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PRELIMINARY DESIGN AND NONLINEAR NUMERICAL ANALYSIS OF AN INFLATABLE OPENOCEAN AQUACULTURE CAGE By JEFFREY SUHEY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004 Copyright 2004 by Jeffrey Suhey This work is dedicated to anyone who strives to challenge himself. ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Dr. Christopher Niezrecki, for his guidance and wisdom. Working on this project has provided me with an opportunity to advance my education and potential. My sincerest thanks go out to Dr. NamHo Kim for his enthusiasm and expertise. I would also like to thank Dr. Ashok Kumar for serving on my defense committee. My thanks go to Dr. Ifju and the members of the Experimental Stress Analysis Lab for their assistance. Srikant Ranj an, Vann Chesney, and JeffLeismer were a great help with my ANSYS installation and licensing issues. I also owe a debt of gratitude to Anne Baumstarck, the Baumstarck family, and Erik Mueller for providing me with temporary lodgings while in Gainesville. And special thanks go to God, my family, and all my friends for their continued support and encouragement. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv L IST O F T A B L E S ......... ............................. ......... ... .............. .. vii LIST OF FIGURES ............. ........ .......... ..... ................... viii ABSTRACT .............. ..................... .......... .............. xii CHAPTER 1 IN TR OD U CTION ............................................... .. ......................... .. 1.1 A quaculture Introduction.. .............................................................. ............... 1 1.2 Inflatable Structure Introduction....................................... .......................... 4 1.3 Inflatable Structure Analyses..................... ............................. 12 1.3.1 Modified Conventional Beam Theory Approach............................... 13 1.3.2 N onlinear A approach ............................................................................14 1.3.3 L inear Shell M ethod ............................ ........................ .... ............... 14 1.3.4 Finite Element Analysis Approach........................ ... ............... 15 1.3.5 Selected A approach ............ .... ...................................... ........ .... ......... 16 2 THEORETICAL DEVELOPMENT ........................................ ...................... 17 2.1 Fluid A nalysis........................................17 2.2 Nonwrinkled Inflatable Theory ................ ... ...................... 19 2.3 Case Specific Nonwrinkled Inflatable Theory .......................................... 21 2.3.1 B ending M om ent ............................................. .. ........................... 21 2.3.2 V ertical D election .............................................................................. 22 2.3.3 W rinkle L ength ............ .... .................................... ...... .......................23 2.4 Conclusion ................................. ............................... ........ 25 3 EXPERIMENTAL PROCEDURE AND RESULTS........................ 26 3.1 Experim mental Setup and Procedure.................................... ....................... 27 3.1.1 Test Sam ple Orientation ....................................................................... 27 3.1.2 Sample Shapes and Correction for Fiber Discontinuity ..........................28 3.1.3 Experim ental Procedure ........................................ ........................ 29 v 3 .2 S am ple Sh ap e C h oice ........................................ ............................................32 3.3 D ata A analysis and Results ............................................................................. 33 4 SIMULATION DEVELOPMENT ................................... .................................... 39 4.1 Verification Model Development and Results ............................................. 40 4.1.1 A applied Loading Conditions ........................................... ............... 41 4.1.2 W ater C current D rag ................................................ ......... ............... 42 4.1.3 Element Selection and Properties...........................................................44 4.1.4 V erification M odel Results...................................................................... 48 4.1.5 V erification M odel Conclusion ...................................... ............... 52 4 .2 C age Sim ulation P process ........................................................... .....................52 4.2.1 Flow D esign M odels........................................................ ............... 53 4.2.2 Elem ent C choice .................................... ................. ......... 53 4.2.3 Loading Conditions ............................................................................. 53 4.3 TopD ow n Flow C age M odel .........................................................................55 4.3.1 A applied L loading C conditions ........................................... .....................56 4 .3.2 B oundary C onditions...................................................................... .. .... 56 4 .4 SideF low C age M odel .............................................................. .....................57 4.4.1 A applied L loading C conditions ........................................... .....................58 4.4.2 B oundary C onditions........................................................ ............... 58 5 SIM U L A TIO N R E SU L T S ........................................ ...................... .....................63 5.1 Sim ulation Strategy .......................................... ............ ...... ........63 5 .2 F ailu re C criteria .............................................................................. 6 4 5.3 Stress Results ................................................................... .........66 5.4 G eom etry D eform ation R results ........................................ ........................ 67 5.5 Sim ulation D election R results ........................................ ......................... 71 5.5.1 Qualitative D election Results................................................................. 71 5.5.2 Maximum Deflection Results ...................................... ...............73 5.6 C age Size R results .............................. .... ...................... .. ...... .... ..... ...... 74 5.7 V velocity R results ......... .. ................ .. .............. ........ .. ...... 77 6 C O N C L U SIO N ......... ......................................................................... ........ .. ..... .. 78 APPENDIX A AN SY S SCRIPT COD E .................................................. ............................... 80 B EX PERIM EN TAL D A TA ......................................................................... .... .... 102 L IST O F R E FE R E N C E S ........................................................................ ................... 104 BIOGRAPHICAL SKETCH ........................................................... ........ 106 LIST OF TABLES Table pge 3.1 Experimental elastic modulus results for three test sample shapes..........................33 3.2 Summary of experimental and adjusted material properties...............................37 5.1 Manufacturer data for VinylFlow commercial drainage tubing ...........................64 5.2 Maximum cage deflections for the topflow orientation.............. ............ 74 5.3 Maximum cage deflections for the sideflow orientation .....................................74 LIST OF FIGURES Figure pge 1.1 Example of a class 1 cage design (Loverich and Forster, 2000).............................3 1.2 Example of a class 2 cage design (Loverich and Forster, 2000).............................3 1.3 Example of a class 3 cage design (Loverich and Forster, 2000).............................4 1.4 Example of a class 4 cage design (Loverich and Forster, 2000).............................4 1.5 'Millennium Arches' in Stockholm, Sweden: entirely selfsupporting inflated structure. (L indstrand, 2000) ........................................ ............... ............... 6 1.6 'Millennium Arches' alternate view (Lindstrand, 2000) .......................................6 1.7 Section of 'Millennium Arches' before installation (Lindstrand, 2000)....................7 1.8 Hawkmoor selfsupporting inflated 'Temprodome': shown with three units linked together, each with dimensions (6 x 4 x 2.8m). (Hawkmoor, 2003).....................7 1.9 Exterior of 'Court TV' inflated dome. 20 ft tall by 40 ft diameter. For scale, note the person in the doorway. (Promotional Design Group, 2001) .............................7 1.10 Interior of 'Court TV' dome showing inflated arch columns. (Promotional Design G ro u p 2 0 0 1) ..............................................................................................................8 1.11 Thinfilm inflated torus used as a support structure for optical reflector (S R S 2 0 0 0 ) ................................................................................................................8 1.12 Inflated thin film with reflective coating to be used as a lightweight deployable antenna, 5 m eter diam eter (SRS, 2000)................................... ....................... 9 1.13 Inflatable Antenna Experiemnt (IAE): 14 meter diameter deployed antenna with inflated support beams (Domheim, 1999) ....... ........ ................................... 9 1.14 IAE Antenna during inflation. Stowed volume is onetenth of deployed volume. (Domheim, 1999) .......................... ....................... 10 1.15 IAE Reflector (14m diameter). For scale, note the person circled in the upper right (L'Garde, 1996) .............. .. ........ .. ...... ......... ......... 10 1.16 Goodyear Corporation inflatable truss radar antenna (Jenkins, 2001)...................11 1.17 L'Garde inflatable solar array (Jenkins, 2001).................................... ............... 11 1.18 ILC Dover, Inc. modular split blanket solar array. Shown deployed and stowed (Jen k in s, 2 0 0 1) ......... ................................................ ............................... 1 1 1.19 Planar deployment solar shade (Jenkins, 2001) ...............................................12 1.20 Inflatable Goodyear Inflatoplane (Jenkins, 2001)..............................................12 2 .1 F low on a vertical tube ............................................................................. ..... .......18 2.2 Flow on an inclined tube ...................................................................... 18 2.4 Drag loading assumed as a distributed force .........................................................22 2.5 Free body diagram of beam section .............................................. ............... 22 3.1 VinylFlow internal fiber layer orientations...................... ..................... 26 3.2 Manufacturer information for Kuriyama VinylFlow series.................................27 3.3 Test sample orientations with respect to continuous tubing ..................................27 3.4 Sample shape B (shown flat and folded)........ ...................... .............. 29 3.5 Sam ple shape C .......................................................................29 3.6 Test equipment: MTI screwdrive tensile test machine (MTI, 2004) ...................31 3.7 Test equipment: Interface 1220AF25k load cell............ .............................31 3.8 Test equipment: Curtis 30k selfcinching grip..................................................... 31 3.9 Example of sample locked in grips before pull test ...........................................32 3.10 Experimental data: five hoop orientation pull samples.......................................35 3.11 Experimental data: five longitudinal orientation pull samples.............................35 3.12 Experimental data: Five hooporientation pull samples grouped together ............36 3.13 Experimental data: Five longitudinalorientation pull samples grouped together..36 3.14 Tested sam ple w ith grip slip m arks................................. ........................ .. ......... 38 4.1 Cage prototype inflated to 12ft diameter, 9ft height............................................39 4.2 Cage prototype deflated to ~70cm ........................................ ....................... 40 4.3 Tubing union to PV C connector........................................ ........................... 40 4.5 Detail of boundary conditions for verification model...........................................41 4.6 Steps in representing internal pressure as pretension forces................................42 4.7 Representative projected area around one surface node ........................................43 4.9 Verification model parameter study: EFS effect on beam deflection ...................46 4.10 Verification model parameter study: Added stiffness effect on beam deflection ...46 4.11 Verification model parameter study: Added stiffness effect on longitudinal beam store ss ............ ............ .. .......... ...... .............................................. . 4 7 4.12 Verification model parameter study: Poisson's ratio effect on beam deflection ....47 4.13 Verification model parameter study: Poisson's ratio effect on longitudinal beam store ss ............ ............ .. .......... ...... .............................................. . 4 8 4.14 Deflection results for the verification model with length of 5m...........................50 4.15 Deflection results for the verification model with length of 10m length.................50 4.16 Deflection results for verification model with 0.5 m/s flow velocity ....................51 4.17 Deviation in deflection results for verification model with 0.5 m/s flow velocity ..51 4.18 Longitudinal bending stress results for verification model with 0.5 m/s flow v elo city ......... ..... ............. .................................... ...........................52 4.19 Two cage simulation flow orientations ........................................ ............... 53 4.20 Control volume analysis around a rigid 45'connector containing pressurized internal fluid ............. .......... .... ................... ............ ............ 54 4.21 Free body diagram of rigid cap with algebraically summed pretension force components ............... ..... .. .......... ........ .. ........ ........ ......... 55 4.22 Force components applied to finite element model of rigid connector....................55 4.23 Topflow boundary condition locations ....................................... ............... 57 4.24 Cage center tube components ........... .. ......... ................... 59 4.25 Expanded view of representative internal cable connections.............................60 4.26 Expanded View of representative mooring cable connections .............................60 4.27 D im tensions of anchor location........................................... .......................... 61 4.28 Sideflow applied loading locations...................................................................... 61 4.29 Sideflow boundary condition locations ...................................... ............... 62 4.30 Expanded view of center tube boundary conditions .............................................62 5.3 Maximum stresses in each simulation and ultimate material strength...................67 5.4 TopFlow deformed geometry at the wrinkle point: View 1...............................68 5.5 TopFlow deformed geometry at the wrinkle point: View 2..................................68 5.6 TopFlow deformed geometry at the wrinkle point: View 3................................69 5.7 SideFlow deformed geometry at the wrinkle point: View 1 ...............................69 5.8 SideFlow deformed geometry at the wrinkle point: View 2...............................70 5.9 Representative rotational effect occurring in sideflow simulation .......................70 5.10 Cage xdiameter net deflection for VF800 tested at flow speed of 1 knot, top flow ......... ..... ............. ..................................... ...........................72 5.11 Maximum zdeflection for VF800 cage tested at flow speed of 1 knot, top flow ......... ..... ............. ..................................... ...........................72 5.12 Cage ydiameter net deflection for VF500 tested at flow speed of 1 knot, sid e flo w ...................................................................... 7 3 5.13 Topflow maximum cage diameters for each tubing material grouped by internal pressure ................ ..................................... ........................... 75 5.14 Sideflow maximum cage diameters for each tubing material grouped by internal pressure ................ ..................................... ........................... 76 5.15 Summary of maximum possible cage size for each material in two flow cases subject to one knot flow velocity VF Series number corresponds to diameter, working pressure, and thickness in Table 5.1 .................................. ............... 76 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science PRELIMINARY DESIGN AND NONLINEAR ANALYSIS OF AN OPENOCEAN AQUACULTURE CAGE By Jeffrey Suhey August 2004 Chair: Dr. Christopher Niezrecki Major Department: Mechanical and Aerospace Engineering A nonlinear finite element analysis is used to predict the static performance of a novel aquaculture cage constructed from inflatedbeam members. Simulations are performed to determine the cage maximum deflections, stresses in the inflated components, and ultimately the maximum possible cage size. Experimental tests are performed on a commercially available anisotropic fabric material to determine longitudinal and hoop elastic moduli. Fluid drag forces on the cage are applied to the inflated members as a constant distributed force that is dependent on the flow speed of the current. Simulations are performed up to 1.6 knots of water current. Two flow cases based on the orientation of the cage with respect to water current are considered. Drag on the surrounding cage nets is excluded from the preliminary design analysis. Within this study, cage members were pressurized from 30 to 80 psi, depending on the material. Important results include successfully merging a linear and nonlinear element to represent tensiononly behavior of an inflated fabric tube. Verification was performed by comparing a simplysupported distributedload inflated beam finite element model with a modified traditional beam analysis. Maximum possible cage sizes are found at constant flow velocity for different materials with varying internal pressure, wall thickness, and tube diameter. Effects of changing flow velocity are found for a specific material case in both flow directions. The results indicate that the largest sized cage for a 1 knot (0.51 m/s) current has a cage diameter of 39 m for the material used in this study. CHAPTER 1 INTRODUCTION The current trend in new inflated structure applications lends itself to studying an underwater application. In this case, a preliminary engineering design analysis of an openocean aquaculture cage is presented. This chapter provides a brief introduction to aquaculture and describes recent developments in inflatable structures. 1.1 Aquaculture Introduction Aquaculture can be described using the following definition: "Aquaculture is defined as the propagation and rearing of aquatic organisms in controlled or selected aquatic environments for any commercial, recreational, or public purpose" (Department of Commerce, 2002). Aquaculture is an agricultural approach that provides an alternative to the other two common methods of obtaining fish; commercial fishing and fish farming with large permanent structures in the ocean or on land (Loverich and Forster, 2000). "Aquaculture is expected to play an increasingly important role in meeting the global demand for fisheries products as the world population continues to expand and fish stocks approach their biological limits" (Fredriksson et al., 2000). The problem is making the use of aquaculture cages economical. A growing need for improved cage designs will hopefully stimulate engineering research to overcome the obstacles now facing aquaculture. Several factors comprise aquaculture cage design. Costs of manufacturing, installation and maintenance, cage stability, durability, and predictability, all play an important role in the design process. One aspect specifically important to aquaculture cage design is the divergence volume. This can be defined as a maximum potential geometric deformation of the cage. Large divergence volumes can cause high fish mortality rates and decrease diver safety. These problems can lead to large economic drawbacks, making the divergence volume a key factor on which to base a design. A variety of cages have been designed to accommodate individual design advantages and disadvantages. These designs have been grouped into four classes that are in current use throughout the aquaculture community. The class 1 cage is the most common in use today and is based on forces due to gravity; weight and buoyancy. The cage typically comprises a floating frame with a weighted net hanging below. This simple design has several problems, including unpredictable failures and high fish mortality. The major disadvantage to this cage design is the large divergence volume. Figure 1.1 shows a typical class 1 cage design. The class 2 cage has anchored rigid truss members used to hold the shape of the netting. Due to the taught netting, there is a significant improvement in the divergence volume compared to the class 1 cage. Similarly, class 3 cages maintain taught netting for this same improvement. Class 3 cages generally have a central truss with a rigid ring connected by ropes and covered in netting. This design allows the cage to maintain its shape without relying on anchors. The deformation of the class 3 cage is governed by the rigidity of its components. It achieves a smaller divergence volume than the class 1 and 2 by providing more supports over the span of the net. Figures 1.2 and 1.3 show examples of typical class 2 and 3 cages, respectively. The Class 4 cage is a completely rigid selfsupported frame covered in netting. Although seemingly ideal due to the very low divergence volume, the costs of material, 3 construction, and installation labor are much higher than of the previous three classes of cage design. Figure 1.4 shows an example of a typical class 4 cage design. .Figue E a c 1 c i i a :,. ?,, :.... ...... : ", : ... I.; .1 .. ,, .. :., ^ 1~ .,'. I 1 Figure 1.1 Example of a class 1 cage design (Loverich and Forster, 2000) Figure 1.2 Example of a class 2 cage design (Loverich and Forster, 2000) I. . N Figure 1.3 Example of a class 3 cage design (Loverich and Forster, 2000) Figure 1.4 Example of a class 4 cage design (Loverich and Forster, 2000) 1.2 Inflatable Structure Introduction Several applications exist for inflatable structures. Applications for inflatable structures have been used successfully in the past. Some familiar applications that are still in use include vehicle tires, surface watercraft (inflated boats, pontoons, hovercraft, etc), and aircraft (blimps, balloons, etc). Within the past few decades, new inflatable applications are being explored and gaining acceptance. Inflatable land structures are completely selfsupported, requiring no solid structural members. These are currently in use in military and architectural designs for tents, hangars, roofs, and small buildings. These applications take advantage of the ^ ^. convenience of inflatable structures to save assembly time and travel weight. "...today, it's well within the bounds of possibility to find a 300 ft long building housing a concert venue, exhibition hall, museum, or spaceage science display, with not a solid supporting member in sight" (Lindstrand, 2000). Figures 1.5 through 1.10 show examples of inflated land structures, with no rigid support members, supported only by internal pressure. "The [Millennium Arches] whole building is 328ft(100m) long, 59ft(18m) wide, and 56ft(17m) high. The main arches are 38ft(11.5m) high, and 59ft(18m) wide while the end arches are 41ft(12.5m) long and the center arch is 164ft(50m) long" (Lindstrand, 2000). The aerospace field is working with inflatables in gossamer structures for space applications such as sunshields, antennas, solar sails, habitats, and structural booms. Space applications take advantage of several features of inflatable structures. "They [inflatable structures] offer large potential reductions in stowed volume, cost, and often weight.... Giant space structures such as 1,000ft. antennas or solar sails may not even be possible with mechanical deployment, but may be doable with inflatable design" (Domheim, 1999). Figures 1.11 through 1.15 show examples of space applications of inflatable structures. A1N W T.. "2Ki, Figure 1.5 'Millennium Arches' in Stockholm, Sweden: entirely selfsupporting inflated structure. (Lindstrand, 2000) Figure 1.6 'Millennium Arches' alternate view (Lindstrand, 2000) Figure 1.7 Section of 'Millennium Arches' before installation (Lindstrand, 2000) .'... .I... ,n" Figure 1.8 Hawkmoor selfsupporting inflated 'Temprodome': shown with three units linked together, each with dimensions (6 x 4 x 2.8m). (Hawkmoor, 2003) Figure 1.9 Exterior of 'Court TV' inflated dome. 20 ft tall by 40 ft diameter. For scale, note the person in the doorway. (Promotional Design Group, 2001) '..'.'.'. .". Figure 1.10 Interior of 'Court TV' dome showing inflated arch columns. (Promotional Design Group, 2001) Figure 1.11 Thinfilm inflated torus used as a support structure for optical reflector (SRS, 2000) Figure 1.12 Inflated thin film with reflective coating to be used as a lightweight deployable antenna, 5 meter diameter (SRS, 2000) Figure 1.13 Inflatable Antenna Experiemnt (IAE): 14 meter diameter deployed antenna with inflated support beams (Domheim, 1999) Figure 1.14 IAE Antenna during inflation. Stowed volume is onetenth of deployed (Dornheim, 1999) Figure 1.15 IAE Reflector (14m diameter). For scale, note the person circled in the upper right (L'Garde, 1996) Figure 1.16 Goodyear Corporation inflatable truss radar antenna (Jenkins, 2001) Figure 1.17 L'Garde inflatable solar array (Jenkins, 2001) Figure 1.18 ILC Dover, Inc. modular split blanket solar array. Shown deployed and stowed (Jenkins, 2001) Figure 1.19 Planar deployment solar shade (Jenkins, 2001) Figure 1.20 Inflatable Goodyear Inflatoplane (Jenkins, 2001) 1.3 Inflatable Structure Analyses This section describes various forms of modeling inflatable structures. A specific emphasis is placed on determining deflection and internal stress behavior of inflatable structures subject to cantilever bending. Also of interest is analysis of the application of fabrics for the inflated material. The basic concept of an inflatable structure is a closed end pressure vessel subject to some applied loading. The internal pressure translates to a force applied over the capped end in the axial direction. This force then creates an axial stress that is able to resist bending and axial stress caused by the applied loading conditions. Chapter 2 more fully explains the derivation of the inflatable theory. 1.3.1 Modified Conventional Beam Theory Approach Experimental data has shown that inflatable beams do not behave according to traditional beam theory. With modification to the traditional beam theory, an accurate model can be made. One study analyzed inflatable beam deflections and stresses for loads between incipient buckling, where bending stress equals axial stress due to pressure, and final collapse. It was shown that initially bending occurred with no wrinkling. When wrinkling occurred, a slack region and a taught region were present. A formulation for the stress was developed showing zero axial stress in the slack region and a portion of the max stress carried in the taught region. By equating expressions for maximum axial stress, a formulation for the relationship between internal pressure and applied tip force was determined. The curvature was determined and integrated twice to find the deflection due to bending. The deflection due to shear was also determined noting that it is negligible when the length of the beam is much larger than the radius of the cross section. The total deflection was then determined as the algebraic sum of the deflections due to bending and shear. Similar results were obtained for a beam subject to distributed load bending (Comer & Levy, 1963). This analysis shows when wrinkles begin to form and where they form. The wrinkled membrane is shown to have some rigidity. The effect of the wrinkling on the bending strength of the beam is determined. Another study applies the modified traditional beam theory approach from Comer and Levy for fabric materials. An experimentally determined modulus of membrane of the fabric material was used in place of Young's modulus. This modulus of membrane is defined as the slope of the experimental stress resultant vs. engineering strain plot, where stress resultant is the force per width of a pulltest sample. Using an approach similar to that of Comer and Levy, the curvature was integrated numerically to determine tip deflection behavior. Experimental results agreed for aspect ratios of l/d>6 (Main et al., 1994). 1.3.2 Nonlinear Approach Since inflatable beams behave nonlinearly, a nonlinear approach, although complex, is the most logical method. Douglas uses the theory of incremental deformations to find the CauchyGreen deformation tensors which can be related to determine the Lagrangian and Eulerian Strains. These lead to a stressdeformation relation. This is all then applied to an inflated beam incorporating the pressure in as a stress. The variation in beam stiffness is plotted as a function of changing internal pressure. Unlike a linear approach, this thorough method takes into account the changes in geometry and changes in material properties (Douglas, 1968). Another study included an explanation of the geometry of fabrics and the interaction between fibers. The general relationship between stress and strain for a fabric material is ultimately shown to be a nonlinear trend (Bulson, 1973). 1.3.3 Linear Shell Method Another method for analyzing inflatable structures is to use a linear shell approach. One study develops a free body diagram of a width of flexed strip of the loaded material. Static equilibrium of forces is then applied, including the applied load and the skin tensions on the width of the material. First, a vertical force summation is made with the application of a small angle approximation. Then, a horizontal force summation is developed and combined with the vertical summation canceling the skin tension terms. The result is an expression analogous to the equation for shearing angular deflection of a solid beam. By comparing these two equations, the internal pressure is found to be analogous to an effective shear modulus. This is used to determine the effect of pressure on the shearing deflection (Topping, 1964). Another source applies continuum mechanics to membranes to obtain several general relations. It defines the kinematics, deformation, strain, strain rate, and stress for membranes. Balance laws are used to derive general mass balance, momentum balance, and energy balance relations (Jenkins, 2000, pp. 4964). To model wrinkling in beams, a tension field model is developed with an approach for application to linear finite element method (Jenkins, 2000, pp 103105). 1.3.4 Finite Element Analysis Approach Another approach is to apply finite element analysis to inflated structures. Finite element analysis can yield accurate results for problems with complex analytical solutions. One study compares results from the modified traditional beam theory with finite element results. An assumption of Brazier's effect is used here. This states that as the tube deforms due to bending and the crosssection becomes flatter, the bending stiffness of the entire beam decreases. This is most likely due to geometry changes that affect the area moment of inertia. This study formulates finite elements of geometrically nonlinear motion of membrane. These allow for changes in length of the elements to account for stretching of the material in the ends but not in the cylinder. Triangle elements are used to model the circular end caps and quad elements are used to model the cylindrical tubing. A pressure relationship is used to apply a force to each element. FEA results match theoretical results from the modified traditional beam approach. An analysis of tubes containing multiple pockets is conducted as well. The behavior of these multicellular inflated beams is shown graphically. By increasing the number of cells, the pressure can be varied throughout the length of the tube and thereby optimize the design (Sakamoto and Natori, 2001). 1.3.5 Selected Approach The modified traditional beam theory will be used in conjunction with a nonlinear finite element model for future analysis. The finite element approach solves complex geometries and provides a visualization of deformed behavior. The modified traditional beam theory will be used to verify the behavior of a simple finite element model. Following verification, the finite element approach will then be used to model complex cage geometries and interactions. Further theoretical development of the modified traditional beam theory is presented in Chapter 2 and the finite element approach is described in Chapter 4. CHAPTER 2 THEORETICAL DEVELOPMENT The development of the theory applied to analytical solutions used in later chapters is described in Chapter 2. A standard fluid flow analysis is first presented, followed by the traditional beam method applied to inflatable theory, which is chosen from the methods presented in Chapter 1. The theory presented focuses on inflated beams before and at the point of wrinkling. The model used for derivation is an internally pressurized circularcylindrical closedend fabric tube. The internal pressure acting over the surface area of the end caps translates to pretension forces in the fabric tube. The essence of the inflatable theory is that bending causes a compressive stress on the underside of the tube opposing the tensile stress caused by the pretension forces. Wrinkling occurs when the compressive portion of bending stress exceeds the pretension stress. In some cases, stress resultant, c*, is used to replace axial stress, c, throughout the analysis, as described in Main's theory, to more accurately model fabric behavior (Main et al., 1994). This also leads to using a resultant elastic modulus, E*, with units of force per unit length. 2.1 Fluid Analysis The fluid analysis is used to determine the total drag force acting on a cylinder for a given external flow condition. The following assumptions are made for the fluid analysis: * Smooth cylinder surface * Constant temperature fluid and material properties at 200C 18 * Steady state flow * Constant crosssectional area In this case, the drag force is caused by friction of fluid current flowing over the tube. The drag force is a function of tube crosssectional area, A, drag coefficient, CD, external fluid density, p, and external flow velocity, V. Equation 2.1 defines the drag force, Fdrag, on a cylinder based on applicable assumptions. Fdr = CApV2 (2.1) 2 The crosssectional area perpendicular to the flow direction is defined, A = Dh, with tube diameter, D, and crosssectional length perpendicular to flow, h. Figures 2.1 and 2.2 define h for a vertical tube and a tube inclined by angle, p, from the horizontal, respectively. Flow Direction h=L 1 Figure 2.1 Flow on a vertical tube Flow Direction h = L sin (p  ^ /^ Figure 2.2 Flow on an inclined tube The coefficient of drag, CD, is determined using the dimensionless parameter, Re, and the approximated range taken from documented experimental data. Equations 2.2 and 2.3 show the Reynold's number relation and the approximated value of CD based on the expected flow velocity range. Re pVD (2.2) CD =1, for 10,000 < Re < 200,000 (2.3) D represents the diameter of the tube. Fluid properties of the flowing current including density and absolute viscosity, [t, are determined from documented values at a temperature of 200 C (Fox & McDonald, 1999). 2.2 Nonwrinkled Inflatable Theory Figure 2.3 shows the assumed stress distribution for a nonwrinkled inflated beam subject to a bending moment. Urn 0 Figure 2.3 Stress distribution in tube subject to bending moment Comer & Levy (1963) defined the stress distribution in this unwrinkled inflated circular cylindrical beam using Equation 2.4. co (1 + cos ) Om(1 cos 0) S= (2.4) 2 2 As shown in Figure 2.3, Gm and ao are the maximum and minimum longitudinal stresses in the circumferential stress distribution, respectively. Equation 2.4 can be rewritten as in Equation 2.5, by diving through by the material thickness. Stress, C, is replaced with stress resultant c*. ( (1+ cos0) o (1 cos)) a= + (2.5) 2 2 Equation 2.6 describes the bending moment, M(x), as a product of the tensile forces, F(O), and the distance, h(O), from the neutral axis of bending. Force is then replaced with the product of stress and area. The area is defined by integrating the fabric thickness, t, over the tube circumference using tube radius, r, and angle, 0, from 0427n. M(x) = F(O)h(O) = (O)h(O)rt dO (2.6) 8=0 o Defining height, h(O) = r cosO, and stress resultant, u (0) = c(O)t, Equation 2.6 can be rewritten using symmetry as Equation 2.7. M(x)= 2J *r2 cos 0dO (2.7) 0 Substituting Equation 2.5 into Equation 2.7 and integrating yields Equation 2.8. Sx 2M((x)28) &,m O 0 (2.8) According to Main (1994), the unwrinkled curvature, pc, is given by Equation 2.9, where E* is the resultant elastic modulus. (2.9) Pc 2rE* Substituting Equation 2.8 into Equation 2.9 and approximating curvature as d y/dx2, where x and y are coordinates, yields Equation 2.10. 1 M(x) d2y = (2.10) pc itrE* dx2 This leads to the final relationship, Equation 2.11, which corresponds to the traditional beam bending relation with I* representing the resultant area moment of inertia. Note the units of resultant area moment of inertia are cubic which corresponds to using the stress resultant, c*, and resultant elastic modulus, E*. d y M(x) I* d2y M (x) ,where I = r' by association (2.11) dx2 E I This differential equation describes the bending behavior of an inflated circular tube. It will be integrated to determine deflection, y, for a specific set of boundary conditions. This theoretical deflection solution is then used to verify simulation deflection results in Chapter 4. 2.3 Case Specific Nonwrinkled Inflatable Theory 2.3.1 Bending Moment Figure 2.4 shows the assumed drag force loading, w, and model boundary conditions, with L representing the tube length. The specific bending moment, M(x), for this situation is derived from free body diagrams of a cut section of the beam as shown in Figure 2.5. Shear, V, and moment, M are shown as reaction forces. Variable x is used as a local coordinate along the tube length. Equations 2.12 and 2.13 show the sum of moments about point A assuming static equilibrium. Fdrag L Internal pressure, P Figure 2.4 Drag loading assumed as a distributed force x x 2 2 wx M tA 4 V wL 2 Figure 2.5 Free body diagram of beam section + MA = wL )x + (wx) +M = 0 (2.12) M(x) = w(Lx x2) (2.13) 2 2.3.2 Vertical Deflection To solve for beam deflections, the specific moment equation, Equation 2.13, is substituted into Equation 2.11 and integrated twice, yielding Equations 2.14 through 2.16. d2y w d y (Lx x2) (2.14) dxC2 2E* mr3 dy w Lx2 dx 2E*Ir3 2 w Lx3 x4 2E*zr3 6 12 With boundary conditions at x = 0, y = 0, and at x and C2, are found to be Equations 2.17 and 2.18 C, =0 + C ) (2.15) (2.16) + C C 2) L, y = 0, constants of integration, Ci (2.17) (2.18) Substituting Equations 2.17 and 2.18 into Equation 2.16 yields the vertical deflection solution shown as Equation 2.19. w Lx3 X4 y2E 6 12 2E* r3 6 12 SL3 12 (2.19) 2.3.3 Wrinkle Length Wrinkle length is the length of the tube at which wrinkling first occurs. This length depends on the moment necessary to initiate wrinkling. Comer and Levy first determine the moment necessary to initiate wrinkling by equating expressions for om (Comer & Levy, 1963). Equation 2.20 describes a force balance between the internal pressure acting over the area of the circular endplate and the longitudinal stress integrated over the circumference of the tube. pr2 = 2 o*r dO (2.20) Substituting Equation 2.5 into Equation 2.20 and integrating yields Equation 2.21, the first expression for om. zpr(1 + cosO ) am co= (2.21) 2t[sin 00, + (r 0,)cos0 ] Substituting Equation 2.5 into the balance of moments, Equation 2.7, yields the second expression for om. 2M(1 + cos 0 ) "tr2 (2r 200 + sin 200 ) Combining Equations 2.21 and 2.22 by eliminating om and setting 00 = 0, yields Equation 2.23, the moment necessary to initiate wrinkling. M = (2.23) 2 Equating the moment to initiate wrinkling, Equation 2.23, with the value of the maximum moment in Equation 2.13 yields Equation 2.24. L wL2 pair3 max M( )w (2.24) 2 8 2 Solving for L and setting the load, w = F/L, gives the general expression for wrinkle length, Equation 2.25. The wrinkle length is the length at which wrinkling first occurs in an inflated tube subject to bending, for a given pressure and flow rate. Lwr 4pR (2.25) F As expected, the wrinkle length will be reduced for high loading force, F, and proportional to the internal pressure and the tube radius. Tube radius, r, is the largest contributing factor to wrinkle length for these conditions. For this specific application, the loading force, F, is the drag force, Fdrag. Since drag force is a function of tube length, Equation 2.25 is further expanded using Equations 2.1 and 2.3 to obtain Equation 2.26. Lw pr2 (2.26) SpV2 Equation 2.26 now shows the wrinkle length as a function of the core contributing variables for this application; internal pressure, tube radius, and external fluid velocity. The wrinkle lengths in Equations 2.25 and 2.26 also depend on the boundary conditions and are derived for the simplysupported conditions shown in Figure 2.4. 2.4 Conclusion The equations derived in this chapter provide a theory for an inflated beam subject to bending. The solution for vertical deflection, Equation 2.19, is used to verify the finite element model developed in Chapters 4 and 5. The wrinkle length is used to verify the final results of a verification model. Verifying a simple model is necessary to justify extending the computer analysis to more complex geometries, not easily solved with traditional analysis. CHAPTER 3 EXPERIMENTAL PROCEDURE AND RESULTS The setup, procedure, and results of the experiments performed to determine the orthotropic material properties of a specific commercial drainage tubing is presented in chapter 3. The Kuriyama VinylFlow series is a composite comprised of a combination of molded PVC layers and synthetic fibers. One fiber layer runs in the longitudinal direction while two other layers are oriented at approximately +30 and 30O from the longitudinal direction as shown in Figure 3.1. Figure 3.2 shows specifications given by the Kuriyama for the different size models of Vinyl Flow. For this application, the desired material properties are the elastic modulus and Poisson's ratio. These material properties are determined experimentally and to be used in a numerical simulation as described in Chapter 4. +30 Longitudinal Direction 30 Figure 3.1 VinylFlow internal fiber layer orientations * P .*eri =l ip .,iS.e ihe .] 5:rnar.ei rnis featuring a bal anced polyester yarn spiral wrap, longitudinal strength member, and homogeneous PVC tube and cover. * Use for general purpose water discharge applications  ideal for use as a drip irrigation supply line. * Smooth tube provides low friction loss (see charts on opposite side). * Lays ; i ;iqr,I i n r ir rnr ii nj ; a I l ui ,c r pressure. * Hc.e ,Tia t.e u"ii,:r o e, l nv r ji 'eaiirrg I.,r *.r,[. ii,  tion applications. * Homogeneous tube and cover construction eliminates separation. * Ultraviolet inhibitors reduce aging and weather checking. Vinylflouw general purpose water discharge hose Nominal Hose Approximate Working Coil Approx. Series Size ID Wall Thickness Pressure Length Weight No. (in.) (in.) (in.) (psi) (ft) (Ibs.coil) VF 150 11, 1.673 .0669 80 300 48 VF 200 2 2.165 .0669 80 300 69 VF 250 21/ 2.598 .0787 80 300 87 VF 300 3 3.130 .0787 70 300 117 VF 400 4 4.134 .0827 70 300 156 VF 500 5 5.039 .0866 40 300 204 VF 600 6 6.181 .0866 50 300 258 VF 800 8 8.169 .1063 45 100/300 130/390 VF 1000 10 10.118 .1181 35 100 181 VF 1200 12 12.126 .1181 30 100 195 VF 1400 14 14.134 .1181 30 100 262 VF 1600 16 16.142 .1181 30 100 310 Figure 3.2 Manufacturer information for Kuriyama VinylFlow series 3.1 Experimental Setup and Procedure 3.1.1 Test Sample Orientation Due to the orthotropic fiber orientation, a value for elastic modulus in both the longitudinal and hoop directions is desired. To achieve this, samples were made to have the fibers aligned in a specific orientation. Figure 3.3 shows the convenient coordinate system chosen and the orientation of each sample. Also shown are Fx and Fy, the tensile forces applied to each orientation sample. Longitudinal, y Fy Hoop,x K Longitudinally Oriented Sample Tubing Coordinate System HoFx Hoop Oriented Sample Figure 3.3 Test sample orientations with respect to continuous tubing 3.1.2 Sample Shapes and Correction for Fiber Discontinuity In order to accurately model the materials' performance, a tensile test sample shape was needed. It was desired to simply use a continuous tube sample but it was not possible to achieve this shape in the hoop direction. It was also inconvenient to test a continuous tube for the large diameter tubes due to the grip size of the tensile test machine. Since all the samples contain discontinuous fibers, it is necessary to determine the numerical effect on elastic modulus. To do this, three different sample shapes are tested and compared to see how the width and continuity of the fibers affect the elastic modulus. The first sample, shape A, is a completely continuous length of tubing with a rectangular shape. The second sample, shape B, is the same as shape A, but differs only with a single longitudinal cut running the entire length of the specimen. The cut severs the continuity of the 30 cross fibers. Figure 3.4 shows the longitudinal cut on the circular crosssection, representing shape B. In order to fit the sample completely within the width of the test grips, samples A and B are folded twice along the longitudinal direction as shown in Figure 3.4. The third sample, shape C, is cut from a template with a shape mimicking a tensile test sample. The ends are designed to fit the width of the grips of the testing device. The center section is designed to taper from the grips to a narrow strip to control break location. Care was taken to keep the tapering smooth to avoid stress concentrations in sharp covers. Figure 3.5 shows an example of shape C. All samples are 23 cm long. Sample C is 2 cm wide at the narrowest section and samples A and B have the same width when folded. Figures are shown with the same standard 5 34" pen for scale. Figure 3.4 Sample shape B (shown flat and folded) Figure 3.5 Sample shape C 3.1.3 Experimental Procedure An MTI 25k screwdriving tensile test machine was used with Curtis 30k self cinching jaw grips to perform the tensile tests. An Interface 1220AF25k load cell was used in conjunction with MTI software to measure force. After observing preliminary tests, a test speed of 0.10 in/min was considered reasonable and chosen for all tests. Figure 3.6 through 3.8 show examples of the test equipment. Figure 3.9 shows an example of a specimen locked in the grips before pulling. A sample is closed in the test grips and pulled at a constant velocity to the initial breaking point. Two marks were made in the middle region of the sample oriented perpendicular to the length. The length between marks and width (width for sample shape C) were measured and recorded intermittently throughout the pull test with digital calipers. The pull force on the specimen is measured and recorded simultaneously with the length and width measurements. The instantaneous axial stress is calculated at each point by dividing the force by the crosssectional area. The crosssectional area is calculated as the product of the width and the thickness of the test strip. The material width and thickness is assumed to be constant in Samples A and B. Only the thickness is assumed constant in Sample C since the varying width was recorded. The instantaneous axial strain, F, is calculated at each point using Equation 3.1. ss c5 5 (3.1) 80 The variable, 6, represents the recorded distance between marks and 60 represents the original distance between marks. This references strain calculations to the original unstressed length. An attempt was made to experimentally determine the material Poisson's ratio by using measurements of the strain in the width and the strain in the length during the uniaxial pull tests. It was discovered, however, that since the cut samples do not have stiffness in compression, and the width decreases with the sample under tension, the width strain is not a valid measure to be used in determining Poisson's ratio. To accurately describe Poisson's ratio, the material must be loaded in two directions at once. Two possible solutions are to use a pull tester capable of pulling in two directions, or to internally pressurize a uniaxial test specimen. Given the complexity of these tests, Poisson's ratio is instead determined in chapter 4, using a parameter study in an ANSYS simulation. Figure 3.6 Test equipment: MTI screwdrive tensile test machine (MTI, 2004) Figure 3.7 Test equipment: Interface 1220AF25k load cell Figure 3.8 Test equipment: Curtis 30k selfcinching grip Figure 3.9 Example of sample locked in grips before pull test 3.2 Sample Shape Choice Using the analysis described in section 3.3, data from the sample shape tests is used to determine material elastic modulus. Five specimens were tested for each sample shape. Table 3.1 shows a summary of the experimentally determined elastic moduli for each sample shape. It can be concluded that neither sample B nor sample C closely represent the full tube structure. This is due to the breakage of the continuous hoop fibers that form the structure of the fabric. Since breaking these fibers is unavoidable in hoop orientation tests, the assumption was made to apply a multiplicative correction factor to all tests made from cut samples. The correction factor is determined from a ratio of the elastic modulus of the continuous tube, sample A, to the elastic modulus of the cut tube, sample B. This correction factor of 1.29 is applied to all future experimental results, including both the longitudinal and hoop oriented samples. Sample shape C is used for all future testing for convenience and predictability. Table 3.1 Experimental elastic modulus results for three test sample shapes Sample Shape and Description Orientation Elastic Modulus (psi) A Folded Continuous Tube Longitudinal 49009 B Folded Cut Tube Longitudinal 38148 C Strip from Template Longitudinal 36098 3.3 Data Analysis and Results This section describes the analysis of the data collected and its application towards the orthotropic material properties. Figures 3.10 and 3.11 show plots of axial stress and axial lengthstrain for VF600 material in hoop and longitudinal orientations, respectively. Five samples per orientation were tested and are plotted with distinct linear trends. In order to interpret one linear trend from all five tests, all the tests are grouped together with one trend as shown in Figures 3.12 and 3.13. Combining five distinct trends is justified by looking at the slope and yintercepts of the trends. The trend from each test has a distinct slope, but they are all close enough to consider within experimental error. The yintercepts between tests, however, are largely different. The yintercept of each trend line indicates an initial load present on the sample before any lengthstrain is recorded. This is explained by the observed behavior of the test grips during the test. The selfcinching grips initially sink in and tighten around the soft material for a brief period of time before a lengthstrain is present in the entire sample. This settling period is interpreted as a preloading on the sample by the load cell, resulting in the yintercept in the data trend. The presence of the yintercept is a result of the specific sample and test apparatus, so it is therefore assumed inconsequential to the experimental material properties. Since the slope is the only valid material property obtained from the data, all five groups can be justifiably analyzed with one trend. The slope of the trend represents the elastic modulus for the corresponding sample orientation. Table 3.2 shows a summary of the experimental material property results. Shown are the experimental and adjusted orthotropic elastic moduli for the hoop and longitudinal orientations. Adjusted elastic moduli are based on the correction factor for breaking the continuity of the fibers, determined in section 3.2. It is interesting to note that the elastic modulus for the longitudinal orientation is much larger than for the hoop orientation since the hoop stress is generally larger than the longitudinal stress. This is best attributed to the fiber layer present in the longitudinal orientation. The material was most likely not designed to be optimized as a pressure vessel, but rather designed for ease of the manufacturing process. * Test 1 * Test 2 Test 3 Test 4 x Test 5 1800 1600 1400 1200 1000 800 600 400 200 0 O.C 4500 4000 3500 3000 2500 S2000 1500 1000 500 0 0.0000 0.0200 0.0400 0.0600 0.0800 Length Strain (inlin) * Test 6 * Test 7 Test 8 Test 9 STest 10 0.1000 0.1200 Figure 3.11 Experimental data: five longitudinal orientation pull samples 000 0.0500 0.1000 0.1500 0.2000 0.250( Length Strain (in/in) Figure 3.10 Experimental data: five hoop orientation pull samples I 0 0 1800 1600 1400 1200 1000 800 600  400 200 0 0.0000 0.0500 0.1000 0.1500 0.2000 Length Strain (in./in.) 0.2500 Figure 3.12 Experimental data: Five hooporientation pull samples grouped together 0.0200 0.0400 0.0600 Length strain (inlin) 0.0800 0.1000 0.1200 Figure 3.13 Experimental data: Five longitudinalorientation pull samples grouped together 4500 4000 3500 3000 2500  2000  1500  1000  500 0 0.0000 Table 3.2 Summary of experimental and adjusted material properties Matea / On Experimental Elastic Adjusted Elastic Material/ Orientation . Modulus (psi) Modulus (psi) VF600/ Hoop 5798.4 7479.9 VF600 / Longitudinal 36098.0 46566.4 3.5 Errors and Accuracy Unavoidable experimental errors occurred in all the tests. Due to the nature of the crosshatched fabric material, inplane and outofplane shape distortions occurred in samples when loaded under tension. The inplane distortions can be described as uneven shifting of the material in the pull direction. The strain measurements were largely affected by the inplane distortion because the measurement lines did not remain perpendicular to the sample length. The outofplane distortions can be described as cupping of the flat, rectangular crosssectional area. This largely affects the width and, in some cases, thickness measurements. Tiny slip marks were observed on some tested samples where the selfcinching grips cut through or slipped on the slick outer material. This would contribute additional error to the strain measurements. Figure 3.14 shows an example of the lines visible on a tested pull sample that indicate slipping. In some cases, manufacturing inconsistencies in the material were exposed when material samples visibly broke in a location other than the expected thinnest portion of the sample. Human error in preparing samples, taking length measurements, and reading the force gage also contributed to experimental error. The wide scattering of the data points in the stress strain curve can also be attributed to the settling period described in section 3.3. These experimental errors all contribute to the wide scattering of the data points in the stressstrain curve. Due to the nature of the composite, flexible material and the test equipment used, the experimental values are considered a valid representation of the material. Pull Direction Grip slippage Grip slippage Normal Grip Wear Figure 3.14 Tested sample with grip slip marks CHAPTER 4 SIMULATION DEVELOPMENT The process of developing a finite element model to represent an actual prototype cage is described in Chapter 4. A simple verification model is first made to explore and verify finite element membrane analysis for an inflatable tube subject to bending. A full cage is then modeled to explore the interaction of all the components. Two flow direction cases are modeled to see the effect of flow direction on stress and deflection in the cage members. Figures 4.1 and 4.2 show a prototype cage inflated and deflated, respectively. Figure 4.3 shows an example of the union between the tubing and a PVC connector using a hose clamp. Figure 4.1 Cage prototype inflated to 12ft diameter, 9ft height lgure 4.2 Cage prototype detlated to /Ucm 4 Hose clamp Flexible tubing PVC connector o A L I Figure 4.3 Tubing union to PVC connector 4.1 Verification Model Development and Results A verification model is designed to establish a basis for further simulations. The verification model is a simply supported tubular beam with a distributed load on top. Figure 4.4 shows the conceptual beam model and distributed loading. The hose clamps are modeled by fixing the x and y displacements of all the nodes along the circumference of each end. In order to avoid rigid body motion in the analysis, the tube is constrained in the z direction. To avoid artificial boundary stress concentrations at the ends, the z displacements of only 2 nodes on the neutral axis of bending are fixed. Figure 4.5 shows the applied boundary conditions and the coordinate system used. Figure 4.4 Conceptual boundary conditions and loading for the verification model z Dz= (one end of tube only) Nodes from outer ring falling on neutral axis of bending Figure 4.5 Detail of boundary conditions for verification model 4.1.1 Applied Loading Conditions The internal pressure is broken up into pressure acting in the radial direction and pressure acting longitudinally. The radial pressure is applied normal to the interior tube surface. The longitudinal pressure is converted to nodal forces, Fi using steps shown in Figure 4.6. Figures 4.6A and 4.6B show the internal pressure, Pi, applied over the area of the circular cap, Acap. Assuming the cap is rigid, the applied pressure translates into an I f I f 1 I I I f I equivalent total axial force, Feq, acting on the cap as shown in Equation 4.2 and Figure 4.6C. This total force is then distributed equally and applied directly to the number of nodes, n, on the outer circular edge at the end of the tube as shown in Equation 4.3 and Figure 4.6D. Feq = P, Acap n Feq = F, 1i=1 F, Feq n (4.1) (4.2) (4.3) Rigid Cap Acap =7r2 Pi ntHF Fabric Tube i=1 (D) Figure 4.6 Steps in representing internal pressure as pretension forces 4.1.2 Water Current Drag Water flowing over the outer surface of the tube creates a drag force. Fluid assumptions and equations are derived in Chapter 2. A programming doloop was Rigid Cap (A) (C) initially implemented into a script file to approximate a parabolic pressure flow distribution by applying individual forces to nodes. Forces are determined from the product of the local pressure and the projected area around the node. Figure 4.7 shows a representative projected area for one node. Applying the pressure distribution to the tensiononly membrane tube in this manner yielded numeric instability. This numeric instability and a need for only global effects of the drag leads to the assumption that local pressure effects of the flow on the tube can be neglected. This simplifies the model by approximating the total drag as an equally distributed load. Figure 4.8 shows an example of the total drag force divided equally and applied to all the nodes along the length of a typical tube. All nodal drag forces are applied in the same global coordinate direction. Xi1 Xi Xi+1 Figure 4.7 Representative projected area around one surface node Equal forces_ _ applied at each node around  circumference Figure 4.8 Representative application of forces to nodes around circumference 4.1.3 Element Selection and Properties The geometry is created in ANSYS by extruding a circle along a linear path. This area is then meshed with rectangular shell elements. The material to be modeled is a thin fabric with orthotropic material properties. Since fabrics have tensile stiffness, but very little compressive stiffness, a thin shell element with optional compressive stiffness is the best choice. The Shell41 element with Keyopt(1) = 2 meets these requirements. It was determined that the zero compressive stiffness caused a numerical instability in the solution of a pressurized beam in drag which resulted in spikes in the deformed shape. Initially, a small elastic foundation stiffness (EFS) was added to the Shell41 elements to attempt to fix the instability. Successive ANSYS simulations varying the EFS showed a large change in mean deflection, shown in Figure 4.9. This large deflection makes choosing a small EFS value unacceptable and eliminates the use of the EFS in all models. In another attempt to remedy the instability, equivalentgeometry Shell63 elements with small elastic modulus were merged to the Shell41 elements to add a small compressive stiffness to the elements. Successive ANSYS simulations varying the added stiffness of the shell63 elements showed an acceptable change in deflection and stress as shown in Figures 4.10 and 4.11. A value of 10,000 Pa is chosen for the added stiffness. Merging the shell63 elements is a preferable choice to fix the instability because the added stiffness has a less interfering effect on the desired results. Implementing this choice eliminates the spikes and smoothes out the deformed shape. The isotropic material properties needed to solve for stress and deflection are elastic modulus and Poisson's ratio. Since the material is orthotropic, hoop elastic modulus, longitudinal elastic modulus, shear modulus, and Poisson's ratio are required. Values for elastic modulus in the hoop and longitudinal directions are found experimentally in Chapter 3. Since Poisson's ratio is not experimentally determined, a value is chosen based on finite element model results. Simulations showed that varying Poisson's ratio had negligible effect on longitudinal stress and deflection. Figure 4.12 shows maximum deflection plotted for various Poisson's ratios determined in an ANSYS model of a cantilevered tube loaded with an end force. Figure 4.13 shows a similar plot for maximum longitudinal stress using the same model. The small variation in stress and deflection justifies assuming a reasonable value of 0.4 for Poisson's ratio. 0.00000E+00 2.00000E03 4.00000E03 6.00000E03 o 8.00000E03 1.00000E02 1.20000E02 1.40000E02 0 20000 40000 60000 80000 100000 120000 1400 EFS Value (Pa) Figure 4.9 Verification model parameter study: EFS effect on beam deflection 0.00E+00 1.00E03 2.00E03 3.00E03 4.00E03 5.00E03 6.00E03 7.00E03 8.00E03 Figure 4.10 0 20000 40000 60000 80000 100000 120000 140000 Added Compressive Stiffness to Shell41 Elements (Pa) Verification model parameter study: Added stiffness effect on beam deflection 00 7.00E+06 6.00E+06 *** * a 5.00E+06 S4.00E+06 0 .00E+06 o J 1.00E+06 0.00E+00 * Maximum Stress * Minimum Stress 0 20000 40000 60000 80000 100000 120000 140000 Added Compressive Stiffness to Shell41 Elements Figure 4.11 Verification model parameter study: Added stiffness effect on longitudinal beam stress 0.025 0.0225 0.02 0.0175 0.015 0.0125 0.01 0.0075 0.005 0.0025 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Poisson's Ratio, vxy (v_hooplongitudinal) Figure 4.12 Verification model parameter study: Poisson's ratio effect on beam deflection c LU ,LL 4 I C. o. t? E x E * * 48 6.00E+06 5.00E+06 * S4.00E+06 S* Maximum Stress 3.00E+06 m Minimum Stress 2.00E+06 S1.00E+06 0.00E+00 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Poisson's Ratio, vxy (v_hooplongitudinal) Figure 4.13 Verification model parameter study: Poisson's ratio effect on longitudinal beam stress 4.1.4 Verification Model Results To verify the model, simulation results are compared to the stress and deflection theoretical solutions derived in Chapter 2. Since flow velocity is proportional to the drag force acting on the beam, flow velocity is used to compare models. The simulations are conducted until the beam reaches failure. A study is done for 5m and 10m tubes. Results from the 5m long tube are shown in Figure 4.14 which correspond to a constant average deviation of 3.22% between theoretical and simulation. Figure 4.15 shows similar results for a 10m tube with a constant average deviation between theoretical and simulation to be 0.567%. These results show that the deflection results between theoretical and simulation stay below an acceptable 5% deviation. Results also show that the deviation decreases as tube length increases. A length study was also performed to see how maximum deflection changed with length. Simulations are performed up to an approximately 10% deflection:length ratio. Figure 4.16 shows a parabolic trend in the simulation and theoretical results. Figure 4.17 plots the deviation between simulation and theoretical results, showing deviation decreasing exponentially as length increases. Deviation drops below 10% above a tube length of approximately 3.5m. The large deviation occurring in tubes shorter than 3.5m is believed to be due to the SaintVenant's principle (Hibbeler, 1997), applicable at the fixed end boundary condition. In this case, the simulated cap restricts radial expansion in the fabric tube, which affects the maximum deflection. A stress simulation is also performed to verify the model behavior. Figure 4.18 shows longitudinal stress results for the verification model. The maximum stress increases, while the minimum stress decreases as expected in bending. A rigid body motion error is received when the minimum longitudinal stress decreases to approximately zero, verifying the tensiononly behavior of the model. This defines failure by wrinkling in this model. 0.00E+00 5.00E02 1.00E01 E 1.50E01 C .2 2.00E01 4 2.50E01 a S3.00E01 S3.50E01 4.00E01 4.50E01 5.00E01 Flowing Water Speed (mph) Figure 4.14 Deflection results for the verification model with length of 5m * Simulation * Theoretical 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Flowing Water Speed (mph) Figure 4.15 Deflection results for the verification model with length of 10m length S I * Simulation * Theoretical S 0.00E+00 2.00E01 4.00E01 E  6.00E01 C a 0 o 8.00E01 9 1.00E+00 > 2 1.20E+00 1.40E+00 1.60E+00 1.80E+00 0.00E+00 1.00E01 2.00E01 3.00E01 4.00E01 5.00E01 6.00E01 7.00E01 8.00E01 9.00E01 1.00E+00 0 1 2 3 4 5 6 Tube Length (m) II 7 8 9 Figure 4.16 Deflection results for verification model with 0.5 m/s flow velocity 140.00% 120.00% = 100.00% 0 G 80.00% c S60.00% O 40.00% 20.00% 0.00% 0 1 2 3 4 5 6 Tube Length (m) 7 8 9 Figure 4.17 Deviation in deflection results for verification model with 0.5 m/s flow velocity * Simulation * Theoretical A A A A A A A A AAA A A.. . 1.40E+07 1.20E+07 1.00E+07 0 S8.00E+06 n Maximum Stress S6.00E+06  ~m Minimum Stress a 4.00E+06 3 2.00E+06 0.OOE+00 2.00E+06 0 2 4 6 8 10 12 14 Tube Length (m) Figure 4.18 Longitudinal bending stress results for verification model with 0.5 m/s flow velocity 4.1.5 Verification Model Conclusion The results from the verification model show the merged shell41 and shell63 elements acceptably model the tensiononly tubing material. The SaintVenant's principle effect is specific to the boundary conditions and geometry of the verification model and is not applied elsewhere. The drag loading and boundary condition applications are acceptable and accurately model the actual conditions. 4.2 Cage Simulation Process The actual tubes, when assembled as a cage, have rigid end caps with hose clamps constraining each end. Based on the geometry of the cage, the tube can flex and extend longitudinally. Internally, the tube contains pressurized air to maintain inflation. The external loading on the tube is a distributed drag force caused by a crossflowing water current. 4.2.1 Flow Design Models With infinite possible current flowdirections for an underwater cage, the number of modeled choices is limited to cages subject to topdown flow and sideflow. Figure 4.19 defines the global coordinate system with sideflow in the positive xdirection and topflow in the negative ydirection. Modeling the entire cage was necessary because unlike the single beam verification model, the boundary conditions that exist at the end of each tube in the full cage structure are complex and interactive among the cage members. \ , / \ TopFlow SideFlow Figure 4.19 Two cage simulation flow orientations 4.2.2 Element Choice All tubes are meshed with merged tensiononly shell41 and shell63 elements in the same stacked element configuration as the verification model determined in Section 4.1.3. The tubular connectors in the cage are rigid relative to the tubing, so shell63 elements are used. LinklO elements model the tensiononly cables that tie together certain connectors throughout the cage. 4.2.3 Loading Conditions Internal pressure in the fabric tubes acts in the radial and longitudinal directions. Pressure acting in the radial direction is applied using the surface element pressure command (sfe) to only the shell41 elements making up the fabric tubes. Since the tubes are not actually capped, as in the verification model, a control volume analysis of the fluid in the rigid connector is used to determine the net effect of the longitudinal pressure in each tube. Figure 4.20 shows a control volume of fluid inside a typical 45'connector. Longitudinal pressure from each tube acts over the entrance area to the connector. The product of the pressure and area is treated as an equivalent force, Fcap, similarly to Equation 4.1 in the analysis of the verification model. Assuming the compressibility of the fluid is negligible, and the fluid is static, the fluid and the connector are approximated as one rigid body. Figure 4.21 shows Figure 4.20 translated into a free body diagram of a typical 45'connector modeled as a rigid body with net x and y component forces. Figure 4.22 shows a finite element model of the same typical 45connector with the net component cap forces divided and applied to all the nodes around the circumference of the center of the connector. The same analysis and nodal force application is used for each connector. r   fluid F f. _^ ~F cap F o ,,_ SFaapy F Fcapx Figure 4.20 Control volume analysis around a rigid 45connector containing pressurized internal fluid Fcap (1 cos 45) cap (sin 45) Figure 4.21 Free body diagram of rigid cap with algebraically summed pretension force components Forces act on nodes around entire circumference Figure 4.22 Force components applied to finite element model of rigid connector 4.3 TopDown Flow Cage Model In the topdown flow model, the center tube is anchored at the top of the cage with flow in the opposite direction. Small frictional drag occurring along the center tube is neglected because it lies in the direction of flow. To save on computing time, the center tube is eliminated from the model and replaced with a single node at the top and bottom of the cage. Although not immediately obvious, symmetry could be not be used to simplify the topdown flow cage model analysis. Dividing the cage along the x, y, or both the x and y axes does not work because the state of bending makes the zdirection boundary conditions along the axis of symmetry unknown. The zdirection boundary conditions also make it impossible to use an axis of symmetry along an axis 45 degrees from the x or y axis. The cable element controlling the zdirection boundary condition lies along this axis of symmetry and can not be split. 4.3.1 Applied Loading Conditions Since external fluid flows over the entire cage, the total fluid drag force on the cage is calculated based on its total crosssectional area perpendicular to the direction of flow. The total force is then divided equally and applied to all nodes of the eight caps and eight tubes. Referring to the coordinate system defined in Figure 4.23, all drag forces act in the negative zdirection for the topdown flow model. 4.3.2 Boundary Conditions Boundary conditions are applied to simulate the conditions acting on the cage as simply as possible while minimizing solution time. The two locations labeled A in Figure 4.23 describe two single nodes at the top and bottom of the cage. These points are placed to obtain a cage diametertoheight ratio of 8:5. Both points are constrained to zero deflection in the x,y, and z directions. Cables on the underside of the cage are slack and do not affect the system, but were included for aesthetics along with the fixed node at the bottom of the cage. Locations labeled B and C in Figure 4.23 describe all nodes around the circumference of a tube at the center of its length. Nodes at location B are constrained to zero deflection in the ydirection. Nodes at location C are constrained to zero deflection in the xdirection. Constraining locations B and C prevents rotational rigid body motion while allowing the individual tubes and cage to expand symmetrically. Z x A C B B ( A C Figure 4.23 Topflow boundary condition locations 4.4 SideFlow Cage Model The sideflow cage starts with the same octagon model as the topdown flow model but includes a model of the center tube, all the cables, and a different anchoring system. Zdirection pretension forces from internal pressure are additionally included at the ends of the center tube in this model. The rigid connectors are pressurized to the same internal pressure as the connecting tubing because successive analysis showed an improvement in the smoothness of the deformation solution around the connections. Symmetry could also not be used in the sideflow cage model analysis. The x axis would be the best choice for axis of symmetry, but the center tube, lying in the z direction, would have to be split longitudinally. Also, the xdirection boundary conditions are unknown on the tubes along the axis of symmetry because the drag flow is causing bending at those points. Figure 4.24 shows the components of the center tube as two flexible membranes and three straight, rigid connectors. The total center tube length is determined by the 8:5 cage diametertoheight ratio. Figure 4.25 shows linkl0 elements connecting nodes in the top, bottom, and center connectors of the center tube to nodes in the top and bottom of the 45degree connectors in the octagon. Cable elements are merged to single nodes on the rigid connector symmetrically as shown. Four tensiononly cables anchor the cage from a single external point. Figure 4.26 shows an example of the linkl0 mooring cable elements merged to one top and bottom node of a 45degree connector. Figure 4.27 shows the anchor point location in the global coordinate system, where L is the length of an individual tube in the octagon. Setting the anchor point x location as a function of tube length maintains an anchor distance from the cage as the cage size is increased. This location is chosen to be reasonably close, while minimizing the ytension component in the anchor cables, which affects the deformation of the cage. 4.4.1 Applied Loading Conditions Drag flow for the sideflow case is approximated as a series of forces applied to specific cage members. Locations A, B, and C in Figure 4.28 describe flexible membrane tubes to which drag is applied. The magnitude of drag on locations A, B, and C is individually based on the respective crosssectional area. Small frictional drag is neglected along location D in Figure 4.28 because the members are parallel to the flow direction. The total drag force for each individual member is divided equally and applied to the nodes corresponding to the respective location. Referring to the coordinate system defined in Figure 4.23, all drag forces act in the positive xdirection for the sideflow model. 4.4.2 Boundary Conditions The location labeled A in Figure 4.29 describes the singlenode anchor point fixed to zero deflection in the x, y and z directions. To allow symmetric expansion and prevent rotational rigid body motion, constraints are placed in two more locations in the cage. Locations labeled B and C in Figure 4.29 describe all nodes around the circumference of a tube at the center of its length. The nodes at location B are constrained to zero deflection in the ydirection, while nodes at location C, further illustrated in Figure 4.30, are fixed to zero deflection in the zdirection. Flexible Membrane Tubing Rigid Connectors Figure 4.24 Cage center tube components Figure 4.25 Expanded view of representative internal cable connections Mooring cables Flexible membrane Rigid tubing connector Figure 4.26 Expanded View of representative mooring cable connections E 3.4L Figure 4.27 Dimensions of anchor location Figure 4.28 Sideflow applied loading locations 'NA Figure 4.29 Sideflow boundary condition locations Flexible membrane tubing Rigid / connector Displacement boundary conditions (nodes of entire circumference) S I I e e Figure 4.30 Expanded view of center tube boundary conditions CHAPTER 5 SIMULATION RESULTS The design criteria and results from the ANSYS cage simulations are presented in Chapter 5. The goal of the study is to determine the maximum cage size possible before failure. Twelve commercially available flexible tubes are tested under top and side flow conditions. The material yielding the largest cage is chosen and subject to an additional velocity simulation. 5.1 Simulation Strategy ANSYS script files are used to parameterize the variables involved in the construction and analysis of each cage. Independent variables were kept constant for all analyses run in this chapter. Independent variables include: * External fluid temperature, viscosity, density * Flexible membrane orthotropic elastic modulii * Rigid connector elastic modulus * Flexible membrane and rigid connector Poisson's ratios * Flexible membrane shear modulus * Cable constant, EA * Rigid Connector geometry Cable length is defined by the cage geometry. Flexible membrane material properties are explained in Chapter 3, while reasonable values are chosen for external fluid, connector, and cable properties. Dependent variables affecting the cage geometry and loading are: * Flexible membrane internal pressure * Flexible membrane tube length * Flexible membrane tube diameter * Flexible membrane thickness * External fluid velocity Internal pressure causes the pretension forces in the flexible membrane tubes. The pretension forces are also based on the area of the tube opening and so are functions of the tube diameter. Drag on a given member is a direct function of tube length and tube diameter. Drag is also a function of the flowing fluid velocity squared. Three of the five independent variables are combined using data from commercially available materials. Table 5.1 presents VinylFlow data including tube diameter, thickness, and working pressure for a given series number. Simulation results make convenient the grouping of VFSeries' by pressure to show approximately constant pressure trends. The three pressure range groups are indicated in Table 5.1. The following simulations vary Vinyl Flow series number, tube length, and external fluid velocity to determine effects on cage performance. Table 5.1 Manufacturer data for VinylFlow commercial drainage tubing Nominal Hose Approximate Working Coil Approx. Pressure Series Size ID Wall Thickness Pressure Length Weight Group No. (in.) (in.) (in.) (psi) (ft.) (lbs.lcoil) (psi) VF 150 111, 1.673 .0669 80 300 48 VF 200 2 2.165 .0669 80 300 69 7080 VF 250 2'/ 2.598 .0787 80 300 87 VF 300 3 3.130 .0787 70 300 117 VF 400 4 4.134 .0827 70 300 156 VF 500 5 5.039 .0866 40 300 204 VF 600 6 6.181 .0866 50 300 258 4050 VF 800 8 8.169 .1063 45 100/300 130/390 VF1000 10 10.118 .1181 35 100 181 VF1200 12 12.126 .1181 30 100 195 3035 VF1400 14 14.134 .1181 30 100 262 VF1600 16 16.142 .1181 30 100 310 5.2 Failure Criteria The failure criteria for both flow cases are the points when wrinkling occurs anywhere in the fabric tubing and when stress exceeds the ultimate strength of the material. Wrinkling is defined here as a compressive stress. Cage wrinkle diameter is defined as the maximum cage diameter before any flexible tube wrinkles due to a compressive stress less than or equal to zero. It is important to consider both the longitudinal and hoop stresses in the analysis. Wrinkling is expected to occur in the longitudinal direction because the combined loading of internal pressure and bending affect the longitudinal stress. Tubes are not expected to wrinkle in the hoop direction because the only loading affecting the hoop stress, internal pressure, creates a tensile hoop stress. The maximum stress will occur in the longitudinal direction because the combined loading of internal pressure and bending affect the longitudinal stress. The hoop stress is affected only by the internal pressure loading. Using the manufacturer's recommended working pressure ensures the hoop stress will not exceed the ultimate strength of the material. Figure 5.2 illustrates a representative case of compressive failure behavior for VF 800 tested at a water speed of 1 knot in topflow. Maximum and minimum longitudinal stresses are shown for various cage sizes. As cage size is increased, maximum stress increases, while minimum stress decreases. Pretension in the tubes keeps both stresses positive, meaning they are always in tension. When the cage size increases to the point that the minimum stress drops below zero, the model fails because the tensiononly elements can not support compressive stress. This is then interpreted by the program as an unconstrained model and results in a rigid body motion error message. Note that the compressive stress could go very slightly negative without failure because of the small compressive strength provided by the merged shell63 elements comprising the tube. Subsequent topflow simulation failures are judged by the rigid body motion error message. The sideflow simulations do not show a smooth trend in the longitudinal stress down to zero as in Figure 5.2, but show a sudden dropoff resulting in a rigid body motion error message. This behavior can be described as a buckling of the structural tubes. The stress behavior is due to changes in the higher order bending occurring in the tubes as cage size changes. Section 5.4 illustrates and further explains this behavior. The sideflow cage failure is therefore also determined by the rigid body motion error message in the simulation. 1.20E+07 Maximum Stress 1.00E+07 Minimum Stress S8.00E+06 S6.00E+06 I 4.00E+06 3 2.00E+06 0.00E+00 0 5 10 15 20 25 30 Cage Diameter (m) Figure 5.2 Simulation results: Bending stresses in a VF800 topflow cage 5.3 Stress Results Figure 5.3 illustrates the maximum longitudinal stresses occurring in each simulation staying below the experimentally determined ultimate strength of the material, 2.6e7 Pa. This ensures the material strength is not exceeded in any simulation. 3.00E+07 2.50E+07 STopFlow Maximum Stress S2.00E+07 m SideFlow g Maximum Stress c 1.50E+07 SUltimate Material o Strength o 1.OOE+07 0 J 5.00E+06 O.OOE+00 VinylFlow Series Number Figure 5.3 Maximum stresses in each simulation and ultimate material strength 5.4 Geometry Deformation Results Figures 5.4 through 5.6 show various views of a typical deformed geometry of a cage subject to topflow at the critical wrinkle point. All eight tubes bend in the negative zdirection with maximum zdeflection occurring at half the tube length. Bending plus the Poisson effect cause the cage diameter to decrease symmetrically throughout the cage. Figures 5.7 through 5.8 show two views of a typical deformed geometry of a cage subject to sideflow at the critical wrinkle point. Note that the net cage xdiameter increases, while the net cage ydiameter decreases. Internal reaction forces occurring through the cable connections cause internal bending moments around the corner connectors of the octagon. Deformation effects of these internal bending moments are visible in the deformed curvature around the connectors and account for multiplemode bending visible in several of the beams. Figure 5.9 illustrates the rotational effect on a representative rigid connector. The different rotational effects occurring throughout the cage contribute to the multiple bending modes and subsequent buckling stress behavior leading to failure. Figure 5.4 TopFlow deformed geometry at the wrinkle point: View 1 Undeformed Deformed Figure 5.5 TopFlow deformed geometry at the wrinkle point: View 2 Undeformed Deformed Figure 5.6 TopFlow deformed geometry at the wrinkle point: View 3 Undeformed Deformed Figure 5.7 SideFlow deformed geometry at the wrinkle point: View 1 ;; Deformed Figure 5.8 SideFlow deformed geometry at the wrinkle point: View 2 Undeformed Deformed Note rotational effect Figure 5.9 Representative rotational effect occurring in sideflow simulation 5.5 Simulation Deflection Results Deflection results are useful in predicting cage behavior. Presented here are qualitative results for a specific material case and maximum deflection results for each material case. Numeric sign associated with deflection is related to the previously defined cage coordinate system. For diameter changes, a negative change indicates a decrease in diameter, while a positive change indicates an increase. 5.5.1 Qualitative Deflection Results Figure 5.10 shows deflection data of the net cage xdiameter deflection for a representative case of the VF800 tested in topflow at a water speed of one knot at various cage diameters. Deflection results are identical for the cage ydiameter based on the symmetry of the geometry and loading. Cage diameter change has a qualitatively linear trend for the topflow cage loading. Figure 5.11 shows data of the maximum z deflection for the same representative case. The data represents the maximum bending deflection occurring in each tube and has an expected qualitatively decreasing parabolic trend as cage size is increased. Figure 5.12 shows cage ydiameter deflection data for the VF500 representative case, but in sideflow. As seen in the geometry deformation of the side flow cage, the cage ydiameter decreases drastically. No clear trend is visible in the smaller xdiameter deflection. This is because the multiple bending modes largely affect the xdiameter cross members and occur differently in each cage size. Xdiameter cage deflections must be considered on a case by case basis for sideflow. S0.5 0 C ,m . S21 .) 2.5 Ca 0 5 10 15 20 25 30 Cage Diameter (m) Figure 5.10 Cage xdiameter net deflection for VF800 tested at flow speed of 1 knot, topflow 0 1 2  3  4 5 0 5 10 15 20 25 30 Cage Diameter (m) Figure 5.11 Maximum zdeflection for VF800 cage tested at flow speed of 1 knot, top flow 0.0 S5 10 15 0.5 E 1.0  > 2.0 01 Z 2.5 3.0 Cage Diameter (m) Figure 5.12 Cage ydiameter net deflection for VF500 tested at flow speed of 1 knot, sideflow 5.5.2 Maximum Deflection Results Two tables are shown to present deflection behavior with respect to the original cage size. Data are summarized in table format because in each case, the maximum deflections occur at different cage sizes which correspond to different drag loadings. With different drag loadings, each case can not be directly compared to determine a graphical trend. Tube diameter and working pressure are included in the tables for comparison between cases. Table 5.2 shows the maximum diameter change and maximum zdeflection with respect to original cage diameter for the topflow case. Diameter deflections are between 410%, while zdeflections are between 1018%. Table 5.3 shows the maximum change in the x and y diameters with respect to original cage diameter for the sideflow case. Xdiameter deflections are small and between 24%, while ydeflections are much larger, at 1422%. Table 5.2 Maximum cage deflections for the topflow orientation Tube diameter Working Maximum cage Ratio of change in cage x Ratio of zdeflection to Series number diameter to original cage (in.) pressure (psi) diameter (m) diameter original cage diameter diameter VF 150 1.673 80 7.2 4.93% 10.78% VF 200 2.165 80 9.6 6.49% 14.98% VF 250 2.598 80 10.8 6.49% 13.53% VF 300 3.130 70 13.2 7.02% 15.83% VF 400 4.134 70 16.8 8.71% 18.78% VF 500 5.039 40 14.4 5.80% 10.57% VF 600 6.181 50 20.4 8.83% 17.37% VF 800 8.169 45 26.4 8.75% 17.47% VF 1000 10.118 35 27.6 7.78% 13.82% VF 1200 12.126 30 30 7.60% 13.59% VF 1400 14.134 30 34.8 8.85% 15.76% VF 1600 16.142 30 39.6 10.10% 17.92% Table 5.3 Maximum ca e deflections for the sideflow orientation Sers Te d r Working Ratio of net x diameter Ratio of net y diameter Series Te dia r pressure aimmae change to original cage change to original cage number (in.) diameter (m) (psi) diameter diameter VF 150 1.673 80 14.4 2.498% 17.346% VF 200 2.165 80 18 2.600% 18.633% VF 250 2.598 80 20.4 2.598% 18.216% VF 300 3.130 70 21.6 2.681% 18.361% VF 400 4.134 70 27.6 3.138% 21.377% VF 500 5.039 40 18 2.569% 14.222% VF 600 6.181 50 21.6 2.731% 17.667% VF 800 8.169 45 25.2 3.150% 18.619% VF 1000 10.118 35 24 2.672% 15.442% VF 1200 12.126 30 25.2 2.779% 14.976% VF 1400 14.134 30 28.8 3.262% 17.278% VF 1600 16.142 30 32.4 3.760% 19.605% 5.6 Cage Size Results Maximum possible cage size is determined by the point at which the cage wrinkles. Figures 5.13 and 5.14 show the maximum cage diameter results by tube diameter for the topflow and sideflow cases respectively. Slight pressure effects are noted between the pressure groups of the topflow case, but are neglected to form one linear increasing trend. The trend generally shows that, independent of pressure, as diameter increases, the maximum possible cage diameter increases for topflow conditions. The sideflow case, however, has significant pressure effects visible in Figure 5.14. Each pressure group is plotted with an increasing linear trend, but a distinct slope and yintercept. Within each pressure group, as diameter increases, the maximum possible cage diameter increases. Also evident is that as pressure increases, the slope of the constant pressure trends becomes steeper. This indicates that for a given diameter, operating at higher pressures increases the maximum possible cage size. Since the flow situations are idealized and true water conditions vary no matter how the cage is anchored, choosing the material for the largest cage is based on the results of both flow cases. Figure 5.15 presents the data from figures 5.13 and 5.14 as a summary of the topflow and sideflow maximum cage diameters possible for all the material models tested. The lesser cage maximum diameter of the two flow cases is the limiting factor for each material. Refer to Table 5.1 for model specific manufacturer data including tube diameter, thickness, and working pressure for each case. 45 40 35 30 slight pressure effect E E E 15 7080 psi m 4050 psi 2 10 3035 psi 5 0 0 2 4 6 8 10 12 14 16 18 Tube Diameter (in) Figure 5.13 Topflow maximum cage diameters for each tubing material grouped by internal pressure 25 20 S15 7080 psi E m 4050 psi 0 10 E 3035 psi 5  0 i i 15  ^ 70 8 p i  0 0 2 4 6 8 10 12 14 16 18 Tube Diameter (in) Figure 5.14 Sideflow maximum cage diameters for each tubing material grouped by internal pressure E 30 0 25 a 20 c 20 o E I 15 E I 10 VinylFlow Series Number Figure 5.15 Summary of maximum possible cage size for each material in two flow cases subject to one knot flow velocity VF Series number corresponds to diameter, working pressure, and thickness in Table 5.1 5.7 Velocity Results VF1600 is chosen because it has the largest limiting cage size of the materials tested. The maximum possible VF1600 cage diameters for topflow and sideflow are averaged and then divided by two to obtain a cage diameter able to withstand higher than one knot flow velocity. This cage diameter, 19.2 m, is used for a velocity test on the cage. The cage withstands the same maximum flow speed of 1.6 knots in both top and side flow cases before wrinkling occurs. In topflow, cage diameter decreases 10.183%, while in sideflow, cage xdiameter increases 4.027%, and cage ydiameter decreases 17.792%. These deflections are reasonable for an inflated flexible membrane cage of this size and correspond to the expected deformation results shown in Section 5.4. CHAPTER 6 CONCLUSION The experimental work performed gave preliminary values of the orthotropic material properties. More advanced test equipment is necessary to model orthotropic fiber tubing to find the directional elastic moduli and Poisson's ratio. Largedeformation strain gages would improve the accuracy in recording the strain used in determining the elastic modulus. A two direction tensile tester would be necessary to experimentally determine Poisson's ratio for the inflatedtube loading case. The values determined were sufficient for the preliminary design and modeling of the orthotropic fiber material. The analysis presented here focused on the inflatable behavior of the structure. The simulation included the inflated tube members, rigid PVC connectors, and cable connectors. It is important to note that the fish netting is excluded from the simulation, but is necessary for a complete analysis. The simulation effectively models the nonlinear behavior of the orthotropic, fabric material by translating the inflatable theory into a finite element model. Preliminary simulations successfully show expected inflatable behavior in fullcage top and side flow models. Preliminary findings warrant further research that models the external netting. Net selection has several biological issues to be addressed such as net biofouling and biologically safe mesh size. Net analysis will potentially require dynamic analysis to model the hydrodynamic interaction between the structural element and the fluid (Tsukrov et al., 2000). Since the cage divergence volume is an important issue, future work should include at least an approximation of the nets. Further simulation work should also include optimizing the inflatable material. The cage material's size and stiffness is expected to offset the increased loading of the nets, making the inflatable aquaculture cage practical and economical. The side and top flow models behave differently, particularly with respect to divergence volume. The internal pressure and tube diameter play the largest role in deciding the maximum possible cage diameter prior to wrinkling. Instead of including a safety factor in the final results, the failure criterion was set to the point at which wrinkling occurred in any tube. Inflatable tubes do not actually fail at the point of wrinkling, but can continue past this point (Main et al., 1994). Future simulations may be able to push the failure criteria by modeling the stiffness behavior of the wrinkled material. This work successfully models one new application of inflatable structures. Nonlinear finite element modeling is shown to be an effective tool in analyzing the behavior of these structures. Inflatables in many applications have a great potential for improving cost, weight, and convenience. This effort adds one step to the progression of inflatable structures, and may encourage their further development and acceptance. APPENDIX A ANSYS SCRIPT CODE The following contains two ANSYS script files used in the design and analysis of the inflated cage. A top flow orientation script is first presented, followed by a side flow orientation script. Many variables used in the analysis are parameterized in the beginning of each script to account for changing conditions. TopFlow /COM, this code has the link10 elements and uses knode command to /COM, connect them to the caps /PREP7 /COM, change colors to white background /RGB,INDEX,100,100,100, 0 /RGB,INDEX, 80, 80, 80,13 /RGB,INDEX, 60, 60, 60,14 /RGB,INDEX, 0, 0, 0,15 /COM, ******************************************* /COM, define geometric tube parameters * /COM, ******************************************* /COM, inner diameter, units = inches ID = 8.169 /COM, material thickness, units = inches thicknessinch = 0.1063 /COM, units = m length = 8 /COM, end cap radius of curvature determines cap length caprad=ID/12 /COM, ************************************** /COM, define element parameters * /COM, ************************************** /COM, number of nodes on quarter circumference n=7 /COM, number of nodes along length m=51 /COM, element depth edepth = length/(m1) /COM, Elastic Modulus in the hoop direction (Pa) Ehoop = 5.157e7 /COM, Elastic Modulus in the longitudinal direction (Pa) Elong = 3.211e8 /COM, Major Poisson's Ratio (xy) poisson = 0.4 /COM, Shear Modulus, Gxy, G HL, units? shearmod = 1.91e7 /COM, Elastic Modulus for Artificial stiffness (small) units=Pa smallmod = 10000 /COM, Elastic Modulus for Cap (large) units=Pa capmod=9e9 /COM,******************************************************* /COM,* define external fluid (water (20 C) parameters * /COM,******************************************************* /COM, appendix A3, external fluid absolute viscosity, kg/ms visc = (1e3) /COM, external fluid density, kg/m^3 dens = (998) /COM, *************************************** /COM, define loading * /COM, *************************************** /COM, internal pressure, psi prespsi = 45 /COM, flowing fluid velocity m/s vel= (0.5144444) /COM, ********************************************** /COM, calculate dependant variables, units * /COM, ********************************************** /COM, angles in degrees *afundeg /COM, cap arc length (m) caplength=0.3 /COM, convert pressure units pressure = (6894.757*prespsi) /COM, convert diameter to radius in meters radius = (ID/2)*0.0254 diam=radius*2 /COM, convert to units = m thickness = thicknessinch*0.0254 /COM, crosssectional area ONE tube, units = m^2 tubecsarea = 2*radius*length /COM, crosssectional area ONE cap, units = m^2 capcsarea=((3.1417*caprad*diam)/4) /COM, crosssectional area ENTIRE cage, units = m^2 totcsarea=((8*tubecsarea)+(8*capcsarea)) /COM, total number of nodes in the entire octagon (cap+tube) ntot=32*(n**2+m*nm3*n+2) /COM, Reynolds number, dimensionless Re = (dens*vel*2*radius)/visc /COM, coefficient of drag, found from Figure 9.13 based on Re cd= 1 /COM, total drag force, units = N drag = 0.5*cd*totcsarea*dens*(vel**2) /COM, force per node equivalent to the internal pressure... /COM, ...acting on the cap, units = N capForce = (pressure*3.1417*radius*radius) /COM,******************************************************* /COM,* Define Elements * /COM,******************************************************* /COM, /COM, 1 Membrane Shell Element /COM, 2 Added Stiffness Membrane Shell Element /COM, 3 Cap Membrane Shell Element /COM, 4 Cable Element ET,1,SHELL41,2 ET,2,SHELL63 ET,3,SHELL63 ET,4,LINK10,,1,0 /COM, Real Constants R,1,thickness R,2,thickness R,3,thickness*10 R,4,0.5.0.3 /COM, Young's Modulus in the hoop direction MP,EX,1, Ehoop MP,EX,2, smallmod MP,EX,3, capmod MP,EX,4, 9e7 /COM, Young's Modulus in the longitudinal direction MP,EY,1, Elong /COM, Major Poisson's Ratio (xy) MP,PRXY, 1,poisson MP,PRXY,2,poisson MP,PRXY,3,poisson /COM, Shear Modulus, Gxy, G HL MP,GXY,1, shearmod /COM,******************************************************* /COM,* CAGE GEOMETRY * /COM,******************************************************* /COM, generate one tube and two 45 degree caps (upper right comer) k,l,length/2,length/2+length*sin(45),0 k,2,length/2+length*cos(45),length/2,0 circle,l,caprad,,,,8 circle,2,caprad,,,,8 1,4,12 circle,4,radius, 12 circle, 12,radius,4 ADRAG,18,19,20,21,,,17 ADRAG,18,19,20,21,,,2 ADRAG,22,23,24,25,,,9 /COM, mesh one tube (upper right comer) lsel,s,line,,17 lsel,a,line,,27 lsel,a,line,,28 lcl a line 30 lsel,a,line,,32 lesize,all,,,m1 TYPE,1 MAT,1 REAL,1 amesh, 1,4,1 /COM, mesh two caps (upper right comer) lsel,s,line,, 18,25,1 lesize,all,,,n1 lsel,s,line,,35 1scl a lile 36 sel a line 18 lsel,a,line,,40 lsel,a,line,,43 lsel,a,line,,44 lsel,a,line,,46 lsel,a,line,,48 lesize,all,,,n1 TYPE,3 MAT,3 REAL,3 amesh,5,12,1 /COM, reflect areas and mesh to create all four covers of octagon arsym,X,all arsym,y,all /COM, generate 2 vertical and 2 horizontal tube areas (sides) k,,length/2,length/2+length*sin(45)+caprad,0 k,,length/2,length/2+length*sin(45)+caprad,0 k,,length/2,length/2length*sin(45)caprad,0 k,,length/2,length/2length*sin(45)caprad,0 k,,length/2+length*cos(45)+caprad,length/2,0 k,,length/2+length*cos(45)+caprad,length/2,0 k,,length/2length*cos(45)caprad,length/2,0 k,,length/2length*cos(45)caprad,length/2,0 1,99,100 1,101,102 1,103,104 1,105,106 ADRAG,63,66,68,69,,,146 ADRAG,95,98,100,101,,,147 ADRAG,104,108,111,113,,,148 ADRAG,72,76,79,81,,,149 /COM, mesh four tubes (sides) lsel,s,line,,51 selaline, 5 lsel,a,line,,55 lsel,a,line,,58 lcl a liic S3 lsel,a,line,,85 lsel,a,line,,87 lsel,a,line,,90 lsel,a,line,,115 lsel,a,line,,117 lsel,a,line,,119 lsel,a,line,,122 lsel,a,line,,146 lsel,a,line,,147 lsel,a,line,,148 lsel,a,line,,149 lsel,a,line,,151 lsel,a,line,,152 lsel,a,line,,154 lsel,a,line,,156 lsel,a,line,,159 lsel,a,line,,160 lsel,a,line,,162 lsel,a,line,,164 lsel,a,line,,167 lsel,a,line,,168 lsel,a,line,,170 lsel,a,line,,172 lsel,a,line,,175 lsel,a,line,,176 lsel,a,line,,178 lsel,a,line,,180 lesize,all,,,m1 TYPE,1 MAT,1 REAL,1 amesh,49,64,1 /COM, regenerate tube areas (sides) and mesh with artificial stiffness lsel,all ADRAG,63,66,68,69,,,146 ADRAG,95,98,100,101,,,147 ADRAG,104,108,111,113,,,148 ADRAG,72,76,79,81,,,149 arsym,X,1,4,1,0,1,0 arsym,y,l,4,1,0,1,0 arsym,X, 13,16,1,0,1,0 arsym,y,13,16,1,0,1,0 TYPE,2 MAT,2 REAL,2 amesh,65,96,1 /COM, merge all nodes, elements, and keypoints nsel,all esel,all nummrg, node, le5 nummrg, kp nummrg, elem, le5 /COM, apply total drag force / total # nodes nsel,all f,all,fz,drag/ntot /COM, apply force boundary conditions esc1l i' pc 1 sfe,all,,pres,,pressure /COM, create cage cables k,200,0,0,((2.4*length)*(5/16)) k,201,0,0,((2.4*length)*(5/16)) TYPE,4 MAT,4 REAL,4 csys,1 nsel,s,loc,y,22.49999,22.50001 csys,0 f,all,fy,(capforce*(1sin(45)))/(4*(n1)) f,all,fx,(capforce*cos(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,203,all 1,203,200,1 lmesh,26 nsel,s,loc,y,22.49999,22.50001 nsel,r,loc,z,radius knode,204,all 1,204,201,1 lmesh,29 csys,1 nsel,s,loc,y,67.49999, 67.50001 csys,0 f,all,fx,(capforce*(1cos(45)))/(4*(n1)) f,all,fy,(capforce*sin(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,205,all 1,205,200,1 lmesh,31 nsel,s,loc,y,67.49999, 67.50001 nsel,r,loc,z,radius knode,206,all 1,206,201,1 lmesh,33 csys,1 nsel,s,loc,y,112.49999, 112.50001 csys,0 f,all,fx,(capforce*(1cos(45)))/(4*(n1)) f,all,fy,(capforce*sin(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,207,all 1,207,200,1 lmesh,70 nsel,s,loc,y,112.49999, 112.50001 nsel,r,loc,z,radius knode,208,all 1,208,201,1 lmesh,74 csys,1 nsel,s,loc,y, 157.49999,157.50001 csys,0 f,all,fy,(capforce*(1sin(45)))/(4*(n1)) f,all,fx,(capforce*cos(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,209,all 1,209,200,1 lmesh,77 nsel,s,loc,y, 157.49999,157.50001 nsel,r,loc,z,radius knode,210,all 1,210,201,1 lmesh,80 csys,1 nsel,s,loc,y,22.49999, 22.50001 csys,0 f,all,fy,(capforce*(1sin(45)))/(4*(n1)) f,all,fx,(capforce*cos(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,211,all 1,211,200,1 Imesh, 102 nsel,s,loc,y,22.49999,22.50001 nsel,r,loc,z,radius knode,212,all 1,212,201,1 Imesh, 106 csys,1 nsel,s,loc,y,67.49999,67.50001 csys,0 f,all,fx,(capforce*(1sin(45)))/(4*(n 1)) f,all,fy,(capforce*cos(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,213,all 1,213,200,1 Imesh, 109 nsel,s,loc,y,67.49999,67.50001 nsel,r,loc,z,radius knode,214,all 1,214,201,1 Imesh, 112 csys,1 nsel,s,loc,y,l 12.49999,112.50001 csys,0 f,all,fx,(capforce*(1sin(45)))/(4*(n1)) f,all,fy,(capforce*cos(45))/(4*(n1)) csys,1 nsel,r,loc,z,radius knode,215,all 1,215,200,1 Imesh,134 nsel,s,loc,y,l 12.49999,112.50001 nsel,r,loc,z,radius knode,216,all 1,216,201,1 Imesh, 138 csys,1 nsel,s,loc,y,157.49999,157.50001 csys,0 f,all,fy,(capforce*(1sin(45)))/(4*(n1)) f,all,fx,(capforce*cos(45))/(4*(n 1)) csys,1 nsel,r,loc,z,radius knode,217,all 1,217,200,1 Imesh, 141 nsel,s,loc,y,157.49999,157.50001 nsel,r,loc,z,radius knode,218,all 1,218,201,1 Imesh, 144 csys,0 87 nsel,s,loc,x,0 d,all,ux,0 nsel,s,loc,y,O d,all,uy,0 dk,200,ux,0 dk,200,uy,0 dk,200,uz,0 dk,201,ux,0 dk,201,uy,0 dk,201,uz,0 nsel,all esel,all nummrg, node nummrg, elem nummrg, kp fini /COM, ************************************ /COM, Solution * /COM, ************************************ /solve nsel,all esel,all autots,on deltim, 1 solve postall cys,0 nsel,s,loc,x,length/2,length/2 nsel,r,loc,y,0,length*5 esln,s,all escl I pi 1 plesol,s,x esel,all nsel,all plnsol,u,x esel,all nsel,all plnsol,u,z SideFlow /PREP7 /COM, ******************************************* /COM, define geometric tube parameters * /COM, ******************************************* /COM, change colors to white background /RGB,INDEX,100,100,100, 0 /RGB,INDEX, 80, 80, 80,13 /RGB,INDEX, 60, 60, 60,14 /RGB,INDEX, 0, 0, 0,15 /COM, inner diameter, units = inches ID = 16.142 /COM, material thickness, units = inches thicknessinch = .1181 /COM, units = m 