computer aided engineering: midterm design project enea...

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 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:


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 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


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:


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)


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)



27

Displacement Under Extreme Weather Loads (Extreme Heat, Configuration 1)


Displacement Under Extreme Weather Loads (Extreme Heat, Configuration 2)


28

Stresses Under Extreme Weather Loads (Extreme Heat, Configuration 1)


Figure 19: Von Mises stress in the cables. Max stress occurs at in the cables [psf]

29


Stresses Under Extreme Weather Loads (Extreme Heat, Configuration 2)


30

Displacement Under Extreme Weather Loads (Extreme Cold, Configuration 1)


Displacement Under Extreme Weather Loads (Extreme Cold, Configuration 2)


31

Stresses Under Extreme Weather Loads (Extreme Cold, Configuration 1)


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


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)


Displacement Under Normal Loads (Worst Case Scenario, Cable Configuration 2)


34

Stress Under Normal Loads (Worst Case Scenario, Cable Configuration 1)


Figure 30: Von Mises stress at the base of the tower

35

Stress Under Normal Loads (Worst Case Scenario, Cable Configuration 2)


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

65

%% 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.

66

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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