220 final report - group 14
TRANSCRIPT
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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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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
materials: Rate: Vector Sum of Displ.-> 0.21532392
$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:
Changeable Input Variables
Material Density 2700
Outer Major Diameter Bottom(B1) 12
Outer Minor Diameter Bottom(A1) 6.40440778
Outer Major Diameter Top(B2) 5.489439541
Outer Minor Diameter Top(A2) 0.154382308
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Thickness Bottom(t1) 0.263244181
Thickness Top(t2) 0.060654316
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
materials: Rate: Vector Sum of Displ.-> 0.248420407
$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:
Changeable Input Variables
Material Density 7820
Outer Major Diameter Bottom(B1) 13.2
Outer Minor Diameter Bottom(A1) 5.114788407
Outer Major Diameter Top(B2) 4.236192117
Outer Minor Diameter Top(A2) 0.062839711
Thickness Bottom(t1) 0.096880375
Thickness Top(t2) 0.020401304
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
materials: Rate: Vector Sum of Displ.-> 0.241172067
$3,221,566.18 $2.76/kg Volume: 4.323E+02
Table 9Results for AluminumRevised Parameters
Changeable Input Variables
Material Density 2700
Outer Major Diameter Bottom(B1) 13.2
Outer Minor Diameter Bottom(A1) 4.950757186
Outer Major Diameter Top(B2) 6.006817514
Outer Minor Diameter Top(A2) 0.150510082
Thickness Bottom(t1) 0.225039063
Thickness Top(t2) 0.056267781
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.