sufficiently far enough away from the resonance peak such that the
transmissibility is considered “low”.
41
Figure 36: FEA modal analysis of tower. Natural frequency = 0.49 [Hz]
As verified by both the finite element and the harmonic oscillator model for a cantilever beam, the tower
has an operating frequency of approximately 0.5 Hz. This frequency oscillates with a magnitude of
approximately 0.06 in., a values that has minimal impact on the tower meeting deflection constraints.
42
Thermal Finite Element Model of Tower
Thermal Effects
Linear expansion of a solid body is given by:
(19) L L ΔTΔ = α 0 where is the linear expansion coefficient. For A514 Steel, the coefficient of thermal expansion α
is . Assuming that the normal temperature of the tower is 60°F (288 K), the height of the.3x10 1 −5 [ 1K ]
tower will grow by 2.6 in. when the environmental temperature is 120°F (322 K). Additionally, the tower
will shrink by approximately 3.0 in. at the lowest temperature expected at the site (-10°F or 250K). The
total height change was determined to be 0.47 ft., or 5.64 in.
It is assumed that the tower is 500 ft tall at 60o F at the time of manufacture and assembly.
The total height change of the tower was determined via thermal analysis in ANSYS APDL by inputting
the same parameters as noted above. The total height change was determined to be 0.47 ft, which is the
same value predicted by manual calculations.
The height change from 60o F to 120o F was determined by hand to be .22 ft (2.6 in) which is the
same result obtained from simulation. The height change from 60o F to -10o F was determined by hand to
be 0.248ft (3.0 in) which is the same result obtained from simulation.
43
Thermal Expansion of Tower
Figure 37: Thermal height change of tower from -10o F to 120o F = 0.47 [ft] = 5.64 [in]
Figure 38: Thermal height change of tower from 60o F to 120o F = 0.22 [ft] = 2.6 [in]
44
Figure 39: Thermal height change of tower from 60o F to -10o F = 0.25 [ft] = 3.0 [in]
45
Base Flange The base flange connection between the tower and the provided concrete foundation is
responsible for holding the tower in place and withstanding all reaction forces induced by loads on the
tower. As such, it is critical that the baseplate remains as rigid as possible under all loading to minimize
deflection and safely handle all stresses as well as to allow for an accurate approximation of the tower as a
cantilever beam with a fixed end. The baseplate was modeled in SolidWorks and analyzed in ANSYS
Workbench.
Figure 40: Whole Model Sketch of Base Flange
46
Figure 41: Detail view of tower hole placement
Figure 42: Base flange cross section [in.]
Due to its rotational symmetry, the baseplate can be simplified to only a fortieth section of the
entire flange and still reflect the true forces through the connection. The loading for the flange connection
was derived from the forces applied to the beam model of the tower. This submodel gives detailed
stresses and deformations around the most stressed bolt holes.
47
Structural Finite Element Model
To ensure that the base plate can hold all of the extreme loads of the tower, a finite element model
was created.
Figure 43: Mesh of base flange segment for structural and thermal models
A mixed mesh was generated in order to better discretize geometry. An inflation mesh was used
around bolt holds, surrounded by a tetrahedral mesh. A mapped mesh was used wherever possible. There
was a focus on creating a mesh with all element aspect ratios below 5.
48
Boundary Conditions
Figure 44: Boundary conditions of flange segment
The loading of the flange segment was determined from the FBD of the tower, and the maximum
stress each bolt is expected to withstand. For more on bolt loading see the “Mounting Hardware” section
below.
Table 20 - Boundary Conditions of Structural Segment Model
Boundary Conditions
Face A Moment contribution from each bolt
Face B One fortieth the weight of the tower, nacelle, and blades
Face C Shear force on segment from wind load
Face D Compression only support from bolts
Faces E & F Faces constrained to stay coplanar with original model
Face G Compression only support representing the ground
49
Deformation of Flange Segment
Figure 45: Deformation of segment of flange [in]
A single segment of the base plate can be modeled as depicted above. A total deformation of
approximately 0.004 in. was determined. This deformation is concentrated on the top surface, particularly
around the bolt holes, and is a vertically directed deformation, as this is an approximation of one smaller
section of the entire base, where the rotational moment is not expressed.
Stress of Flange Segment
Figure 46: Von Mises stress through segment of flange [psf]
50
The base plate design has an estimated factor of safety of 3.2, which is higher than the target of
1.65, but necessary for the connection hardware FOS of 4.
In verifying the dimensions of the base flange with a 1/40 model, a full ring section can be
modeled with the same dimensions and precise meshing to evaluate the response of the entire base to
loads on the tower. The arc in the center between holes is the contact point between the base flange and
tower, which will be affixed the same way tower sections are welded to each other. The 39 ft. base section
of the tower will be welded directly onto the flange.
Figure 47: Partial view of full ring mesh
The mesh, a composition of tetrahedral and mapped hex meshes, is composed of 788,500
elements and allows for maximum precision in stress and deformation analysis. Extra refinement is
applied to the washer areas around holes to ensure the most accurate results at the regions of highest
expected reaction forces and stresses.
Figure 48: Zoomed view of sections of interest
51
Figure 49: Loading cases on full flange model
The loading of the flange was determined from the FBD of the entire tower, taking reaction forces
and moments into consideration while also accounting for compressive support provided by bolts,
washers, and the concrete foundation.
Table 21 - Boundary Conditions of Structural Segment Model
Boundary Conditions
Face A Weights of the tower, nacelle, and blades
Face B Shear force imposed on base by wind loads on tower
Faces C, F & G Compression only support from the ground, bolts, and washers
Face E Equivalent moment from all tower loads about the tower base
Component H Gravitational body acceleration
Component I Fixed surface to avoid rigid body displacement
52
Figure 50: Deformations resulting from loading on entire ring
The deformation pattern matches expectations based on the given loading and boundary
conditions: small deformations in areas of compression and around the bolts, and a larger deformation in
areas between the bolts experiencing tension. The maximum deformation experienced by the flange is 1.5
E-3 ft.
53
Figure 51: Von Mises stresses resulting from loading on entire flange
The maximum stress recorded on the flange of the tower under the given loading conditions is
5.48 E06 psf. This gives a FOS of 2.9, which is well above the required 1.65 for base elements. While this
FOS would normally call for a smaller sizing of the base to line up more accurately with the requirements,
this higher FOS is necessary to achieve a bolt connection FOS of 4. This model also lines up very well
with data from the 1/40 section of the base flange. The above loading also shows regions of higher stress
around areas in tension as opposed to compression, which is logically sound, as the flange members in
tension will also experience an interaction with tensioned bolt heads connected to the concrete foundation
below. These simulations validate both safety and reliability of chosen dimensions for the base.
Assuming the welded connections can be approximated as a smooth connection in one uniform
piece of metal, these maximum stresses ensure that the connection between the base and tower is strong
enough to withstand all expected loading.
54
Thermal Model
To ensure that the flange does not experience any excessive stresses due to thermal expansion, a
thermal model was developed. A linear approximation was used to determine the radial expansion of the
tower and flange. To better understand the volumetric expansion, a finite element model was used.
Linear Thermal Expansion
The radial thermal expansion of the tower and the flange must be considered in sizing of the bolt
holes. The center of the bolt holes in the steel shifts approximately 0.12 in. in either direction from normal
operating temperatures to extreme temperatures. However, this significant thermal expansion does not
account for the thermal expansion of the concrete, which expands at a similar rate to steel. The thermal
expansion of the concrete allows for a standard 3 in. loose fit clearance hole to be used. The maximum
deviation of the hole location when temperatures experience maximum fluctuation is 0.05 in, meaning the
distance from the center of the tower to the center of the bolt hole grows or shrinks by ~0.05 in.
Figure 52: Radial thermal displacement of mounting holes
55
Thermal Finite Element Model
To accurately display this thermal expansion, a finite element model is implemented.
Boundary Conditions
Figure 53: Boundary conditions of thermal model
A transient thermal model is used to apply a change in temperature from 60o F to -10o F. This
temperature differential was chosen as it represents the extreme conditions farthest from normal operating
conditions. The solution of the transient thermal model was then put into a static structural model where
the flange segment was constrained to only increase radially and away from the ground.
Table 22 - Table of Boundary Conditions for Thermal Submodel
56
Boundary Conditions Thermal Expansion/Contraction
Face A Face constrained to stay coplanar with original model
Face B Face constrained to stay coplanar with original model
Face C Face constrained to not displace into the ground
Face D Face of tower body projected onto ground fixed to act as reference
point for expansion of section
Initial Temperature 60°F
Maximum Difference Temperature -70°F
Thermal Contraction from Operating Conditions to Extreme Temperatures
Figure 54: Thermal contraction of flange segment [+ is inwards contraction] [in]
The finite element model gives a good visualization for the volumetric expansion of the flange. A
linear estimate of the maximum deformation is 0.006 in. This hand calculation lines up similarly to the
finite element model.
(19) L L ΔT 1.3x10 )(11.3")(38.889 K) .006 in Δ = α 0 = ( −5 = 0
57
Mounting Hardware
The flange is mounted to the ground using eighty 3 in. diameter bolts. The number and diameter
of the bolts was selected based on numerous material, geometric, and loading constraints. Researching
material options for high strength bolts led to a material choice of A354 BD Steel which has a tensile
strength of 150,000 psi. Additionally, A354 BD Steel bolts are relatively common and cheap to
manufacture when compared with less common materials.
Using the loads from preliminary calculations, it was determined that the bolts must restrain the
tower from a force of approximately 600,000 lbf in shear and a moment of 2.2 E08 ft*lbs. Using the
generic formula for bolt tension in a flange (seen below) it was determined that the maximum bolt tension
must not exceed approximately 250,000 lbf. All mounting bolts have a FOS of approximately 4.1.
(20) olt T ension B = M
2R+ R(sin( )) ∑n
14 n
360 2
With an approximate radius of 18.5 feet, it was determined that eighty bolts would be required to
prevent the tower from dislodging from the concrete support structure. This was done by solving the
above equation for “n,” the number of bolts in the flange. The maximum force in each bolt was
determined by finding the maximum force a bolt of this geometry and material can withstand to achieve a
factor of safety of 4.1.
Design Time Estimate & Conclusion The total design time for this structure was estimated to be 212 hours. This estimate includes
preliminary calculations, CAD modeling, and simulation time.
Once again, RotorStorm© surpassed client expectations and delivered a well thought out product
that minimized overall build costs with a focus on safety and a robust model that can be applied in almost
any conditions around the world. The tapered beam with cable supports proved to be an excellent design
concept that drastically reduced material costs and allowed for design requirements to be met and
exceeded.
Certain factors were assumed to be negligible; all relevant calculations were incorporated into the
design. One future consideration would be to model the bolted connections of the tower, and rather than
manufacture the beam out of a single long beam, segment the tower to facilitate manufacturability.
58
Appendices
Appendix A: Variables & Parameters
Table 23 - Table of used Variables
Symbol Description
L Length of tower
α Coefficient of thermal expansion
T Temperature
F Applied Force
E Modulus of Elasticity
I Second Moment of Area (Moment of Inertia)
d Diameter
t Thickness
A Area
Q First Moment of Area
ρ Density
Cd Drag Coefficient
P Power
σ Normal Stress
τ Shear Stress
δ Deflection
w Wind Beam Load
Rair Gas Constant for Air
V Shear Stress
59
Appendix B: Formulae Used
- Area moment of inertia of an annular cross section I = 4π ( ) )( 2
d 4 − ( 2d − t 4)
- Area of an annular cross section π A = ( ) )( 2d 2 − ( 2
d − t 2)
Q = 32 ( ) )( 2
d 3 − ( 2d − t 3)
ρ = PR Tair
(Wind Load)ρ Vw = 21 * d * Cd
2
(Torque from power generation)M P ower = ωP ower
σ = IMy
σ = AP
τ = ItV Q
otal StressT = IMy + A
P + ItV Q
σAllowable = F OSσyield
Elastic stretch of cable, where G=0.00000779 for a 1x19 cableE =D2F G
60
Appendix C: Initial Modeling
The initial diameter of the tower was limited to 39 ft. in order to ensure sufficient clearance with
the 21 foot offset plane of the blades. To allow for ease of manufacturing, the wall thickness was limited
to 2.5 in. Therefore, the initial tower design consisted of a 39 ft constant diameter, 2.5 in. thick (38.66 ft
inner diameter) structure with annular cross section. This initial design was favored for its simplicity in
manual calculation and computer modeling.
The results indicated that such a tower design would not meet the customer’s requirements for
maximum deflection under extreme operating conditions, and the weight of the tower was a concern
coming in at around 2500 tons. With the tower being quite heavy, increasing the wall thickness was
unreasonable. In order to meet deflection requirements, however, some extra structure was required.
Two options were considered: two perpendicular internal steel plates that would traverse up the
entire length of the tower, or external supporting cables to help reduce deflection in the tower. The cable
solution was determined to be more effective at reducing deflection and lighter than steel plates. The steel
plates would act to increase the cross sectional moment of inertia, however the increase in the moment of
inertia was one order of magnitude less than the tower itself. Additionally, if 2 in. thick perpendicular
plates were added, the tower would weigh an additional 750 tons. Two inch thick plates were chosen as
the comparison to steel cables since they approach the limit of manufacturability and would increase the
area moment of inertia as much as possible within the design constraints.
As a result, steel cables were selected as a lighter and more feasible alternative. Additionally, a
tapered tower design was implemented to save weight. Although this would reduce the stiffness of the
tower, it would allow for necessary weight reduction, and the resulting increase in deflections would be
mitigated by the supporting cables. A final design with a tapered tower and cables was selected based on
these criteria.
Initial hand calculations were run on a straight 39 ft diameter tower with 2 in. thick walls. To
verify the accuracy of the FEA model, the hand calculated deflections and stresses were compared to the
simulated values for the same tower dimensions in APDL. Since the values were very close, we gained
confidence in the FEA model. Next, a tapered tower was modeled in APDL and results were verified by
assuming a straight annular beam with 2 in. thick walls and a diameter equal to the average diameter of
the tapered tower. Again, the results matched closely, and confidence in the model was established. In
order to meet deflection requirements, cables were introduced. In order to have clearance with the blades,
and for maximum effectiveness of the cable without using excessive amounts of land for the cables, the
cables were placed 250 ft up the tower, and anchored 250 ft away from the central axis of the tower on the
ground. In order to verify the model with cables, reaction forces on the tower with the given loads and an
assumed tension were determined. The tower was modeled with the assumed tension acting as a force on
61
the tower to determine the reaction forces on the tower. The results for reaction forces from hand
calculations were compared with the reaction forces determined by the FEA model. These results were
very similar. Then, the tapered tower was modeled with tension only link elements. The tension in the
cable that was determined by APDL was used to hand calculate the reaction forces with the simulated
value for tension. This value was compared to the reaction forces produced by APDL, and the results
were very similar. Although not a direct solution to the problem, the problem is inherently
difficult/impossible to solve, being an indeterminate load case after introducing the cables. Therefore,
some values need to be assumed in order to solve this problem, and the hand calculated values agree with
the simulated values, so some confidence was established with the model in that sense.
The tower is designed to have a rotating nacelle that aligns the blades to be perpendicular to
incoming wind at all times. Additionally, the blades on the wind turbine lock up when winds speeds go
above 60 mph. Finally, the plane the blades lie in was simulated as lying a distance of 21 ft from the
central axis of the tower, however, this plane may be moved further away if needed to allow for more
clearance. Without consideration for stresses on other components of the wind turbine, offsetting this
plane to be 21 ft from the outer surface of the tower (at a minimum) would have a negligible impact on
stresses and deflection of the tower, and would even result in a slight decrease of these values. Therefore,
a minimum clearance of 2ft of the blades with the tower is obtained even if the plane of the blades lies 21
ft away from the central axis of the tower.
Finally, the featured tower is gradually tapered from a cross section diameter of 39 feet at the
base to 35 feet at the top. The wall of the tower is 2 in. thick throughout the height of the tower to allow
for manufacturability and easier transport for onsite assembly.
Additionally, it can be assumed that the blades are splayed away from the tower to allow for more
clearance if needed. Under maximum wind load, the blades could be designed to flex to the point that all
three blades lie in a vertical plane parallel to the tower’s central axis as depicted in the image below. This
is to allow for clearance of the blades with the tower during operation. Loads are still modeled as acting
perpendicular to the length of the tower to maintain simulation of the worst case scenario.
Figure 55: Assumption of blade deflection
62
Appendix D: MATLAB Scripts for Preliminary Calculations
% Normal Operating Conditions = NOC
% Extreme Weather Conditions = EWC
clc; clear all;
%% Beam Dimensions
d = 39; %ft t = 2/12; %ft
h = 500; %ft
%% Section Properties
I = (pi/4)*((d/2)^4-(((d/2)-t))^4) %ft^4 A_Tower = pi()*((d/2)^2-((d/2-t))^2) %ft^2 Q = (2/3)*((d/2)^3-((d/2)-t)^3); % ft^3
%% Material Properties
E = 3.89*10^9; %psf
%% Wind Load Parameters
Cd_Blade = 0.82; Cd_Tower = 1.2; %for a long cylinder mph_to_ftpersec = 1.46667; %ft/s per mph V_NOC = 60*mph_to_ftpersec; %ft/s V_EWC = 105*mph_to_ftpersec %ft/s T = -10:120; %Degrees Farenheit T_Kelvin = (T - 32) .* 5/9 + 273.15; %K Pressure = 101325; %Pa R = 287.05; %kJ/kg K den_SI_to_Imp = 1/515.379; %slug/ft^3 per kg/m^3 den_air = (Pressure/(R*T_Kelvin(1)))*den_SI_to_Imp %slug/ft^3 Ax = ((39+35)/2)*h; %ft^2
%% Gravity
g = 32.2; %ft/s^2
%% Tower parameters
den_steel = 490*0.031081; %slug/ft^3 b = 21; %ft Distance from hub to blades d_blades = 450; %ft d_hub = 18; %ft L_blade = (d_blades-d_hub)/2 %ft L_CM_Blade = L_blade/3; %ft width_blade = L_blade/12 %ft A_Blade = width_blade*L_blade; %ft^2
%% Paramters for moment from power generation
Power = 9.25*1341.02*550 %ft*lbf/s rpm_to_rad_s = 2*pi()/60; %rad/s per rpm rpm = 6:12; w = rpm .* rpm_to_rad_s; %rad/s
63
newton_m2ft_lbs = 0.737562; %% Design Criteria
Stress_FOS = 1.65; Buckling_FOS = 4; Yield_Stress = 1.58*10^7; %psf Normal_Operating_Deflection = h/500; %ft Extreme_Operating_Deflection = h/370; %ft
%% Calculate Loads
W_Nacelle = 185000; %lbf W_Blade = 12500; %lbf W_Tower = den_steel*A_Tower*h*g %lbf
% NOC
P_Wind_Blade_NOC = .5*Cd_Blade*den_air*V_NOC^2; %lbf/ft^2 = slug/(ft*s^2) P_Wind_Tower_NOC = .5*Cd_Tower*den_air*V_NOC^2; %lbf/ft^2 = slug/(ft*s^2) F_Wind_Blade_NOC = P_Wind_Blade_NOC*3*A_Blade %lbf = slug*ft/s^2 F_Wind_Tower_NOC = P_Wind_Tower_NOC*Ax; %lbf = slug*ft/s^2 wind_load__tower_NOC = d*.5*den_air*Cd_Tower*V_NOC^2 M_Wind = (Power/w(end))*newton_m2ft_lbs %ft*lb
% EWC
P_Wind_Blade_EWC = .5*Cd_Blade*den_air*V_EWC^2; %lbf/ft^2 = slug/(ft*s^2) P_Wind_Tower_EWC = .5*Cd_Tower*den_air*V_EWC^2; %lbf/ft^2 = slug/(ft*s^2) F_Wind_Blade_EWC = P_Wind_Blade_EWC*3*A_Blade %lbf = slug*ft/s^2 F_Wind_Tower_EWC = P_Wind_Tower_EWC*Ax; %lbf = slug*ft/s^2 wind_load_tower_EWC = (Ax/h)*.5*den_air*Cd_Tower*V_EWC^2 % Blades can lock up, so no torque acting on the tower from power generation
%% Does the diameter and thickness work for normal operating conditions?
%% Calculate Internal Forces for Normal Operating Conditions
P_NOC = W_Tower + W_Nacelle + 3*W_Blade; %lbf Shear_NOC = F_Wind_Tower_NOC + F_Wind_Blade_NOC; %lbf M_NOC = -3*b*W_Blade + F_Wind_Blade_NOC*h + M_Wind + F_Wind_Tower_NOC*h/2; %ft*lbf
%% Calculate Stresses for Normal Operating Conditions
Bending_Stress_NOC = M_NOC*(d/2)/I; %lbf/ft^2 Axial_Stress_NOC = P_NOC/A_Tower; %lbf/ft^2 Shear_Stress_NOC = Shear_NOC*Q/(I*t); %lbf/ft^2
Total_Stress_NOC = Bending_Stress_NOC + Axial_Stress_NOC + Shear_Stress_NOC %lbf/ft^2 %% Stress Failure
Allowable_Stress_NOC = Yield_Stress/Stress_FOS
if Total_Stress_NOC <= Allowable_Stress_NOC Result_Stress_NOC = 'Stress passed NOC'; else
Result_Stress_NOC = 'Stress not passed NOC'; end
%% Check for Buckling
Critical_Buckling_Load_NOC = (W_Tower + W_Nacelle + 3*W_Blade);
64
P_Buckling = .25*(pi())^2*E*I/h^2; fos = P_Buckling/Critical_Buckling_Load_NOC
if Critical_Buckling_Load_NOC > P_Buckling*Buckling_FOS Result_Buckling_NOC = 'Buckling not passed NOC'; else
Result_Buckling_NOC = 'Buckling passed NOC'; end
%% Does the diameter and thickness work for Extreme Weather Conditions?
%% Calculate Internal Forces for Normal Operating Conditions
P_EWC = W_Tower + W_Nacelle + 3*W_Blade %lbf Shear_EWC = F_Wind_Tower_EWC + F_Wind_Blade_EWC %lbf M_EWC = -3*b*W_Blade + F_Wind_Blade_EWC*h + F_Wind_Tower_EWC*h/2 %ft*lbf
%% Calculate Stresses for Extreme Weather Conditions
Bending_Stress_EWC = M_EWC*(d/2)/I; %lbf/ft^2 Axial_Stress_EWC = P_EWC/A_Tower; %lbf/ft^2 Shear_Stress_EWC = Shear_EWC*Q/(I*t); %lbf/ft^2
Total_Stress_EWC = Bending_Stress_EWC + Axial_Stress_EWC + Shear_Stress_EWC %lbf/ft^2 %% Stress Failure
Allowable_Stress_EWC = Yield_Stress/Stress_FOS;
if Total_Stress_EWC <= Allowable_Stress_EWC Result_Stress_EWC = 'Stress passed EWC'; else
Result_Stress_EWC = 'Stress not passed EWC'; end
%% Check for Buckling
Critical_Buckling_Load_EWC = (W_Tower + W_Nacelle + 3*W_Blade); P_Buckling = .25*(pi())^2*E*I/h^2;
fos1 = P_Buckling/Critical_Buckling_Load_EWC
if Critical_Buckling_Load_EWC > P_Buckling*Buckling_FOS Result__Buckling_EWC = 'Buckling not passsed EWC'; else
Result__Buckling_EWC = 'Buckling passed EWC'; end
%% Max Deflection NOC
Deflection_Max_NOC = F_Wind_Blade_NOC*h^3/(3*E*I) + wind_load__tower_NOC*h^4/(8*E*I) + 3*b*W_Blade*h^2/(2*E*I) + M_Wind*h^2/(2*E*I) %(F_Wind_Tower_NOC*(h/2)^2/(6*E*I))*(3*h-h/2)
if Deflection_Max_NOC < Normal_Operating_Deflection fprintf('Deflection passed NOC \n'); else
fprintf('Deflection not passed NOC \n'); end
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%% Max Deflection EWC
Deflection_Max_EWC = F_Wind_Blade_EWC*h^3/(3*E*I) + wind_load_tower_EWC*h^4/(8*E*I) + 3*b*W_Blade*h^2/(2*E*I) %(F_Wind_Tower_EWC*(h/2)^2/(6*E*I))*(3*h-h/2)
if Deflection_Max_EWC < Extreme_Operating_Deflection fprintf('Deflection passed EWC \n'); else
fprintf('Deflection not passed EWC \n'); end
I_Required = F_Wind_Blade_EWC*h^3/(3*E*Extreme_Operating_Deflection) + wind_load_tower_EWC*h^4/(8*E*Extreme_Operating_Deflection) + 3*b*W_Blade*h^2/(2*E*Extreme_Operating_Deflection);
%% Thermal Expansion
coeff_thermal_expansion = 1.3*10^-5; %1/K
dt = T_Kelvin(end) - T_Kelvin(1) dh = coeff_thermal_expansion*h*dt
%% Natural Frequency of the Tower
wn = (1/(2*pi))*sqrt(3*E*I/((.23*(4720000/g)*h^3)))% + (W_Nacelle/g + 3*W_Blade/g))*h^3)) %Hz
Check_for_natural_frequency = w./wn./(2*pi);
Works Cited
1. Moaveni, Saeed. Finite Element Analysis: Theory and Application with ANSYS. Pearson, Inc.,
2015.
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2. Denholm, Paul, Maureen Hand, Maddalena Jackson, and Sean Ong. Land-Use Requirements of
Modern Wind Power Plants in the United States , August 2009.
https://www.nrel.gov/docs/fy09osti/45834.pdf.
3. Choudhury, Mahbuboor & Quayyum, Shahriar & Amanat, Khan. (2008). Modeling and Analysis
of a Bolted Flanged Pipe Joint Subjected to Bending.
4. Tafheem, Zasiah & Amanat, Khan. (2015). Finite element investigation on the behavior of bolted
flanged steel pipe joint subject to bending. Journal of Civil Engineering, The Institution of
Engineers Bangladesh (IEB). 43. 79-91.
5. EngineeringToolBox,(2004).DragCoefficient.Available at
https://www.engineeringtoolbox.com/drag-coefficient-d_627.html, 29 Oct 2019
6. Avallone, Eugene A., and Theodore Baumeister. Mark's Standard Handbook for Mechanical
Engineers. McGraw-Hill, 1996.
7. “Beam Deflection Formulae.” Iowa State University,
home.eng.iastate.edu/~shermanp/STAT447/STAT%20Articles/Beam_Deflection_Formulae.pdf.
8. Structural Applications of Steel Cables for Buildings: ASCE/SEI 19-16. American Society of
Civil Engineers, 2016.
9. “A514 Steel.” MakeItFrom.com, 8 Nov. 2019,
www.makeitfrom.com/material-properties/ASTM-A514-Alloy-Steel.
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