220 final report - group 14

8/2/2019 220 Final Report - Group 14

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Author(s): Braden Forbes, Eric Kyfiuk, Mitch Fitzgibbon

Filename: 220 Final Report - Group 14

Date: 3-Aug-2010

Total Pages: 22

MECH 220GROUP 14FINAL PROJECT

REPORT -WIND TURBINE DESIGN

DEPARTMENT OF MECHANICAL ENGINEERING


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Mech 220 Group 14 Final Project Report - Wind Turbine Design Date: 8/3/2010

Abstract

This document describes the wind turbine tower design process,

including methods used and insights encountered. It includes towerdimensions and other numerical qualities with supporting calculations

and it specifies the location where the safety-factor is encountered.

Using Excel, a simulation was done that optimized the changeable

dimensions within specified limits to produce the most inexpensive

tower while staying within performance constraints.

The safety-factor-adjusted maximum stress and maximum deflection

were both encountered with the optimized solution. A SolidWorks

model was made and tested to verify and visually augment the

simulation. The design was experimented with to find the marginal

benefit of increasing the base diameter limits, and it was found thatincreasing the major diameter of the bottom of the tower would be

economical.

The expectation for location of maximum shear stress was near the

bottom of the tower, but the location was found to be near to the top of

the tower. The large weight of the nacelle in comparison to the mass

of the optimized tower was determined to be the reason for this.


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Table of Contents

1.1 Goals ................................................................................................................... 1

1.2 Problem Definition ............................................................................................. 1

2 Method ....................................................................................................................... 3

2.1 Geometric and Physical Properties ..................................................................... 3

2.1.1 Finding Radii and Thickness ...................................................................... 3

2.1.2 Finding Area, Volume, Moments of Inertia ............................................... 3

2.2 Forces and Moments ........................................................................................... 4

2.3 Analyzing Tower Twist ...................................................................................... 5

2.4 Determining Tower Deflections ......................................................................... 5

2.5 parameterization of and Finding Coordinates Around Cross Section ............. 6

2.6 Calculating Normal Stress .................................................................................. 6

2.7 Calculating Shear Stress ..................................................................................... 7

2.8 Calculating Max Shear Stress ............................................................................. 7

2.9 Material Characteristics ...................................................................................... 8

3 Simulation Results ..................................................................................................... 9

3.1 Steel .................................................................................................................... 9

3.2 Aluminum ......................................................................................................... 10

3.3 SolidWorks Simulation Verification and Visual Results ................................. 11

3.3.1 Stresses ..................................................................................................... 12

3.3.2 Displacements .......................................................................................... 15

4 Expanded Constraints Experiment .......................................................................... 15

4.1 Steel .................................................................................................................. 15

4.2 Aluminum ......................................................................................................... 16

5 Conclusion ............................................................................................................... 17

6 AppendixRough Calculations .............................................................................. 18

Table of Figures

Figure 1 - Tower Criteria with Cross-Section and Coordinate System ............................ 2Figure 2 - SolidWorks Model ......................................................................................... 11

Figure 3 - Tower Stresses: Trimetric View ..................................................................... 12

Figure 4 - Tower Stresses: Side View ............................................................................. 13

Figure 5 - Tower Stresses: Bottom View ........................................................................ 14

Figure 6Displacements: Side View ............................................................................. 15


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List of Tables

Table 1Material Properties for Steel ............................................................................. 8

Table 2 - Material Properties for Aluminum .................................................................... 9

Table 3Results for SteelOriginal Parameters ............................................................ 9Table 4Dimensions for Steel ResultsOriginal Parameters ....................................... 10

Table 5Results for AluminumOriginal Parameters ................................................. 10

Table 6Dimensions for Results for AluminumOriginal Parameters ....................... 11

Table 7Results for SteelRevised Parameters .......................................................... 16

Table 8Dimensions for Results for SteelRevised Parameters ................................. 16

Table 9Results for AluminumRevised Parameters .................................................. 17

Table 10Dimensions for Results for AluminumRevised Parameters ..................... 17


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1.1 GOALS

Goals of the final project were to:

Gain a better understanding of the theory learned in class by applying it to a

practical problem.

Gain experience in optimization by using Microsoft Excels Solver tool.

1.2 PROBLEM DEFINITION

The purpose of the project was to obtain the optimal dimensions for the tower of a

wind turbine concept. The design is angled downwind at 18 degrees, to minimize the

effect of the towers wake on the rotors, and the base can rotate and lock to keep the

tower aligned with the wind. The tower was to be hollow and have an elliptical cross

section that tapered up its length. The thickness of the walls of the tower was to be

constant around each cross section, but vary up the length of the tower. The tower was

loaded with two discreet forces on the top of the tower and two distributed forces along

the length of the tower. The weight of the nacelle/rotor and the weight of the tower itself

were also considered. The design of the tower, the forces, and the given dimensions can

be seen inFigure 1 - Tower Criteria with Cross-Section and Coordinate System.

The design variable that was to be minimized was the mass of the tower, and six

dimensions were to be varied to obtain this optimal design; these included the major and

minor diameter and wall thickness at each end of the tower. The major diameter could

not exceed 12 meters at the base, and 10 meters at the top. The material and all of its

associated properties was another variable that could be changed. Other constraints

included the deflection at the top of the tower, which could not exceed 20 cm, and the

angle of twist at the top of the tower, which could not exceed 3 degrees. The final

constraint was that the stress in the tower could not exceed the maximum yield strength

of the given material, including a factor of safety of 2.


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Figure 1 - Tower Criteria with Cross-Section and Coordinate System


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Where A and B were the outer major and minor radii, respectively, and a and b were

the inner major and minor radii, respectively.

Integrating the formula for area, we found the formula for the volume of the tower

above any cut, and also used it in Excel. It bears mentioning that the volume is done

analytically in the formula sheet (not numerically). This was done to obtain a precise

result for weight acting at each cross-sectional area.

The moments of inertia about the x and z axes were also needed for future

calculations. These were found to be:

Where A, B, a, and b were the same variables as when finding area.

2.2 FORCES AND MOMENTS

The next step in the project was to decompose the numerous forces acting on the

tower into a set of 3 principle stresses and 3 moments. This was done by summing the

components of each force along each axis, and by calculating the moments that result by

moving every force to the principle coordinate system. The coordinate system we used

can be seen in Figure 1 - Tower Criteria with Cross-Section and Coordinate System.

The weight of the nacelle and the tower produced moments about the x axis, as well

as shear forces along the z axis, and normal stresses along the y axis. The discreet force

S and the distributed force St created shear stresses along the x axis and moments about

the z axis. Also, the discreet force S was the only force to create a moment about the y

axis, despite our initial hypothesis that the distributed load St would also do the same.

The discreet force T and the distributed load Tt both created shear stresses in the z axis,

normal forces in the y axis, and moments about the x axis. The calculations done to

obtain these resultant forces and moments can be seen in the Appendix.


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2.3 ANALYZING TOWER TWIST

As the cross section of the tower was not constant, the formula for angle of twist of a

circular cross section could not be used. Instead, several values needed to be found,

including the major and minor radii of the centreline, as well as the circumference, and

the area enclosed by the centreline. When analyzing the twist angle about the central y-

axis of the tower, we decided to use a Riemann sum approach. We did this by assuming

a constant cross section for one metre sections, finding the twist of the section, and

adding the angle of twist of the previous section. Although this did not give an exact

answer, it was a good approximation. The formula used to calculate the twist angle over

one metre can be found in the Appendix.

2.4 DETERMINING TOWER DEFLECTIONS

Before determining the deflections of the tower in the x and z directions, we first

needed to find the slope of the tower, which varied along its length. To find the slope of

the beam in the x and z directions, we used a Riemann sum instead of doing a complex

integral, which gave us a fairly accurate approximation of the slope. We summed the

slopes from the bottom of the tower to give the largest slope at the top of the tower.

After finding the slopes, we had to integrate the values again to find the

displacement in the x and z directions. Again, instead of performing a complex

integration, we used a Riemann sum approximation. To do this, we multiplied the slope

of the tower at a given cross section by the length of the section (our tower was divided

into 1m sections), and adding the deflection of the previous section. The deflections

were summed from the bottom, giving the largest deflections at the top of the tower.

Although we could have performed a more accurate approximation by other methods,

the error attributed to using Riemann sums was more than covered by the factor of

safety we used.


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2.5 PARAMETERIZATION OF AND FINDING COORDINATES AROUND CROSS SECTION

Calculating the normal and shear stress at several points around each cross section

required us to know the coordinates of each point with respect to the x and z axes. To

find these coordinates, it was necessary to create a parameter of, which we called t.

Where a and b are the major and minor radii, respectively. Finding the coordinates of

a single point along the circumference of the ellipse was done with the following

formulas:

2.6 CALCULATING NORMAL STRESS

There were three forces that had to be considered when calculating the normal stress

at each point around a cross section: the normal force along the y axis, and the moments

about the x axis and z axis. The normal force along the y axis created compressive stress

at every point, but the two moments had the potential to create compressive or tensile

stress, depending on which point is selected. Because of this, care had to be taken in

order to get the correct signs out of each component of stress.

Finding the stress caused by the normal force along the y axis was straightforward,

and could be done with the force over area formula. Finding the stress caused by the

two moments was slightly more complicated, and required the following formulas:

The results we obtained showed compressive stress on the downwind side of the tower,

and tensile stress on the upwind side, which was expected, as shown in Figure 4 -

Tower Stresses: Side View.


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2.7 CALCULATING SHEAR STRESS

When calculating the shear stress at each point around every cross section, there were

three forces to consider. The first was the torque about the y axis, T, and the others were

the shear forces along the x and z axes, Vx and Vz. Calculating the shear stress causedby the torque T was fairly straight forward, and was done with the shear stress formula

for closed thin walled sections:

Where tis the thickness of the tower wall, and is the area enclosed by half of the

thickness. Calculating the shear stress caused by the two shear forces was much more

complicated, and required shear stress in thinwalled members analysis. The formula

used was

Where Vis the shear force along a given axis,Iis the moment of inertia about the

perpendicular axis, and Q is the first moment of area about the perpendicular axis. tis

the thickness of the wall of the tower, but it was multiplied by 2 due to the symmetry of

the towers cross section. Calculating the Q values proved to be quite difficult because

of the complexity of ellipses, and as a result they are a likely source of error. To find

them, we had to determine the area of the tower wall bound by the point we were

analyzing and a point symmetric about the axis of the shear force. We then had to find

the centroid of this same section of area. Luckily, several variables cancelled out in the

calculation and reduced the workload, but the formulas were still relatively complex.

To find the overall shear stress, we added the shear stress caused by the three different

forces, with T and Vz making positive shears and Vx creating a negative shear. The

calculations for the shear stress can be found in the Appendix.

2.8 CALCULATING MAX SHEAR STRESS

Knowing the max shear stress at each point around every cross section is critical,

because the yield strength of ductile materials directly depends on it. We were able to

find the max shear stress at each point by applying Mohrs circle with the normal and


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shear stress at each point. The max shear stress was equal to the radius of the circle,

which could be calculated using the Pythagorean theorem:

2.9 MATERIAL CHARACTERISTICS

The simulation was done with two practical materials: Steel and 6061-T6 Aluminum.

Steel was the first material tested for tower suitability in the analysis. Its properties, as

entered in the simulation, are as follows:

Material Properties

Density Y G

Yield

Strength w/SF

7820 2E+11 7.930E+10 9.450E+07

Table 1Material Properties for Steel

A specific grade of steel wasnt selected, but rather the median value of yield strength

(according to Wolfram Alpha) was used. One notable characteristic of steel in

comparison to aluminum is its continued good performance after repeated loading. If

the elastic region is not exceeded, it wont fail by fatigue. Aluminum, on the other hand,

will fail due to fatigue after repeated loading, so the tower will have to be replaced if

aluminum is used to construct it. Steel, however, must be protected from oxidation

(rust), so either active or passive cathodic protection must be implemented in such a

case. Painting the tower might be expensive.

The aluminum selected for simulation is a precipitation-hardening alloy that contains

mostly magnesium and silicon additions. This alloy is common and easy to weld, which

made it an ideal selection for the tower application. The other properties of the materials

(Youngs andrigidity moduli and mass) dont depend on the grade. Below are the

properties of aluminum as entered in the simulation.
http://en.wikipedia.org/wiki/Precipitation_hardeninghttp://en.wikipedia.org/wiki/Precipitation_hardening


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Material Properties

Density Y G

Yield

Strength w/

SF

2700 7.000E+10 2.600E+10 1.375E+07

Table 2 - Material Properties for Aluminum

3 SIMULATION RESULTS

3.1 STEEL

When the Excel solver was used to optimize the wind turbine tower design with steel as

the material, the following results were produced:

Results - Steel

Maximum Shear

Stress Mass

Twist

Angle

Displacement

x Displacement z

9.450E+07 1.206E+06 3.300E-02 1.247E-01 2.000E-01

Cost of

materials: Rate: Vector Sum of Displ.-> 0.235675637

$965,107.92 $0.80/kg Volume: 1.543E+02

Table 3

Results for Steel

Original Parameters

The dimensions used to produce these results are as follows.

Changeable Input Variables

Material Density 7820

Outer Major Diameter Bottom(B1) 12

Outer Minor Diameter Bottom(A1) 5.473547724

Outer Major Diameter Top(B2) 4.615307818

Outer Minor Diameter Top(A2) 0.061136375

Thickness Bottom(t1) 0.111019609

Thickness Top(t2) 0.019015941


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Table 4Dimensions for Steel ResultsOriginal Parameters

The design reaches constraints in two places: It deflects the maximum amount of 20 cm

in the z-direction (in the direction along the length of the nacelle, as expected) and it

encounters the maximum allowable shear stress with safety factor 2 (at the top of the

tower). The least amount of material that could be used to construct a tower that obeys

the constraints set out in this assignment is 1200 tonnes of steel. At $0.80/kg, that costs

about 1 million dollars. However, the cost of materials in an application like this would

be dwarfed by the cost of manufacturing, given the varying thickness and major and

minor diameters.

3.2 ALUMINUM

When the Excel solver was used to optimize the wind turbine tower design withaluminum as the material, the following results were produced:

Results

Maximum Shear

Stress Mass

Twist

Angle

Displacement

x Displacement z

1.375E+07 1.261E+06 7.822E-03 7.977E-02 2.000E-01

Cost of


$3,479,538.68 $2.76/kg Volume: 4.669E+02

Table 5Results for AluminumOriginal Parameters

The dimensions used to produce these results are shown:



Outer Major Diameter Bottom(B1) 12





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Table 6Dimensions for Results for AluminumOriginal Parameters

The design once again reaches constraints in two places: It deflects the maximumamount of 20 cm in the z-direction (in the direction along the length of the nacelle, as

expected) and it encounters the maximum allowable shear stress with safety factor 2 (at

the top of the tower). This time, the least possible amount of material is 1300 tonnes of

aluminum. At $2.76/kg, that costs about 3 1/2 million dollars. Again, the cost of

materials in an application like this would be dwarfed by the cost of manufacturing, but

the cost of materials is still greater than what it would be if steel were used. It bears

mentioning that the volume of material used in the case of aluminum is greater (i.e. the

tower dimensions are greater), but the mass is quite a bit less. The high cost of

producing aluminum is the reason it is still more economical to use steel. Also, it takes

less steel to construct the tower than it does aluminum. Aluminum might be practical for

a smaller turbine, for which the maximum stresses dont encounter the yield strength of

steel.

3.3 SOLIDWORKS SIMULATION VERIFICATION AND VISUAL RESULTS

Figure 2 - SolidWorks Model


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Once steel was selected as the optimal building material, a SolidWorks simulation was

done to verify Excel results and gain visual appreciation for the distribution of stresses

and various elements deflections. The SolidWorks output was quite similar to that of

Excel, and the differences that were observed can be attributed to differences in

simulation method (e.g. St. Venants principle was used in our simulation but probably

not for SolidWorks) and the fact that only forces that acted at the nacelle (i.e. the

nacelles weight and the side and front wind forces) were included in the SolidWorks

simulation, in order to cut down on simulation time.

3.3.1 STRESSESSee the following graphical illustrations ofthe designs stresses in response to loading

at the nacelle.

Figure 3 - Tower Stresses: Trimetric View


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The loading directions applied can be seen well in 3D in Figure 3 - Tower Stresses:Trimetric View.

Figure 4 - Tower Stresses: Side View

It was anticipated that the maximum stress would occur in a lower spot on the tower,

closer to the base. However, careful simulation in both SolidWorks and Excel produced

maximum stresses close to the top of the tower, at around 11 m from the top. It was

concluded that this is because the weight of the nacelle as given in the criteria literally

outweighs the contribution of the mass of the tower itself. If a heavier material were

used or a lighter nacelle were specified, the location of the maximum stress might well

move to the region we first expected.


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The maximum compressive and tensile stresses were manually located as occurring at 0

degrees and 180 degrees, respectively. By our convention, 0 degrees occurs at the

downwind edge of the tower.

Figure 5 - Tower Stresses: Bottom View

The nature of stresses induced in a member due to bending shows up quite well in

Figure 5 - Tower Stresses: Bottom View. As indicated by the accompanying von Mises

stress color scale, green indicates greater stress concentration than blue. The tension

occurring in the back portion of the tower and the compression in the overhanging


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front portion are revealed in the green contrasting with the blue. The increasing

stresses going up the tower can also be seen in this image.

3.3.2 DISPLACEMENTS

Figure 6Displacements: Side View

The displacement at the top of the tower is greatest and this was expected, as shown in

Figure 6Displacements: Side View.

4 EXPANDED CONSTRAINTS EXPERIMENT

4.1 STEEL

The simulation was performed again for steel with the minimum diameters at the base

expanded by 10% as instructed, and the results changed. This was expected, because the

maximum value of major diameter had been encountered in the steel simulation under

the original parameters. Therefore, making the minimum allowable dimensions bigger

allowed for a more optimal solution with respect to mass and cost of material.


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Results Steel Expanded

Maximum Shear

Stress Mass

Twist

Angle

Displacement

x Displacement z

9.450E+07 1.119E+06 3.424E-02 1.474E-01 2.000E-01

Cost of


$895,446.11 $0.80/kg Volume: 1.431E+02

Table 7Results for SteelRevised Parameters

When the parameters were relaxed, the cost went down by about $100, 000. The

dimensions used to produce the revised result are shown in the following table:



Outer Major Diameter Bottom(B1) 13.2






Table 8Dimensions for Results for SteelRevised Parameters

4.2 ALUMINUM

When the constraint was relaxed for aluminum, the following new results were

produced. The result was similar to the steel result.


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Results Aluminum Expanded

Maximum Shear

Stress Mass

Twist

Angle

Displacement

x Displacement z

1.375E+07 1.167E+06 9.074E-03 1.348E-01 2.000E-01

Cost of


$3,221,566.18 $2.76/kg Volume: 4.323E+02

Table 9Results for AluminumRevised Parameters



Outer Major Diameter Bottom(B1) 13.2






Table 10Dimensions for Results for AluminumRevised Parameters

5 CONCLUSION

In order to solve this project, we had to combine much of the knowledge that was

gained in this course and use it to solve a complex engineering design problem. It gave

us an idea of the types of design projects we, as engineers, will encounter in our careers.Using Microsoft Excel to design this tower also proved very useful. Once functions

were developed to define variables based on the tower dimensions, (deflections, shears,

and normal stresses) it was quite simple to use Excels solver tool to minimize the mass

of the tower.


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Based on our results, the optimal tower for this application would be a steel tower with

a mass of 1200 tonnes. The optimal aluminum tower has a greater mass, and the cost of

materials would be far greater, due to the high price of aluminum. Steel also has the

advantage of not failing due to fatigue over a long period of time, whereas the face-

centred cubic structure of aluminum would cause the tower to plastically deform and

fail in fatigue over a long period of time, even with shear stresses acting on it under the

yield strength.

Although we used several approximations during the course of this project including

using Riemann sums and a slightly approximated value of Q as well as the Excel

solvers error, our large factor of safety more than accounts for this.

It was discovered that both stiffness and strength requirements were important in the

tower design, as the maximum constraints on the yield strength and z displacements

were simultaneously reached. However, the constraints on the angle of twist and x

displacement were not reached. The maximum shearing stress was encountered near the

top of the tower, as the mass of the nacelle far outweighed the mass of the tower at that

point.

Because the maximum constraint on the major diameter of the bottom of the tower was

reached for both steel and aluminum, the design was modified when we increased the

constraint by 10%. After doing so and using the Excel solver again, we found that the

revised optimal tower design also reached the maximum constraint on the bottom major

diameter. The overall mass of the tower was decreased, however, because the thickness

of the tower walls had also decreased, reducing the total volume. For the steel tower, the

mass was reduced by 7.21% when the major diameter constraint was increased,

resulting in material cost savings of nearly $70,000. For the aluminum tower, the mass

was reduced by 7.45% when the major diameter constraint was increased, resulting in

material cost savings of over $250,000.

6 APPENDIXROUGH CALCULATIONS

See attached work done to find Q-values, resultant forces, shear and normal stresses,

moments of inertia, centerline-bounded area, and analytic results for volume and area

formulae.

220 final report - group 14

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