Modeling and Finite Element Analysis of a Standard Concrete Canoe

AN ABSTRACT OF THE THESIS OF Stephanie S. Stache for the degree of Honors Baccalaureate of Science in Civil Engineering presented on September 4, 2013. Title: Modeling and Finite Element Analysis of a Standard Concrete Canoe.

Abstract approved:

Thomas Miller

This thesis project provides a guiding framework for the structural analysis portion of the National Concrete Canoe Competition (NCCC). Step-by-step instructions on how to 3-dimensionally model a “standard” sized concrete canoe in AutoCAD Civil

3D, import the model into SAP2000, and proceed to analyze the canoe with the

SAP2000 finite element analysis program are provided. Various hand calculations, reality checks, comparisons to the actual performance of previous canoes, and comparisons with similar concrete canoe analyses are used to justify the applicability of the outlined procedure. Recommendations for future research and how the analysis should affect mix design and paddler location to prevent structural failure are also offered.

Key words: concrete canoe, finite element analysis, SAP2000, AutoCAD Civil 3D, 3D model

Corresponding e-mail address: [email protected]

Modeling and Finite Element Analysis of a Standard Concrete Canoe

Stephanie S. Stache

A PROJECT

submitted to

Oregon State University

University Honors College

in partial fulfillment of the requirements for the degree of

Honors Baccalaureate of Science in Civil Engineering (Honors Scholar)

Presented September 4, 2013 Commencement June 2014

TABLE OF CONTENTS

1. Introduction ...... 1

2. Literature Review ...... 3

2.1 Das and Shaik (2010) ...... 3 2.2 Gilbert, Ooi, and Engberg (2006) ...... 3 2.3 Muzenski and Klett (2010) ...... 4

3. Methodology ...... 5

3.1 Materials ...... 5 3.2 Methods...... 6

4. Results ...... 7

4.1 Beaver Believer Canoe ...... 7 4.2 Hood Canoe ...... 14 4.3 General Canoe ...... 21

5. Discussion of Results ...... 25

5.1 Introduction ...... 25 5.2 Comparison of Models ...... 25 5.3 Comparison to Actual Canoes ...... 29 5.4 Additional Considerations ...... 34

6. Conclusion...... 36 6.1 Summary of Results ...... 36 6.2 Summary of Contributions ...... 36 6.3 Future Research ...... 38

7. Bibliography ...... 40

8. Appendices ...... 41

LIST OF FIGURES

Figure 1: Maximum Stresses for BB with 2-Paddlers ...... 9

Figure 2: Minimum Stresses for BB with 2-Paddlers ...... 10

Figure 3: Maximum Stresses for BB with 4-Paddlers ...... 12

Figure 4: Minimum Stresses for BB with 4-Paddlers ...... 13

Figure 5: Maximum Stresses for Hood with 2-Paddlers ...... 16

Figure 6: Minimum Stresses for Hood with 2-Paddlers...... 17

Figure 7: Maximum Stresses for Hood with 4-Paddlers ...... 19

Figure 8: Minimum Stresses for Hood with 4-Paddlers...... 20

Figure 9: SMax Stresses for Critical Area Load at the Side ...... 23

Figure 10: SMin Stresses for Critical Area Load at the Side ...... 24

Figure 11: Crack in the Middle Surface Inside of BB ...... 30

Figure 12: Reinforcement in Tension ...... 31

Figure 13: Punch-through Failure of Hood Canoe from Above ...... 32

Figure 14: Punch-through Failure of Hood Canoe from the Bottom ...... 33

Figure 15: Tensile Failure of Hood Canoe ...... 34

List of Tables

Table 1: Weights of the BB Canoe ...... 7

Table 2: Iterative Sum of Reactions for BB with 2-Paddlers ...... 8

Table 3: Stresses for BB with 2-Paddlers (psi)...... 8

Table 4: Iterative Sum of Reactions for BB with 4-Paddlers ...... 11

Table 5: Stresses for BB with 4-Paddlers (psi)...... 11

Table 6: Weights of the Hood Canoe ...... 14

Table 7: Iterative Sum of Reactions for Hood with 2-Paddlers ...... 15

Table 8: Stresses for Hood with 2-Paddlers (psi) ...... 15

Table 9: Iterative Sum of Reactions for Hood with 4-Paddlers ...... 18

Table 10: Stresses for Hood with 4-Paddlers (psi) ...... 18

Table 11: Stresses with 4-Paddlers nearer the Canoe Ends (psi) ...... 21

Table 12: Stresses with 4-Paddlers with Knees Farther Apart (psi) ...... 21

Table 13: Maximum Stresses from 1 lb Loads at Critical Locations ...... 22

Table 14: Comparison of Strengths and Stresses ...... 26

Table 15: Differences between Surface Area, Thickness, and Volume ...... 28

LIST OF APPENDICES

Appendix A: Tutorial for Primary Analysis ...... 42 Appendix B: Hand Calculations ...... 101 Appendix C: Standard Canoe Section Coordinates with Section Area Calculations ...... 107 Appendix D: Copyright Release ...... 119

LIST OF APPENDIX B FIGURES Figure B1: Canoe Simplified as a Simply Supported Beam from Essam (2005) ...... 102

Figure B2: Simple Analysis from Essam (2005) ...... 103

Figure B3: Precise Analysis from Essam (2005) ...... 105

LIST OF ACRONYMS ASCE American Society of Civil Engineers

BB Beaver Believer, indicates a “standard” shaped canoe

FEA Finite Element Analysis

NCCC National Concrete Canoe Competition

OSU Oregon State University

ACKNOWLEDGMENTS My deepest thanks go to those special few who supported me throughout my education of civil engineering, mainly by challenging me and urging me to succeed.

These people most notably include my mother (Michelle Lu), dearest friend (Jordan

Baird), and my mentor (Dr. Thomas Miller). Thank you for challenging and simultaneously urging me to succeed.

Further acknowledgments must go to the efforts of the tutorial proofreaders:

Garrison Stache and Brett Voyles. Both noted major flaws in the tutorial, so their contribution was vital. Thank you so much for volunteering yourselves as test subjects.

Additional thanks to the generous donations of the scholarships that made my education, and this thesis, much more feasible. More specifically, thanks go to the

Donald & Elizabeth Abott, Glen W. Holcomb Honorary Endowed, J & N Johnson,

Loosley, Oregon State University Engineering Dean’s, and Oregon State University

Diversity scholarships. This thesis is only the start of how I can repay your generosity.

MODELING AND FINITE ELEMENT ANALYSIS OF A STANDARD CONCRETE CANOE

1. INTRODUCTION

The National Concrete Canoe Competition (NCCC) is a national competition (actually practiced in several countries) to demonstrate civil engineering excellence within universities. In the U.S. the NCCC is held by the American Society of Civil Engineers

(ASCE). Since the establishment of the U.S. competition in the 1960s, Oregon State

University (OSU) has only qualified for the national level of the competition once, in

2012, partly due to its almost complete lack of structural analysis before 2010.

Unlike other civil engineering endeavors, the structural analysis for the competition is a closely guarded secret of most NCCC teams. Thus, public guides on the structural analysis for the NCCC are incomplete or non-specific. General approaches to analysis for the competition are outlined in several articles, but the focus is mainly on the results of the structural analysis. An additional difficulty to developing the structural analysis is that each analysis must be specially tailored to the specific properties of each individual canoe, such as the dimensions and mix design. In order to simplify the complex shapes required for the canoes a finite element analysis

(FEA) is typically used.

2011 was when I first started working on the three-dimensional structural analysis under the mentorship of Dr. Thomas Miller. Prior to 2011 the structural analysis component of the OSU concrete canoe competition consisted of a simplified 2

calculation assuming the canoe was half of the shell of a cylinder. An obvious improvement to drastically increase the accuracy of the analysis was the development of a scaled three-dimensional model. This model was created in

AutoCAD. In 2011 this model was a “standard” concrete canoe, based on the coordinates for a maximum canoe cross-section, as specified by ASCE on their NCCC website. In 2012 this model was scaled down to reflect a two foot decrease in canoe length implemented by the OSU NCCC captains. The tutorial outlined in Appendix A reflects the standard concrete canoe shape, though the applicability of the structural analysis will be tested by comparing the 2011 and 2012 models to the behavior of the actual canoes.

SAP20000 was the program used to complete the structural analysis. This program was able to import the three-dimensional model from AutoCAD and analyze the canoe as a finite element model. The finite element model of the canoe was loaded to obtain the resulting stresses in relation to each defining coordinate. It is vital that future civil engineers, such as students that participate in civil engineering competitions, are able to understand and perform structural analysis that is a key component to any structural engineering design. The purpose of this thesis is to develop a comprehensive methodology that will provide a guiding framework for the structural analysis portion of the NCCC.

2. LITERATURE REVIEW

Three separate analyses conducted for the concrete canoe competition were consulted in the Discussion of Results section.

2.1 Das and Shaik (2010)

FEA was accomplished with the ABAQUS program with a 3D model consisting of the middle cross-sections of the canoe. The model was loaded with a single paddler and the weight of the concrete at that section, with fixed constraints in all directions on the bottom of the model. A unique feature of this FEA is the use of tetrahedral meshes to create an accurate model and develop stresses.

2.2 Gilbert, Ooi, and Engberg (2006)

This FEA examined dynamic behavior with variations in the shape of a concrete canoe. Dynamic behaviors studied included the natural frequency, mode shapes, and damping effects of the canoe in motion. An important consideration for this FEA was the determination of material properties, such as elastic modulus and Poisson’s ratio, which is noted in Appendix A.

2.3 Muzenski and Klett (2010)

The ANSYS program was used to conduct an FEA on a 3D model using the same shape as the “standard” canoe used in this thesis, although the model consists of only half of the canoe. A significant difference in this model from the FEA in this thesis is the addition of motion forces, as discussed in Conclusions section 6.3.2.

Three different paddler load cases were considered in the FEA: no paddler, two paddler, three paddler, and four paddler. In addition, paddling benches were included in the model for each load case. Since the canoe designs discussed in this thesis did not include paddling benches, loads from the paddlers were applied to these benches rather than through the paddler knees used to model paddling loads in this thesis. This FEA also provided a Young’s Modulus and Poisson’s Ratio discussed in Appendix A.

3. METHODOLOGY

3.1 Materials

3.1.1 AutoCAD Civil 3D

Computer aided design (CAD) is commonly used in both education and in the civil engineering profession. AutoCAD is one of the most commonly used drafting programs in the nation, while AutoCAD Civil 3D is a specialized version of AutoCAD designed to assist civil engineers with 3-dimensional plans. To ease reading, the rest of the thesis shall refer to AutoCAD Civil 3D as “AutoCAD”.

3.1.2 SAP2000

Primary analysis was completed using SAP2000, a finite element analysis (FEA) program. The program was chosen due to its common use in the structural engineering field and in the OSU curriculum, and thus would be familiar to the OSU

NCCC team. This program is required in several classes, two of which are required in the OSU civil engineering undergraduate program. Thus, this thesis expects users will be familiar with this program.

3.1.3 Hand Calculations

Hand calculations were used as a “reality check” to ensure the loads, and thus results, are reasonable. Three different methods to estimate the pressure loading on the canoe are presented in Appendix B.

3.2 Methods

Detailed and systematic instructions for the primary analysis, only with racing loads, are located in the attached tutorial; see Appendix A. In addition, explanations for specific steps can be located in Appendix A as the boxed text.

4. RESULTS

4.1 Beaver Believer Canoe

The Beaver Believer (BB) canoe has a “standard” shape, with a total surface area of approximate 67 square feet. Buoyant forces for each of the race load cases on the BB canoe correspond to the weights shown in Table 1.

Table 1: Weights of the BB Canoe

Weight of Canoe 475 lbs Weight of Canoe and 2-Paddlers 835 lbs Weight of Canoe and 4-Paddlers 1135 lbs

4.1.1 2-Paddler Load

The iterative process from step 4.3 in Appendix A to determine the buoyant force area load for the 2-paddler men’s race is shown in Table 2, where the sum of the reactions must correspond to the weight in Table 1. Table 3 shows that the largest stresses in the top face of the canoe were in compression, while the largest stresses in the bottom face were in tension. The maximum tension stresses in the 2-paddler race (Table 3) were greater than the maximum tension stresses in the 4-paddler race shown in Table 5. The model shows that the greatest stresses occur at the paddler locations, as seen in Figure 1 and Figure 2.

Table 2: Iterative Sum of Reactions for BB with 2-Paddlers

Area Load Sum of Reactions (psi) (lb) 0.1 469 0.2 939 0.15 703 0.16 751 0.17 798 0.18 846 0.175 822 0.176 826 0.177 830 0.178 835

Table 3: Stresses for BB with 2-Paddlers (psi)

Max. Top Min. Top Max. Bottom Min. Bottom Tension 52.52 20.65 210.2 108.8 Compression 162.2 69.79 44.77 18.86

Figure 1: Maximum Stresses for BB with 2-Paddlers

Figure 2: Minimum Stresses for BB with 2-Paddlers

4.1.2 4-Paddler Load

The iterative process to determine the buoyant force area load for the 4-paddler men’s race is shown in Table 4, where the sum of the reactions must correspond to the weight in Table 1. Table 5 shows that the largest stresses in the top face of the canoe were in compression, while the largest stresses in the bottom face were in tension. The maximum tension stresses in the 4-paddler race (Table 5) were less than the maximum tension stresses in the 2-paddler race shown in Table 3. The values of the compression stresses in the top for the 4-paddler race were greater than the values of the top compressive stresses for the 2-paddler race shown in

Table 3. The model shows that the greatest stresses occur at the paddler locations, as seen in Figure 3 and Figure 4.

Table 4: Iterative Sum of Reactions for BB with 4-Paddlers

Area Load Sum of Reactions (psi) (lb) 0.2 965 0.3 1448 0.25 1207 0.24 1159 0.23 1110 0.235 1135

Table 5: Stresses for BB with 4-Paddlers (psi)

Max. Top Min. Top Max. Bottom Min. Bottom Tension 54.96 11.63 204.4 90.12 Compression 166.9 75.42 65.63 10.24

Figure 3: Maximum Stresses for BB with 4-Paddlers

Figure 4: Minimum Stresses for BB with 4-Paddlers

4.2 Hood Canoe

This canoe was scaled down from the standard canoe size, so that the length was decreased by 2 feet and total surface area was approximately 54 square feet.

Buoyant forces for each of the race load cases on the Hood canoe correspond to the weights shown in Table 6.

Table 6: Weights of the Hood Canoe

Weight of Canoe 230 lbs Weight of Canoe and 2-Paddlers 590 lbs Weight of Canoe and 4-Paddlers 890 lbs

4.2.1 2-Paddler Load

The iterative process to determine the buoyant force area load for the 2-paddler men’s race is shown in Table 7, where the sum of the reactions must correspond to the weight in Table 6. Table 8 shows that the largest stresses in the top face of the canoe were in compression, while the largest stresses in the bottom face were in tension. The maximum bottom stresses for the 2-paddler race were greater than the stresses in the 4-paddler race shown in Table 10. The model shows that the greatest stresses occur at the paddler locations, as seen in Figure 5 and Figure 6.

Table 7: Iterative Sum of Reactions for Hood with 2-Paddlers

Area Load Sum of Reactions (psi) (lb) 0.1 388 0.2 1166 0.15 582 0.16 621 0.155 601 0.154 596 0.153 594 0.152 588

Table 8: Stresses for Hood with 2-Paddlers (psi)

Max. Top Min. Top Max. Bottom Min. Bottom Tension 59.78 19.69 262.6 130.4 Compression 201.1 84.16 54.28 26.95

Figure 5: Maximum Stresses for Hood with 2-Paddlers

Figure 6: Minimum Stresses for Hood with 2-Paddlers

4.2.2 4-Paddler Load

The iterative process to determine the buoyant force area load for the 4-paddler men’s race is shown in Table 9, where the sum of the reactions must correspond to the weight in Table 6. Table 10 shows that the largest stresses in the top face of the canoe were in compression, while the largest stresses in the bottom face were in tension. The values of the tension stresses in the top for the 4-paddler race were greater than the values of the top stresses in the 2-paddler race shown in Table 8.

All the models show that the greatest stresses occur at the paddler locations, as seen in Figure 7 and Figure 8.

Table 9: Iterative Sum of Reactions for Hood with 4-Paddlers

Area Load Sum of Reactions (psi) (lb) 0.2 795 0.3 1192 0.25 993 0.24 954 0.23 914 0.22 874 0.225 894 0.224 890

Table 10: Stresses for Hood with 4-Paddlers (psi)

Max. Top Min. Top Max. Bottom Min. Bottom Tension 37.21 0.6226 262.4 114.5 Compression 214.0 97.43 45.29 1.396

Figure 7: Maximum Stresses for Hood with 4-Paddlers

Figure 8: Minimum Stresses for Hood with 4-Paddlers

4.3 General Canoe

The results shown below correspond to the 4-paddler load with a standard canoe.

4.3.1 Paddlers nearer the Ends

Each paddler was moved one cross-section (12 inches) closer to the nearest canoe end. All of the values for the stresses in Table 11 exceeded the corresponding values in Table 5, which contains the results from the general model in Appendix A.

Table 11: Stresses with 4-Paddlers nearer the Canoe Ends (psi)

Max. Top Min. Top Max. Bottom Min. Bottom Tension 61.38 13.70 248.7 120.2 Compression 197.6 86.60 70.55 11.31

4.3.2 Paddler Knees Spread Farther Apart

Each paddler “knee”, approximated by a pinned-support, was moved further from the centerline of the canoe by one shell section (approximately one inch away). All values in Table 12 were less than the corresponding values in Table 5, except for the minimum bottom compression.

Table 12: Stresses with 4-Paddlers with Knees Farther Apart (psi)

Max. Top Min. Top Max. Bottom Min. Bottom Tension 50.86 9.50 199.2 81.1 Compression 155.8 63.14 62.74 11.77

4.3.3 Critical Areas

The term “critical areas” is used to describe the areas that produce the largest

stresses when loaded with a 1 lb point load. Locations of the loads in Table 13 are

based on their labels in SAP2000, as shown in Section 6 of Appendix A. The labels of

the five critical areas are: SIDE, BOW, STERN, MID, and CORNER. The “in” refers to a

load pointed inward in relation to the canoe, such as a load on the outside of the

canoe pressing the canoe walls towards each other. Similarly, “out” refers to a load

which points away from the canoe, such as a load attempting to push the canoe

walls away from each other.

Table 13 shows the maximum stresses among all of the critical area loads examined.

This table shows the maximum stresses occurred due to a load at the side. Figure 9

and Figure 10 both show that the greatest stresses occur at the loading site (the top

of the canoe side in the middle). The figures also show that additional large stresses

occur in the bow.

Table 13: Maximum Stresses from 1 lb Loads at Critical Locations

Max. Top Min. Top Max. Bottom Min. Bottom Tension Critical Area SIDE in MID in SIDE out MID out Tension Stress (psi) 2.175 0.4420 1.802 0.7661 Compression Critical Area SIDE out MID out SIDE in MID in Compression Stress (psi) 2.175 0.4420 1.802 0.7661

Figure 9: SMax Stresses for Critical Area Load at the Side

Figure 10: SMin Stresses for Critical Area Load at the Side

5. DISCUSSION OF RESULTS

5.1 Introduction

This thesis developed a comprehensive methodology, found in Appendix A, that provides a guiding framework for the structural analysis portion of the NCCC.

Performance of the actual canoes was related to the analysis results from a corresponding 3-D model.

5.2 Comparison of Models

This section discusses the differences and similarities between the 3-D models. For instance, Figure 5 through Figure 8 show that the greatest stresses occur at the paddlers’ knees. Thus, it is important to reinforce these areas with fibers, mesh, or increasing the concrete thickness. The latter option may be explored using the analysis, which could be used to determine the optimal thickness in relation to stresses. Since this FEA may be used prior to construction, the average thickness specified in the model may not be correct. Thus, it may be necessary to test the sensitivity of the stress results resulting in a slight change in the model thickness before trying to optimize the model thickness.

5.2.1 2-Paddler and 4-Paddler Load Cases

Comparison of the 2-paddler and 4-paddler cases for both canoes has led to a vital conclusion: concrete canoes are likely to fail in tension. The tension stresses in both canoe bottoms were the maximum stresses for both canoes. In addition, the strength of concrete in tension is substantially less than its strength in compression. Thus, it is likely that these canoes will fail in tension. This conclusion is supported by Table

14, where the maximum compressive stresses are less than half of the design compressive strengths while the tensile strengths are exceeded by the maximum tensile stresses.

Table 14: Comparison of Strengths and Stresses

Canoe BB Hood Design Compression Strength (psi) 1240 1000 Max. Compression Stress (psi) 167 214 Tested Tensile Strength (psi) 130 241 Max. Tensile Stress (psi) 210 263

For each canoe both of the 4-paddler cases had the largest (compression) stresses in the top face, while both the 2-paddler canoe cases had the largest (tension) stresses in the bottom face. This result was confirmed with an FEA using the ANSYS program, where the maximum tensile strength was 350 psi (Muzenski & Klett, 2010, p. 3). The

2-paddler case had the greatest tension stresses in each canoe. Since the canoes are likely to fail in tension, this means that the 2-paddler load case is the controlling load case, despite the 4-paddler load case being subject to greater buoyancy

pressure. One likely explanation for this is that the additional paddlers in the middle of the canoe decrease the upwards tension by causing the canoe to deflect downwards.

The literature also supports the stresses found by the analysis. An FEA conducted with a single cross-section and loaded by a single male paddler yielded a maximum tensile stress of 55.6 psi (Das & Shaik, 2010, p. 5). Another FEA yielded a maximum stress of approximately 350 psi with the 2-paddler loading, with the addition of motion stresses and paddling benches (Muzenski & Klett, 2010, p. 3). The maximum stress for the 4-paddler load case was about 170 psi (Muzenski & Klett, 2010, p. 3).

All of these stresses are within an order of magnitude of the maximum stresses found in this analysis.

5.2.2 BB and Hood

All of the values for the maximum stresses in Hood were greater than the stresses from the corresponding load case with BB. Although Hood weighed 245 pounds less than BB, and thus experienced less buoyancy pressure, Hood also had a significant decrease in surface area and thickness from BB. It is likely that the decreased surface area and thickness decreased the cross-section, and thus increased the Hood stresses enough to become greater than the stresses in BB. Table 15 shows the difference in surface areas, thickness, and volume of the concrete.

Table 15: Differences between Surface Area, Thickness, and Volume

Canoe BB Hood Surface Area (ft2) 66.7 54.1 Average Thickness (in) 1.4 1.1 Volume (ft3) 7.8 5.0

5.2.3 Critical Areas

By using unit loads in different areas and comparing the resulting stresses, the critical areas were determined. Avoidance of loading these critical areas is especially important during the de-molding procedure and transportation of the canoe, since these are the instances when an impact to a critical area is most likely to occur.

Since the largest stresses occurred due to critical loads at the top of the canoe side in the middle, this area should be avoided. Additional large tension stresses occurred from loading the middle of the canoe bottom, so caution should be used around this area as well. Depending on the amount of additional weight, it may be advantageous to add extra reinforcement in these areas.

5.2.4 Paddlers nearer the Ends

The stresses for the BB 4-paddler load case with paddlers nearer the ends resulted in larger stresses, so the original models used were not the most conservative.

However, the models are based on the actual positions of the paddlers during the

races, so it is more realistic to use the models as specified. If necessary, the paddler positions may also be limited during the races to prevent failure from the increased stresses. There may also be a physical limitation imposed on the paddlers from flotation blocks at the ends that are typically incorporated into the canoe construction.

5.2.5 Paddler Knees Spread Farther Apart

The stresses for the BB 4-paddler load case with the paddlers’ knees spread further apart resulted in smaller stresses, so the models used were conservative in this regard. This result make logical sense, since more of the paddler load will be spread over greater area. However, it is also likely that the estimates of the stresses will increase the more the paddlers’ knees come together. In other words, it is necessary to consult with the paddling team to strongly encourage them to not put their knees together to avoid overstresses from this loading case.

5.3 Comparison to Actual Canoes

This section discusses the differences and similarities between the performance of the actual canoes and the analysis results from the corresponding models.

Applicability of the models, through comparison with actual canoe behaviors during competition, is also discussed in this section.

5.3.1 BB

Every load case of the analysis shows that compression is the greatest stress in the top faces, while tension is the greatest in the bottom. This result concurs with the BB canoe behavior. The result that maximum stress was located at the bottom was also confirmed in a separate FEA done with the ANSYS program (Muzenski & Klett, 2010, p. 3). Once the canoes experienced water pressure the canoe ends lifted and were pushed together. Thus, the middle surface inside of the canoes was visibly put into compression, since small cracks visible prior to immersion were less apparent after due to the cracked sides being pushed together by compression. However, an example of a crack, due to tension stresses in the top occurs while de-molding, as shown in Figure 11. Following this logic, it becomes apparent that if the top face of the canoe is in compression, then the bottom face will be in tension.

Figure 11: Crack in the Middle Surface Inside of BB

Despite the maximum tensile stress exceeding the tested tensile strength for the BB concrete, BB did not experience tensile failure. This is most likely due to the reinforcement, since the reinforcement visibly held together tensile cracks in the concrete, as documented in Figure 12.

Figure 12: Reinforcement in Tension

5.3.2 Hood

Hood experienced a “punch through” failure due to a paddler’s knee breaking through the bottom of the canoe, as seen in Figure 13. This failure was not prevented by the reinforcement because the knee pressed through an overlap splice in the reinforcement. The top panel of the overlap is visible in Figure 13, while the bottom panel is visible in Figure 14. Since only one knee broke through, it is unlikely that this failure was caused by an increase in stresses due to the paddler knees being put together (as discussed in section 5.2.5). The position of the paddler’s knee

also went through the canoe at a location approximated in the model, so failure due to an increase in stresses from paddlers nearer the ends of the canoe is also unlikely.

To prevent punch through failures in overlapped sections the reinforcement meshes should be completely surrounded by concrete, rather than letting the meshes rest atop each other. In other words, both sides of the reinforcement meshes should be bonded to concrete rather than letting the meshes directly contact and overlap each other. This failure demonstrates that the analysis must consider constructability in order to remain accurate.

Figure 13: Punch-through Failure of Hood Canoe from Above

Figure 14: Punch-through Failure of Hood Canoe from the Bottom

After the punch through failure, a secondary failure from tensile stresses occurred, as shown in Figure 15. The decrease in cross-section from the punch through combined with the larger stresses in Hood compared to BB is likely the cause of the secondary failure. Since stress results from force over an area, the decrease in the surface area of Hood probably exceeded the decrease in water pressure loading

(due to decreased canoe weight). Larger stresses may help explain why Hood failed whereas BB did not. In this way, the analysis could possibly function as a way to optimize canoe shape with racing stresses. For instance, various shapes could be modeled with FEA. The shape with the minimum stresses may then be chosen for construction.

Figure 15: Tensile Failure of Hood Canoe

5.4 Additional Considerations

Additional considerations based on the results, and recommendations for future analysis and testing are discussed.

5.4.1 Minimum Tensile Strength

There is an additional need to consider minimum tensile strength of concrete to prevent cracking. Tensile strength for the canoes was found with the split tensile test; however, an additional test for the modulus of rupture may be useful. One should reinforce specific critical areas, and determine a baseline tensile strength required for the concrete in areas that will not be reinforced.

5.4.2 Minimum Compression Strength

Although the analysis in the tutorial was conducted after the canoe was constructed, an analysis should really be performed prior to mix design in order to specify a minimum compressive strength. The compressive strength of the concrete in a canoe should exceed the estimated compressive stresses.

5.4.3 Hand Calculations

The hand calculations in Appendix B estimate the pressure loading on the side of a standard canoe. According to the simplified method by Muzenski and Klett (2010) the pressure loading should be 0.325 psi. The next simplified method states that the pressure loading should be 0.315 with 4-paddlers and 0.232 psi with 2-paddlers. All of the hand calculation estimates have the same order of magnitude as the final pressure loadings shown in Table 2, Table 4, Table 7, and Table 9. Since the hand calculations in B1.1 and B1.2 exceeded the actual loadings used, this suggests these hand calculations are conservative.

Lastly, the method in B1.3 yielded 0.158 psi for 4-paddlers and 0.134 psi for 2- paddlers. Both these estimates are less than the loadings used in the analysis. When the B1.3 calculations were redone to determine a radius of pipe that would correspond to the pressure loadings used in the analysis, a radius of 6 inches (as opposed to the 10 inches shown in B1.3) was necessary for both load cases.

6. CONCLUSION

6.1 Summary of Results

The following are significant points made in the Discussion of the Results section:

 Additional reinforcement is needed at paddler locations due to high stresses  Additional reinforcement should be put in critical areas  Concrete canoes are likely to fail from tensile stresses  Minimum tensile strength of the concrete should be specified by the analysis  Maximum tensile stresses occur with the 2-paddler load case  Hood probably had greater stresses due to its shorter length  Critical areas that should not be loaded are the tops of the canoe sides in the middle  Another critical area that should not be loaded is the middle of the canoe bottom  Paddlers should try to stay near the middle of the canoe to decrease upwards tensile stresses  Paddlers should avoid bringing their knees together; this increases stresses  Tension stresses occur on the bottom of the canoe during racing  Reinforcement is necessary when tensile stresses exceed concrete tensile strength  Overlapped reinforcing mesh should bond to concrete on all sides to be effective  Analysis must consider constructability to be accurate  Design compressive strength should exceed the compressive stresses from analysis  Modulus of rupture for the concrete should be tested

6.2 Summary of Contributions

This is a summary of the new knowledge offered by this thesis, organized from most to least important.

6.2.1 Finite Element Analysis Applicable to Actual Concrete Canoes

The development of an analysis directly applicable to actual concrete canoe performance establishes the value of the tutorial. In addition, users are presented with multiple ways to understand and check this FEA, which may also be applied to other FEAs.

6.2.2 Tutorial Document

Provision of a document with step-by-step instructions allows students to navigate an unfamiliar analysis procedure on their own, so teams will not need to plan an actual handoff or train successors.

6.2.3 Integration of 3D Modeling into Analysis

Advancements in modeling and analysis software have encouraged the integration of 3D models and analysis in many professional civil engineering projects. Guiding students through this integration may become increasingly necessary for career preparation. Additionally, the adjustment of a 3D model in an analysis program to increase the accuracy of results to reflect an actual structure may be necessary for engineering projects involving existing structures, such as repairs.

6.3 Future Research

Several possible uses of this analysis were noted in the Discussion of Results, including:

 Optimization of canoe thickness with racing-induced stresses  Optimization of canoe shape with racing induced-stresses

6.3.1 De-molding and Transportation Load Cases

Although critical areas were identified, a more involved analysis of the specific cases should be conducted to prevent failure. Thus, I would recommend that further analysis be done for the transportation and de-molding load cases. Both load cases are highly dependent on the specific techniques and materials involved with each process. Consultation and planning with the captains will be required to ascertain the correct way to support the model. For instance, a line of roller supports could be used to model the straps of the metal frame carrier constructed for short-distance travel (such as loading or entering the water).

A necessary consideration of the de-molding process is the type of mold. A male mold can be flipped and de-molded while the tops of the walls are supported by the floor. In this instance, roller supports at the tops of the canoe walls in the model may be sufficient to accurately analyze the support conditions of the de-molding process.

A female mold typically involves the removal of the mold that supports the canoe while the mold is stripped away, until the canoe rests on its bottom.

6.3.2 Motion

The analysis also neglects motion. More specifically, the increase in water pressure due to motion and drag forces. It is recommended that motion forces should increase the total water pressure on the bow by 75%, the sides by 20%, and the stern by 5% (Muzenski & Klett, 2010, p. 2). Additional analysis should be done to determine the effect of varying the water pressure distribution, to model possible changes from movement during the races. More importantly, an increase in the stresses due to motion shifts the location of the maximum stresses (Muzenski &

Klett, 2010, p. 3). Thus, the location of additional reinforcement may need to be modified.

6.3.3 Reinforcement

SAP2000 has the capability of analyzing composite materials to determine final stresses and deformations. However, the material properties of both the concrete and the reinforcement mesh must be known and manually input into SAP2000.

Thus, the tensile strength of reinforcement should be tested prior to analysis. An additional test of a composite of concrete with reinforcement may also be helpful.

7. BIBLIOGRAPHY

ACI Committee 318. (2011). ACI 318-11 Building Code Requirements for Structural Concrete. Farmington Hills, MI: American Concrete Institute.

Carasquillo, R. L., Nilson, A. H., & Slate, F. O. (1981, May-June). "Properties of High Strength Concrete Subjected to Short Term Loads". Journal of the American Concrete Institute, 78(3), 171-178.

Das, G., & Shaik, S. (2010). Optimization of the Design of a Concrete Canoe using Finite Element Analysis. American Society for Engineering Education.

Essam, A. (2005, December 27). E-book. Retrieved November 25, 2012, from AHM531: www.ahm531.com/E- book/Uploaded/THE%20WHOLE%20THING.doc Gilbert, J. A., Ooi, T. K., & Engberg, R. C. (2006). Modal Analysis of a Lightweight Concrete Canoe. Concrete Canoe Magazine, pp. 27-33. Muzenski, S., & Klett, J. (2010). Finite Element Analysis of a Concrete Canoe. University of Wisconsin-Milwaukee, Advanced Manufacturing and Design Laboratory. Milwaukee, WI: University of Wisconsin-Milwaukee. Narayanan, N., & Ramamurthy, K. (2000, April). Structure and properties of aerated concrete: a review. Cement & Concrete Composites, 22, 321-329. Retrieved March 2012, from https://www.ownerbuilderbook.com/forum/files/forums/19679-1.pdf

Wight, J. K., & MacGregor, J. G. (2011). Reinforced Concrete: Mechanics and Design (6 ed.). Prentice Hall.

8. APPENDICES

APPENDIX A: TUTORIAL FOR PRIMARY ANALYSIS

TABLE OF CONTENTS 1. Introduction ...... 44 1.1 Description of Contents ...... 44 1.2 Expected Requirements of Users ...... 44 1.3 Explanation of Special Formats ...... 45

2. Draft the 3D Model ...... 47 2.1 Create cross-sections ...... 47 2.2 Create 3D surfaces ...... 49 2.3 Make 3D surfaces exportable to SAP2000 ...... 52

3. Refine the Model for Analysis...... 54 3.1 Import 3D model ...... 54 3.2 Standardize direction of sections ...... 56 3.3 Define material ...... 59 3.4 Define Section Properties...... 63 3.5 Estimate Volume and Weight ...... 66 3.6 Refine Section Properties ...... 71

4. Analyze Load Case with 4 Paddlers ...... 74 4.1 Create file...... 74 4.2 Define the supports ...... 74 4.3 Load canoe with buoyant force ...... 75 4.4 Format Excel Results ...... 82 4.5 View Stresses ...... 83

5. Analyze Load Case with 2 Paddlers ...... 87 5.1 Create file...... 87 5.2 Define the supports ...... 87 5.3 Load canoe with buoyant force ...... 88 5.4 Format Excel Results ...... 89

6. Analyze Critical Areas ...... 90 6.1 Create Analysis File ...... 90 6.2 Create the Critical Area Loads ...... 90 6.3 Analyze the Critical Load Cases ...... 96 6.4 Format Excel Results ...... 97

7. Conclusion...... 99

8. Bibliography ...... 100

1. INTRODUCTION

1.1 Description of Contents This tutorial features systematic instructions on how to: 1. Create a 3D model of a “standard” canoe in AutoCAD 2012 2. Export 3D model from AutoCAD 2012 into SAP2000v.15.1.0 3. Define load cases 4. Analyze load cases 5. Interpret results for canoe design 6. Check with reality and hand calculations

To make this tutorial easier to follow, it uses the same inputs from the 2012 canoe Beaver Believer (BB).

1.2 Expected Requirements of Users It is expected that users of this tutorial will have basic experience with AutoCAD, SAP2000, and Excel. At the very least, users should already know how to:  Create a layer in AutoCAD  Select AutoCAD and SAP2000 objects, especially with a selection box  Zoom in and out of AutoCAD and SAP2000  Save or do a “Save as”

If these functions are not known by the user, please refer to the Help sections in the respective programs by opening the programs, pressing the “F1” button on the keyboard, doing a search for the keywords, and reading through the provided articles. Here are some recommendations:  Try to understand each section before starting, especially the boxed portions  Save regularly and continuously, in addition to the saves required by the instructions

1.3 Explanation of Special Formats Specific actions, such as typing in a command, are indicated by different text formats. Please note these formats before starting the tutorial:  Boxed text explains the reasoning behind the methodology o Inputs based on actual values from the mix and canoe construction are boxed o For example, the explanation for the canoe wall section thickness looks like this:

Average thickness of the BB walls was 1.5 inches after construction.

 Drop down menu selections will be in bold and separated by  o For example, File  Save indicates:

 Non-letter buttons to be typed are simply italicized o For example, Enter indicates pressing on the keyboard button marked “Enter  Sections for drop-down menus will not be in bold, but will be in “ ” followed by  o For example, “Files of type:”  Drawing (*.dwg) indicates:

 Typed commands will be located between ‘ ’ o Typed commands avoid having to locate specific tool buttons in AutoCAD o Commands are capitalized to make them easier to see, not because they must be input as uppercase letters o For example, ‘PL’ indicates typing “P” and then “L”, and will start a 2D polyline the same way this button would:

2. DRAFT THE 3D MODEL

2.1 Create cross-sections ASCE maximum cross-section coordinates are available for export into Excel at: http://www.asce.org/concretecanoe/rules-regulations/ 2.1.1 To see the coordinates, click on the link under “Resources” called “Standard NCCC Hull Design Coordinates” and open the excel file 2.1.2 After opening AutoCAD and creating a new file, create a new current layer called “Frame” 2.1.3 Make sure Dynamic Input uses Absolute coordinates a. ‘DYNPICOORDS’ b. Enter c. ‘1’ d. Enter 2.1.4 ‘3DPOLY’ 2.1.5 Type in the x-coordinate provided by the “Standard NCCC Hull Design Coordinates” excel file (x-coordinates are located under the “station (x)” column) 2.1.6 ‘,’ 2.1.7 Type in the provided y-coordinate (located under the “offset (y)” column) 2.1.8 ‘,’ 2.1.9 Type in the provided z-coordinate (located under the “height (z)” column) 2.1.10 Enter 2.1.11 Continue steps 2.1.5 – 2.1.10 until all the coordinates for that station have been input 2.1.12 Enter 2.1.13 ‘MIRROR’ 2.1.14 Select all of the lines 2.1.15 Enter 2.1.16 ‘0,0’ 2.1.17 Enter 2.1.18 ‘1,0’ 2.1.19 Enter 2.1.20 Make sure “N” is typed into the “Erase source objects?” query before pressing Enter

Once you are done, you should have something similar to this (the uppermost line is not necessary, and has been added by snapping a polyline to the top of each cross-section):

Although the distance between the middle cross-sections is significantly smaller than the distances between the rest of the cross-sections, this should not affect the results of the model, since the sections will be joined to act as one shell and the area loading shall be used.

2.2 Create 3D surfaces 2.2.1 Create a new current layer called “Mesh” 2.2.2 ‘SURFTAB1’ (defines the number of sections created between cross- sections) 2.2.3 ‘12’

You can type in a greater number to increase the accuracy of the mesh. For this model, a number less than 20 should provide adequate accuracy (the greater the number, the greater the number of planar surfaces to be analyzed, which will increase the amount of time taken for analysis).

2.2.4 ‘RULESURF’ 2.2.5 Select 2 consecutive half-cross-section lines a. Be consistent with the general location you click to select the lines. For instance, if you select the 1st line near the top then you need to select the 2nd line near the top:

You should end up with smooth mesh, such as this:

b. Selecting the top of the 1st line and bottom of the 2nd line leads to this:

2.2.6 Continue steps 2.2.3 – 2.2.4 with alternating sections, like this:

After the last section you the model should look similar to this:

2.2.7 Select all the mesh sections a. You can either click on each one separately or b. Quick select by layer c. ‘QSELECT’ d. “Properties:”  Layer e. “Value:”  Mesh

2.2.8 Mirror the mesh a. ‘MIRROR’ b. ‘0,0’ c. Enter d. ‘1,0’ e. Make sure “N” is typed into the “Erase source objects?” query before pressing Enter

2.3 Make 3D surfaces exportable to SAP2000 2.3.1 Create a new current layer called “Shell” 2.3.2 Select all mesh sections (see step 2.2.7) 2.3.3 Duplicate the sections selected in step 2.3.2 into the “Shell” layer by copying and pasting them 2.3.4 Turn off the “Mesh” layer 2.3.5 ‘EXPLODE’ will make each rectangle in the mesh its own separate shell section 2.3.6 Resolve conflict sections (between end meshes and regular section), once you zoom in they will look like 2 triangles:

a. Zoom in and select the triangles (indicated by the black arrows in the picture above) at one end of the canoe b. Delete c. Select the sections directly above where the triangles used to be d. Click on the blue section point that used to be at the triangle tip e. Move the point directly down to where the triangle base was. The sections should now look like this:

2.3.7 Repeat step 2.3.6 for the triangles on the other end of the canoe 2.3.8 Save the drawing 2.3.9 Hold down Ctrl, Shift, and S to do a “Save As” 2.3.10 “Files of type:”  AutoCAD 2010 DXF (*.dxf) to save as an exportable AutoCAD file:

3. REFINE THE MODEL FOR ANALYSIS

3.1 Import 3D model 3.1.1 Create new file a. Open SAP b. File  New Model… c. “Initialize Model from Defaults with Units”  lb, in, F d. Click on “Blank” under “Select Template”

3.1.2 Import .dxf a. File  Import  AutoCAD .dxf File… b. “Units”  lb, in, F

c. Enter d. “Shells”  Shell

e. Click “OK”

3.1.3 Save the file

3.2 Standardize direction of sections 3.2.1 Show local axes of each section a. Display  Show Misc Assigns  Area… b. Click on “Local Axes”

c. Click “OK”

Now something similar to this should be displayed:

3.2.2 Select the sections with the blue arrow (z direction) pointing outward 3.2.3 Reverse the local axes a. Assign  Area  Reverse Local 3 b. Click on “Keep Assigns in Same Local Orientation”

3.3 Define material 3.3.1 Define  Materials 3.3.2 Click on “Add New Material…”

3.3.3 “Material Type”  Concrete and “Grade”  f’c 3000 psi Lightweight

3.3.4 Click “OK”

3.3.5 Select the newly created material “3000PsiLW” and click on “Modify/Show Material…”

3.3.6 Change the applicable material properties according to the mix design a. “Material Name and Display Color”  type in a new name b. “Units”  lb, ft, F c. “Weight per Unit Volume”  type in unit weight of mix

BB mix unit weight was 61.8 pcf.

d. “Units”  Kip, in, F e. “Modulus of Elasticity, E”  type in value

BB compressive strength was 1240 psi, so the Modulus of Elasticity used was 519 ksi (Narayanan & Ramamurthy, 2000, p. 325). This value is also similar to the elastic modulus of 510 ksi used in a separate analysis (Gilbert, Ooi, & Engberg, 2006, p. 31). A Young’s Modulus of 500 ksi is described as “conservative” by a separate analysis (Muzenski & Klett, 2010, p. 2).

f. “Poisson’s Ratio, U”  type in “0.18”

This value was based on a recommended value (Wight & MacGregor, 2011). A separate analysis used a similar value of 0.2 (Muzenski & Klett, 2010, p. 2). This value differs from the Poisson’s ratio of 0.137 used in a separate finite element analysis, though the concrete was only 0.34 inches thick (Gilbert, Ooi, & Engberg, 2006, p. 31).

Shear modulus was 75 ksi for a separate finite element analysis for concrete sides 0.34 inches thick (Gilbert, Ooi, & Engberg, 2006, p. 31).

g. “Units”  lb, in, F h. “Specified Concrete Compressive Strength, f’c”  type in value from mix design

BB design strength was 1240 psi.

3.3.7 Click “OK” 3.3.8 Click “OK” on the “Define Materials” menu

3.4 Define Section Properties 3.4.1 Define  Section Properties  Area Sections… 3.4.2 Click “Add New Section…”

3.4.3 Input Shell Section Data a. “Section Name”  rename the section (such as “Canoe”) b. “Type”  click on “Shell – Thin” c. “Material Name”  select the newly defined material (from last step) d. “Thickness”  type in the planned average thickness for both “Membrane” and “Bending”

Average thickness of the BB walls was 1.5 inches after construction.

e. Click “OK” f. Click “OK” on the “Area Sections” menu

3.4.4 Assign section properties to the canoe a. Select all of the canoe b. Assign  Area  Sections c. “Sections”  select your newly created Section (I used “Canoe”)

d. Click “OK”

3.5 Estimate Volume and Weight These are estimates to tell you how accurate your analysis is, as well as give you an idea of how exact the construction was. The values calculated here should not be put in to the final report. Instead, physically weigh the entire canoe and use this value for the weight required in the report. 3.5.1 Export the section areas a. File  Export  SAP2000 MS Excel Spreadsheet .xls File… b. Expand “Connectivity Data” (click on the plus symbol next to it) c. Expand “Object Connectivity” d. Click on “Table: Connectivity - Area” e. “Options”  check the box for “Open File After Export” f. Click “OK”

g. Save the file h. Click “Save”

3.5.2 Set-Up Excel a. Copy the “Connectivity – Area” tab b. Rename the copy as “Volume & Wt” c. Go to the “Volume & Wt” tab d. Insert 3 rows under the title of the table, so it looks like this:

3.5.3 Get rid of the background color of the new rows 3.5.4 Type in the following:

3.5.5 Calculate Total Surface Area a. Sum the numbers in the “AreaArea” column

b. Divide the value from 4.2.7 by 144 to get the value in square feet

3.5.6 Estimate Average Wall Thickness a. Input the probable thickness of the canoe

Average thickness of the BB walls was 1.5 inches after construction.

b. Divide the value from 4.2.2 by 12 to get the value in feet

3.5.7 Calculate Total Volume a. Multiply the values for the Total Area and the Thickness to get Total Volume, for each respective unit, like so:

b. Or multiply the “in^2” Total Area and the “in” Thickness value to get the “in^3” Total Volume, then divide the “in^3” Total Volume Value by 1728 to get the “ft^3” Total Volume value

3.5.8 Estimate Weight a. Input the probable unit weight of the concrete for “Mix Unit Weight”

BB had a mix unit weight of 61.8 pcf.

b. Multiply the value for “Mix Unit Weight” with the value for “Total Volume” in cubic feet

c. Input the actual weight

Actual weight of BB was 375 lbs.

3.5.9 Compare Estimated and Actual Weights a. Take the absolute difference between the values for “Est. Weight” and “Actual Weight”

b. Divide the value from 4.2.8 by the value for “Actual Weight”

c. Format the value from 4.2.9 into a percentage

According to the 2012 National Concrete Canoe Competition rules the allowable difference between the actual canoe weight and the estimated weight in the report is 10 lbs, so you can skip step 3.6 if your weights differ by less than 10 lbs.

3.6 Refine Section Properties 3.6.1 Calculate the Actual Average Thickness a. Copy the “Volume & Wt” tab from step 3.5 b. Rename the copy “Volume & Wt & Thickness” c. Change the thickness in 0.1 inch increments until the % difference is minimized (or there is less than 10 lbs of difference between the weights)

3.6.2 Go to the SAP2000 model saved after step 3.4 3.6.3 Define  Section Properties  Area Sections… 3.6.4 Click “Add New Section…”

3.6.5 Input Shell Section Data a. “Section Name”  rename the section b. “Type”  click on “Shell – Thin” c. “Material Name”  select your newly defined material (from last step) d. “Thickness”  type in the planned average thickness for both “Membrane” and “Bending” (approximate weight was obtained with a thickness of 1.4 inches)

e. Click “OK” f. Click “OK” on the “Area Sections” menu

3.6.6 Assign section properties to the canoe a. Select all of the canoe b. Assign  Area  Sections c. “Sections”  select your newly created Section

d. Click “OK”

4. ANALYZE LOAD CASE WITH 4 PADDLERS

4.1 Create file 4.1.1 Open the refined model in SAP2000 4.1.2 Save as a new file (such as “Race_4”)

4.2 Define the supports 4.2.1 Select joints at the likely locations for the paddlers’ knees

4.2.2 Assign  Joint  Restraints… 4.2.3 Click on the pin symbol, or check every translation

4.2.4 Repeat steps 4.2.1– 4.2.3 for the paddlers on the other end of the canoe

4.3 Load canoe with buoyant force 4.3.1 Define load pattern a. Define  Load Patterns… b. “Load Pattern Name”  type in “WATER” and “Type”  OTHER c. Click on “Add New Load Pattern” d. Click “OK”

4.3.2 Select the entire canoe 4.3.3 Apply buoyant force on the canoe a. Assign  Area Loads  Uniform (Shell)… b. “Load Pattern Name”  WATER (make sure units are in “lb, in, F”) c. “Load”  type in arbitrary load (remember, it is in psi) d. “Coord System”  Area Local e. “Direction”  3 f. Make sure “Replace Existing Loads” is selected

A separate analysis used a similar pressure of 0.217 psi (Muzenski & Klett, 2010, p. 2).

4.3.4 Check direction of buoyant force (force should push canoe sides inward) a. Display  Show Load Assigns  Area… b. “Load Pattern Name”  WATER c. Click on “Uniform Load Resultants” d. Check that the forces look similar to this:

4.3.5 Run the buoyant force as the only load case a. Analyze  Set Load Cases to Run… b. Click on “DEAD” c. Click “Run/Do Not Run Case” so that “Action”  “Do not Run”

d. Repeat c after clicking on “MODAL”

e. Click on “Run Now” (you can get to “Run Now” by clicking the triangular button):

4.3.6 Display the reactions a. Display  Show Forces/Stresses  Joints b. “Case/Combo”  “Case/Combo Name”  WATER c. “Type”  Click on “Show Results as Arrows” d. Click “OK”

4.3.7 Export the reactions a. File  Export  SAP2000 MS Excel Spreadsheet .xls File… b. Expand “Joint Output” (click on the plus symbol next to it) c. Expand “Reactions” d. Click on “Table: Joint Reactions” e. “Options”  check the box for “Open File After Export” f. Click “OK”

g. Save the file h. Click “Save”

4.3.8 Check the buoyant force a. Go to the excel file and sum the “F3” column

b. See if the absolute value of the “F3” sum is equal to the total weight

Total weight is the weight of the canoe and the paddlers. For the 4- paddler load case there are 2 male paddlers and 2 female paddlers. Male paddlers can be approximated as 180 lb each, while female paddlers are approximately 150 lb each. BB was weighed as 475 lb, so the total weight for the 4-paddler race was approximately 1135 lb.

c. Close the excel file

4.3.9 Repeat steps 4.3.3– 4.3.8 (you can skip steps 4.3.4 and 4.3.6) until the “F3” sum equals the total weight

4.3.10 Export the resulting stresses a. File  Export  SAP2000 MS Excel Spreadsheet .xls File… b. Expand “Element Output” c. Expand “Area Output” d. Click on “Table: Element Stresses – Area Shells” e. “Options”  check the box for “Open File After Export” f. Click “OK” g. Save the file separate from the exported reaction table

4.4 Format Excel Results 4.4.1 Copy the “Element Stresses – Area Shells” tab 4.4.2 Rename the copy “RESULTS” 4.4.3 Under the first row insert two rows and clear their background color 4.4.4 Find the maximum for the “SMaxTop” column in a cell above 4.4.5 Find the minimum for the “SMaxTop” column in the cell under the maximum 4.4.6 Repeat 4.4.4 – 4.4.5 for the columns labeled “SMinTop”, “SMaxBot”, and “SMinBot” If you want, you may hide all the other columns with stresses by selecting and then right clicking and selecting “Hide”. The excel tab should now look similar to this:

4.5 View Stresses 4.5.1 Display  Show Forces/Stresses  Shells… 4.5.2 “Component Type”  Check “Shell Stresses” 4.5.3 “Output Type”  Check “Absolute Maximum” 4.5.4 “Component”  Check “SMax”

4.5.5 Save a screenshot of the canoe in 3D view, which should look similar to the following:

4.5.6 Display  Show Forces/Stresses  Shells… 4.5.7 “Component Type”  Check “Shell Stresses” 4.5.8 “Output Type”  Check “Absolute Maximum” 4.5.9 “Component”  Check “SMin”

4.5.10 Save the screenshot, which should look similar to the following:

5. ANALYZE LOAD CASE WITH 2 PADDLERS

5.1 Create file 5.1.1 Open the refined model in SAP2000 5.1.2 Save as a new file (such as “Race_2”)

5.2 Define the supports 5.2.1 Select joints at the likely locations for the paddlers knees

5.2.2 Assign  Joint  Restraints… 5.2.3 Click on the pin symbol, or check every translation

5.2.4 Repeat steps 5.2.1 – 5.2.3 for the paddlers on the other end of the canoe

5.3 Load canoe with buoyant force 5.3.1 Define load pattern a. Define  Load Patterns… b. “Load Pattern Name”  type in “WATER” and “Type”  OTHER c. Click on “Add New Load Pattern” d. Click “OK”

5.3.2 Select approximately 80% up the sides of the canoe a. Left click the screen below the canoe and beyond the end of the canoe. b. The selection box should exclude the very top line of sections in the middle of the canoe:

Approximately 80% of BB was submerged for the 2-paddler races. Selecting only 80% up the canoe sides is more accurate, since there is less weight to be displaced by buoyant forces (the canoe will float higher in the water).

5.3.3 Repeat steps 4.3.3– 4.3.8 (see the section about the Load Case with 4 Paddlers)

5.3.4 Reset the pressure load on the canoe a. Assign  Area Loads  Uniform (Shell)… b. “Load Pattern Name”  WATER (make sure units are in “lb, in, F”) c. “Load”  0 d. “Coord System”  Area Local e. “Direction”  3 f. Make sure “Replace Existing Loads” is selected

5.3.5 Repeat steps 4.3.3– 4.3.8 and step 5.3.4 (you can skip steps 4.3.4 and 4.3.6) until the “F3” sum equals the total weight 5.3.6 Repeat step 4.3.10

5.4 Format Excel Results 5.4.1 Copy the “Element Stresses – Area Shells” tab 5.4.2 Rename the copy “RESULTS” 5.4.3 Under the first row insert two rows and clear their background color 5.4.4 Find the minimum for the “SMaxTop” column in a cell above 5.4.5 Find the maximum for the “SMaxTop” column in the cell under the minimum 5.4.6 Repeat steps 5.4.4 –5.4.5 for the columns labeled “SMinTop”, “SMaxBot”, and “SMinBot” If you want, you may hide all the other columns with stresses by selecting and then right clicking and selecting “Hide” The excel tab should now look similar to this:

6. ANALYZE CRITICAL AREAS

6.1 Create Analysis File 6.1.1 Open the refined model from section 3 in SAP2000 6.1.2 Save as a new file (such as “Critical_Areas”)

6.2 Create the Critical Area Loads 6.2.1 Define the load patterns a. Define  Load Patterns… b. “Load Pattern Name”  type in “SIDE in” c. “Type”  OTHER d. Click on “Add New Load Pattern” e. Click “OK” f. Repeat 6.2.1b. – 6.2.1e. by typing in the following new load patterns for 6.2.1b.:  “SIDE out”  “BOW in”  “BOW out”  “STERN in”  “STERN out”  “MID in”  “MID out”  “CORNER in”  “CORNER out”

6.2.2 Load the critical area a. Select the joint at the top of one of the canoe sides in the middle b. Assign  Area Loads  Joint Forces c. “Load Pattern Name”  SIDE in d. “Force 2”  “1” e. “Coord System”  Local f. Click “OK”

6.2.3 Check the critical area loading a. Display  Show Load Assigns  Joint… b. “Load Pattern Name”  SIDE in c. Click on “Uniform Load Resultants” d. Check that the forces look similar to this:

6.2.4 Repeat 6.2.2. – 6.2.3 with the “SIDE out” load case and typing “-1” for 6.2.2d.

6.2.5 Repeat 6.2.2. – 6.2.3 with the “BOW in” load case with these changes:  6.2.2a.  Select the joint at the top of the bow (the end further from the middle)  6.2.2c.  “Load Pattern Name”  BOW in  6.2.2d.  “Force 1”  “1”  6.2.3b.  “Load Pattern Name”  BOW in  6.2.3d.  Check that the forces look similar to this:

6.2.6 Repeat 6.2.2. – 6.2.3 with the “BOW out” load case and typing “-1” for 6.2.2d.

6.2.7 Repeat 6.2.2. – 6.2.3 with the “STERN in” load case with these changes:  6.2.2a.  Select the joint at the top of the stern (the end closer to the middle)  6.2.2c.  “Load Pattern Name”  STERN in  6.2.2d.  “Force 1”  “-1”  6.2.3b.  “Load Pattern Name”  STERN in  6.2.3d.  Check that the forces look similar to this:

6.2.8 Repeat 6.2.2. – 6.2.3 with the “STERN out” load case and typing “1” for 6.2.2d.

6.2.9 Repeat 6.2.2. – 6.2.3 with the “MID in” load case with these changes:  6.2.2a.  Select the joint in the middle of the canoe bottom  6.2.2c.  “Load Pattern Name”  MID in  6.2.2d.  “Force 3”  “1”  6.2.3b.  “Load Pattern Name”  MID in  6.2.3d.  Check that the forces look similar to this:

6.2.10 Repeat 6.2.2. – 6.2.3 with the “MID out” load case and typing “-1” for 6.2.2d.

6.2.11 Repeat 6.2.2. – 6.2.3 with the “CORNER in” load case with these changes:  6.2.2a.  Select the joint between the “STERN in” and the “MID in” joints  6.2.2c.  “Load Pattern Name”  CORNER in  6.2.2d.  “Force 3”  “1”  6.2.3b.  “Load Pattern Name”  CORNER in  6.2.3d.  Check that the forces look similar to this:

6.2.12 Repeat 6.2.2. – 6.2.3 with the “CORNER out” load case and typing “-1” for 6.2.2d.

6.3 Analyze the Critical Load Cases 6.3.1 Run the buoyant force as the only load case a. Analyze  Set Load Cases to Run… b. Click on “DEAD” c. Click “Run/Do Not Run Case” so that “Action”  “Do not Run” d. Repeat c. after clicking on “MODAL” e. Click on “Run Now” (you can get to “Run Now” by clicking the triangular button):

6.3.2 Export the resulting stresses a. File  Export  SAP2000 MS Excel Spreadsheet .xls File… b. Expand “Element Output” c. Expand “Area Output” d. Click on “Table: Element Stresses – Area Shells” e. “Options”  check the box for “Open File After Export” f. Click “OK” g. Save the file separate from the exported reaction table

6.4 Format Excel Results 6.4.1 Copy the “Element Stresses – Area Shells” tab 6.4.2 Rename the copy “RESULTS” 6.4.3 Under the first row insert four rows and clear their background color 6.4.4 Find the minimum for the “SMaxTop” column in a cell above 6.4.5 Find the maximum for the “SMaxTop” column in the cell under the minimum 6.4.6 Repeat 5.4.4 – 5.4.5 for the columns labeled “SMinTop”, “SMaxBot”, and “SMinBot”

If you want, you may hide all the other columns with stresses by selecting the columns, right clicking, and then selecting “Hide”.

6.4.7 Determine largest absolute value of the stresses a. Compare the absolute values of the minimum and maximum stresses b. In the cell below the each column of the minimum and maximum stresses type in the equation in the function line, like so:

6.4.8 Determine the critical area loads responsible for the largest stresses a. Copy the “OutputCase” column b. Paste the column copy after the last stress column c. Under the cells from 6.4.7 lookup the “OutputCase” value associated with the absolute largest stress with the following function:

7. CONCLUSION For the specific results and analysis, please refer to the Modeling and Finite Element Analysis of a Standard Concrete Canoe thesis. In particular, note the Results, Discussion of Results, and Conclusions sections.

100

8. BIBLIOGRAPHY ACI Committee 318. (2011). ACI 318-11 Building Code Requirements for Structural Concrete. Farmington Hills, MI: American Concrete Institute.

Carasquillo, R. L., Nilson, A. H., & Slate, F. O. (1981, May-June). "Properties of High Strength Concrete Subjected to Short Term Loads". Journal of the American Concrete Institute, 78(3), 171-178.

Das, G., & Shaik, S. (2010). Optimization of the Design of a Concrete Canoe using Finite Element Analysis. American Society for Engineering Education.

Essam, A. (2005, December 27). E-book. Retrieved November 25, 2012, from AHM531: www.ahm531.com/E- book/Uploaded/THE%20WHOLE%20THING.doc

Gilbert, J. A., Ooi, T. K., & Engberg, R. C. (2006). Modal Analysis of a Lightweight Concrete Canoe. Concrete Canoe Magazine, pp. 27-33. Muzenski, S., & Klett, J. (2010). Finite Element Analysis of a Concrete Canoe. University of Wisconsin-Milwaukee, Advanced Manufacturing and Design Laboratory. Milwaukee, WI: University of Wisconsin-Milwaukee. Narayanan, N., & Ramamurthy, K. (2000, April). Structure and properties of aerated concrete: a review. Cement & Concrete Composites, 22, 321-329. Retrieved March 2012, from https://www.ownerbuilderbook.com/forum/files/forums/19679-1.pdf

Wight, J. K., & MacGregor, J. G. (2011). Reinforced Concrete: Mechanics and Design (6 ed.). Prentice Hall.

101

APPENDIX B: HAND CALCULATIONS

B1.1 Method from Muzenski and Klett (2010):

( )( )

( ) ( )

102

B1.2 Method from Essam (2005):

This method treat the canoe as a beam in two approaches: simplified and precise.

The simplified analysis models the canoe as a beam with an average width and a uniform distributed loading, while the precise analysis uses a varying cross- sectional area and varied buoyancy force all along the canoe.

Figure B1: Canoe Simplified as a Simply Supported Beam from Essam (2005)

103

B1.2.1 Simple Analysis Method:

The method here follows the method in the figure from the Essam (2005) text, except for my addition of the determination of the pressure loading.

Figure B2: Simple Analysis from Essam (2005)

Determination of Point Loads for 4-Paddler Case: 1 man = 180 lb 1 woman = 150 lb ½ canoe = 475 lb/2 = 237.5 lb Sum = 567.5 lb Point Load = 567.5 lb

Determination of Buoyancy Distributed Load for 4-Paddler Case: 2 men = 360 lb 2 women = 300 lb 1 canoe = 475 lb Sum = 1135 lb Distributed Load = 1135 lb/15 ft = 75.67 lb/ft

Determination of Pressure Load: Distributed Area Load = (Distributed Load)/(Average Width, calculated with excel section coordinates in Appendix C) = (75.67 lb/ft)(ft/12)/(20 in) Distributed Area Load = 0.315 psi

104

Determination of Diagram Extremes: Max. Shear = 567.5 lb Max. Moment = ½ (Max. Shear)(Length, from Appendix C coordinates/2) Max. Moment = ½ (567.5 lb)(15 ft/2) = 2128.125 lb-ft

Determination of Point Loads for 2-Paddler Case: 1 man = 180 lb ½ canoe = 475 lb/2 = 237.5 lb Sum = 417.5 lb Point Load = 417.5 lb

Determination of Buoyancy Distributed Load for 2-Paddler Case: 2 men = 360 lb 1 canoe = 475 lb Sum = 835 lb Distributed Load = 835 lb/15 ft = 55.67 lb/ft

Determination of Diagram Extremes: Max. Shear = 417.5 lb Max. Moment = ½ (Max. Shear)(Length, from Appendix C coordinates/2) Max. Moment = ½ (567.5 lb)(15 ft/2) = 1565.625 lb-ft

105

B1.2.2 Precise Analysis Method:

This section demonstrates the basic idea for the method, and has been left incomplete at the same point Essam (2005) left off.

Figure B3: Precise Analysis from Essam (2005)

Determination of Incremental Canoe Weight: Incremental Canoe Weight = (Canoe Weight)/(# Increments) Incremental Canoe Weight = 475 lb/40 inc = 11.875 lb/inc.

Determination of Incremental Buoyancy Force: Displaced Water Volume = (Submerged Cross-Section Area)(Increment Width), from figure Displaced Water Volume = (85 in2)(6 in) = 510 in3 = 0.295 ft3 Incremental Buoyancy Force = (Displaced Water Volume)(Unit Weight of Water) = (0.295 ft3)(62.4 lb/ft3) = 18.408 lb Incremental Buoyancy Force = 18.4 lb

106

B1.3 Simplified Canoe Shape Analysis:

The canoe shape is simplified as half a pipe.

Determine Simplified Canoe Dimensions: Pipe Radius = R = (Average Width, calculated with Appendix C coordinates)/2 = 20 in./2 Pipe Radius = 10 in Pipe Thickness = T = average canoe thickness Pipe Thickness = 1.4 in Canoe Length = L Canoe Length = 20 ft

Steps to Determine Depth of Wetted Canoe with 4-Paddler: 1) Guess Depth of Wetted Canoe = D 2) Determine Weight of Canoe and Paddlers = W Weight of Canoe and Paddlers = 1135 lb 3) Calculate Central Angle = ϴ Central Angle = 2arccos((R – D)/R) 4) Calculate Wetted Cross-Sectional Area = K Wetted Cross-Sectional Area = R2(ϴ - sinϴ)/2 5) Calculate Volume of Displaced Water = V Volume of Displaced Water = K*L 6) Calculate Weight of Displaced Water = V(Unit Weight of Water) Weight of Displaced Water = K*L*(62.4 pcf) 7) Repeat steps 1) – 6) until Weight of Displaced Water = W

Estimate Buoyant Pressure with 4-Paddler: From steps 1) – 7): D = 8.7 in Wetted Surface Area = ϴ*R*L + 2K = 2.88(10 in)(20 ft*12 in/ft) + 2(131.13 in2) Wetted Surface Area = 7176 in2 Buoyant Pressure = W/(Wetted Surface Area) = 1135 lb/6914 in2 Buoyant Pressure = 0.158 psi

Estimate Buoyant Pressure with 4-Paddler: From steps 1) – 7): D = 6.9 in Wetted Surface Area = 2.51(10 in)(20 ft*12 in/ft) + 2(96.087 in2) Wetted Surface Area = 6219 in2 Buoyant Pressure = 835 lb/6219 in2 Buoyant Pressure = 0.134 psi

107

APPENDIX C: STANDARD CANOE SECTION COORDINATES WITH SECTION AREA CALCULATIONS

Bow Stem Cum. Cum. Cum. station (x) offset (y) height (z) ∆ x ∆ z Area Area Volume Wt. 6.9417 0.0000 4.0000 (sq. in.) (sq. ft) (cu. ft) (lb) 4.2854 0.0000 4.5000 2.6563 0.5000 0.6641 3.0027 0.0000 5.0000 1.2827 0.5000 2.3750 0.0211 0.0211 1.32 2.3664 0.0000 5.5000 0.6363 0.5000 2.6250 0.0165 0.0165 1.03 2.0141 0.0000 6.0000 0.3523 0.5000 2.8750 0.0347 0.0347 2.17 1.7602 0.0000 6.5000 0.2539 0.5000 3.1250 0.0547 0.0547 3.41 1.5602 0.0000 7.0000 0.2000 0.5000 3.3750 0.0764 0.0764 4.77 1.2596 0.0000 8.0000 0.3006 1.0000 7.5000 0.0998 0.0998 6.23 1.0271 0.0000 9.0000 0.2325 1.0000 8.5000 0.1519 0.1519 9.48 0.8349 0.0000 10.0000 0.1922 1.0000 9.5000 0.2109 0.2109 13.16 0.6647 0.0000 11.0000 0.1702 1.0000 10.5000 0.2769 0.2769 17.28 0.5067 0.0000 12.0000 0.1580 1.0000 11.5000 0.3498 0.3498 21.83 0.3756 0.0000 13.0000 0.1311 1.0000 12.5000 0.4297 0.4297 26.81 0.2496 0.0000 14.0000 0.1260 1.0000 13.5000 0.5165 0.5165 32.23 0.1242 0.0000 15.0000 0.1254 1.0000 14.5000 0.6102 0.6102 38.08 0.0000 0.0000 16.0000 0.1242 1.0000 15.5000 0.7109 0.7109 44.36

Stern Stem Cum. Cum. Cum. station (x) offset (y) height (z) ∆ x ∆ z Area Area Volume Wt. 231.9500 0.0000 2.5000 (sq. in.) (sq. ft) (cu. ft) (lb) 235.5752 0.0000 3.0000 3.6252 0.5000 0.9063 237.1369 0.0000 3.5000 1.5617 0.5000 1.6250 0.0176 0.0176 1.10 237.7704 0.0000 4.0000 0.6335 0.5000 1.8750 0.0113 0.0113 0.70 238.1538 0.0000 4.5000 0.3834 0.5000 2.1250 0.0243 0.0243 1.52 238.4662 0.0000 5.0000 0.3124 0.5000 2.3750 0.0391 0.0391 2.44 238.7083 0.0000 5.5000 0.2421 0.5000 2.6250 0.0556 0.0556 3.47 238.8678 0.0000 6.0000 0.1595 0.5000 2.8750 0.0738 0.0738 4.60 239.0099 0.0000 6.5000 0.1421 0.5000 3.1250 0.0938 0.0938 5.85 239.1411 0.0000 7.0000 0.1312 0.5000 3.3750 0.1155 0.1155 7.20 239.3414 0.0000 8.0000 0.2003 1.0000 7.5000 0.1389 0.1389 8.67 239.5140 0.0000 9.0000 0.1726 1.0000 8.5000 0.1910 0.1910 11.92 239.6222 0.0000 10.0000 0.1082 1.0000 9.5000 0.2500 0.2500 15.60 239.7206 0.0000 11.0000 0.0984 1.0000 10.5000 0.3160 0.3160 19.72 239.8148 0.0000 12.0000 0.0942 1.0000 11.5000 0.3889 0.3889 24.27 239.9074 0.0000 13.0000 0.0926 1.0000 12.5000 0.4688 0.4688 29.25 240.0000 0.0000 14.0000 0.0926 1.0000 13.5000 0.5556 0.5556 34.67

108

Station 1 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 12.0000 0.0000 3.5247 (sq. in.) (sq. ft) (cu. ft) (lb) 12.0000 0.7783 4.0000 0.7783 0.4753 0.1850 12.0000 1.0772 4.5000 0.2989 0.5000 0.4639 0.0045 0.0045 0.28 12.0000 1.2769 5.0000 0.1997 0.5000 0.5885 0.0086 0.0086 0.54 12.0000 1.4197 5.5000 0.1428 0.5000 0.6742 0.0133 0.0133 0.83 12.0000 1.5219 6.0000 0.1022 0.5000 0.7354 0.0184 0.0184 1.15 12.0000 1.5913 6.5000 0.0694 0.5000 0.7783 0.0238 0.0238 1.48 12.0000 1.6357 7.0000 0.0444 0.5000 0.8068 0.0294 0.0294 1.83 12.0000 1.7182 8.0000 0.0825 1.0000 1.6770 0.0410 0.0410 2.56 12.0000 1.8008 9.0000 0.0826 1.0000 1.7595 0.0533 0.0533 3.32 12.0000 1.8833 10.0000 0.0825 1.0000 1.8421 0.0660 0.0660 4.12 12.0000 1.9659 11.0000 0.0826 1.0000 1.9246 0.0794 0.0794 4.96 12.0000 2.0484 12.0000 0.0825 1.0000 2.0072 0.0933 0.0933 5.82 12.0000 2.1310 13.0000 0.0826 1.0000 2.0897 0.1079 0.1079 6.73 12.0000 2.2135 14.0000 0.0825 1.0000 2.1723 0.1229 0.1229 7.67 12.0000 2.2961 15.0000 0.0826 1.0000 2.2548 0.1386 0.1386 8.65 12.0000 2.3580 15.7500 0.0619 0.7500 1.7453 0.1507 0.1507 9.41

Station 2 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 24.0000 0.0000 2.8323 (sq. in.) (sq. ft) (cu. ft) (lb) 24.0000 1.1006 3.0000 1.1006 0.1677 0.0923 24.0000 2.1248 3.5000 1.0242 0.5000 0.8064 0.0062 0.0062 0.39 24.0000 2.7124 4.0000 0.5876 0.5000 1.2093 0.0146 0.0146 0.91 24.0000 3.1207 4.5000 0.4083 0.5000 1.4583 0.0248 0.0248 1.55 24.0000 3.4145 5.0000 0.2938 0.5000 1.6338 0.0361 0.0361 2.25 24.0000 3.6219 5.5000 0.2074 0.5000 1.7591 0.0483 0.0483 3.02 24.0000 3.7571 6.0000 0.1352 0.5000 1.8448 0.0611 0.0611 3.82 24.0000 3.8310 6.5000 0.0739 0.5000 1.8970 0.0743 0.0743 4.64 24.0000 3.8779 7.0000 0.0469 0.5000 1.9272 0.0877 0.0877 5.47 24.0000 3.9765 8.0000 0.0986 1.0000 3.9272 0.1150 0.1150 7.17 24.0000 4.0751 9.0000 0.0986 1.0000 4.0258 0.1429 0.1429 8.92 24.0000 4.1737 10.0000 0.0986 1.0000 4.1244 0.1716 0.1716 10.71 24.0000 4.2723 11.0000 0.0986 1.0000 4.2230 0.2009 0.2009 12.54 24.0000 4.3709 12.0000 0.0986 1.0000 4.3216 0.2309 0.2309 14.41 24.0000 4.4695 13.0000 0.0986 1.0000 4.4202 0.2616 0.2616 16.32 24.0000 4.5681 14.0000 0.0986 1.0000 4.5188 0.2930 0.2930 18.28 24.0000 4.6667 15.0000 0.0986 1.0000 4.6174 0.3250 0.3250 20.28 24.0000 4.7160 15.5000 0.0493 0.5000 2.3457 0.3413 0.3413 21.30

Station 3 Cum. Cum. Cum.

109

station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 36.0000 0.0000 2.2509 (sq. in.) (sq. ft) (cu. ft) (lb) 36.0000 2.0968 2.5000 2.0968 0.2491 0.2612 36.0000 3.5169 3.0000 1.4201 0.5000 1.4034 0.0116 0.0116 0.72 36.0000 4.3820 3.5000 0.8651 0.5000 1.9747 0.0253 0.0253 1.58 36.0000 4.9897 4.0000 0.6077 0.5000 2.3429 0.0415 0.0415 2.59 36.0000 5.4271 4.5000 0.4374 0.5000 2.6042 0.0596 0.0596 3.72 36.0000 5.7334 5.0000 0.3063 0.5000 2.7901 0.0790 0.0790 4.93 36.0000 5.9288 5.5000 0.1954 0.5000 2.9156 0.0993 0.0993 6.19 36.0000 6.0247 6.0000 0.0959 0.5000 2.9884 0.1200 0.1200 7.49 36.0000 6.0814 6.5000 0.0567 0.5000 3.0265 0.1410 0.1410 8.80 36.0000 6.1381 7.0000 0.0567 0.5000 3.0549 0.1622 0.1622 10.12 36.0000 6.2516 8.0000 0.1135 1.0000 6.1949 0.2053 0.2053 12.81 36.0000 6.3650 9.0000 0.1134 1.0000 6.3083 0.2491 0.2491 15.54 36.0000 6.4784 10.0000 0.1134 1.0000 6.4217 0.2937 0.2937 18.32 36.0000 6.5919 11.0000 0.1135 1.0000 6.5352 0.3390 0.3390 21.16 36.0000 6.7053 12.0000 0.1134 1.0000 6.6486 0.3852 0.3852 24.04 36.0000 6.8188 13.0000 0.1135 1.0000 6.7621 0.4322 0.4322 26.97 36.0000 6.9322 14.0000 0.1134 1.0000 6.8755 0.4799 0.4799 29.95 36.0000 7.0456 15.0000 0.1134 1.0000 6.9889 0.5285 0.5285 32.98 36.0000 7.0740 15.2500 0.0284 0.2500 1.7650 0.5407 0.5407 33.74

Station 4 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 48.0000 0.0000 1.7366 (sq. in.) (sq. ft) (cu. ft) (lb) 48.0000 2.8955 2.0000 2.8955 0.2634 0.3813 48.0000 4.7675 2.5000 1.8720 0.5000 1.9158 0.0160 0.0160 1.00 48.0000 5.9174 3.0000 1.1499 0.5000 2.6712 0.0345 0.0345 2.15 48.0000 6.7265 3.5000 0.8091 0.5000 3.1610 0.0565 0.0565 3.52 48.0000 7.3089 4.0000 0.5824 0.5000 3.5089 0.0808 0.0808 5.04 48.0000 7.7160 4.5000 0.4071 0.5000 3.7562 0.1069 0.1069 6.67 48.0000 7.9746 5.0000 0.2586 0.5000 3.9227 0.1341 0.1341 8.37 48.0000 8.0992 5.5000 0.1246 0.5000 4.0185 0.1621 0.1621 10.11 48.0000 8.1614 6.0000 0.0622 0.5000 4.0652 0.1903 0.1903 11.87 48.0000 8.2235 6.5000 0.0621 0.5000 4.0962 0.2187 0.2187 13.65 48.0000 8.2856 7.0000 0.0621 0.5000 4.1273 0.2474 0.2474 15.44 48.0000 8.4099 8.0000 0.1243 1.0000 8.3478 0.3054 0.3054 19.05 48.0000 8.5342 9.0000 0.1243 1.0000 8.4721 0.3642 0.3642 22.73 48.0000 8.6585 10.0000 0.1243 1.0000 8.5964 0.4239 0.4239 26.45 48.0000 8.7828 11.0000 0.1243 1.0000 8.7207 0.4845 0.4845 30.23 48.0000 8.9071 12.0000 0.1243 1.0000 8.8450 0.5459 0.5459 34.06 48.0000 9.0313 13.0000 0.1242 1.0000 8.9692 0.6082 0.6082 37.95 48.0000 9.1556 14.0000 0.1243 1.0000 9.0935 0.6713 0.6713 41.89 48.0000 9.2799 15.0000 0.1243 1.0000 9.2178 0.7353 0.7353 45.88

110

Station 5 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 60.0000 0.0000 1.2894 (sq. in.) (sq. ft) (cu. ft) (lb) 60.0000 3.1623 1.5000 3.1623 0.2106 0.3330 60.0000 5.6193 2.0000 2.4570 0.5000 2.1954 0.0176 0.0176 1.10 60.0000 7.0785 2.5000 1.4592 0.5000 3.1745 0.0396 0.0396 2.47 60.0000 8.0985 3.0000 1.0200 0.5000 3.7943 0.0660 0.0660 4.12 60.0000 8.8328 3.5000 0.7343 0.5000 4.2328 0.0953 0.0953 5.95 60.0000 9.3489 4.0000 0.5161 0.5000 4.5454 0.1269 0.1269 7.92 60.0000 9.6819 4.5000 0.3330 0.5000 4.7577 0.1600 0.1600 9.98 60.0000 9.8502 5.0000 0.1683 0.5000 4.8830 0.1939 0.1939 12.10 60.0000 9.9185 5.5000 0.0683 0.5000 4.9422 0.2282 0.2282 14.24 60.0000 9.9851 6.0000 0.0666 0.5000 4.9759 0.2627 0.2627 16.39 60.0000 10.0516 6.5000 0.0665 0.5000 5.0092 0.2975 0.2975 18.57 60.0000 10.1182 7.0000 0.0666 0.5000 5.0425 0.3325 0.3325 20.75 60.0000 10.2513 8.0000 0.1331 1.0000 10.1848 0.4033 0.4033 25.16 60.0000 10.3845 9.0000 0.1332 1.0000 10.3179 0.4749 0.4749 29.63 60.0000 10.5176 10.0000 0.1331 1.0000 10.4511 0.5475 0.5475 34.16 60.0000 10.6507 11.0000 0.1331 1.0000 10.5842 0.6210 0.6210 38.75 60.0000 10.7839 12.0000 0.1332 1.0000 10.7173 0.6954 0.6954 43.39 60.0000 10.9170 13.0000 0.1331 1.0000 10.8505 0.7708 0.7708 48.10 60.0000 11.0501 14.0000 0.1331 1.0000 10.9836 0.8470 0.8470 52.86 60.0000 11.1500 14.7500 0.0999 0.7500 8.3250 0.9049 0.9049 56.46

Station 6 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 72.0000 0.0000 0.9091 (sq. in.) (sq. ft) (cu. ft) (lb) 72.0000 2.4012 1.0000 2.4012 0.0909 0.1091 72.0000 5.9256 1.5000 3.5244 0.5000 2.0817 0.0152 0.0152 0.95 72.0000 7.7749 2.0000 1.8493 0.5000 3.4251 0.0390 0.0390 2.43 72.0000 9.0429 2.5000 1.2680 0.5000 4.2045 0.0682 0.0682 4.26 72.0000 9.9544 3.0000 0.9115 0.5000 4.7493 0.1012 0.1012 6.31 72.0000 10.6017 3.5000 0.6473 0.5000 5.1390 0.1369 0.1369 8.54 72.0000 11.0315 4.0000 0.4298 0.5000 5.4083 0.1744 0.1744 10.88 72.0000 11.2685 4.5000 0.2370 0.5000 5.5750 0.2131 0.2131 13.30 72.0000 11.3546 5.0000 0.0861 0.5000 5.6558 0.2524 0.2524 15.75 72.0000 11.4248 5.5000 0.0702 0.5000 5.6949 0.2920 0.2920 18.22 72.0000 11.4950 6.0000 0.0702 0.5000 5.7300 0.3318 0.3318 20.70 72.0000 11.5652 6.5000 0.0702 0.5000 5.7651 0.3718 0.3718 23.20 72.0000 11.6354 7.0000 0.0702 0.5000 5.8002 0.4121 0.4121 25.71 72.0000 11.7758 8.0000 0.1404 1.0000 11.7056 0.4934 0.4934 30.79 72.0000 11.9162 9.0000 0.1404 1.0000 11.8460 0.5756 0.5756 35.92 72.0000 12.0566 10.0000 0.1404 1.0000 11.9864 0.6589 0.6589 41.11

111

72.0000 12.1970 11.0000 0.1404 1.0000 12.1268 0.7431 0.7431 46.37 72.0000 12.3374 12.0000 0.1404 1.0000 12.2672 0.8283 0.8283 51.68 72.0000 12.4778 13.0000 0.1404 1.0000 12.4076 0.9144 0.9144 57.06 72.0000 12.6182 14.0000 0.1404 1.0000 12.5480 1.0016 1.0016 62.50 72.0000 12.6884 14.5000 0.0702 0.5000 6.3267 1.0455 1.0455 65.24

Station 7 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 84.0000 0.0000 0.5957 (sq. in.) (sq. ft) (cu. ft) (lb) 84.0000 5.4638 1.0000 5.4638 0.4043 1.1045 84.0000 7.8979 1.5000 2.4341 0.5000 3.3404 0.0309 0.0309 1.93 84.0000 9.4890 2.0000 1.5911 0.5000 4.3467 0.0611 0.0611 3.81 84.0000 10.6296 2.5000 1.1406 0.5000 5.0297 0.0960 0.0960 5.99 84.0000 11.4362 3.0000 0.8066 0.5000 5.5165 0.1343 0.1343 8.38 84.0000 11.9945 3.5000 0.5583 0.5000 5.8577 0.1750 0.1750 10.92 84.0000 12.3324 4.0000 0.3379 0.5000 6.0817 0.2172 0.2172 13.55 84.0000 12.4712 4.5000 0.1388 0.5000 6.2009 0.2603 0.2603 16.24 84.0000 12.5444 5.0000 0.0732 0.5000 6.2539 0.3037 0.3037 18.95 84.0000 12.6176 5.5000 0.0732 0.5000 6.2905 0.3474 0.3474 21.68 84.0000 12.6908 6.0000 0.0732 0.5000 6.3271 0.3913 0.3913 24.42 84.0000 12.7640 6.5000 0.0732 0.5000 6.3637 0.4355 0.4355 27.18 84.0000 12.8372 7.0000 0.0732 0.5000 6.4003 0.4800 0.4800 29.95 84.0000 12.9836 8.0000 0.1464 1.0000 12.9104 0.5696 0.5696 35.54 84.0000 13.1300 9.0000 0.1464 1.0000 13.0568 0.6603 0.6603 41.20 84.0000 13.2764 10.0000 0.1464 1.0000 13.2032 0.7520 0.7520 46.92 84.0000 13.4228 11.0000 0.1464 1.0000 13.3496 0.8447 0.8447 52.71 84.0000 13.5692 12.0000 0.1464 1.0000 13.4960 0.9384 0.9384 58.56 84.0000 13.7156 13.0000 0.1464 1.0000 13.6424 1.0331 1.0331 64.47 84.0000 13.8620 14.0000 0.1464 1.0000 13.7888 1.1289 1.1289 70.44 84.0000 13.8986 14.2500 0.0366 0.2500 3.4701 1.1530 1.1530 71.95

Station 8 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 96.0000 0.0000 0.3493 (sq. in.) (sq. ft) (cu. ft) (lb) 96.0000 3.6188 0.5000 3.6188 0.1507 0.2727 96.0000 7.2762 1.0000 3.6574 0.5000 2.7238 0.0208 0.0208 1.30 96.0000 9.3411 1.5000 2.0649 0.5000 4.1543 0.0497 0.0497 3.10 96.0000 10.7718 2.0000 1.4307 0.5000 5.0282 0.0846 0.0846 5.28 96.0000 11.8014 2.5000 1.0296 0.5000 5.6433 0.1238 0.1238 7.72 96.0000 12.5291 3.0000 0.7277 0.5000 6.0826 0.1660 0.1660 10.36 96.0000 13.0057 3.5000 0.4766 0.5000 6.3837 0.2103 0.2103 13.13 96.0000 13.2584 4.0000 0.2527 0.5000 6.5660 0.2559 0.2559 15.97 96.0000 13.3458 4.5000 0.0874 0.5000 6.6511 0.3021 0.3021 18.85 96.0000 13.4214 5.0000 0.0756 0.5000 6.6918 0.3486 0.3486 21.75

112

96.0000 13.4971 5.5000 0.0757 0.5000 6.7296 0.3953 0.3953 24.67 96.0000 13.5727 6.0000 0.0756 0.5000 6.7675 0.4423 0.4423 27.60 96.0000 13.6484 6.5000 0.0757 0.5000 6.8053 0.4896 0.4896 30.55 96.0000 13.7241 7.0000 0.0757 0.5000 6.8431 0.5371 0.5371 33.52 96.0000 13.8754 8.0000 0.1513 1.0000 13.7998 0.6329 0.6329 39.50 96.0000 14.0267 9.0000 0.1513 1.0000 13.9511 0.7298 0.7298 45.54 96.0000 14.1780 10.0000 0.1513 1.0000 14.1024 0.8278 0.8278 51.65 96.0000 14.3293 11.0000 0.1513 1.0000 14.2537 0.9267 0.9267 57.83 96.0000 14.4807 12.0000 0.1514 1.0000 14.4050 1.0268 1.0268 64.07 96.0000 14.6320 13.0000 0.1513 1.0000 14.5564 1.1279 1.1279 70.38 96.0000 14.7833 14.0000 0.1513 1.0000 14.7077 1.2300 1.2300 76.75

Station 9 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 108.0000 0.0000 0.1697 (sq. in.) (sq. ft) (cu. ft) (lb) 108.0000 5.5081 0.5000 5.5081 0.3303 0.9097 108.0000 8.4436 1.0000 2.9355 0.5000 3.4879 0.0305 0.0305 1.91 108.0000 10.3083 1.5000 1.8647 0.5000 4.6880 0.0631 0.0631 3.94 108.0000 11.6292 2.0000 1.3209 0.5000 5.4844 0.1012 0.1012 6.31 108.0000 12.5789 2.5000 0.9497 0.5000 6.0520 0.1432 0.1432 8.94 108.0000 13.2375 3.0000 0.6586 0.5000 6.4541 0.1880 0.1880 11.73 108.0000 13.6472 3.5000 0.4097 0.5000 6.7212 0.2347 0.2347 14.65 108.0000 13.8301 4.0000 0.1829 0.5000 6.8693 0.2824 0.2824 17.62 108.0000 13.9059 4.5000 0.0758 0.5000 6.9340 0.3306 0.3306 20.63 108.0000 13.9816 5.0000 0.0757 0.5000 6.9719 0.3790 0.3790 23.65 108.0000 14.0573 5.5000 0.0757 0.5000 7.0097 0.4277 0.4277 26.69 108.0000 14.1330 6.0000 0.0757 0.5000 7.0476 0.4766 0.4766 29.74 108.0000 14.2087 6.5000 0.0757 0.5000 7.0854 0.5258 0.5258 32.81 108.0000 14.2844 7.0000 0.0757 0.5000 7.1233 0.5753 0.5753 35.90 108.0000 14.4358 8.0000 0.1514 1.0000 14.3601 0.6750 0.6750 42.12 108.0000 14.5873 9.0000 0.1515 1.0000 14.5116 0.7758 0.7758 48.41 108.0000 14.7387 10.0000 0.1514 1.0000 14.6630 0.8776 0.8776 54.76 108.0000 14.8901 11.0000 0.1514 1.0000 14.8144 0.9805 0.9805 61.18 108.0000 15.0415 12.0000 0.1514 1.0000 14.9658 1.0844 1.0844 67.67 108.0000 15.1930 13.0000 0.1515 1.0000 15.1173 1.1894 1.1894 74.22 108.0000 15.3444 14.0000 0.1514 1.0000 15.2687 1.2954 1.2954 80.83

Station 10 (midship) Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 120.0000 0.0000 0.0570 (sq. in.) (sq. ft) (cu. ft) (lb) 120.0000 6.4372 0.5000 6.4372 0.4430 1.4258 120.0000 9.0758 1.0000 2.6386 0.5000 3.8783 0.0368 0.0066 0.41 120.0000 10.8219 1.5000 1.7461 0.5000 4.9744 0.0714 0.0128 0.80 120.0000 12.0693 2.0000 1.2474 0.5000 5.7228 0.1111 0.0199 1.24

113

120.0000 12.9628 2.5000 0.8935 0.5000 6.2580 0.1546 0.0277 1.73 120.0000 13.5725 3.0000 0.6097 0.5000 6.6338 0.2006 0.0360 2.25 120.0000 13.9356 3.5000 0.3631 0.5000 6.8770 0.2484 0.0446 2.78 120.0000 14.0770 4.0000 0.1414 0.5000 7.0032 0.2970 0.0533 3.33 120.0000 14.1523 4.5000 0.0753 0.5000 7.0573 0.3460 0.0621 3.88 120.0000 14.2276 5.0000 0.0753 0.5000 7.0950 0.3953 0.0710 4.43 120.0000 14.3029 5.5000 0.0753 0.5000 7.1326 0.4448 0.0798 4.98 120.0000 14.3782 6.0000 0.0753 0.5000 7.1703 0.4946 0.0888 5.54 120.0000 14.4535 6.5000 0.0753 0.5000 7.2079 0.5447 0.0978 6.10 120.0000 14.5288 7.0000 0.0753 0.5000 7.2456 0.5950 0.1068 6.66 120.0000 14.6794 8.0000 0.1506 1.0000 14.6041 0.6964 0.1250 7.80 120.0000 14.8301 9.0000 0.1507 1.0000 14.7548 0.7989 0.1434 8.95 120.0000 14.9807 10.0000 0.1506 1.0000 14.9054 0.9024 0.1620 10.11 120.0000 15.1313 11.0000 0.1506 1.0000 15.0560 1.0070 0.1807 11.28 120.0000 15.2819 12.0000 0.1506 1.0000 15.2066 1.1126 0.1997 12.46 120.0000 15.4325 13.0000 0.1506 1.0000 15.3572 1.2192 0.2188 13.65 120.0000 15.5831 14.0000 0.1506 1.0000 15.5078 1.3269 0.2382 14.86

Station 10.2 (max. width) Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 122.1538 0.0000 0.0440 (sq. in.) (sq. ft) (cu. ft) (lb) 122.1538 6.5296 0.5000 6.5296 0.4560 1.4887 122.1538 9.1357 1.0000 2.6061 0.5000 3.9163 0.0375 0.0308 1.92 122.1538 10.8669 1.5000 1.7312 0.5000 5.0007 0.0723 0.0593 3.70 122.1538 12.1046 2.0000 1.2377 0.5000 5.7429 0.1121 0.0920 5.74 122.1538 12.9906 2.5000 0.8860 0.5000 6.2738 0.1557 0.1278 7.97 122.1538 13.5939 3.0000 0.6033 0.5000 6.6461 0.2019 0.1656 10.34 122.1538 13.9512 3.5000 0.3573 0.5000 6.8863 0.2497 0.2049 12.78 122.1538 14.0882 4.0000 0.1370 0.5000 7.0099 0.2984 0.2448 15.28 122.1538 14.1634 4.5000 0.0752 0.5000 7.0629 0.3474 0.2851 17.79 122.1538 14.2386 5.0000 0.0752 0.5000 7.1005 0.3967 0.3255 20.31 122.1538 14.3138 5.5000 0.0752 0.5000 7.1381 0.4463 0.3662 22.85 122.1538 14.3890 6.0000 0.0752 0.5000 7.1757 0.4961 0.4071 25.40 122.1538 14.4641 6.5000 0.0751 0.5000 7.2133 0.5462 0.4482 27.97 122.1538 14.5393 7.0000 0.0752 0.5000 7.2509 0.5966 0.4895 30.54 122.1538 14.6897 8.0000 0.1504 1.0000 14.6145 0.6981 0.5728 35.74 122.1538 14.8401 9.0000 0.1504 1.0000 14.7649 0.8006 0.6569 40.99 122.1538 14.9904 10.0000 0.1503 1.0000 14.9153 0.9042 0.7419 46.29 122.1538 15.1408 11.0000 0.1504 1.0000 15.0656 1.0088 0.8277 51.65 122.1538 15.2912 12.0000 0.1504 1.0000 15.2160 1.1145 0.9144 57.06 122.1538 15.4415 13.0000 0.1503 1.0000 15.3664 1.2212 1.0020 62.52 122.1538 15.5919 14.0000 0.1504 1.0000 15.5167 1.3289 1.0904 68.04

114

Station 11 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 132.0000 0.0000 0.0000 (sq. in.) (sq. ft) (cu. ft) (lb) 132.0000 6.7777 0.5000 6.7777 0.5000 1.6944 132.0000 9.2601 1.0000 2.4824 0.5000 4.0095 0.0396 0.0396 2.47 132.0000 10.9287 1.5000 1.6686 0.5000 5.0472 0.0747 0.0747 4.66 132.0000 12.1244 2.0000 1.1957 0.5000 5.7633 0.1147 0.1147 7.16 132.0000 12.9783 2.5000 0.8539 0.5000 6.2757 0.1583 0.1583 9.88 132.0000 13.5554 3.0000 0.5771 0.5000 6.6334 0.2043 0.2043 12.75 132.0000 13.8902 3.5000 0.3348 0.5000 6.8614 0.2520 0.2520 15.72 132.0000 14.0126 4.0000 0.1224 0.5000 6.9757 0.3004 0.3004 18.75 132.0000 14.0870 4.5000 0.0744 0.5000 7.0249 0.3492 0.3492 21.79 132.0000 14.1614 5.0000 0.0744 0.5000 7.0621 0.3982 0.3982 24.85 132.0000 14.2357 5.5000 0.0743 0.5000 7.0993 0.4475 0.4475 27.93 132.0000 14.3101 6.0000 0.0744 0.5000 7.1365 0.4971 0.4971 31.02 132.0000 14.3845 6.5000 0.0744 0.5000 7.1737 0.5469 0.5469 34.13 132.0000 14.4588 7.0000 0.0743 0.5000 7.2108 0.5970 0.5970 37.25 132.0000 14.6076 8.0000 0.1488 1.0000 14.5332 0.6979 0.6979 43.55 132.0000 14.7563 9.0000 0.1487 1.0000 14.6820 0.7999 0.7999 49.91 132.0000 14.9051 10.0000 0.1488 1.0000 14.8307 0.9029 0.9029 56.34 132.0000 15.0538 11.0000 0.1487 1.0000 14.9795 1.0069 1.0069 62.83 132.0000 15.2025 12.0000 0.1487 1.0000 15.1282 1.1120 1.1120 69.39 132.0000 15.3513 13.0000 0.1488 1.0000 15.2769 1.2180 1.2180 76.01 132.0000 15.5000 14.0000 0.1487 1.0000 15.4257 1.3252 1.3252 82.69

Station 12 1.0000 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 144.0000 0.0000 0.0322 (sq. in.) (sq. ft) (cu. ft) (lb) 144.0000 6.3936 0.5000 6.3936 0.4678 1.4955 144.0000 8.8857 1.0000 2.4921 0.5000 3.8198 0.0369 0.0369 2.30 144.0000 10.5467 1.5000 1.6610 0.5000 4.8581 0.0706 0.0706 4.41 144.0000 11.7350 2.0000 1.1883 0.5000 5.5704 0.1093 0.1093 6.82 144.0000 12.5851 2.5000 0.8501 0.5000 6.0800 0.1516 0.1516 9.46 144.0000 13.1629 3.0000 0.5778 0.5000 6.4370 0.1963 0.1963 12.25 144.0000 13.5032 3.5000 0.3403 0.5000 6.6665 0.2426 0.2426 15.14 144.0000 13.6321 4.0000 0.1289 0.5000 6.7838 0.2897 0.2897 18.07 144.0000 13.7053 4.5000 0.0732 0.5000 6.8344 0.3371 0.3371 21.04 144.0000 13.7784 5.0000 0.0731 0.5000 6.8709 0.3848 0.3848 24.01 144.0000 13.8515 5.5000 0.0731 0.5000 6.9075 0.4328 0.4328 27.01 144.0000 13.9247 6.0000 0.0732 0.5000 6.9441 0.4810 0.4810 30.02 144.0000 13.9978 6.5000 0.0731 0.5000 6.9806 0.5295 0.5295 33.04 144.0000 14.0709 7.0000 0.0731 0.5000 7.0172 0.5782 0.5782 36.08 144.0000 14.2172 8.0000 0.1463 1.0000 14.1441 0.6765 0.6765 42.21 144.0000 14.3635 9.0000 0.1463 1.0000 14.2904 0.7757 0.7757 48.40

115

144.0000 14.5097 10.0000 0.1462 1.0000 14.4366 0.8759 0.8759 54.66 144.0000 14.6560 11.0000 0.1463 1.0000 14.5829 0.9772 0.9772 60.98 144.0000 14.8023 12.0000 0.1463 1.0000 14.7292 1.0795 1.0795 67.36 144.0000 14.9485 13.0000 0.1462 1.0000 14.8754 1.1828 1.1828 73.81 144.0000 15.0948 14.0000 0.1463 1.0000 15.0217 1.2871 1.2871 80.32

Station 13 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 156.0000 0.0000 0.1200 (sq. in.) (sq. ft) (cu. ft) (lb) 156.0000 5.5069 0.5000 5.5069 0.3800 1.0463 156.0000 8.1007 1.0000 2.5938 0.5000 3.4019 0.0309 0.0309 1.93 156.0000 9.8150 1.5000 1.7143 0.5000 4.4789 0.0620 0.0620 3.87 156.0000 10.9772 2.0000 1.1622 0.5000 5.1981 0.0981 0.0981 6.12 156.0000 11.8357 2.5000 0.8585 0.5000 5.7032 0.1377 0.1377 8.59 156.0000 12.4271 3.0000 0.5914 0.5000 6.0657 0.1798 0.1798 11.22 156.0000 12.7885 3.5000 0.3614 0.5000 6.3039 0.2236 0.2236 13.95 156.0000 12.9404 4.0000 0.1519 0.5000 6.4322 0.2683 0.2683 16.74 156.0000 13.0117 4.5000 0.0713 0.5000 6.4880 0.3133 0.3133 19.55 156.0000 13.0830 5.0000 0.0713 0.5000 6.5237 0.3586 0.3586 22.38 156.0000 13.1543 5.5000 0.0713 0.5000 6.5593 0.4042 0.4042 25.22 156.0000 13.2256 6.0000 0.0713 0.5000 6.5950 0.4500 0.4500 28.08 156.0000 13.2969 6.5000 0.0713 0.5000 6.6306 0.4960 0.4960 30.95 156.0000 13.3683 7.0000 0.0714 0.5000 6.6663 0.5423 0.5423 33.84 156.0000 13.5109 8.0000 0.1426 1.0000 13.4396 0.6356 0.6356 39.66 156.0000 13.6535 9.0000 0.1426 1.0000 13.5822 0.7300 0.7300 45.55 156.0000 13.7962 10.0000 0.1427 1.0000 13.7249 0.8253 0.8253 51.50 156.0000 13.9388 11.0000 0.1426 1.0000 13.8675 0.9216 0.9216 57.51 156.0000 14.0814 12.0000 0.1426 1.0000 14.0101 1.0189 1.0189 63.58 156.0000 14.2241 13.0000 0.1427 1.0000 14.1528 1.1172 1.1172 69.71 156.0000 14.3667 14.0000 0.1426 1.0000 14.2954 1.2164 1.2164 75.91

Station 14 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 168.0000 0.0000 0.2748 (sq. in.) (sq. ft) (cu. ft) (lb) 168.0000 3.9566 0.5000 3.9566 0.2252 0.4455 168.0000 6.8680 1.0000 2.9114 0.5000 2.7062 0.0219 0.0219 1.37 168.0000 8.6148 1.5000 1.7468 0.5000 3.8707 0.0488 0.0488 3.04 168.0000 9.8381 2.0000 1.2233 0.5000 4.6132 0.0808 0.0808 5.04 168.0000 10.7188 2.5000 0.8807 0.5000 5.1392 0.1165 0.1165 7.27 168.0000 11.3369 3.0000 0.6181 0.5000 5.5139 0.1548 0.1548 9.66 168.0000 11.7341 3.5000 0.3972 0.5000 5.7678 0.1948 0.1948 12.16 168.0000 11.9323 4.0000 0.1982 0.5000 5.9166 0.2359 0.2359 14.72 168.0000 12.0044 4.5000 0.0721 0.5000 5.9842 0.2775 0.2775 17.31 168.0000 12.0733 5.0000 0.0689 0.5000 6.0194 0.3193 0.3193 19.92

116

168.0000 12.1422 5.5000 0.0689 0.5000 6.0539 0.3613 0.3613 22.55 168.0000 12.2112 6.0000 0.0690 0.5000 6.0884 0.4036 0.4036 25.18 168.0000 12.2801 6.5000 0.0689 0.5000 6.1228 0.4461 0.4461 27.84 168.0000 12.3490 7.0000 0.0689 0.5000 6.1573 0.4889 0.4889 30.51 168.0000 12.4869 8.0000 0.1379 1.0000 12.4180 0.5751 0.5751 35.89 168.0000 12.6248 9.0000 0.1379 1.0000 12.5559 0.6623 0.6623 41.33 168.0000 12.7626 10.0000 0.1378 1.0000 12.6937 0.7505 0.7505 46.83 168.0000 12.9005 11.0000 0.1379 1.0000 12.8316 0.8396 0.8396 52.39 168.0000 13.0384 12.0000 0.1379 1.0000 12.9695 0.9296 0.9296 58.01 168.0000 13.1762 13.0000 0.1378 1.0000 13.1073 1.0207 1.0207 63.69 168.0000 13.3141 14.0000 0.1379 1.0000 13.2452 1.1126 1.1126 69.43

Station 15 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 180.0000 0.0000 0.4964 (sq. in.) (sq. ft) (cu. ft) (lb) 180.0000 0.4526 0.5000 0.4526 0.0036 0.0008 180.0000 5.1826 1.0000 4.7300 0.5000 1.4088 0.0098 0.0098 0.61 180.0000 7.0680 1.5000 1.8854 0.5000 3.0627 0.0311 0.0311 1.94 180.0000 8.3364 2.0000 1.2684 0.5000 3.8511 0.0578 0.0578 3.61 180.0000 9.2454 2.5000 0.9090 0.5000 4.3955 0.0883 0.0883 5.51 180.0000 9.8946 3.0000 0.6492 0.5000 4.7850 0.1216 0.1216 7.58 180.0000 10.3330 3.5000 0.4384 0.5000 5.0569 0.1567 0.1567 9.78 180.0000 10.5868 4.0000 0.2538 0.5000 5.2300 0.1930 0.1930 12.04 180.0000 10.6828 4.5000 0.0960 0.5000 5.3174 0.2299 0.2299 14.35 180.0000 10.7487 5.0000 0.0659 0.5000 5.3579 0.2671 0.2671 16.67 180.0000 10.8146 5.5000 0.0659 0.5000 5.3908 0.3046 0.3046 19.00 180.0000 10.8805 6.0000 0.0659 0.5000 5.4238 0.3422 0.3422 21.35 180.0000 10.9463 6.5000 0.0658 0.5000 5.4567 0.3801 0.3801 23.72 180.0000 11.0122 7.0000 0.0659 0.5000 5.4896 0.4182 0.4182 26.10 180.0000 11.1440 8.0000 0.1318 1.0000 11.0781 0.4952 0.4952 30.90 180.0000 11.2758 9.0000 0.1318 1.0000 11.2099 0.5730 0.5730 35.76 180.0000 11.4076 10.0000 0.1318 1.0000 11.3417 0.6518 0.6518 40.67 180.0000 11.5394 11.0000 0.1318 1.0000 11.4735 0.7315 0.7315 45.64 180.0000 11.6711 12.0000 0.1317 1.0000 11.6053 0.8121 0.8121 50.67 180.0000 11.8029 13.0000 0.1318 1.0000 11.7370 0.8936 0.8936 55.76 180.0000 11.9347 14.0000 0.1318 1.0000 11.8688 0.9760 0.9760 60.90

Station 16 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 192.0000 0.0000 0.7849 (sq. in.) (sq. ft) (cu. ft) (lb) 192.0000 2.9338 1.0000 2.9338 0.2151 0.3155 192.0000 5.1746 1.5000 2.2408 0.5000 2.0271 0.0163 0.0163 1.02 192.0000 6.5097 2.0000 1.3351 0.5000 2.9211 0.0366 0.0366 2.28 192.0000 7.4434 2.5000 0.9337 0.5000 3.4883 0.0608 0.0608 3.79

117

192.0000 8.1157 3.0000 0.6723 0.5000 3.8898 0.0878 0.0878 5.48 192.0000 8.5880 3.5000 0.4723 0.5000 4.1759 0.1168 0.1168 7.29 192.0000 8.8923 4.0000 0.3043 0.5000 4.3701 0.1471 0.1471 9.18 192.0000 9.0456 4.5000 0.1533 0.5000 4.4845 0.1783 0.1783 11.12 192.0000 9.1089 5.0000 0.0633 0.5000 4.5386 0.2098 0.2098 13.09 192.0000 9.1709 5.5000 0.0620 0.5000 4.5700 0.2415 0.2415 15.07 192.0000 9.2329 6.0000 0.0620 0.5000 4.6010 0.2735 0.2735 17.07 192.0000 9.2950 6.5000 0.0621 0.5000 4.6320 0.3057 0.3057 19.07 192.0000 9.3570 7.0000 0.0620 0.5000 4.6630 0.3380 0.3380 21.09 192.0000 9.4811 8.0000 0.1241 1.0000 9.4191 0.4034 0.4034 25.17 192.0000 9.6051 9.0000 0.1240 1.0000 9.5431 0.4697 0.4697 29.31 192.0000 9.7292 10.0000 0.1241 1.0000 9.6672 0.5368 0.5368 33.50 192.0000 9.8533 11.0000 0.1241 1.0000 9.7913 0.6048 0.6048 37.74 192.0000 9.9774 12.0000 0.1241 1.0000 9.9154 0.6737 0.6737 42.04 192.0000 10.1014 13.0000 0.1240 1.0000 10.0394 0.7434 0.7434 46.39 192.0000 10.2255 14.0000 0.1241 1.0000 10.1635 0.8140 0.8140 50.79

Station 17 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 204.0000 0.0000 1.1403 (sq. in.) (sq. ft) (cu. ft) (lb) 204.0000 2.9654 1.5000 2.9654 0.3597 0.5333 204.0000 4.4319 2.0000 1.4665 0.5000 1.8493 0.0165 0.0165 1.03 204.0000 5.3750 2.5000 0.9431 0.5000 2.4517 0.0336 0.0336 2.09 204.0000 6.0447 3.0000 0.6697 0.5000 2.8549 0.0534 0.0534 3.33 204.0000 6.5259 3.5000 0.4812 0.5000 3.1427 0.0752 0.0752 4.69 204.0000 6.8583 4.0000 0.3324 0.5000 3.3461 0.0985 0.0985 6.14 204.0000 7.0629 4.5000 0.2046 0.5000 3.4803 0.1226 0.1226 7.65 204.0000 7.1538 5.0000 0.0909 0.5000 3.5542 0.1473 0.1473 9.19 204.0000 7.2109 5.5000 0.0571 0.5000 3.5912 0.1722 0.1722 10.75 204.0000 7.2681 6.0000 0.0572 0.5000 3.6198 0.1974 0.1974 12.32 204.0000 7.3252 6.5000 0.0571 0.5000 3.6483 0.2227 0.2227 13.90 204.0000 7.3824 7.0000 0.0572 0.5000 3.6769 0.2483 0.2483 15.49 204.0000 7.4967 8.0000 0.1143 1.0000 7.4396 0.2999 0.2999 18.71 204.0000 7.6110 9.0000 0.1143 1.0000 7.5539 0.3524 0.3524 21.99 204.0000 7.7253 10.0000 0.1143 1.0000 7.6682 0.4056 0.4056 25.31 204.0000 7.8396 11.0000 0.1143 1.0000 7.7825 0.4597 0.4597 28.68 204.0000 7.9539 12.0000 0.1143 1.0000 7.8968 0.5145 0.5145 32.11 204.0000 8.0682 13.0000 0.1143 1.0000 8.0111 0.5701 0.5701 35.58 204.0000 8.1825 14.0000 0.1143 1.0000 8.1254 0.6266 0.6266 39.10

Station 18 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 216.0000 0.0000 1.5627 (sq. in.) (sq. ft) (cu. ft) (lb) 216.0000 2.2393 2.0000 2.2393 0.4373 0.4896

118

216.0000 3.1682 2.5000 0.9289 0.5000 1.3519 0.0128 0.0128 0.80 216.0000 3.7819 3.0000 0.6137 0.5000 1.7375 0.0249 0.0249 1.55 216.0000 4.2201 3.5000 0.4382 0.5000 2.0005 0.0387 0.0387 2.42 216.0000 4.5341 4.0000 0.3140 0.5000 2.1886 0.0539 0.0539 3.37 216.0000 4.7485 4.5000 0.2144 0.5000 2.3207 0.0701 0.0701 4.37 216.0000 4.8765 5.0000 0.1280 0.5000 2.4063 0.0868 0.0868 5.41 216.0000 4.9359 5.5000 0.0594 0.5000 2.4531 0.1038 0.1038 6.48 216.0000 4.9868 6.0000 0.0509 0.5000 2.4807 0.1210 0.1210 7.55 216.0000 5.0377 6.5000 0.0509 0.5000 2.5061 0.1384 0.1384 8.64 216.0000 5.0886 7.0000 0.0509 0.5000 2.5316 0.1560 0.1560 9.74 216.0000 5.1904 8.0000 0.1018 1.0000 5.1395 0.1917 0.1917 11.96 216.0000 5.2922 9.0000 0.1018 1.0000 5.2413 0.2281 0.2281 14.23 216.0000 5.3940 10.0000 0.1018 1.0000 5.3431 0.2652 0.2652 16.55 216.0000 5.4957 11.0000 0.1017 1.0000 5.4449 0.3030 0.3030 18.91 216.0000 5.5975 12.0000 0.1018 1.0000 5.5466 0.3415 0.3415 21.31 216.0000 5.6993 13.0000 0.1018 1.0000 5.6484 0.3808 0.3808 23.76 216.0000 5.8011 14.0000 0.1018 1.0000 5.7502 0.4207 0.4207 26.25

Station 19 Cum. Cum. Cum. station (x) offset (y) height (z) ∆ y ∆ z Area Area Volume Wt. 228.0000 0.0000 2.1638 (sq. in.) (sq. ft) (cu. ft) (lb) 228.0000 0.9529 2.5000 0.9529 0.3362 0.1602 228.0000 1.4530 3.0000 0.5001 0.5000 0.6015 0.0053 0.0017 0.11 228.0000 1.7715 3.5000 0.3185 0.5000 0.8061 0.0109 0.0036 0.22 228.0000 1.9973 4.0000 0.2258 0.5000 0.9422 0.0174 0.0057 0.36 228.0000 2.1595 4.5000 0.1622 0.5000 1.0392 0.0246 0.0081 0.51 228.0000 2.2720 5.0000 0.1125 0.5000 1.1079 0.0323 0.0106 0.66 228.0000 2.3418 5.5000 0.0698 0.5000 1.1535 0.0404 0.0133 0.83 228.0000 2.3855 6.0000 0.0437 0.5000 1.1818 0.0486 0.0160 1.00 228.0000 2.4293 6.5000 0.0438 0.5000 1.2037 0.0569 0.0187 1.17 228.0000 2.4718 7.0000 0.0425 0.5000 1.2253 0.0654 0.0215 1.34 228.0000 2.5581 8.0000 0.0863 1.0000 2.5150 0.0829 0.0273 1.70 228.0000 2.6444 9.0000 0.0863 1.0000 2.6013 0.1010 0.0332 2.07 228.0000 2.7307 10.0000 0.0863 1.0000 2.6876 0.1196 0.0394 2.46 228.0000 2.8169 11.0000 0.0862 1.0000 2.7738 0.1389 0.0457 2.85 228.0000 2.9032 12.0000 0.0863 1.0000 2.8601 0.1587 0.0523 3.26 228.0000 2.9895 13.0000 0.0863 1.0000 2.9464 0.1792 0.0590 3.68 228.0000 3.0758 14.0000 0.0863 1.0000 3.0327 0.2003 0.0659 4.11

119

APPENDIX D: COPYRIGHT RELEASE

University Honors College Copyright Release Form

We are planning to release this Honors Thesis in one or more electronic forms. I grant the right to publish my thesis / my abstract (circle one) entitled,

Modeling and Finite Element Analysis of a Standard Concrete Canoe , in the Honors College OSU Library’s Digital Repository (D-Space), and its employee the nonexclusive license to archive and make accessible, under conditions specified below.

The right extends to any format in which this publication may appear, including but not limited to print and electronic formats. Electronic formats include but are not limited to various computer platforms, application data formats, and subsets of this publication.

I, as the Author, retain all other rights to my thesis, including the right to republish my thesis all or part in other publications.

I certify that all aspects of my thesis which may be derivative have been properly cited, and I have not plagiarized anyone else’s work. I further certify that I have proper permission to use any cited work which is not included in my thesis which exceeds the Fair Use Clause of the United States Copyright Law, such as graphs or photographs borrowed from other articles or persons.

Signature:

Printed Name: Stephanie S. Stache

Date: August 15, 2013