ANSYS Workbench Verification Manual

ANSYS Workbench Verification ManualANSYS, Inc. Release 18.1 Southpointe April 2017 2600 ANSYS Drive Canonsburg, PA 15317 ANSYS, Inc. and [email protected] ANSYS Europe, Ltd. are UL http://www.ansys.com registered ISO (T) 724-746-3304 9001: 2008 (F) 724-514-9494 companies. Copyright and Trademark Information © 2017 ANSYS, Inc. Unauthorized use, distribution or duplication is prohibited. ANSYS, ANSYS Workbench, AUTODYN, CFX, FLUENT and any and all ANSYS, Inc. brand, product, service and feature names, logos and slogans are registered trademarks or trademarks of ANSYS, Inc. or its subsidiaries located in the United States or other countries. ICEM CFD is a trademark used by ANSYS, Inc. under license. CFX is a trademark of Sony Corporation in Japan. All other brand, product, service and feature names or trademarks are the property of their respective owners. FLEMlm and FLEXnet are trademarks of Flexera Software LLC. 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Table of Contents Introduction ............................................................................................................................................... 1 Overview ............................................................................................................................................... 1 AIM Test Cases ........................................................................................................................................ 2 Index of Test Cases .................................................................................................................................. 2 I. DesignModeler Descriptions ................................................................................................................... 9 1. VMDM001: Extrude, Chamfer, and Blend Features ............................................................................... 11 2. VMDM002: Cylinder using Revolve, Sweep, Extrude, and Skin-Loft ....................................................... 13 3. VMDM003: Extrude, Revolve, Skin-Loft, and Sweep .............................................................................. 15 4. VMDM004: Extrude, Revolve, Skin-Loft, and Sweep .............................................................................. 17 II. SpaceClaim Descriptions ...................................................................................................................... 19 SCGEO1. VMSC_Geometry001: Alter Model Using Sketch, Pull, and Fill .................................................... 21 SCGEO2. VMSC_Geometry002: Alter Model Using Split Face, Split Body and Mirror ................................. 25 SCGEO3. VMSC_Geometry003: Alter Model using Linear, Circular, and Two-Dimensional Patterns ............ 31 SCGEO4. VMSC_Geometry004: Alter Model Using Revolve, Chamfer, and Offset Faces ............................. 37 SCGEO5. VMSC_Geometry005: Alter Model Using Blend and Project ...................................................... 43 SCGEO6. VMSC_Geometry006: Alter Model Using Sheet Metal Tools and Operations .............................. 47 SCPREP1. VMSC_Prepare001: Midsurface, Extend, Imprint, and Spot Weld ............................................... 55 SCPREP2. VMSC_Prepare002: Volume Extract and Enclosure ................................................................... 57 SCPREP3. VMSC_Prepare003: Beam Create, Extract, and Orient ............................................................... 63 SCPREP4. VMSC_Prepare004: Remove Rounds, Interference, Faces, and Short Edges ................................ 65 SCREP1. VMSC_Repair001: Stitch, Gap, and Missing Faces ....................................................................... 69 SCREP2. VMSC_Repair002: Merge Faces, Small Faces, Simplify, Straighten ................................................ 71 SCREP3. VMSC_Repair003: Split Edges, Extra Edges, and Duplicates ........................................................ 77 III. Mechanical Application Descriptions .................................................................................................. 81 1. VMMECH001: Statically Indeterminate Reaction Force Analysis ........................................................... 83 2. VMMECH002: Rectangular Plate with Circular Hole Subjected to Tensile Loading ................................. 87 3. VMMECH003: Modal Analysis of Annular Plate .................................................................................... 89 4. VMMECH004: Viscoplastic Analysis of a Body (Shear Deformation) ...................................................... 93 5. VMMECH005: Heat Transfer in a Composite Wall ................................................................................. 95 6. VMMECH006: Heater with Nonlinear Conductivity .............................................................................. 97 7. VMMECH007: Thermal Stress in a Bar with Temperature Dependent Conductivity ................................ 99 8. VMMECH008: Heat Transfer from a Cooling Spine ............................................................................. 103 9. VMMECH009: Stress Tool for Long Bar With Compressive Load .......................................................... 107 10. VMMECH010: Modal Analysis of a Rectangular Plate ....................................................................... 109 11. VMMECH011: Large Deflection of a Circular Plate With Uniform Pressure ......................................... 111 12. VMMECH012: Buckling of a Stepped Rod ........................................................................................ 113 13. VMMECH013: Buckling of a Circular Arch ........................................................................................ 115 14. VMMECH014: Harmonic Response of a Single Degree of Freedom System ....................................... 117 15. VMMECH015: Harmonic Response of Two Storied Building Under Transverse Loading ..................... 119 16. VMMECH016: Fatigue Tool with Non-Proportional Loading for Normal Stress ................................... 121 17. VMMECH017: Thermal Stress Analysis with Remote Force and Thermal Loading .............................. 123 18. VMMECH018: A Bar Subjected to Tensile Load with Inertia Relief ..................................................... 125 19. VMMECH019: Mixed Model Subjected to Bending Loads with Solution Combination ....................... 127 20. VMMECH020: Modal Analysis for Beams ......................................................................................... 129 21. VMMECH021: Buckling Analysis of Beams ....................................................................................... 131 22. VMMECH022: Structural Analysis with Advanced Contact Options ................................................... 133 23. VMMECH023: Curved Beam Assembly with Multiple Loads ............................................................. 135 24. VMMECH024: Harmonic Response of a Single Degree of Freedom System for Beams ....................... 137 25. VMMECH025: Stresses Due to Shrink Fit Between Two Cylinders ...................................................... 139 26. VMMECH026: Fatigue Analysis of a Rectangular Plate Subjected to Edge Moment ........................... 143 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. iii Workbench Verification Manual 27. VMMECH027: Thermal Analysis for Shells with Heat Flow and Given Temperature ............................ 147 28. VMMECH028: Bolt Pretension Load Applied on a Semi-Cylindrical Face ........................................... 149 29. VMMECH029: Elasto-Plastic Analysis of a Rectangular Beam ............................................................ 151 30. VMMECH030: Bending of Long Plate Subjected to Moment - Plane Strain Model ............................. 153 31. VMMECH031: Long Bar With Uniform Force and Stress Tool - Plane Stress Model .............................. 155 32.VMMECH032: Radial Flow due to Internal Heat Generation in a Copper Disk - Axisymmetric Model ... 157 33. VMMECH033: Spring Mass System Subjected to Enforced Motion with Displacement Base Excita- tions ................................................................................................................................................... 161 34. VMMECH034: Rubber Cylinder Pressed Between Two Plates ............................................................ 163 35. VMMECH035: Thermal Stress in a Bar With Radiation ...................................................................... 165 36. VMMECH036: Thermal Stress Analysis of a Rotating Bar Using Temperature Dependant Density ....... 169 37. VMMECH037: Cooling of a Spherical Body ...................................................................................... 171 38. VMMECH038: Crashing Blocks Simulation with Transient Structural Analysis .................................... 173 39. VMMECH039: Transient Response of a Spring-Mass System ............................................................. 175 40. VMMECH040: Deflection of Beam using Symmetry and Anti-Symmetry ........................................... 177 41. VMMECH041: T-Stress Calculation for a Crack in a Plate Using Pre-Meshed Crack .............................. 179 42. VMMECH042: Hydrostatic Pressure Applied on a Square Bar Fully and Partially Submerged in a Flu- id ....................................................................................................................................................... 181 43. VMMECH043: Fundamental Frequency of a Simply-Supported Beam .............................................. 183 44. VMMECH044: Thermally Loaded Support Structure ......................................................................... 185 45. VMMECH045: Laterally Loaded Tapered Support Structure .............................................................. 187 46. VMMECH046: Pinched Cylinder ...................................................................................................... 189 47. VMMECH047: Plastic Compression of a Pipe Assembly .................................................................... 191 48. VMMECH048: Bending of a Tee-Shaped Beam ................................................................................. 193 49. VMMECH049: Combined Bending and Torsion of Beam ................................................................... 195 50. VMMECH050: Cylindrical Shell Under Pressure ................................................................................ 197 51. VMMECH051: Bending of a Circular Plate Using Axisymmetric Elements .......................................... 201 52. VMMECH052: Velocity of Pistons for Trunnion Mechanism ............................................................... 205 53. VMMECH053: Simple Pendulum With SHM Motion .......................................................................... 207 54. VMMECH054: Spinning Single Pendulum ........................................................................................ 209 55. VMMECH055: Projector Mechanism - Finding the Acceleration of a Point ......................................... 213 56. VMMECH056: Coriolis Component of Acceleration (Rotary Engine Problem) .................................... 215 57. VMMECH057: Calculation of Velocity of Slider and Force by Collar ................................................... 217 58. VMMECH058: Reverse Four Bar Linkage Mechanism ........................................................................ 219 59. VMMECH059: Bending of a Solid Beam (Plane Elements) ................................................................. 221 60. VMMECH060: Crank Slot Joint Simulation with Flexible Dynamic Analysis ........................................ 223 61. VMMECH061: Out-of-Plane Bending of a Curved Bar ....................................................................... 227 62. VMMECH062: Stresses in a Long Cylinder ........................................................................................ 229 63. VMMECH063: Large Deflection of a Cantilever ................................................................................ 233 64. VMMECH064: Small Deflection of a Belleville Spring ........................................................................ 235 65. VMMECH065: Thermal Expansion to Close a Gap at a Rigid Surface .................................................. 237 66. VMMECH066: Bending of a Tapered Plate ........................................................................................ 239 67. VMMECH067: Elongation of a Solid Tapered Bar .............................................................................. 241 68. VMMECH068: Plastic Loading of a Thick Walled Cylinder .................................................................. 245 69. VMMECH069: Barrel Vault Roof Under Self Weight ........................................................................... 247 70. VMMECH070: Hyperelastic Thick Cylinder Under Internal Pressure ................................................... 249 71. VMMECH071: Centerline Temperature of a Heat Generating Wire .................................................... 251 72. VMMECH072: Thermal Stresses in a Long Cylinder ........................................................................... 255 73. VMMECH073: Modal Analysis of a Cyclic Symmetric Annular Plate ................................................... 259 74. VMMECH074: Tension/Compression Only Springs ........................................................................... 263 75. VMMECH075: Harmonic Response of Two-Story Building under Transverse Loading ........................ 265 76. VMMECH076: Elongation of a Tapered Shell With Variable Thickness ............................................... 267 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information iv of ANSYS, Inc. and its subsidiaries and affiliates. Workbench Verification Manual 77. VMMECH077: Heat Transfer in a Bar with Variable Sheet Thickness .................................................. 269 78. VMMECH078: Gasket Material Under Uniaxial Compression Loading-3-D Analysis ........................... 271 79. VMMECH079: Natural Frequency of a Motor-Generator ................................................................... 275 80. VMMECH080: Transient Response of a Spring-Mass System ............................................................. 277 81. VMMECH081: Statically Indeterminate Reaction Force Analysis ........................................................ 279 82. VMMECH082: Fracture Mechanics Stress for a Crack in a Plate .......................................................... 283 83. VMMECH083: Transient Response to a Step Excitation ..................................................................... 285 84. VMMECH084: Mullins Effect on a Rubber Tube Model Subjected to Tension Loading ........................ 289 85. VMMECH085: Bending of a Composite Beam .................................................................................. 291 86. VMMECH086: Stress Concentration at a Hole in a Plate .................................................................... 293 87. VMMECH087: Campbell Diagrams and Critical Speeds Using Symmetric Orthotropic Bearings ......... 297 88. VMMECH088: Harmonic Response of a Guitar String ....................................................................... 301 89. VMMECH089: Delamination Analysis of a Double Cantilever Beam Using Contact-Based Debond- ing ..................................................................................................................................................... 303 90. VMMECH090: Delamination Analysis of a Double Cantilever Beam Using Interface Delamination ..... 305 91. VMMECH091: Unbalanced Harmonic Response of a Shaft Carrying Single Rotor with Damping ........ 307 92. VMMECH092: Convection Treatment Problem for a Hollow Cylinder with Fluid Flow ........................ 311 93. VMMECH093: C*-Integral Calculation for a Single-Edge Cracked Plate Using Pre-Meshed Crack ........ 313 94.VMMECH094: Residual Vector in Stand-Alone and Linked Mode-Superposition Harmonic Analysis .... 315 95. VMMECH095: 2-D Double Cantilever Beam Problem ....................................................................... 317 96. VMMECH096: 2-D Fracture Problem Under Thermal Loading ........................................................... 319 97. VMMECH097: Inclined Crack in 2-D Plate Under Uniform Tension Loading ....................................... 321 98. VMMECH098: 2-D End Notched Flexure Problem ............................................................................ 323 99. VMMECH099: Mode I Crack Growth Analysis of DCB Using Interface Delamination ........................... 325 IV. Design Exploration Descriptions ....................................................................................................... 327 1. VMDX001: Optimization of L-Shaped Cantilever Beam Under Axial Load ........................................... 329 2. VMDX002: Optimization of Bar with Temperature-Dependent Conductivity ....................................... 331 3. VMDX003: Optimization of Water Tank Column for Mass and Natural Frequency ................................ 333 4. VMDX004: Optimization of Frequency for a Plate With Simple Support at all Vertices ......................... 337 5. VMDX005: Optimization of Buckling Load Multiplier With CAD Parameters and Young's Modulus ...... 339 V. Explicit Dynamics Descriptions ........................................................................................................... 343 1. EXDVM2: Propagation of Shock and Rarefaction Waves in a Shock Tube ............................................ 345 1. EXDVM3: Flow of Gas Past an Infinite Two-Dimensional Wedge ......................................................... 349 1. EXDVM4: Regular and Mach Deflections Off a Two-Dimensional Wedge ............................................ 353 1. EXDVM6: 3-D Taylor Cylinder Impact ................................................................................................. 359 1. EXDVM7: 2-D Taylor Cylinder Impact ................................................................................................. 363 VI. Aqwa Descriptions ............................................................................................................................. 367 1. AQVM1: Hydrostatic Evaluation of a Floating Inverted Pyramid ......................................................... 369 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. v Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information vi of ANSYS, Inc. and its subsidiaries and affiliates. Introduction The following topics are discussed in this chapter: Overview AIM Test Cases Index of Test Cases Overview This manual presents a collection of test cases that demonstrate a number of the capabilities of the Workbench analysis environment. The available tests are engineering problems that provide independent verification, usually a closed form equation. Many of them are classical engineering problems. The solutions for the test cases have been verified, however, certain differences may exist with regard to the references. These differences have been examined and are considered acceptable. The workbench analyses employ a balance between accuracy and solution time. Improved results can be obtained in some cases by employing a more refined finite element mesh but requires longer solution times. For the tests, an error rate of 3% or less has been the goal. These tests using Microsoft Windows 7 Enterprise 64-bit . These results are reported in the test docu- mentation. Slightly different results may be obtained when different processor types or operating systems are used. The tests contained in this manual are a partial subset of the full set of tests that are run by ANSYS developers to ensure a high degree of quality for the Workbench product. The verification of the Workbench product is conducted in accordance with the written procedures that form a part of an overall Quality Assurance program at ANSYS, Inc. You are encouraged to use these tests as starting points when exploring new Workbench features. Geometries, material properties, loads, and output results can easily be changed and the solution re- peated. As a result, the tests offer a quick introduction to new features with which you may be unfamil- iar. Some test cases will require different licenses, such as DesignModeler, Emag, or Design Exploration. If you do not have the available licenses, you may not be able to reproduce the results. The Educational version of Workbench should be able to solve most of these tests. License limitations are not applicable to Workbench Education version but problem size may restrict the solution of some of the tests. The archive files for each of the Verification Manual tests are available at the Customer Portal. Download the ANSYS Workbench Verification Manual Archive Files. These zipped archives provide all of the necessary elements for running a test, including geometry parts, material files, and workbench databases. To open a test case in Workbench, locate the archive and import it into Workbench. You can use these tests to verify that your hardware is executing the ANSYS Workbench tests correctly. The results in the databases can be cleared and the tests solved multiple times. The test results should be checked against the verified results in the documentation for each test. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 1 Introduction ANSYS, Inc. offers the Workbench Verification and Validation package for users that must perform system validation. This package automates the process of test execution and report generation. If you are interested in contracting for such services contact the ANSYS, Inc. Quality Assurance Group. AIM Test Cases A subset of test cases within this manual are supported by ANSYS AIM. These cases are divided into ANSYS Mechanical and ANSYS AIM specific parts. The following test cases are supported by ANSYS AIM: VMMECH001 (p. 83) - Statically Indeterminate Reaction Force Analysis VMMECH002 (p. 87) - Rectangular Plate with Circular Hole Subjected to Tensile Loading VMMECH003 (p. 89) - Modal Analysis of Annular Plate VMMECH005 (p. 95) - Heat Transfer in a Composite Wall VMMECH007 (p. 99) - Thermal Stress in a Bar with Temperature Dependent Conductivity VMMECH006 (p. 97) - Heater with Nonlinear Conductivity VMMECH008 (p. 103) - Heat Transfer from a Cooling Spine VMMECH018 (p. 125) - A Bar Subjected to Tensile Load with Inertia Relief VMMECH025 (p. 139) - Stresses Due to Shrink Fit Between Two Cylinders VMMECH026 (p. 143) - Fatigue Analysis of a Rectangular Plate Subjected to Edge Moment VMMECH027 (p. 123) - Thermal Stress Analysis with Remote Force and Thermal Loading VMMECH032 (p. 157) - Radial Flow due to Internal Heat Generation in a Copper Disk VMMECH035 (p. 165) - Thermal Stress in a Bar with Radiation VMMECH037 (p. 171) - Cooling of a Spherical Body VMMECH050 (p. 197) - Cylindrical Shell Under Pressure VMMECH062 (p. 229) - Stresses in a Long Cylinder VMMECH067 (p. 241) - Elongation of a Solid Tapered Bar VMMECH071 (p. 251) - Centerline Temperature of a Heat Generating Wire VMMECH072 (p. 255) - Thermal Stresses in a Long Cylinder VMMECH073 (p. 259) - Modal Analysis of a Cyclic Symmetric Annular Plate Index of Test Cases Test Case Number Solver Element Type Analysis Type Solution Options VMMECH001 (p. 83) ANSYS Solid Static Linear AIM Structural ANSYS Mechanical VMMECH002 (p. 87) ANSYS Solid Static Linear AIM Structural ANSYS Mechanical VMMECH003 (p. 89) ANSYS Solid Modal Free Vibration Mechanical VMMECH004 (p. 93) ANSYS Solid Structural Nonlinear, Mechanical Viscoplastic Materials VMMECH005 (p. 95) ANSYS Solid Static Thermal Linear AIM Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 2 of ANSYS, Inc. and its subsidiaries and affiliates. Index of Test Cases Test Case Number Solver Element Type Analysis Type Solution Options ANSYS Mechanical VMMECH006 (p. 97) ANSYS Solid Static Thermal Nonlinear AIM ANSYS Mechanical VMMECH007 (p. 99) ANSYS Solid Static Nonlinear Mechanical Structural Thermal Stress VMMECH008 (p. 103) ANSYS Solid Static Thermal Linear AIM ANSYS Mechanical VMMECH009 (p. 107) ANSYS Solid Static Linear Mechanical Structural VMMECH010 (p. 109) ANSYS Shell Modal Free Vibration Mechanical VMMECH011 (p. 111) ANSYS Shell Static Nonlinear, Mechanical Structural Large Deformation VMMECH012 (p. 113) ANSYS Solid Buckling Mechanical VMMECH013 (p. 115) ANSYS Shell Buckling Mechanical VMMECH014 (p. 117) ANSYS Solid Harmonic Mechanical VMMECH015 (p. 119) ANSYS Solid Harmonic Mechanical VMMECH016 (p. 121) ANSYS Solid Static Fatigue Mechanical Structural VMMECH017 (p. 123) ANSYS Solid Static Linear Thermal Mechanical Structural Stress VMMECH018 (p. 125) ANSYS Solid Static Linear, Inertia Mechanical Structural relief VMMECH019 (p. 127) ANSYS Beam Static Linear Mechanical Structural Shell VMMECH020 (p. 129) ANSYS Beam Modal Mechanical VMMECH021 (p. 131) ANSYS Beam Buckling Mechanical VMMECH022 (p. 133) ANSYS Solid Static Nonlinear, Mechanical Structural Contact VMMECH023 (p. 135) ANSYS Beam Static Linear Mechanical Structural Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 3 Introduction Test Case Number Solver Element Type Analysis Type Solution Options VMMECH024 (p. 137) ANSYS Beam Harmonic Mechanical VMMECH025 (p. 139) ANSYS Solid Static Linear Mechanical Structural VMMECH026 (p. 143) ANSYS Shell Static Fatigue Mechanical Structural VMMECH027 (p. 147) ANSYS Shell Static Linear Thermal Mechanical Structural Stress VMMECH028 (p. 149) ANSYS Solid Static Mechanical Structural VMMECH029 (p. 151) ANSYS Solid Static Nonlinear, Mechanical Structural Plastic Materials VMMECH030 (p. 153) ANSYS 2-D Solid, Static Mechanical Plane Strain Structural VMMECH031 (p. 155) ANSYS 2-D Solid, Static Mechanical Plane Stress Structural VMMECH032 (p. 157) ANSYS 2-D Solid, Static Linear Thermal Mechanical Axisymmetric Structural Stress VMMECH033 (p. 161) ANSYS Solid Static Electromagnetic Mechanical Structural VMMECH034 (p. 163) ANSYS Solid Static Nonlinear, Mechanical Structural Large Deformation VMMECH035 (p. 165) ANSYS Solid Coupled AIM (Static Thermal ANSYS and Static Mechanical Stress) VMMECH036 (p. 169) ANSYS Solid Static Sequence Mechanical Structural Loading VMMECH037 (p. 171) ANSYS 2-D Solid, Transient Mechanical Axisymmetric Thermal VMMECH038 (p. 173) ANSYS Solid Transient Flexible Mechanical Structural Dynamic VMMECH039 (p. 175) ANSYS Solid Transient Flexible Mechanical Structural Dynamic Spring VMMECH040 (p. 177) ANSYS Beam Static Mechanical Structural VMMECH041 (p. 179) ANSYS Solid Static Electromagnetic Mechanical Structural VMMECH042 (p. 181) ANSYS Solid Static Hydrostatic Mechanical Structural Fluid VMMECH043 (p. 183) ANSYS Beam Modal Mechanical Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 4 of ANSYS, Inc. and its subsidiaries and affiliates. Index of Test Cases Test Case Number Solver Element Type Analysis Type Solution Options VMMECH044 (p. 185) ANSYS Beam Static Linear Thermal Mechanical Structural Stress VMMECH045 (p. 187) ANSYS Shell Static Mechanical Structural VMMECH046 (p. 189) ANSYS Shell Static Mechanical Structural VMMECH047 (p. 191) ANSYS 2-D Solid, Static Nonlinear, Mechanical Axisymmetric Structural Plastic Materials VMMECH048 (p. 193) ANSYS Beam Static Mechanical Structural VMMECH049 (p. 195) ANSYS Beam Static Mechanical Structural VMMECH050 (p. 197) ANSYS Axisymmetric Static Mechanical Shell Structural VMMECH051 (p. 201) ANSYS Axisymmetric Static Mechanical Shell Structural VMMECH052 (p. 205) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH042 (p. 181) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH054 (p. 209) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH055 (p. 213) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH056 (p. 215) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH057 (p. 217) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH058 (p. 219) ANSYS Multipoint Rigid Dynamic Mechanical Constraint VMMECH059 (p. 221) ANSYS 2-D Plane Static Mechanical Stress Shell Structural VMMECH060 (p. 223) ANSYS Solid Transient Flexible Mechanical Structural Dynamic Multipoint Constraint VMMECH061 (p. 227) ANSYS Beam Static Mechanical Structural VMMECH062 (p. 229) ANSYS Axisymmetric Static Mechanical Shell Structural VMMECH063 (p. 233) ANSYS Shell Static Nonlinear, Mechanical Structural Large Deformation Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 5 Introduction Test Case Number Solver Element Type Analysis Type Solution Options VMMECH064 (p. 235) ANSYS Shell Static Mechanical Structural VMMECH065 (p. 237) ANSYS Solid Static Linear Thermal Mechanical Structural Stress Shell VMMECH066 (p. 239) ANSYS Shell Static Mechanical Structural VMMECH067 (p. 241) ANSYS Solid Static AIM Structural ANSYS Mechanical VMMECH068 (p. 245) ANSYS 2-D Solid, Static Nonlinear, Mechanical Plane Strain Structural Plastic Materials VMMECH069 (p. 247) ANSYS Shell Static Mechanical Structural VMMECH070 (p. 249) ANSYS 2-D Solid Static Nonlinear, Mechanical Structural Large Deformation VMMECH071 (p. 251) ANSYS 2-D Thermal Static Thermal Mechanical Solid VMMECH072 (p. 255) ANSYS 2-D Thermal Static Linear Thermal Mechanical Solid Structural Stress VMMECH073 (p. 259) ANSYS Solid Modal AIM ANSYS Mechanical VMMECH074 (p. 263) ANSYS Solid Rigid Body Mechanical Dynamics Spring VMMECH075 (p. 265) ANSYS Solid Harmonic Mechanical VMMECH076 (p. 267) ANSYS Shell Static Mechanical Structural VMMECH077 (p. 269) ANSYS Thermal Shell Static Thermal Mechanical VMMECH078 (p. 271) ANSYS 3-D Solid Static Mechanical Structural 3-D Gasket VMMECH079 (p. 275) ANSYS Pipe Modal Mechanical VMMECH080 (p. 277) ANSYS Spring Transient Mode-Superposition Mechanical Dynamic Mass VMMECH081 (p. 279) ANSYS Pipe Modal Mechanical Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 6 of ANSYS, Inc. and its subsidiaries and affiliates. Index of Test Cases Test Case Number Solver Element Type Analysis Type Solution Options Mass Spectral VMMECH082 (p. 283) ANSYS Solid Static Fracture Mechanical Structural Mechanics VMMECH083 (p. 285) ANSYS Spring, Mass Transient Mode-Superposition Mechanical Dynamic VMMECH084 (p. 289) ANSYS Solid Static Nonlinear, Mechanical Structural Hyperelastic VMMECH085 (p. 291) ANSYS Solid Static Composite Mechanical Structural Material VMMECH086 (p. 293) ANSYS Solid Static Mechanical Structural Submodeling (2D-2D) VMMECH087 (p. 297) ANSYS Line Body Modal Mechanical Point Mass Bearing Connection VMMECH088 (p. 301) ANSYS Beam Static Linear Mechanical Structural Perturbation Modal Harmonic VMMECH089 (p. 303) ANSYS Solid Static Contact-Based Mechanical Structural Debonding VMMECH090 (p. 305) ANSYS Solid Static Interface Mechanical Structural Delamination VMMECH091 (p. 307) ANSYS Beam Harmonic Mechanical VMMECH092 (p. 311) ANSYS Coupled Static Mechanical Thermal Pipe Structural Thermal Surface Thermal Solid VMMECH093 (p. 313) ANSYS Solid Static Pre-Meshed Mechanical Structural Crack VMMECH094 (p. 315) ANSYS Spring-Damper Modal Mechanical Structural Harmonic Mass VMMECH095 (p. 317) ANSYS Solid Static Pre-Meshed Mechanical Structural Crack Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 7 Introduction Test Case Number Solver Element Type Analysis Type Solution Options VMMECH096 (p. 319) ANSYS Solid Static Mechanical Structural VMMECH097 (p. 321) ANSYS Solid Static Pre-Meshed Mechanical Structural Crack VMMECH098 (p. 323) ANSYS Solid Static Pre-Meshed Mechanical Structural Crack VMMECH099 (p. 325) ANSYS Solid Static Mechanical Structural Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 8 of ANSYS, Inc. and its subsidiaries and affiliates. Part I: DesignModeler Descriptions VMDM001: Extrude, Chamfer, and Blend Features Overview Feature: Extrude, Chamfer, and Blend Drawing Units: Millimeter Test Case Create a Model using Extrude, Chamfer, and Blend features. A polygonal area is extruded 60 mm. A rectangular area of 30 mm x 40 mm [having a circular area of radius 6 mm subtracted] is extruded to 20 mm. Both resultant solids form one solid geometry. A rect- angular area (24 mm x 5 mm) is subtracted from the solid. Two rectangular areas (40 mm x 10 mm) are extruded 10 mm and subtracted from solid. Two rectangular areas (25 mm x 40 mm) are extruded 40 mm and subtracted from solid. A Chamfer (10 mm x 10 mm) is given to 4 edges on the resultant solid. Four Oval areas are extruded and subtracted from Solid. Fillet (Radius 5 mm) is given to 4 edges using Blend Feature. Verify Volume of the resultant geometry. Figure 1: Final Model after creating Extrude, Chamfer, and Blend Calculations 1. Volume of Solid after extruding Polygonal Area: v1 = 264000 mm3. 2. Volume of rectangular area having circular hole: v2 = 21738.05 mm3. Net Volume = V = v1 + v2 = 285738.05 mm3. 3. Volume of rectangular (24mm x 5mm) solid extruded 30mm using Cut Material = 3600 – 565.5 = 3034.5 mm3. Net volume V = 285738.05 – 3034.5 = 282703.5 mm3. 4. Volume of two rectangular areas each 40mm x 10mm extruded 10mm = 8000 mm3. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 11 VMDM001 Net volume V = 282703.5 – 8000 = 274703.5 mm3. 5. Volume of two rectangular areas 25mm x 40mm extruded 40mm = 80000 mm3. Net volume V = 274703.5 – 80000 = 194703.5 mm3. 6. Volume of four solids added due to Chamfer = 4 x 500 = 2000 mm3 Net volume V = 194703.5 + 2000 = 196703.5 mm3. 7. Volume of four oval areas extruded 10 mm = 7141.6 mm3. Net volume V = 196703.5 - 7141.6 = 189561.9 mm3. 8. Volume of 4 solids subtracted due to Blend of radius 5 mm = 429.2 mm3. Hence Net volume of final Solid body = V = 189561.9 – 429.2 = 189132.7 mm3. Results Comparison Results Target DesignModeler Error (%) 3 Volume (mm ) 189132.7437 189132.7436 -3.158 x 10-8 Surface Area (mm2) 44261.61 44261.29 0.001 Number of Faces 52 52 0 Number of Bodies 1 1 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 12 of ANSYS, Inc. and its subsidiaries and affiliates. VMDM002: Cylinder using Revolve, Sweep, Extrude, and Skin-Loft Overview Feature: Revolve, Sweep, Extrude, and Skin-Loft Drawing Units: Millimeter Test Case Create a Model using Revolve, Sweep, Extrude, and Skin-Loft features. A Rectangular area (100 mm x 30 mm) is revolved about Z-Axis in 3600 to form a Cylinder. A circular area of radius 30 mm is swept 100 mm using Sweep feature. A circular area of radius 30 mm is extruded 100 mm. A solid cylinder is created using Skin-Loft feature between two coaxial circular areas each of radius 30 mm and 100 mm apart. Verify Volume of the resultant geometry. Figure 2: Final Model after creating Revolve, Sweep, Extrude, and Skin-Loft Calculations 1. Volume of Cylinder created after Revolving Rectangular area (100 mm x 30 mm) = v1 = 282743.3 mm3. 2. Volume of Cylinder created when a circular area (Radius 30mm) is swept 100 mm = v2 = 282743.3 mm3. Net Volume = V = v1 + v2 = 282743.3 + 282743.3 = 565486.6 mm3. 3. Volume of Cylinder after extruding a circular area (Radius 30 mm) 100 mm = 282743.3 mm3. Net Volume = V = 565486.6 + 282743.3 = 848229.9 mm3. 4. Volume of Cylinder created after using Skin-Loft feature between two circular areas of Radius 30 mm and 100 mm apart. = 282743.3 mm3. Net Volume of the final Cylinder = 848229.9 + 282743.3 = 1130973.2 mm3. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 13 VMDM002 Results Comparison Results Target DesignModeler Error (%) 3 Volume (mm ) 1130973.36 1130973.36 2.585 x 10-8 Surface Area (mm2) 81053.09 81053.09 3.228 x 10-9 Number of Faces 3 3 0 Number of Bodies 1 1 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 14 of ANSYS, Inc. and its subsidiaries and affiliates. VMDM003: Extrude, Revolve, Skin-Loft, and Sweep Overview Feature: Extrude, Revolve, Skin-Loft, and Sweep Drawing Units: Millimeter Test Case Create a Model using Extrude, Revolve, Skin-Loft, and Sweep. A rectangular area (103 mm x 88 mm) is extruded 100 mm to form a solid box. A circular area of radius 25 mm is revolved 900 using Revolve feature and keeping Thin/Surface option to Yes and 3 mm Inward and Outward Thickness. A solid is subtracted using Skin-Loft feature between two square areas (each of side 25 mm) and 100 mm apart. The two solid bodies are frozen using Freeze feature. A circular area of radius 25 mm is swept using Sweep feature and keeping Thin/Surface option to Yes and 3 mm Inward and Outward Thickness. Thus a total of 4 geometries are created. Verify the volume of the resulting geometry. Figure 3: Final Model after creating Extrude, Revolve, Skin-Loft and Sweep Calculations 1. Volume of rectangular (103 mm x 88 mm) solid extruded 100mm = 906400 mm3. 2. Volume of solid after revolving circular area of Radius 25 mm through 900 = 29639.6 mm3. Net Volume of solid box, Va = 906400 - 29639.6 = 876760.3 mm3. 3. Volume of additional body created due to Revolve feature = Vb= 11134.15 mm3. 4. Volume of the rectangular box cut after Skin-Loft between two square areas each of side 25 mm = 62500 mm3. Net Volume of solid box becomes Va = 876760.3 – 62500 = 814260.3 mm3. 5. Volume of additional two bodies created due to Sweep feature: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 15 VMDM003 • Vc = 47123.9 mm3 and Vd = 28352.8 mm3. • And total volume that gets subtracted from box due to Sweep Feature = 75476.7 mm3. • Hence Net volume of box, Va = 814260.3 - 75476.7 = 738783.6 mm3. • Sum of volumes of all four bodies = Va+Vb+Vc+Vd = 738783.6 + 11134.15 + 47123.9 +28352.8 = 825394.4 mm3. Results Comparison Results Target DesignModeler Error (%) 3 Volume (mm ) 825394.4289 825394.4917 7.614 x 10-6 Surface Area (mm2) 101719.466 101719.951 4.773 x 10-4 Number of Faces 22 22 0 Number of Bodies 4 4 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 16 of ANSYS, Inc. and its subsidiaries and affiliates. VMDM004: Extrude, Revolve, Skin-Loft, and Sweep Overview Feature: Extrude, Revolve, Skin-Loft, and Sweep Drawing Units: Millimeter Test Case Create a model using Sketch, Plane, Extrude, Revolve, and Loft features in walled conditions. The total volume is created using the following procedures: • Lofted volume (VL): a hexagon denoted as Sketch 1 with an area of 9038.6 mm2 is lofted through a circular profile with an area of 5026.55 mm2 (Sketch 2) and finally through another circular profile with an area of 11309.7 mm2 (Sketch 3). This results in a lofted body with a volume of 1.3764 x 106 mm3. • Shell volume (VS): a ring with an outer radius of 60 mm and an inner radius or 50 mm is extruded 60 mm and connected to the lofted body. This results in a shell with a volume of 2.0735 x 105 mm3. • Dome volume (VD): a circular arc that is oriented perpendicular to the extrusion is revolved 360 degrees to create a dome with a hollow cylindrical cap in the middle, and this body is placed on top of the shell. This results in a dome with a volume of 3.7913 x 105 mm3. • Removed volume (VR): a cylindrical volume of 6.28319 x 104 mm3 is removed from the total body by placing a circle centered at the X,Y origin and extruding it along the Z-axis. Verify the volume of the resulting geometry. Figure 4: Final Model After Creating Extrude, Revolve, and Loft Features in Walled Conditions Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 17 VMDM004 Calculations 1. Volume of dome (VD) = 3.7913 x 106 mm3 2. Volume of the shell (VS) = 2.0735 x 105 mm3 3. Volume of the loft (VL) = 1.3764 x 106 mm3 4. Volume of the removed cylinder (VR) = 6.28319 x 104 mm3 5. Total volume of solid body (VT) = 3.7913 x 106 + 2.0735 x 105 + 1.3764 x 106 - 6.28319 x 104 = 1.9001 x 106 mm3 Results Comparison Results Target DesignModeler Error (%) 3 Volume of Loft (mm ) 1376268.34 1376421.068 0.011 Surface Area of Loft 80661.4953 80664.4587 0.004 (mm2) Volume of Extrusion & 1583598.45 1583764.766 0.011 Loft (mm3) Surface Area of 122130.078 122133.411 0.003 Extrusion & Loft (mm2) Volume of Revolution, 1962720.78 1962889.52 0.009 Extrusion, and Loft (mm3) Surface Area of 153706.1303 153709.4544 0.002 Revolution, Extrusion, and Loft (mm2) Volume of Total Body 1899888.93 1900057.67 0.009 (mm3) Surface Area of Total 165644.1824 165647.5065 0.002 Body (mm2) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 18 of ANSYS, Inc. and its subsidiaries and affiliates. Part II: SpaceClaim Descriptions VMSC_Geometry001: Alter Model Using Sketch, Pull, and Fill Overview Feature: Sketch, Pull, and Fill Drawing Units: Millimeter Test Case A rectangular sketch of 50 mm x 35 mm and a circle with radius of 5 mm is drawn on the top face. The sketches are pulled for a distance of 20 mm. A cylindrical hole of radius 5 mm and depth of 20 mm is filled. The result is verified after each step using surface area and volume validations. Figure 5: Final Modal after Sketch, Pull, and Fill Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 21 VMSC_Geometry001 Figure 6: Original Model Results Comparison Figure 7: Model After Using Sketch Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 22 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry001 Area of rectangle = 50 mm x 35 mm = 1750 mm2 Area of circle = x 5 mm x 5 mm = 78.5398 mm2 Result Target SCDM Error (%) Area of rectangular 0.00175 0.00175 0.0 sketch (m2) Area of circular sketch 7.8539 x 10-5 7.8540 x 10-5 0.0 (m2) Figure 8: Model After Using Pull Volume added by pulling rectangular sketch = 50 mm x 35 mm x 20 mm = 35000 mm3 Volume added by pulling circular sketch = x 5 mm x 5 mm x 20 mm = 1570.7963 mm3 Original volme = 68429.2037 mm3 Final volume = 68429.2037 + 35000 + 1570.7963 = 105000 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.000105 0.000105 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 23 VMSC_Geometry001 Figure 9: Model After Using Fill Volume added by filling cylindrical hole = x 5 mm x 5 mm x 20 mm = 1570.7963 mm3 Volume after pull = 105000 mm3 Final volume = 105000 + 1570.7963 = 106570.7963 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.0001065707963 0.0001065707963 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 24 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry002: Alter Model Using Split Face, Split Body and Mirror Overview Feature: Split Face, Split Body, Mirror Drawing Units: Millimeter Test Case The top face of a rectangular model is split using opposite corner edges as reference. The body is then split into two using the central vertical face. The smaller body is removed using the Split Body tool. The same face is used as a mirror plane to mirror the geometry. The result is verified after each step using surface area or volume validations. Figure 10: Final Model After Split Face, Split Body, and Mirror Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 25 VMSC_Geometry002 Figure 11: Original Model Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 26 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry002 Results Comparison Figure 12: Model After Using Split Face Area of triangle after Split Face operation = 0.5 mm x 50 mm x 35 mm = 875 mm2. Result Target SCDM Error (%) Area of triangular sketch 0.000875 0.000875 0.0 (m2) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 27 VMSC_Geometry002 Figure 13: Model After Using Split Body Volume removed by Split Body operation = 50 mm x 35 mm x 20 mm = 35000 mm3 Original volume = 105000 mm3 Final volume = 105000 - 35000 = 70000 mm3 Result Target SCDM Error (%) 3 -5 -5 Total volume (m ) 7 x 10 7 x 10 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 28 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry002 Figure 14: Model After Using Mirror Volume after Split Body operation = 70000 mm3 Volume added due to Mirror operation = 70000 mm3 Final volume = 70000 + 70000 = 140000 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.00014 0.00014 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 29 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 30 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry003: Alter Model using Linear, Circular, and Two-Dimensional Patterns Overview Feature: Linear, Circular, and Two-Dimensional Patterns Drawing Units: Millimeters Test Case A linear pattern of 7 protrusions is created. Next, a circular pattern with equidistant angles and a total of 8 protrusions is made. Finally, a two-dimensional pattern with an x- and y-offset of 10 mm is made. The result is verified after each step using volume validations. Figure 15: Final Model After Using Linear, Circular, and Two-Dimensional Features Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 31 VMSC_Geometry003 Figure 16: Original Model Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 32 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry003 Results Comparison Figure 17: Model After Using Linear Pattern Volume added by linear pattern operation = 6 x 5 mm x 5 mm x 20 mm = 3000 mm3 Original volume of block (left) = 70500 mm3 Final volume = 70500 + 3000 = 73500 mm3 Result Target SCDM Error (%) 3 -5 -5 Total volume (m ) 7.35 x 10 7.35 x 10 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 33 VMSC_Geometry003 Figure 18: Model After Using Circular Pattern Volume added by Circular Pattern operation = x 2.5 mm x 2.5 mm x 20 mm x 7 = 2748.8935 mm3 Original volume of circular block (middle) = 39662.2126 mm3 Final volume = 39662.2126 + 2748.8935 = 42411.106 mm3 Result Target SCDM Error (%) 3 -5 -5 Total volume (m ) 4.2411106 x 10 4.2411501 x 10 9.31 x 10-4 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 34 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry003 Figure 19: Model After Using Two Dimensional Pattern Volume added by Two Dimensional Pattern operation = 13 x 5 mm x 5 mm x 20 mm = 6500 mm3 Original volume of block (right) = 70500 mm3 Final volume = 70500 + 6500 = 77000 mm3 Result Target SCDM Error (%) 3 -5 -5 Total volume (m ) 7.7 x 10 7.7 x 10 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 35 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 36 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry004: Alter Model Using Revolve, Chamfer, and Offset Faces Overview Feature: Revolve, Chamfer, Offset Faces Drawing Units: Millimeter Test Case The bottom face of a rectangular recess at the top-right of the model is revolved. A chamfer with sides of 3 mm and 4mm each is made on the top-left portion of the solid. An offset condition is specified for the protruding rectangular extrusions. They are pull for a distance of 20 mm to show the offset relation- ship. The result is verified after each stop using volume validations. Figure 20: Final Model After Revolve, Chamfer, and Offset Faces Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 37 VMSC_Geometry004 Figure 21: Original Model Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 38 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry004 Results Comparison Figure 22: Model After Using Revolve Volume added by revolving rectangular face = = 15707.9632 mm3 Original volume = 148375 mm3 Final volume = 148375 + 15707.9632 = 164082.9632 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.0001640829632 0.0001640830254 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 39 VMSC_Geometry004 Figure 23: Model After Using Chamfer Volume removed by Chamfer operation = (0.5 x 3 x 4 x 50) = 300 mm3 Volume at the end of Revolve operation = 164082.9632 mm3 Final volume = 164082.9632 - 300 = 163782.9632 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.0001537829632 0.0001637826218 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 40 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry004 Figure 24: Model After Using Offset Faces Volume added by extruding protrusions = 2 x 15 x 15 x 20 = 9000 mm3 Volume at the end of the Chamfer operation = 163782.9632 mm3 Final volume = 163782.9632 + 9000 = 172782.9632 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.0001727829632 0.001727833025 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 41 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 42 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry005: Alter Model Using Blend and Project Overview Feature: Blend, Project Drawing Units: Millimeter Test Case A rectangular surface of 50 x 35 mm is blended into a face of the same dimension, making a solid block. A circular surface is projected onto the face of the solid. The result is verified after each step using either surface area or volume validations. Figure 25: Final Model After Blend and Project Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 43 VMSC_Geometry005 Figure 26: Original Model Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 44 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry005 Results Comparison Figure 27: Model After Using Blend Volume added by pulling rectangular sketch = 50 x 35 x 20 = 35000 mm3 Original volume = 70000 mm3 Final volume = 70000 + 35000 = 105000 mm3 Result Target SCDM Error (%) 3 Total volume (m ) 0.000105 0.000105 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 45 VMSC_Geometry005 Figure 28: Model After Using Project Surface area of circular surface = x 5 x 5 = 78.5398 mm2 Result Target SCDM Error (%) 2 -5 -5 Surface area (m ) 7.85398 x 10 7.85398 x 10 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 46 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry006: Alter Model Using Sheet Metal Tools and Operations Overview Feature: Junction-Specific Pull, Form, Bend, Split, Double Wall Drawing Units: Millimeter Test Case Import a model, pull the left edge by 30 mm after specifying the resulting junction to be sharp. Create two forms, including one circular punch and one rectangular knock-out. Bend the sheet metal at 20 mm from the right-most edge. Split the created bend such that the edge length on the right is 20 mm. Create a double wall on the left-most pulled face. The result is verified after each step using volume or edge length calculations. Figure 29: Final Model After Using Sheet Metal Tools and Features Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 47 VMSC_Geometry006 Figure 30: Original Model Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 48 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry006 Results Comparison Figure 31: Model After Using Pull Volume added by pulling edge = 31 x 50 x 1 = 1550 mm3 Original volume = 3500 mm3 Final volume = 3500 + 1550 = 5050 mm3 Result Target SCDM Error (%) 3 -6 -6 Total volume (m ) 5.05 x 10 5.05 x 10 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 49 VMSC_Geometry006 Figure 32: Model After Using Form Volume removed by circular punch = x 2.5 x 2.5 = 19.6349 mm3 Volume after Pull operation = 5050 mm3 Final volume = 5050 - 19.6349 = 5030.3650 mm3 Result Target SCDM Error (%) 3 -6 -6 Total volume (m ) 5.030365 x 10 5.030365 x 10 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 50 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry006 Figure 33: Model After Using Bend Volume added due to bend = 117.81 - (50 x 2) = 17.81 mm3 Volume after Form operation = 5030.3550 mm3 Final volume = 5030.3650 + 17.81 = 5048.175 mm3 Result Target SCDM Error (%) 3 -6 -6 Total volume (m ) 5.048175 x 10 5.048184 x 10 0.2 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 51 VMSC_Geometry006 Figure 34: Model After Using Split Edge length of the shorter split region = 20 mm Result Target SCDM Error (%) Edge length (m) 0.02 0.02 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 52 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Geometry006 Figure 35: Model After Using Double Wall Volume added by double wall feature = 78.54 + 1550 = 1628.54 mm3 Volume after Split operation = 5048.175 - (0.1 x 19) - (0.2375) = 5046.0375 mm3 Final volume = 5046.0375 + 1628.54 = 6674.5775 mm3 Result Target SCDM Error (%) 3 -6 -6 Total volume (m ) 6.6745775 x 10 6.6745469 x 10 0.5 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 53 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 54 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare001: Midsurface, Extend, Imprint, and Spot Weld Overview Feature: Midsurface, Extend, Imprint, and Spot Weld Drawing Units: Millimeter Test Case A solid model is midsurfaced and then the solid is hidden (midsurface thickness should be equal to solid thickness). 8 midsurface bodies are created. 5 untouched small midsurfaces are extended (surface area made larger). 3 of the extended midsurfaces are imprinted and 2 are spot welded onto the largest midsurface. Verify the existence of the midsurfaces and spot welds, and verify the geometry change due to the ex- tending and imprinting operations. Note The target and SCDM values of surface area are measured from within SCDM. The target values of surface area are measured before extending the midsurfaces, and the SCDM values of surface area are measured after extending them. This part of the verification is to make sure the surface area does increase after the Extend operation. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 55 VMSC_Prepare001 Figure 36: Final Model After Midsurface, Extend, Imprint and Spot Weld Operations Results Comparison Results Target SCDM Error (%) Number of Midsurface Bodies 8 8 0 Thickness of Midsurfaces 12.7 12.699999998851515 0.00000001 Surface Area of Midsuface4 after extending > 5008.05794534485 - 4536.02590874 Surface Area of Midsuface5 after extending > 4910.4983497921181 - 4449.17233383 Surface Area of Midsuface6 after extending > 4739.78817217394 - 4296.94659955 Surface Area of Midsuface7 after extending > 4492.93401173876 - 4076.50156109 Surface Area of Midsuface8 after extending > 4089.6292714564277 - 3715.85600571 Edge Count of Misdurface2 after 13 13 0 imprinting Number of SpotWelds 2 2 0 Number of Points for Each SpotWeld 8 8 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 56 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare002: Volume Extract and Enclosure Overview Feature: Volume Extract and Enclosure Drawing Units: Millimeter Test Case This test case is divided into several scenarios, with results for each. Scenario 1 Extract volume from an oil pan model, and verify the existence of the extracted volume. Figure 37: Extract Volume from Oil Pan Scenario 2 Modify the oil pan model by enlarging an internal cylindrical surface, then update the extracted volume. Verify the updated surface area of the counterpart of the cylindrical surface on the extracted volume. The target value is measured from the cylindrical surface of the oil pan model and the SCDM value is measured from the updated cylindrical surface (mating surface) of the extracted volume. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 57 VMSC_Prepare002 Figure 38: Updated Extracted Volume Scenario 3 Create an enclosure volume from the cooling block object, then verify the existence of the enclosure volume. The target surface count is counted using the cooling block object and the SCDM surface count is counted using the enclosure volume. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 58 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare002 Figure 39: Enclosure Volume from the Cooling Block Object Scenario 4 Modify the cooling block object by cutting a cylindrical voume from the bottom base. Update the en- closure volume, verifying extra surfaces of the update volume. The target surface area is measured from the cooling block object and the SCDM surface area is measured from the enclosure volume (mating surfaces). Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 59 VMSC_Prepare002 Figure 40: Enclosure Volume from the Cooling Block Object Results Comparison Scenario 1 Results Target SCDM Error (%) Name of the extracted Volume Volume N/A volume Is it a closed body True True N/A Scenario 2 Results Target SCDM Error (%) Surface area of the changed cylindrical 7964.59 7964.5892573257434 9.32E-6 surface Scenario 3 Results Target SCDM Error (%) Name of the Enclosure Enclosure Enclosure N/A volume Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 60 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare002 Results Target SCDM Error (%) Number of Face Count 44 44 0 Scenario 4 Results Target SCDM Error (%) Type of the 1st new surface on Enclosure volume Plane Plane N/A Surface area of the planar surface 28.27 28.274333882308154 1.53E-2 Type of the 2nd new surface on Enclosure volume Cylinder Cylinder N/A Surface area of the cylindrical surface 9.05 9.0477868423385 2.21 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 61 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 62 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare003: Beam Create, Extract, and Orient Overview Feature: Beam Create, Extract, and Orient Drawing Units: Millimeter Test Case This test case is divided into several scenarios, with results for each. Scenario 1 Create beams with circular profile and extract beams from existing solid. Verify the existence of the extracted volume. Figure 41: Created Beams and Extracted Beams Scenario 2 Orient one beam by changing its Section Anchor to Location, then move the section toward Y direction for 99.64 mm. Verify the section location of the oriented beam. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 63 VMSC_Prepare003 Figure 42: Orient Beam Results Comparison Scenario 1 Results Target SCDM Error (%) Number of SketchCurve with Beam properties 8 8 0 Number of Beams with section name of Circle 4 4 0 Number of Beams with section name of Extracted Profile1 1 1 0 Number of Beams with section name of Extracted Profile2 2 2 0 Number of Beams with section name of Extracted Profile3 1 1 0 Scenario 2 Results Target SCDM Error (%) Anchor Location of Oriented 99.64 99.64 0 Beam Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 64 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare004: Remove Rounds, Interference, Faces, and Short Edges Overview Feature: Remove Rounds, Interference, Faces, and Short Edges Drawing Units: Millimeter Test Case This test case is divided into several scenarios, with results for each. Scenario 1 Remove the rounds as shown in the figure, then verify the absence of these round faces. Figure 43: Remove Rounds Scenario 2 Remove the interference volume from the smaller object, then verify the cylindrical surface area and the circular edge length. The target values are measured from the larger object before removing inter- ference, the SCDM values are measured from the mating feature of the smaller object after removing interference. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 65 VMSC_Prepare004 Figure 44: Remove Interference Volume from Smaller Objects Scenario 3 Remove the round faces by using the Remove Faces tool, then verify the existence of the edge loop. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 66 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Prepare004 Figure 45: Enclosure Volume from the Cooling Block Scenario 4 Remove the tiny edge using the Short Edge tool, then verify the absence of the tiny edge. Figure 46: Remove the Tiny Edge Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 67 VMSC_Prepare004 Results Comparison Scenario 1 Results Target SCDM Error (%) Number of box selected round 0 0 0 faces Scenario 2 Results Target SCDM Error (%) Surface area of the cylindrical surface on the 7206.06 7206.0553515527412 6.45E-5 changed object Edge length of the circular edge on the changed 39.906007969873911 39.906007969873911 0 object Scenario 3 Results Target SCDM Error (%) Number of newly created 30 30 0 edges Scenario 4 Results Target SCDM Error (%) Number of box selected 0 0 0 edges Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 68 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Repair001: Stitch, Gap, and Missing Faces Overview Feature: Stitch, Gap, and Missing Faces Drawing Units: Millimeter Test Case Use the Stitch tool to stitch all the surface bodies into one single surface body, then use the Gap tool to fill the gaps. Use the Missing Faces tool to generate missing faces. The surface body will be solidified into a solid body. Verify the existence of the solid body. Figure 47: Solidify Surface Bodies into a Single Solid Body Results Comparison Results Target SCDM Error (%) Number of solid 1 1 0 body Is it a closed body True True N/A Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 69 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 70 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Repair002: Merge Faces, Small Faces, Simplify, Straighten Overview Feature: Merge Faces, Small Faces, Simplify, Straighten Drawing Units: Millimeter Test Case This test case is divided into several scenarios, with results for each. Scenario 1 Merge adjacent faces together, then verify the absence of the merged (removed) faces. Figure 48: Merge Faces into Adjacent Faces Scenario 2 Use the Small Faces tool to detect any small faces to remove, then verify the absence of the previous small faces. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 71 VMSC_Repair002 Figure 49: Detect and Fix Small Faces Scenario 3 Use the Simplify tool to simplify the red face only (see the figure below). The Spline face is simplifed into a Sphere face. Verify the Surface Type of the simplified face. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 72 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Repair002 Figure 50: Simplify the Red Face Scenario 4 Straighten the faces that are not exactly perpendicular to the adjacent faces, then verify the length of the face edges. The target value (baseline) was interactively measured in SCDM after confirming the straighten operation created perpendicular faces. Figure 51: Straighten the Highlighted Faces Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 73 VMSC_Repair002 Figure 52: Baseline Measurement After Confirming Straighten Operation (Right Edge) Figure 53: Baseline Measurement After Confirming Straighten Operation (Left Edge) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 74 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Repair002 Results Comparison Scenario 1 Results Target SCDM Error (%) Number of box selected 0 0 0 faces Scenario 2 Results Target SCDM Error (%) Number of box selected 0 0 0 faces Scenario 3 Results Target SCDM Error (%) Surface Type of the simplified Sphere Sphere N/A face Scenario 4 Results Target SCDM Error (%) Vertical edge length on both 0.1778 ≈0.1778 ≈0 sides Note In SC journal files, you will find edge length in meters. The values are all very close to the interactively measured baseline. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 75 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 76 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Repair003: Split Edges, Extra Edges, and Duplicates Overview Feature: Split Edges, Extra Edges, and Duplicates Drawing Units: Millimeter Test Case This test case is divided into several scenarios, with results for each. Scenario 1 Use the Split Edges tool to find and fix the edges that are supposed to be combined together, thus reducing the edge numbers. Verify the edge count after using the tool. The target value was measured before the fix, so the SCDM value after the fix should be smaller. Figure 54: Find and Fix the Split Edges Scenario 2 Use the Extra Edges tool to detect and remove the extra eduge, then verify the removal. The target value was measured before the fix, so the SCDM value after the fix should be smaller. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 77 VMSC_Repair003 Figure 55: Detect and Fix the Extra Edge Scenario 3 Use the Duplicate tool to find and remove duplicate surfaces. Verify the number of surface bodies after the fix to be zero. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 78 of ANSYS, Inc. and its subsidiaries and affiliates. VMSC_Repair003 Figure 56: Find and Remove Duplicate Surfaces Results Comparison Scenario 1 Results Target SCDM Error (%) Number of edges after < 1338 N/A fix 1348 Scenario 2 Results Target SCDM Error (%) Number of edges after < 1337 N/A fix 1338 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 79 VMSC_Repair003 Scenario 3 Results Target SCDM Error (%) Number of solid bodies after fix 4 4 0 Number of surface bodies after 0 0 0 fix Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 80 of ANSYS, Inc. and its subsidiaries and affiliates. Part III: Mechanical Application Descriptions VMMECH001: Statically Indeterminate Reaction Force Analysis Overview Reference: S. Timoshenko, Strength of Materials, Part 1, Elementary Theory and Problems, 3rd Edition, CBS Publishers and Distributors, pg. 22 and 26 Solver(s): ANSYS AIM ANSYS Mechanical Analysis Linear Static Structural Analysis Type(s): Element Solid Type(s): Test Case An assembly of three prismatic bars is supported at both end faces and is axially loaded with forces F1 and F2. Force F1 is applied on the face between Parts 2 and 3 and F2 is applied on the face between Parts 1 and 2. Apply advanced mesh control with element size of 0.5”. Find reaction forces in the Y direction at the fixed supports. Material Properties Geometric Properties Loading E = 2.9008e7 psi Cross section of Force F1 = ν = 0.3 all parts = 1” x -1000 (Y ρ = 0.28383 lbm/in3 1” direction) Length of Part Force F2 = -500 1 = 4" (Y direction) Length of Part 2 = 3" Length of Part 3 = 3” Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 83 VMMECH001 Figure 57: ANSYS Mechanical Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Y Reaction Force at Top 900 901.14 0.127 Fixed Support (lbf ) Y Reaction Force at Bottom 600 598.86 -0.190 Fixed Support (lbf ) Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 84 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH001 Figure 58: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Y Reaction Force at Top 900 901.09 0.121 Fixed Support (lbf ) Y Reaction Force at Bottom 600 598.91 0.182 Fixed Support (lbf ) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 85 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 86 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH002: Rectangular Plate with Circular Hole Subjected to Tensile Loading Overview Reference: J. E. Shigley, Mechanical Engineering Design, McGraw-Hill, 1st Edition, 1986, Table A-23, Figure A-23-1, pg. 673 Solver(s) ANSYS AIM ANSYS Mechanical Analysis Linear Static Structural Analysis Type(s): Element Solid Type(s): Test Case A rectangular plate with a circular hole is fixed along one of the end faces and a tensile pressure load is applied on the opposite face. A convergence with an allowable change of 10% is applied to account for the stress concentration near the hole. The Maximum Refinement Loops is set to 2 and the Refinement mesh control is added on the cylindrical surfaces of the hole with Refinement = 1. Find the Maximum Normal Stress in the x direction on the cylindrical surfaces of the hole. Material Properties Geometric Properties Loading E = 1000 Pa Length = 15 m Pressure = -100 ν=0 Width = 5 m Pa Thickness = 1 m Hole radius = 0.5 m Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 59: ANSYS Mechanical Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 87 VMMECH002 Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Maximum Normal X Stress 312.5 314.8 0.736 (Pa) Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 60: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum Normal X Stress 312.5 310.755 -0.559 (Pa) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 88 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH003: Modal Analysis of Annular Plate Overview Reference: R. J. Blevins, Formula for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company Inc., 1979, Table 11-2, Case 4, pg. 247 Solver(s) ANSYS AIM ANSYS Mechanical Analysis Free Vibration Analysis Type(s): Element Solid Type(s): Test Case An assembly of three annular plates has cylindrical support (fixed in the radial, tangential, and axial directions) applied on the cylindrical surface of the hole. Sizing control with element size of 0.5” is applied to the cylindrical surface of the hole. Find the first six modes of natural frequencies. Material Properties Geometric Properties Loading E = 2.9008e7 psi Inner diameter ν = 0.3 of inner plate = ρ = 0.28383 lbm/in3 20" Inner diameter of middle plate = 28" Inner diameter of outer plate = 34" Outer diameter of outer plate = 40" Thickness of all plates = 1" Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 89 VMMECH003 Figure 61: ANSYS Mechanical Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) 1st Frequency Mode (Hz) 310.911 310.14 -0.25 2nd Frequency Mode (Hz) 318.086 315.54 -0.80 3rd Frequency Mode (Hz) 318.086 315.59 -0.78 4th Frequency Mode (Hz) 351.569 346.79 -1.36 5th Frequency Mode (Hz) 351.569 347.07 -1.28 6th Frequency Mode (Hz) 442.451 437.17 -1.19 Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 90 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH003 Figure 62: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) 1st Frequency Mode (Hz) 310.911 310.194 -0.231 2nd Frequency Mode (Hz) 318.086 315.539 -0.800 3rd Frequency Mode (Hz) 318.086 315.674 -0.758 4th Frequency Mode (Hz) 351.569 346.725 -1.378 5th Frequency Mode (Hz) 351.569 346.814 -1.353 6th Frequency Mode (Hz) 442.451 435.321 -1.611 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 91 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 92 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH004: Viscoplastic Analysis of a Body (Shear Deformation) Overview Reference: B. Lwo and G. M. Eggert, "An Implicit Stress Update Algorithm Using a Plastic Predictor". Submitted to Computer Methods in Applied Mechanics and Engineering, January 1991. Solver(s) ANSYS Mechanical Analysis Nonlinear Structural Analysis Type(s): Element Solid Type(s): Test Case A cubic shaped body made up of a viscoplastic material obeying Anand's law undergoes uniaxial shear deformation at a constant rate of 0.01 cm/s. The temperature of the body is maintained at 400°C. Find the shear load (Fx) required to maintain the deformation rate of 0.01 cm/sec at time equal to 20 seconds. Figure 63: Schematic Material Properties Geometric Properties Loading Ex (Young's Modulus) = h = 1 cm Temp = 400°C 60.6 GPa thickness = 1 = 673°K (Poisson's Ratio) = cm Velocity 0.4999 (x-direction) = So = 29.7 MPa 0.01 cm/sec @ Q/R = 21.08999E3 K y = 1 cm Time = 20 sec A = 1.91E7 s-1 = 7.0 m = 0.23348 ho = 1115.6 MPa = 18.92 MPa = 0.07049 a = 1.3 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 93 VMMECH004 Results Comparison Results Target Mechanical Error (%) Fx, N -845.00 -791.76 -6.3 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 94 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH005: Heat Transfer in a Composite Wall Overview Reference: F. Kreith, Principles of Heat Transfer, Harper and Row Publisher, 3rd Edition, 1976, Example 2-5, pg. 39 Solver(s): ANSYS AIM ANSYS Mechanical Analysis Linear Static Thermal Analysis Type(s): Element Solid Type(s): Test Case A furnace wall consists of two layers: fire brick and insulating brick. The temperature inside the furnace is 3000°F (Tf) and the inner surface convection coefficient is 3.333 x 10-3 BTU/s ft2°F (hf). The ambient temperature is 80°F (Ta) and the outer surface convection coefficient is 5.556 x 10-4 BTU/s ft2°F (ha). Find the Temperature Distribution. Material Properties Geometric Properties Loading Fire brick wall: k = Cross-section = -4 1" x 1" 2.222 x 10 BTU/s ft °F Fire brick wall Insulating wall: k = thickness = 9" Insulating wall 2.778 x 10-5 BTU/s thickness = 5" ft °F Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 64: ANSYS Mechanical Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Minimum Temperature (°F) 336 336.68 0.202 Maximum Temperature (°F) 2957 2957.2 0.007 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 95 VMMECH005 Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 65: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Minimum Temperature (°F) 336 336.68 0.202 Maximum Temperature (°F) 2957 2957.2 0.007 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 96 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH006: Heater with Nonlinear Conductivity Overview Reference: Vedat S. Arpaci, Conduction Heat Transfer, Addison-Wesley Book Series, 1966, pg. 130 Solver(s): ANSYS AIM ANSYS Mechanical Analysis Type(s): Nonlinear Static Thermal Analysis Element Type(s): Solid Test Case A liquid is boiled using the front face of a flat electric heater plate. The boiling temperature of the liquid is 212°F. The rear face of the heater is insulated. The internal energy generated electrically may be as- sumed to be uniform and is applied as internal heat generation. Find the maximum temperature and maximum total heat flux. Material Properties Geometric Properties Loading k = [0.01375 * (1 + 0.001 T)] Radius = 3.937” Front face BTU/s in°F Thickness = 1” temperature = Temperature Conductivity 212°F (°F) (BTU/s in°F) Internal heat 32 1.419e-002 generation = 10 1000 2.75e-002 BTU/s in3 Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 66: ANSYS Mechanical Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 97 VMMECH006 Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Maximum Temperature (°F) 476 480.58 0.96 Maximum Total Heat Flux 10 9.9998 -0.002 (BTU/s in2) Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 67: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum Temperature (°F) 476 478.49 0.524 Maximum Total Heat Flux 10 9.914 -0.858 (BTU/s in2) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 98 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH007:Thermal Stress in a Bar with Temperature Dependent Conductivity Overview Reference: Any basic Heat Transfer book Solver(s): ANSYS AIM ANSYS Mechanical Analysis Nonlinear Thermal Stress Analysis Type(s): Element Solid Type(s): Test Case A long bar has thermal conductivity that varies with temperature. The bar is constrained at both ends by frictionless surfaces. A temperature of T °C is applied at one end of the bar (End A). The reference temperature is 5°C. At the other end, a constant convection of h W/m2 °C is applied. The ambient temperature is 5 °C. Advanced mesh control with element size of 2 m is applied. Find the following: • Minimum temperature • Maximum thermal strain in z direction (on the two end faces) • Maximum deformation in z direction • Maximum heat flux in z direction at z = 20 m Temperature at a distance z from rear face is given by: Thermal strain in the z-direction in the bar is given by: Deformation in the z-direction is given by: Heat flux in the z-direction is given by: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 99 VMMECH007 Material Properties Geometric Properties Loading E = 2 x 1011 Pa Length = 20 m Rear face ν=0 Width = 2 m temperature T Breadth = 2 m = 100 °C α = 1.5 x 10-5 / °C Film Coefficient k = 0.038 * (1 + 0.00582 * T) W/m h = 0.005 W/m2 °C °C Ambient Temperature Conductivity temperature = (°C) (W/m °C) 5 °C 5 3.91 x 10-2 Reference 800 0.215 temperature = 5 °C Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 68: ANSYS Mechanical Schematic Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 100 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH007 Figure 69: ANSYS AIM Structural Schematic Figure 70: ANSYS AIM Thermal Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 101 VMMECH007 Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Minimum Temperature (°C) 38.02 38.014 -0.016 Maximum Thermal strain (z 0.000495 0.0004952 0.042 = 20) (m/m) Maximum Thermal strain (z 0.001425 0.001425 0.000 = 0) (m/m) Maximum Z Deformation 0.00232 0.002341 0.905 (m) Maximum Z Heat Flux (z = 0.165 0.16507 0.042 20) (W/m2) Results Comparison for ANSYS AIM Results Target AIM Error (%) Minimum Temperature (°C) 38.02 38.009 -0.029 Maximum Thermal strain (z = 20) 0.000495 0.0004951 0.0272 (m/m) Maximum Thermal strain (z = 0) (m/m) 0.001425 0.001425 0.0 Maximum Z Deformation (m) 0.00232 0.00234 1.11 Maximum Z Heat Flux (z = 20) (W/m2) 0.165 0.1652 0.172 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 102 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH008: Heat Transfer from a Cooling Spine Overview Reference: Kreith, F., Principles of Heat Transfer, Harper and Row, 3rd Edition, 1976, Equation 2-44a, pg. 59, Equation 2–45, pg. 60 Solver(s): ANSYS AIM ANSYS Mechanical Analysis Linear Static Thermal Analysis Type(s): Element Solid Type(s): Test Case A steel cooling spine of cross-sectional area A and length L extend from a wall that is maintained at temperature Tw. The surface convection coefficient between the spine and the surrounding air is h, the air temper is Ta, and the tip of the spine is insulated. Apply advanced mesh control with element size of 0.025'. Find the heat conducted by the spine and the temperature of the tip. Figure 71: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 103 VMMECH008 Figure 72: Model Setup for ANSYS AIM Material Properties Geometric Properties Loading E = 4.177 x 109 psf Cross section = Tw = 100°F ν = 0.3 1.2” x 1.2” Ta = 0°F Thermal L = 8” h = 2.778 x 10-4 conductivity k = -3 BTU/s ft2 °F 9.71 x 10 BTU/s ft °F Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Temperature of the Tip (°F) 79.0344 79.078 0.055 -3 Heat Conducted by the 6.364 x10 6.3614 x -0.041 Spine (Heat Reaction) 10-3 (BTU/s) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 104 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH008 Results Comparison for ANSYS AIM Results Target AIM Error (%) Temperature of the Tip (°F) 79.0344 79.08 0.055 -3 -3 Heat Conducted by the 6.364 x 10 6.361 x 10 -0.041 Spine (Heat Reaction) (BTU/s) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 105 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 106 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH009: Stress Tool for Long Bar With Compressive Load Overview Reference: Any basic Strength of Materials book Solver(s): ANSYS Mechanical Analysis Linear Static Structural Analysis Type(s): Element Solid Type(s): Test Case A multibody of four bars connected end to end has one of the end faces fixed and a pressure is applied to the opposite face as given below. The multibody is used to nullify the numerical noise near the contact regions. Find the maximum equivalent stress for the whole multibody and the safety factor for each part using the maximum equivalent stress theory with tensile yield limit. Figure 73: Schematic Material Properties Material E (Pa) ν Tensile Yield (Pa) Part 1 1.93e11 0 2.07e8 Part 2 7.1e10 0 2.8e8 Part 3 2e11 0 2.5e8 Part 4 1.1e11 0 2.8e8 Geometric Properties Loading Part 1: 2 m x 2 Pressure = 2.5e8 mx3m Pa Part 2: 2 m x 2 m x 10 m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 107 VMMECH009 Part 3: 2 m x 2 mx5m Part 4: 2 m x 2 mx2m Results Comparison Results Target Mechanical Error (%) Maximum Equivalent Stress 2.5e8 2.5e8 0.000 (Pa) Safety Factor for Part 1 0.828 0.828 0.000 Safety Factor for Part 2 1.12 1.12 0.000 Safety Factor for Part 3 1 1 0.000 Safety Factor for Part 4 1.12 1.12 0.000 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 108 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH010: Modal Analysis of a Rectangular Plate Overview Reference: Blevins, Formula for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company Inc., 1979, Table 11-4, Case 11, pg. 256 Solver(s): ANSYS Mechanical Analysis Free Vibration Analysis Type(s): Element Shell Type(s): Test Case A rectangular plate is simply supported on both the smaller edges and fixed on one of the longer edges as shown below. Sizing mesh control with element size of 6.5 mm is applied on all the edges to get accurate results. Find the first five modes of natural frequency. Figure 74: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Length = 0.25 ν = 0.3 m ρ = 7850 kg/m3 Width = 0.1 m Thickness = 0.005 m Results Comparison Results Target Mechanical Error (%) 1st Frequency Mode (Hz) 595.7 590.03 -0.952 2nd Frequency Mode (Hz) 1129.55 1118.4 -0.987 3rd Frequency Mode (Hz) 2051.79 2038.1 -0.667 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 109 VMMECH010 Results Target Mechanical Error (%) 4th Frequency Mode (Hz) 2906.73 2879.3 -0.944 5th Frequency Mode (Hz) 3366.48 3350 -0.489 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 110 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH011: Large Deflection of a Circular Plate With Uniform Pressure Overview Reference: Timoshenko S.P., Woinowsky-Krieger S., Theory of Plates and Shells, McGraw-Hill, 2nd Edition, Article 97, equation 232, pg. 401 Solver(s): ANSYS Mechanical Analysis Nonlinear Structural Analysis (Large Deformation On) Type(s): Element Shell Type(s): Test Case A circular plate is subjected to a uniform pressure on its flat surface. The circular edge of the plate is fixed. To get accurate results, apply sizing control with element size of 5 mm on the circular edge. Find the total deformation at the center of the plate. Figure 75: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Radius = 0.25 Pressure = ν = 0.3 m 6585.18 Pa Thickness = 0.0025 m Results Comparison Results Target Mechanical Error (%) Total deformation (m) 0.00125 0.0012344 -1.248 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 111 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 112 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH012: Buckling of a Stepped Rod Overview Reference: Warren C. Young, Roark's Formulas for Stress & Strains, McGraw Hill, 6th Edition, Table 34, Case 2a, pg. 672 Solver(s): ANSYS Mechanical Analysis Buckling Analysis Type(s): Element Solid Type(s): Test Case A stepped rod is fixed at one end face. It is axially loaded by two forces: a tensile load at the free end and a compressive load on the flat step face at the junction of the two cross sections. To get accurate results, apply sizing control with element size of 6.5 mm. Find the Load Multiplier for the First Buckling Mode. Figure 76: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Larger diameter Force at free ν = 0.3 = 0.011982 m end = 1000 N Smaller Force at the flat diameter = step face = 0.010 m -2000 N Length of Both forces are larger diameter in the z = 0.2 m direction Length of smaller diameter = 0.1 m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 113 VMMECH012 Results Comparison Results Target Mechanical Error (%) Load Multiplier 22.5 22.749 1.1263 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 114 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH013: Buckling of a Circular Arch Overview Reference: Warren C. Young, Roark's Formulas for Stress Strains, McGraw Hill, 6th Edition, Table 34, Case 10, pg. 679 Solver(s): ANSYS Mechanical Analysis Buckling Analysis Type(s): Element Shell Type(s): Test Case A circular arch of a rectangular cross section (details given below) is subjected to a pressure load as shown below. Both the straight edges of the arch are fixed. Find the Load Multiplier for the first buckling mode. Figure 77: Schematic Material Properties Geometric Properties Loading E = 2e5 MPa Arch Pressure = 1 ν=0 cross-section = MPa 5 mm x 50 mm Mean radius of arch = 50 mm Included angle = 90° Results Comparison Results Target Mechanical Error (%) Load Multiplier 544 546.07 0.4 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 115 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 116 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH014: Harmonic Response of a Single Degree of Freedom System Overview Reference: Any basic Vibration Analysis book Solver(s): ANSYS Mechanical Analysis Harmonic Analysis Type(s): Element Solid Type(s): Test Case An assembly where four cylinders represent massless springs in series and a point mass simulates a spring mass system. The flat end face of the cylinder (Shaft 1) is fixed. Harmonic force is applied on the end face of another cylinder (Shaft 4) as shown below. Find the z directional Deformation Frequency Response of the system on the face to which force is applied for the frequency range of 0 to 500 Hz for the following scenarios using Mode-Superposition. Solution intervals = 20. • Scenario 1: Damping ratio = 0 • Scenario 2: Damping ratio = 0.05 Figure 78: Schematic Material Properties Material E (Pa) ν ρ (kg/m3) Shaft 1 1.1 x 1011 0.34 1 x 10-8 Shaft 2 1.1 x 1011 0.34 1 x 10-8 Shaft 3 4.5 x 1010 0.35 1 x 10-8 Shaft 4 4.5 x 1010 0.35 1 x 10-8 Geometric Properties Loading Each cylinder: Force = 1 x 107 N Diameter = 20 mm (Z-direction) Length = 50 mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 117 VMMECH014 Point Mass = 3.1044 Kg Results Comparison Results Target Mechanical Error (%) Maximum Amplitude 0.1404 0.1412 0.5698 without damping (m) Phase angle without 180 180 0.000 damping (degrees) Maximum Amplitude with 0.14 0.14078 0.557 damping (m) Phase angle with damping 175.6 175.58 -0.011 (degrees) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 118 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH015: Harmonic Response of Two Storied Building Under Transverse Loading Overview Reference: W. T. Thomson, Theory of Vibration with Applications, 3rd Edition, 1999, Example 6.4-1, pg. 166 Solver(s): ANSYS Mechanical Analysis Harmonic Analysis Type(s): Element Solid Type(s): Test Case A two-story building has two columns (2K and K) constituting stiffness elements and two slabs (2M and M) constituting mass elements. The material of the columns is assigned negligible density so as to make them as massless springs. The slabs are allowed to move only in the y direction by applying frictionless supports on all the faces of the slabs in the y direction. The end face of the column (2K) is fixed and a harmonic force is applied on the face of the slab (M) as shown in the figure below. Find the y directional Deformation Frequency Response of the system at 70 Hz on each of the vertices as shown below for the frequency range of 0 to 500 Hz using Mode-Superposition. Use Solution intervals = 50. Figure 79: Schematic Material Properties Material E (Pa) ν ρ (kg/m3) Block 2 2e18 0.3 7850 Shaft 2 4.5e10 0.35 1e-8 Block 1 2e18 0.3 15700 Shaft 1 9e10 0.35 1e-8 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 119 VMMECH015 Geometric Loading Properties Force = -1e5 N (y Block 1 and 2: direction) 40 mm x 40 mm x 40 mm Shaft 1 and 2: 20 mm x 20 mm x 200 mm Results Comparison Results Target Mechanical Error (%) Maximum Amplitude for 0.20853 0.2118 1.568 vertex A (m) Maximum Amplitude for 0.074902 0.075832 1.241 vertex B (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 120 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH016: Fatigue Tool with Non-Proportional Loading for Normal Stress Overview Reference: Any basic Machine Design book Solver(s): ANSYS Mechanical Analysis Fatigue Analysis Type(s): Element Solid Type(s): Test Case A bar of rectangular cross section has the following loading scenarios. • Scenario 1: One of the end faces is fixed and a force is applied on the opposite face as shown below in Figure 80: Scenario 1 (p. 121). • Scenario 2: Frictionless support is applied to all the faces of the three standard planes (faces not seen in Figure 81: Scenario 2 (p. 121)) and a pressure load is applied on the opposite faces in positive y- and z-directions. Find the life, damage, and safety factor for the normal stresses in the x, y, and z directions for non- proportional fatigue using the Soderberg theory. Use a design life of 1e6 cycles, a fatigue strength factor or 1, a scale factor of 1, and 1 for coefficients of both the environments under Solution Combination. Figure 80: Scenario 1 Figure 81: Scenario 2 Material Properties E = 2e11 Pa ν = 0.3 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 121 VMMECH016 Material Properties Ultimate Tensile Strength = 4.6e8 Pa Yield Tensile Strength = 3.5e8 Pa Endurance Strength = 2.2998e6 Pa Number of Alternating Stress Cycles (Pa) 1000 4.6e8 1e6 2.2998e6 Geometric Loading Properties Scenario 1: Force Bar: 20 m x 1 m = 2e6 N x 1m (y-direction) Scenario 2: Pressure = -1e8 Pa Analysis Non-proportional fatigue uses the corresponding results from the two scenarios as the maximum and minimum stresses for fatigue calculations. The fatigue calculations use standard formulae for the Soderberg theory. Results Comparison Results Target Mechanical Error (%) Stress Component - Component Life 3335.1049 3329.9 -0.156 X Damage 299.8406 300.31 0.157 Safety 0.019 0.019025 0.132 Factor Stress Component - Component Life 14765.7874 14653 -0.764 Y Damage 67.724 68.247 0.772 Safety 0.04569 0.045378 -0.683 Factor Stress Component - Component Life 14765.7874 14766 0.001 Z Damage 67.724 67.725 0.001 Safety 0.04569 0.045696 0.013 Factor Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 122 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH017: Thermal Stress Analysis with Remote Force and Thermal Loading Overview Reference: Any basic Strength of Materials book Solver(s): ANSYS AIM ANSYS Mechanical Analysis Linear Thermal Stress Analysis Type(s): Element Solid Type(s): Test Case A cylindrical rod assembly of four cylinders connected end to end has frictionless support applied on all the cylindrical surfaces and both the flat end faces are fixed. Other thermal and structural loads are as shown below. Find the Deformation in the x direction of the contact surface on which the remote force is applied. To get accurate results apply a global element size of 1.5 m. Material Properties Geometric Properties Loading E = 2 x 1011 Pa Diameter = 2 m Given temperature ν=0 Lengths of (End A) = 1000°C cylinders in Given temperature α = 1.2 x 10-5/°C order from End (End B) = 0°C A: 2 m, 5 m, 10 Remote force = 1 m, and 3 m. x 1010 N applied on the contact surface at a distance 7 m from end A. Location of remote force = (7,0,0) m Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 82: ANSYS Mechanical Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 123 VMMECH017 Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Maximum X Deformation 0.101815 0.10025 -1.5 (m) Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 83: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum X Deformation 0.101815 0.10237 0.546 (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 124 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH018: A Bar Subjected to Tensile Load with Inertia Relief Overview Reference: Any basic Strength of Materials book Solver(s): ANSYS AIM ANSYS Mechanical Analysis Linear Static Structural Analysis (Inertia Relief On) Type(s): Element Solid Type(s): Test Case A long bar assembly is fixed at one end and subjected to a tensile force at the other end as shown below. Turn on Inertia Relief. Find the deformation in the z direction Analysis where: L = total length of bar A = cross-section m = mass Material Properties Geometric Properties Loading E = 2 x 1011 Pa Cross-Section = Force P = 2 x 105 ν = 0.3 2mx2m N (positive z Lengths of bars ρ = 7850 kg/m3 direction) in order from End A: 2 m, 5 m, 10 m, and 3 m. Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 125 VMMECH018 Figure 84: ANSYS Mechanical Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) -6 Maximum Z Deformation 2.5 x 10 2.5043 x 0.172 (m) 10-6 Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 85: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) -6 Maximum Z Deformation 2.5 x 10 2.4757 x -0.968 (m) 10-6 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 126 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH019: Mixed Model Subjected to Bending Loads with Solution Combination Overview Reference: Any basic Strength of Materials book Solver(s): ANSYS Mechanical Analysis Linear Static Structural Analysis Type(s): Element Beam and Shell Type(s): Test Case A mixed model (shell and beam) has one shell edge fixed as shown below. Bending loads are applied on the free vertex of the beam as given below. Apply a global element size of 80 mm to get accurate results. • Scenario 1: Only a force load. • Scenario 2: Only a moment load. Find the deformation in the y direction under Solution Combination with the coefficients for both the environments set to 1. Figure 86: Scenario 1 Figure 87: Scenario 2 Material Properties Geometric Properties Loading E = 2e5 Pa Shell = 160 mm Force F = -10 N (y ν=0 x 500 mm x 10 direction) mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 127 VMMECH019 Material Properties Geometric Properties Loading Beam Moment M = rectangular -4035 Nmm @ cross section = z-axis 10 mm x 10 mm Beam length = 500 mm Analysis where: I = total bending length of the mixed model I = moment of inertia of the beam cross-section Results Comparison Results Target Mechanical Error (%) Maximum Y-Deformation -7.18742 -7.2542 0.929 (mm) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 128 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH020: Modal Analysis for Beams Overview Reference: Any basic Vibration Analysis book Solver(s): ANSYS Mechanical Analysis Modal Analysis Type(s): Element Beam Type(s): Test Case Two collinear beams form a spring mass system. The density of the longer beam is kept very low so that it acts as a massless spring and the smaller beam acts as a mass. The end vertex of the longer beam (acting as a spring) is fixed. The cross section details are as shown below. Find the natural frequency of the axial mode. Figure 88: Cross Section Details for Both Beams Figure 89: Schematic Material Properties Material E (Pa) ν ρ (kg/m3) Spring 1.1e11 0.34 1e-8 Mass 2e11 0 7.85e5 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 129 VMMECH020 Geometric Properties Loading Spring beam length = 500 mm Mass beam length = 5 mm Results Comparison Results Target Mechanical Error (%) Natural Frequency of Axial 1188.6 1190.5 0.160 Mode (Hz) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 130 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH021: Buckling Analysis of Beams Overview Reference: Warren C. Young, Roark's Formulas for Stress and Strains, McGraw Hill, 6th Edition, Table 34, Case 3a, pg. 675 Solver(s): ANSYS Mechanical Analysis Buckling Analysis Type(s): Element Beam Type(s): Test Case A beam fixed at one end and is subjected to two compressive forces. One of the forces is applied on a portion of the beam of length 50 mm (L1) from the fixed end and the other is applied on the free vertex, as shown below. Find the load multiplier for the first buckling mode. Figure 90: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa L1 = 50 mm Force on L1 = ν = 0.3 Total length = -1000 N (x 200 mm direction) Rectangular Force on free cross section = vertex = -1000 N (x 10 mm x 10 direction) mm Results Comparison Results Target Mechanical Error (%) Load Multiplier 10.2397 10.198 -0.407 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 131 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 132 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH022: Structural Analysis with Advanced Contact Options Overview Reference: Any basic Strength of Material book Solver(s): ANSYS Mechanical Analysis Nonlinear Static Structural Analysis Type(s): Element Solid Type(s): Test Case An assembly of two parts with a gap has a Frictionless Contact defined between the two parts. The end faces of both the parts are fixed and a given displacement is applied on the contact surface of Part 1 as shown below. Find the Normal stress and Directional deformation - both in the z direction for each part for the following scenarios: • Scenario 1: Interface treatment - adjust to touch. • Scenario 2: Interface treatment - add offset. Offset = 0 m. • Scenario 3: Interface treatment - add offset. Offset = 0.001 m. • Scenario 4: Interface treatment - add offset. Offset = -0.001 m. Validate all of the above scenarios for Augmented Lagrange and Pure Penalty formulations. Figure 91: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Gap = 0.0005 m Given ν=0 Dimensions for displacement = (0, each part: 0.1 0, 0.0006) m m x 0.1 m x 0.5m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 133 VMMECH022 Results Comparison The same results are obtained for both Augmented Lagrange and Pure Penalty formulations. Results Target MechanicalError (%) Adjust To Touch Maximum directional z 6e-4 6e-4 0.000 deformation Part 1 (m) Maximum directional z 6e-4 5.9786e-4 -0.357 deformation Part 2 (m) Maximum normal z stress 2.4e8 2.4e8 0.000 Part 1 (Pa) Maximum normal z stress -2.4e8 -2.3915e8 -0.354 Part 2 (Pa) Add Offset. Offset = 0 m Maximum directional z 6e-4 6e-4 0.000 deformation Part 1 (m) Maximum directional z 1e-4 0.99644e-4 -0.356 deformation Part 2 (m) Maximum normal z stress 2.4e8 2.4e8 0.000 Part 1 (Pa) Maximum normal z stress -4e7 -3.9858e7 -0.355 Part 2 (Pa) Add Offset. Offset = Maximum directional z 6e-4 6e-4 0.000 0.001 m deformation Part 1 (m) Maximum directional z 1.1e-3 1.0961e-3 -0.355 deformation Part 2 (m) Maximum normal z stress 2.4e8 2.4e8 0.000 Part 1 (Pa) Maximum normal z stress -4.4e8 -4.3843e8 -0.357 Part 2 (Pa) Add Offset. Offset = Maximum directional z 6e-4 6e-4 0.000 -0.001 m deformation Part 1 (m) Maximum directional z 0 0 0.000 deformation Part 2 (m) Maximum normal z stress 2.4e8 2.4e8 0.000 Part 1 (Pa) Maximum normal z stress 0 0 0 Part 2 (Pa) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 134 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH023: Curved Beam Assembly with Multiple Loads Overview Reference: Any basic Strength of Materials book Solver(s): ANSYS Mechanical Analysis Linear Static Structural Analysis Type(s): Element Beam Type(s): Test Case An assembly of two curved beams, each having an included angle of 45°, has a square cross-section. It is fixed at one end and at the free end a Force F and a Moment M are applied. Also, a UDL of "w " N / mm is applied on both the beams. Use a global element size of 30 mm to get accurate results. See the figure below for details. Find the deformation of the free end in the y direction. Figure 92: Schematic Equivalent Loading: Material Properties Geometric Properties Loading Beam 1: For each beam: Force F = -1000 N E1 = 1.1e5 MPa Cross-section = (y direction) ν1 = 0 10 mm x 10 Moment M = mm -10000 Nmm (about z-axis) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 135 VMMECH023 Material Properties Geometric Properties Loading ρ1 = 8.3e-6 Radius r = 105 UDL w = -5 N/mm kg/mm 3 mm (y direction) on Beam 2: Included angle both beams E2 = 2e5 MPa = 45° This UDL is applied as an edge force ν2 = 0 on each beam ρ2 = 7.85e-6 with magnitude = kg/mm3 -5 (2 x 3.14 x 105) / 8 = -412.334 N Analysis The deflection in the y direction is in the direction of the applied force F and is given by: where: δ = deflection at free end in the y direction I = moment of inertia of the cross-section of both beams Results Comparison Results Target Mechanical Error (%) Minimum Y Deformation -8.416664 -8.4688 0.619 (mm) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 136 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH024: Harmonic Response of a Single Degree of Freedom System for Beams Overview Reference: Any basic Vibration Analysis book Solver(s): ANSYS Mechanical Analysis Harmonic Analysis Type(s): Element Beam Type(s): Test Case Two collinear beams form a spring-mass system. The density of the longer beam is kept very low so that it acts as a massless spring and the smaller beam acts as a mass. The end vertex of the longer beam (acting as a spring) is fixed. A Harmonic force F is applied on the free vertex of the shorter beam in z direction. Both beams have hollow circular cross-sections, as indicated below. • Scenario 1: Damping ratio = 0 • Scenario 2: Damping ratio = 0.05 Find the z directional deformation of the vertex where force is applied at frequency F = 500 Hz for the above scenarios with solution intervals = 25 and a frequency range of 0 to 2000 Hz. Use both Mode Superposition and Full Method. Figure 93: Schematic Material Properties Material E ν ρ (Pa) (kg/m3) Spring 1.1e11 0.34 1e-8 Mass 2e11 0 7.85e5 Geometric Properties Loading Cross-section of Harmonic force F each beam: = 1 e6 N Outer radius = (z-direction) 10 mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 137 VMMECH024 Geometric Properties Loading Inner radius = 5 mm Length of longer beam = 100 mm Length of shorter beam = 5 mm Results Comparison Results Target MechanicalError (%) Mode-Superposition Maximum z directional 4.11332 4.078 x -0.859 deformation without damping x 10-3 10-3 (m) Maximum z directional 4.11252 4.0765 x -0.876 deformation with damping (m) x 10-3 10-3 Full Method Maximum z directional 4.11332 4.1132 x -0.003 deformation without damping x 10-3 10-3 (m) Maximum z directional 4.11252 4.1022 x -0.251 deformation with damping (m) x 10-3 10-3 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 138 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH025: Stresses Due to Shrink Fit Between Two Cylinders Overview Reference: Stephen P. Timoshenko, Strength of Materials, Part 2 - Advanced Theory and Problems, 3rd Edition, pg. 208-214 Solver(s): ANSYS AIM ANSYS Mechanical Analysis Linear Static Structural Analysis Type(s): Element Solid Type(s): Test Case One hollow cylinder is shrink fitted inside another. Both cylinders have length L and both the flat faces of each cylinder are constrained in the axial direction. They are free to move in radial and tangential directions. An internal pressure of P is applied on the inner surface of the inner cylinder. To get accurate results, apply a global element size of 0.8 inches. Find the maximum tangential stresses in both cylinders. Material Properties Geometric Properties Loading Both cylinders Inner Cylinder: P = 30000 psi are made of ri = 4” the same ro = 6.005” material Ri = 6” E = 3 x 107 psi Ro = 8” ν=0 Length of both ρ = 0.28383 cylinders = 5” lbm/in3 Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 139 VMMECH025 Figure 94: Schematic Note Tangential stresses can be obtained in the Mechanical application using a cylindrical coordin- ate system. To simulate interference, set Contact Type to Rough with interface treatment set to add offset with Offset = 0. Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 140 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH025 Figure 95: ANSYS AIM Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Maximum normal y stress, inner 35396.67 35758 1.020802 cylinder (psi) Maximum normal y stress, outer 42281.09 42297 0.037629 cylinder (psi) Note Here y corresponds to θ direction of a cylindrical coordinate system. Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum Normal Y Stress, Inner Cylinder (psi) 35396.67 35824.9 1.210 Maximum Normal Y Stress, Outer Cylinder 42281.09 42327.45 0.110 (psi) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 141 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 142 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH026: Fatigue Analysis of a Rectangular Plate Subjected to Edge Moment Overview Reference: Any standard Machine Design and Strength of Materials book Solver(s): ANSYS AIM ANSYS Mechanical Analysis Fatigue Analysis Type(s): Element Shell Type(s): Test Case A plate of length L, width W, and thickness T is fixed along the width on one edge and a moment M is applied on the opposite edge about the z-axis. Find the maximum Bending Stress (Normal X Stress) and maximum Total Deformation of the plate. Also find the part life and the factor of safety using Goodman, Soderberg, & Gerber criteria. Use the x-stress component. Consider load type as fully reversed and a Design Life of 1e6 cycles, Fatigue Strength factor of 1, and Scale factor of 1. Figure 96: Schematic for ANSYS Mechanical Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 143 VMMECH026 Figure 97: Schematic for ANSYS AIM Material Properties Geometric Properties Loading E = 2 x 1011 Pa Length L = 12 x Moment M = 0.15 Nm -3 (counterclockwise @ z-axis) ν = 0.0 10 m Ultimate tensile Width W = 1 x strength = 1.29 x 109 10-3 m Pa Thickness T = 1 x Endurance strength = 10-3 m 8 1.38 x 10 Pa Yield Strenth = 2.5 x 108 Pa No. of Alternating Cycles Stresses (Pa) 1000 1.08 x 109 1 x 106 1.38 x 108 Results Comparison for ANSYS Mechanical Results Target MechanicalError (%) Maximum normal x-stress (Pa) 9 x108 9 x 108 0.000 Maximum total deformation (m) 6.48 x 6.4984 x 0.284 10-4 10-4 SN-Goodman Safety factor 0.1533 0.15333 0.020 Life 1844.3 1844.4 0.005 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 144 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH026 Results Target MechanicalError (%) SN-Soderberg Safety factor 0.1533 0.15333 0.020 Life 1844.3 1844.4 0.005 SN-Gerber Safety Factor 0.1533 0.15333 0.020 Life 1844.3 1844.4 0.005 Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum normal x-stress (Pa) 9 x108 9 x 108 0.000 Maximum total deformation (m) 6.48 x 6.4976 x 0.272 10-4 10-4 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 145 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 146 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH027:Thermal Analysis for Shells with Heat Flow and Given Temperature Overview Reference: Any standard Thermal Analysis book Solver(s): ANSYS Mechanical Analysis Thermal Stress Analysis Type(s): Element Shell Type(s): Test Case A plate of length (L), width (W), and thickness (T) is fixed along the width on one edge and heat flow (Q) is applied on the same edge. The opposite edge is subjected to a temperature of 20 °C. Ambient temperature is 20 °C. To get accurate results, apply a sizing control with element size = 2.5e-2 m. Find the maximum temperature, maximum total heat flux, maximum total deformation, and heat reaction at the given temperature. Figure 98: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Length L = 0.2 Heat flow Q = 5 W ν = 0.0 m Given Temperature Coefficient of Width W = 0.05 = 20°C thermal m expansion α = Thickness T = 1.2e-5/°C 0.005 m Thermal conductivity k = 60.5 W/m°C Analysis Heat Reaction = -(Total heat generated) Heat flow due to conduction is given by: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 147 VMMECH027 where: Th = maximum temperature T1 = given temperature Total heat flux is: Temperature at a variable distance z from the fixed support is given by: Thermal deformation in the z-direction is given by: Results Comparison Results Target MechanicalError (%) Maximum Temperature (°C) 86.1157 86.116 0.000 2 Maximum Total Heat Flux (W/m ) 2e4 2e4 0.000 Maximum Total Deformation (m) 7.93386e-5 7.9958e-5 0.781 Heat Reaction (W) -5 -5 0.000 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 148 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH028: Bolt Pretension Load Applied on a Semi-Cylindrical Face Overview Reference: Any standard Strength of Materials book Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Solid Type(s): Test Case A semi-cylinder is fixed at both the end faces. The longitudinal faces have frictionless support. A bolt pretension load is applied on the semi-cylindrical face. To get accurate results, apply sizing control with element size of 0.01 m. Find the Z directional deformation and the adjustment reaction due to the bolt pretension load. Figure 99: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Length L = 1 m Pretension as ν = 0.0 Diameter D = preload = 19.635 0.05 m N (equal to adjustment of 1e-7 m) Analysis The bolt pretension load applied as a preload is distributed equally to both halves of the bar. Therefore the z-directional deformation due to pretension is given by: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 149 VMMECH028 Results Comparison Results Target MechanicalError (%) Minimum z-directional deformation -5.00E-08 -5.0002E-080.004 (m) Maximum z-directional deformation 5.00E-08 4.9502E-08 -0.996 (m) Adjustment Reaction (m) 1.00E-07 1.00E-07 0.000 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 150 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH029: Elasto-Plastic Analysis of a Rectangular Beam Overview Reference: Timoshenko S., Strength of Materials, Part II, Advanced Theory and Problems, Third Edition, Article 64, pp. 349 Solver(s): ANSYS Mechanical Analysis Static Plastic Analysis Type(s): Element Solid Type(s): Test Case A rectangular beam is loaded in pure bending. For an elastic-perfectly-plastic stress-strain behavior, show that the beam remains elastic at M = Myp = σypbh2 / 6 and becomes completely plastic at M = Mult = 1.5 Myp. To get accurate results, set the advanced mesh control element size to 0.5 inches. Figure 100: Stress-Strain Curve Figure 101: Schematic Material Properties Geometric Properties Loading E = 3e7 psi Length L = 10” M = 1.0 Myp to 1.5 ν = 0.0 Width b = 1” Myp σyp = 36000 psi Height h = 2” (Myp = 24000 lbf - in) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 151 VMMECH029 Analysis The load is applied in three increments: M1 = 24000 lbf-in, M2 = 30000 lbf-in, and M3 = 36000 lbf-in. Results Comparison M/Myp Target Mechanical Error (%) State Equivalent State Equivalent Stress (psi) Stress (psi) 1 fully 36000 fully 36059 0.164 elastic elastic 1.25 elastic-plastic 36000 elastic-plastic 36288 0.800 1.5 plastic solution not plastic solution not - converged converged Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 152 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH030: Bending of Long Plate Subjected to Moment - Plane Strain Model Overview Reference: Any standard Strength of Materials book Solver(s): ANSYS Mechanical Analysis Plane Strain Analysis Type(s): Element 2D Structural Solid Type(s): Test Case A long, rectangular plate is fixed along the longitudinal face and the opposite face is subjected to a moment of 5000 lbf-in about the z-axis. To get accurate results, set the advanced mesh control element size to 0.5 inches. Find X normal stress at a distance of 0.5 inches from the fixed support. Also find total deformation and reaction moment. Figure 102: Schematic Material Properties Geometric Properties Loading E = 2.9e7 psi Length L = Moment M = 5000 ν = 0.0 1000” lbf-in Width W = 40” Thickness T = 1” Analysis Since the loading is uniform and in one plane (the x-y plane), the above problem can be analyzed as a plane strain problem. Therefore, the moment applied will be per unit length (5000/1000 = 5 lbf-in). Analysis takes into account the unit length in the z-direction. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 153 VMMECH030 Figure 103: Plane Strain Model (analyzing any cross section (40” x 1”) along the length) Results Comparison Results Target Mechanical Error (%) Normal Stress 30 30 0.000 Maximum Normal Stress in 30 30 0.000 the X-Direction (psi) Maximum Total Deformation 0.1655e-2 0.16553e-2 0.018 (in) Reaction Moment (lbf-in) -5 -5 0.000 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 154 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH031: Long Bar With Uniform Force and Stress Tool - Plane Stress Model Overview Reference: Any standard Strength of Materials book Solver(s): ANSYS Mechanical Analysis Type(s): Plane Stress Analysis Element Type(s): 2D Structural Solid Test Case A long, rectangular bar assembly is fixed at one of the faces and the opposite face is subjected to a compressive force. To get accurate results, set the advanced mesh control element size to 1 m. Find the maximum equivalent stress for the whole assembly and safety factor, safety margin, and safety ratio for the first and last part using the maximum equivalent stress theory with Tensile Yield Limit. Figure 104: Schematic Material Properties Material E (Pa) ν Tensile Yield (Pa) Part 1 1.93e11 0 2.07e8 Part 2 7.1e10 0 2.8e8 Part 3 2e11 0 2.5e8 Part 4 1.1e11 0 2.8e8 Geometric Properties Loading Part 1: 2 mx2mx Force = 1e9 N in 3m the negative Part 2: 2 mx2mx x-direction 10 m Part 3: 2 mx2mx 5m Part 4: 2 mx2mx 2m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 155 VMMECH031 Analysis Since the loading is uniform and in one plane, the above problem can be analyzed as a plane stress problem. Analysis is done considering thickness of 2 m along z-direction Figure 105: Plane Stress Model (Analyzing any cross section along Z) Results Comparison Results Target Mechanical Error (%) Maximum Equivalent Stress (Pa) 2.5e8 2.5e8 0.000 Part 1 Safety Factor 0.828 0.828 0.000 Safety Margin -0.172 -0.172 0.000 Safety Ratio 1.207 1.2077 0.058 Part 4 Safety Factor 1.12 1.12 0.000 Safety Margin 0.12 0.12 0.000 Safety Ratio 0.892 0.89286 0.096 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 156 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH032: Radial Flow due to Internal Heat Generation in a Copper Disk - Axisymmetric Model Overview Reference: Any basic Heat Transfer book Solver(s): ANSYS AIM ANSYS Mechanical Analysis Axisymmetric Analysis Type(s): Element 2D Structural Solid Type(s): Test Case A copper disk with thickness t and radii Ri and Ro is insulated on the flat faces. It has a heat-generating copper coaxial cable (of radius Ri) passing through its center. The cable delivers a total heat flow of Q to the disk. The surrounding air is at a temperature of To with convective film coefficient h. To get ac- curate results, set the advanced mesh control element size to 0.002 m. Find the disk temperature and heat flux at inner and outer radii. Figure 106: Schematic Material Properties Geometric Properties Loading E = 1.1 x 1011 Pa Ri = 10 mm Q = 100 W (Internal ν = 0.34 Ro = 60 mm Heat Generation = Thermal t = 8 mm 39788735.77 W/m3) conductivity k = Film coefficient h = 401.0 W/m-°C 1105 W/m2-°C Surrounding temperature To = 0°C Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 157 VMMECH032 Analysis Assumptions and Modeling Notes for ANSYS Mechanical Because the geometry and loading are symmetric about the y-axis, the above problem can be analyzed as an axisymmetric problem in ANSYS Mechanical. Figure 107: Axisymmetric Model Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 108: Model in ANSYS AIM Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Maximum Temperature (°C) 38.9 38.896 -0.010 Minimum Temperature (°C) 30 30.007 0.023 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 158 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH032 Results Target Mechanical Error (%) 2 Maximum Heat Flux (W/m ) 198943 197840 -0.554 2 Minimum Heat Flux (W/m ) 33157 33151 -0.018 Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum Disk Temperature 38.9 38.895 -0.011 (C) Minimum Disk Temperature 30.0 30.007 0.022 (C) Maximum Heat Flux (W/m2) 198943 198913 -0.015 Minimum Heat Flux (W/m2) 33157 33147 -0.031 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 159 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 160 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH033: Spring Mass System Subjected to Enforced Motion with Displacement Base Excitations Overview Reference: Thompson, W.T., Theory of Vibration with Applications, 3rd Edition, Chapter 3, pp. 63-65, 1999 Solver(s): ANSYS Mechanical Analysis Harmonic Analysis Type(s): Element Spring-Damper Type(s): Surface Test Case A vehicle has a mass of 500 kg (applied as distributed mass) and the total spring constant of its suspen- sion system is 19600 N/m. The profile of the road is approximated as a sine wave of amplitude 10 mm and a wavelength of 1.5 m. Determine the amplitude of oscillations of the mass: • When driven at critical speed and having damping factor of 0.5 • When driven at 50 km/h and having damping factor of 0.4 Figure 109: Schematic Material Properties Geometric Properties Loading E = 1 x 1013 Pa Mass of vehicle, m = Sinusoidal base excitation of -20 3 500 kg amplitude Y = 10 mm and = 1 x 10 kg/m wavelength = 1.5 m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 161 VMMECH033 Material Properties Geometric Properties Loading Stiffness of spring, K = 19600 N/m Analysis Natural circular frequency of the system, Critical damping coefficient, Damping ratio, Circular frequency of forced vibration, Absolute amplitude of vibration, Relative amplitude of vibration, Absolute phase angle, Relative phase angle, where m = mass of vehicle V = speed of vehicle Y = amplitude of sine wave = wavelength C = damping coefficient Results Comparison Results Target Mechanical Error (%) = 14.142 mm = 14.142 mm 0% Critical speed ( = 0.99647 Hz), damping 0% = -45° or 135° = -45° factor ( ) of 0.5 = 10.00 mm = 10.00 mm 0% = -90° or 90° = -90° 0% 50 km/h ( = 9.26 Hz), = 0.876 mm = 0.875 mm 0% damping factor ( ) of = -87.32° or -92.68° = -92.68° 0% 0.4 = 10.079 mm = 10.079 mm 0% -4.98° or -175.02° = -175.03° 0% Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 162 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH034: Rubber Cylinder Pressed Between Two Plates Overview Reference: T. Tussman, K.J. Bathe, "A Finite Element Formulation for Nonlinear Incompressible Elastic and Inelastic Analysis", Computers and Structures, Vol. 26 Nos 1/2, 1987, pp. 357-409 Solver(s): ANSYS Mechanical Analysis Nonlinear Static Structural Analysis (Large Deformation ON) Type(s): Element Solid Type(s): Test Case A rubber cylinder is pressed between two rigid plates using a maximum imposed displacement of δmax. Determine the total deformation. Figure 110: Schematic Material Properties Geometric Properties Loading Solid1: Solid1: Displacement in Y direction = -0.1 m E = 2 x 1011 Pa 0.05 m x 0.01 m x 0.4 ν = 0.3 m ρ = 7850 kg/m3 Solid2: Mooney-Rivlin Constants Solid2: Quarter Circular Cylinder C10 = 2.93 x 105 Pa Radius = 0.2 m Length = 0.05 m C01 = 1.77 x 105 Pa Incompressibility Parameter D1 1/Pa = 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 163 VMMECH034 Analysis Due to geometric and loading symmetry, the analysis can be performed using one quarter of the cross section. • Frictionless supports are applied on 3 faces (X = 0, Z = 0 and Z = 0.05 m). • Given displacement of 0.1 m is applied on the top surface. • The bottom surface of Solid1 is completely fixed. • Frictionless Contact with Contact stiffness factor of 100 is used to simulate the rigid target. • Augmented Lagrange is used for Contact formulation. Results Comparison Results Target Mechanical Error (%) Total Deformation (m) 0.165285 0.16528 -0.003025 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 164 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH035: Thermal Stress in a Bar With Radiation Overview Reference: Any Basic Heat transfer and Strength of Materials book Solver(s): ANSYS AIM ANSYS Mechanical Analysis Coupled Analysis (Static Thermal and Static Stress) Type(s): Element Solid Type(s): Test Case Heat of magnitude 2500 W and Heat Flux of magnitude 625 W/m2 is flowing through a long bar (2 x 2 x 20) m in an axial direction, and radiating out from the other face having emissivity 0.3; Ambient temperature is maintained at 20 °C. Find the following: • Temperatures on End Faces. • Thermal strain and Directional deformation and Normal Stress in Z direction if both the end faces have frictionless supports and Reference temperature of 22°C. Analysis (Heat flowing through body) Q = (Heat Flow) + (Heat Flux * Area) = 5000 W (Heat flowing through body) = (Heat Conducted through body) = (Heat Radiated out of the Surface) i.e. Q = Qr =QC = 5000 W. Heat Radiated out of the body gives T2 = 260.16 °C. Heat Conducted through the body gives T1 = 673.38 °C. Thermal strain is given by: The compressive stress introduced is given by: Temperature at a distance z from the face with higher temperature is given by: Only half-length is considered for calculating deformation, since deformation is symmetric: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 165 VMMECH035 Material Properties Geometric Properties Loading E = 2.0 x 1011 Pa Part 1: 2 mx2mx Heat Flow = 2500W on Part v=0 2m 4 Part 2: 2 mx2mx Heat Flux = 625 W/m2 on α = 1.2 x 10-5 1/°C 5m Part 4 k = 60.5 W/m°C Part 3: 2 mx2mx Radiation = 20 °C, 0.3 10 m Part 4: 2 mx2mx 3m Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 111: Schematic in ANSYS Mechanical Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Temperature on Part 4 (°C) 673.38 673.49 0.016 Temperature on Part 1 (°C) 260.16 260.15 -0.004 Maximum Thermal Strain 7.81656 x 7.8179 x 0.017 (m/m) 10-3 10-3 Minimum Thermal Strain 2.85792 x 0.0028578 -0.004 (m/m) 10-3 Normal Stress in Z direction -1.067448 x -1.0183 x -4.6 (Pa) 109 109 Directional Deformation in -0.0123966 -0.012572 1.4 Z direction (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 166 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH035 Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 112: Schematic in ANSYS AIM Results Comparison for ANSYS AIM Results Target AIM Error (%) Temperature on Part 4 (°C) 673.38 673.48749 -3.730 x 10-4 Temperature on Part 1 (°C) 260.15 260.26 0.044 Maximum Thermal Strain 7.81656 x 7.8179 x 0.017 (m/m) 10-3 10-3 Minimum Thermal Strain 2.85792 x 0.0028592 0.044 (m/m) 10-3 Directional Deformation in -0.0123966 -0.012562 1.334 Z direction (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 167 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 168 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH036: Thermal Stress Analysis of a Rotating Bar Using Temperature Dependant Density Overview Reference: Any Basic Strength of Materials book Solver(s): ANSYS Mechanical Analysis Static Stress Analysis (Sequence Loading) Type(s): Element Solid Type(s): Test Case A Bar (2 m x 2m x 20m) with one end fixed and with a rotational velocity about X axis at location (1, 1, 0) is subjected to a Uniform Temperature (Thermal Condition Load) in three steps. For all the steps, Reference Temperature is 0°C. Frictionless Support is applied on all the longitudinal faces. Figure 113: Schematic Material Properties Geometric Loading Properties E = 1 x 106 Pa Part 1: Rotational Velocity (rad/s) in -5 2mx2 steps: α = 1 x 10 1/°C m x 20 ν=0 1. (1, 0, 0) m Temperature °C Density kg/m3 2. (0.5, 0, 0) 50 30 100 60 3. (0.25, 0, 0) 150 90 Thermal Condition °C 1. 50°C 2. 100°C 3. 150°C Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 169 VMMECH036 Analysis Results Comparison Results Target Mechanical Error (%) Equivalent Stress (Pa) Step 1 6500 6502.6 0.040 Step 2 4000 4001.3 0.032 Step 3 2625 2625.5 0.019 Total Deformation (m) Step 1 0.09 0.09 0 Step 2 0.06 0.06 0 Step 3 0.045 0.045 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 170 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH037: Cooling of a Spherical Body Overview Reference: F. Kreith, "Principles of Heat Transfer", 2nd Printing, International Textbook Co., Scranton, PA, 1959, pg. 143, ex. 4-5. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Transient Thermal Analysis Type(s): Element Plane Type(s): Test Case Determine the temperature at the center of a spherical body, initially at a temperature T0, when exposed to an environment having a temperature Te for a period of 6 hours (21600 s). The surface convection coefficient is h. • Initial temperature, T0 = 65 °F • Surface temperature, Te = 25°F • Convection coefficient h = 5.5556e-4 BTU/s-ft2-°F • Time, t = 21600 seconds • Radius of the sphere ro = 2 in = 1/6 ft Material Properties Geometric Properties Loading K = (1/3) BTU/hr-ft-°F Quarter Circular lamina Convection applied 3 Radius = 0.16667 ft on Edge = ρ = 62 lb/ft c = 1.075 Btu/lb-°F 5.5556e-4 BTU/s-ft2-°F Ambient Temperature for Convection = 25°F Analysis Assumptions and Modeling Notes for ANSYS Mechanical Since the problem is axisymmetric, only a 2-D quarter model is used in ANSYS Mechanical. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 171 VMMECH037 Figure 114: Schematic for ANSYS Mechanical Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Figure 115: Schematic for ANSYS AIM Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Temperature at the center 28 28.688 2.457 of body after 21600s (°F) Results Comparison for ANSYS AIM Results Target AIM Error (%) Temperature at the center of body after 21600s 28 28.5 1.79 (°F) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 172 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH038: Crashing Blocks Simulation with Transient Structural Analysis Overview Reference: Any basic kinematics book Solver(s): ANSYS Mechanical Analysis Flexible Dynamic Analysis Type(s): Element Solid Type(s): Test Case Three blocks are resting on a base. A left block of mass 2.355 x 10-4 kg is given a constant initial velocity of 100 mm/sec to collide with a middle block of mass 1.1775 x 10-4 kg. Frictionless supports are applied as shown in the figure and also on the bottom faces of the left and middle blocks. The right block is fixed using a fixed support and the base is fixed by applying a fixed joint. Find the velocity of both the moving blocks after impact. Figure 116: Schematic Material Properties Geometric Properties Loading E = 2 x 105 MPa Left block = 3 mm x 2 Left block initial ν = 0.3 mm x 5 mm velocity = 100 Middle block = 2.5 mm mm/s (X direction) ρ = 7.85 x 10-6 kg/mm3 x 2 mm x 3 mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 173 VMMECH038 Material Properties Geometric Properties Loading Right block =3 mm x 6 mm x 4 mm Base = 3 mm x 8.607 mm x 75.15 mm Analysis For a perfectly elastic collision between the blocks, the following equations are true: (1) where: and = mass of the left and middle blocks in kg and = initial and final velocity of the left block in mm/sec = initial velocity of the middle block in mm/sec = final velocity of the middle block after impact in mm/sec Results Comparison Results Target Mechanical Error (%) Velocity of the left block 33.3 33.636 1.009 after impact (mm/sec) Velocity of the middle block 133.4 132.73 -0.502 after impact (mm/sec) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 174 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH039: Transient Response of a Spring-Mass System Overview Reference: R. K. Vierck, Vibration Analysis, 2nd Edition, Harper & Row Publishers, New York, NY, 1979, sec. 5-8. Solver(s): ANSYS Mechanical Analysis Flexible Dynamic Analysis Type(s): Element Solid and Spring Type(s): Test Case A system containing two masses, m1 and m2, and two springs of stiffness k1 and k2 is subjected to a pulse load F(t) on mass 1. Determine the displacement response of the system for the load history shown. Figure 117: Schematic Material Properties Geometric Properties Loading E = 2 x 1011 Pa 2 Blocks = 2 m x 2 m x F0 = 50 N γ = 0.3 2m td = 1.8 sec Length of L1 spring = ρ = 0.25 kg/m3 6m k1 = 6 N/m Length of L2 spring = k2 = 16 N/m 7m m1 = 2 kg Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 175 VMMECH039 Material Properties Geometric Properties Loading m2 = 2kg Analysis Assumptions and Modeling Notes The step drop is modeled as F0 = 50 N @ t = 1.8 sec and F0 = 0 N @ t = 1.80001 sec. Results Comparison Results Target Mechanical Error (%) Y1, m (@ t = 1.3s) 14.48 14.335 -1.001 Y2, m (@ t = 1.3s) 3.99 3.915 -1.880 Y1, m (@ t = 2.4s) 18.32 18.506 1.015 Y2, m (@ t = 2.4s) 6.14 6.1939 0.878 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 176 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH040: Deflection of Beam using Symmetry and Anti-Symmetry Overview Reference: Any Basic Strength of Materials Book Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Beam Type(s): Test Case A long bar 1m X 1m X 24m with simply supported ends is subjected to lateral load of 1000 N at a distance of 8m from one end. Find Deformation at the 8m from simply Supported end. Scenario 1: Considering Symmetry Scenario 2: Considering Anti-Symmetry Figure 118: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Bar = 1m x 1m x 24m Force = -1000 N γ=0 (Y-direction) at 8m ρ = 0.001 kg/m3 from Simply Supported end Analysis Scenario 1: Considering Symmetry Scenario 2: Considering Anti-Symmetry Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 177 VMMECH040 Results Comparison Results Target Mechanical Error (%) Scenario 1: Directional -2.569e-5 -2.5695e-5 0.019 Deformation in Y-direction (m) Scenario 2: Directional -1.70662e-6 -1.7383e-6 1.856 Deformation in Y-direction (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 178 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH041: T-Stress Calculation for a Crack in a Plate Using Pre-Meshed Crack Overview Reference: Fett, T., Stress Intensity Factors, T-Stresses, Weight Functions, Institute of Ceramics in Mechanical Solver(s): ANSYS Mechanical Analysis Type(s): Static Analysis Element Type(s): Solid Test Case A rectangular plate with a center crack is subjected to an end tensile load . Symmetry boundary con- ditions are considered and T-Stress is determined using Pre-Meshed Crack. Figure 119: Two-Dimensional Fracture Problem Sketch Material Properties Geometric Properties Loading Young's Modulus (E) = 207000 MPa a = 10 mm Tensile stress ( ) = 100 Coefficient of friction ( ) = 0.3 a / W = 0.2 MPa H / W = 0.75 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 179 VMMECH041 Analysis Assumptions and Modeling Notes The problem is solved using 2-D solid elements with plane-strain behavior. A one-quarter plate is modeled and symmetric boundary conditions are considered. The crack tip nodes and the number of paths surrounding the crack tip nodes are defined using Pre-Meshed Crack. The plate is subjected to a tensile stress and the T-Stress is computed for the crack tip nodes. The crack front and the path sur- rounding the crack front are defined using Pre-Meshed Crack. Results Comparison Result Target Mechanical Error (%) T-Stress -110 -110.760 -0.691 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 180 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH042: Hydrostatic Pressure Applied on a Square Bar Fully and Partially Submerged in a Fluid Overview Reference: Any Basic Strength of Materials Book Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Solid Type(s): Test Case Long bar 20m x 2m x 2m is immersed in a fluid and is fixed at one end. Fluid density is 1000 kg/m3 and Hydrostatic acceleration is 10 m/s2 in negative Z direction. Hydrostatic pressure is applied on a longitudinal face normal to X-axis at different locations as given in the scenarios below. Find normal stress in Z direction of square bar. Scenario 1: Square bar is partially immersed in the fluid up to 15 m in Z direction from the fixed support. Scenario 2: Square bar is fully immersed in the fluid up to 25 m in Z direction from the fixed support Figure 120: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Long bar = 20m x 2m Hydrostatic γ=0 x 2m Pressure ρ = 7850 kg/m3 Acceleration = -10 m/s2 (Z direction) Surface Location: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 181 VMMECH042 Material Properties Geometric Properties Loading Scenario 1: (2,1,5) m Scenario 2: (2,1,-5) m Analysis Scenario 1: Partially Submerged (Pressure distribution in triangular form) Pressure distribution on square bar in triangular form, one end is maximum and other end is zero Pressure on square bar = P = ρ x g x h Load per meter is w = P x L Maximum bending moment = Normal stress = Bending stress = Maximum bending moment / Sectional Modulus Scenario 2: Fully Submerged (Pressure distribution in trapezoidal form) Maximum bending moment = where: W1 = Maximum Load per meter (@ 25m) W2 = Minimum Load per meter (@ 5m) Normal stress = Bending stress = Maximum bending moment / Sectional Modulus Results Comparison Results Target Mechanical Error (%) Normal Stress (Partially 8.4375e6 8529300 -1.088 Submerged) (Pa) Normal Stress (Fully 3.50e7 3.5241e7 0.689 Submerged) (Pa) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 182 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH043: Fundamental Frequency of a Simply-Supported Beam Overview Reference: W. T. Thompson, Vibration Theory and Applications, 2nd Printing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965, pg. 18, ex. 1.5-1 Solver(s): ANSYS Mechanical Analysis Modal Analysis Type(s): Element Beam Type(s): Test Case Determine the fundamental frequency f of a simply-supported beam of length = 80 in and uniform cross-section A = 4 in2 as shown below. Figure 121: Schematic Material Properties Geometric Properties Loading E = 3 x 107 psi = 80 in ρ=0.2836 lb/in3 A = 4 in2 h = 2 in I = 1.3333 in4 Results Comparison Results Target Mechanical Error (%) Frequency (Hz) 28.766 28.613 -0.532 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 183 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 184 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH044: Thermally Loaded Support Structure Overview Reference: S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 30, problem 9. Solver(s): ANSYS Mechanical Analysis Linear Thermal Stress Analysis Type(s): Element Beam Type(s): Test Case An assembly of three vertical wires has a rigid horizontal beam on which a vertically downward force Q is acting. Length of the wires is 20 in, the spacing between the wires is 10 in and the reference tem- perature is 70 °F. The entire assembly is subjected to a temperature rise of ΔT. Find the stresses in the copper and steel wire of the structure shown below. The wires have a cross-sectional area of A. Figure 122: Schematic Material Properties Geometric Properties Loading VMSIM044_material_rigid: A = 0.1 in2 Q = 4000 lb (Y direction) Er = 3e16 psi ΔT = 10 °F νr = 0 VMSIM044_material_copper: Ec = 1.6e7 psi νc = 0 αc = 9.2e-6 / °F VMSIM044_material_steel: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 185 VMMECH044 Material Properties Geometric Properties Loading Es = 3e7 psi νs = 0 αs = 7e-6 / °F Results Comparison Results Target Mechanical Error (%) Stress in steel (psi) 19695 19695 0.00 Stress in copper (psi) 10152 10152 0.00 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 186 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH045: Laterally Loaded Tapered Support Structure Overview Reference: S. H. Crandall, N. C. Dahl, An Introduction to the Mechanics of Solids, McGraw-Hill Book Co., Inc., New York, NY, 1959, pg. 342, problem 7.18. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Shell Type(s): Test Case A cantilever beam of thickness t and length has a depth which tapers uniformly from d at the tip to 3d at the wall. It is loaded by a force F at the tip, as shown. Find the maximum bending stress at the mid-length (X = ). Figure 123: Schematic Material Properties Geometric Properties Loading Es = 3 x 107 psi = 50 in F = 4000 lb (Y νs=0 d = 3 in direction) t = 2 in Results Comparison Results Target Mechanical Error (%) Bending stress at mid length (psi) 8333 8309.1 -0.2868 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 187 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 188 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH046: Pinched Cylinder Overview Reference: R. D. Cook, Concepts and Applications of Finite Element Analysis, 2nd Edition, John Wiley and Sons, Inc., New York, NY, 1981, pp. 284-287 H. Takemoto, R. D. Cook, "Some Modifications of an Isoparametric Shell Element", International Journal for Numerical Methods in Engineering, Vol. 7 No. 3, 1973. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Shell Type(s): Test Case A thin-walled cylinder is pinched by a force F at the middle of the cylinder length. Determine the radial displacement δ at the point where F is applied. The ends of the cylinder are free edges. A one-eighth symmetry model is used. One-fourth of the load is applied due to symmetry. Figure 124: Schematic Material Properties Geometric Properties Loading Es = 10.5e6 psi = 10.35 in F = 100 lbf (Y ν s = 0.3125 r = 4.953 in direction) t = 0.094 in Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 189 VMMECH046 Analysis Due to symmetrical boundary and loading conditions, one-eighth model is used and one-fourth of the load is applied. Results Comparison Results Target Mechanical Error (%) Deflection (in) -0.1139 -0.11391 -0.00878 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 190 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH047: Plastic Compression of a Pipe Assembly Overview Reference: S. H. Crandall, N. C. Dahl, An Introduction to the Mechanics of Solids, McGraw-Hill Book Co., Inc., New York, NY, 1959, pg. 180, ex. 5.1. Solver(s): ANSYS Mechanical Analysis Plastic Structural Analysis Type(s): Element Axisymmetric Type(s): Test Case Two coaxial tubes, the inner one of 1020 CR steel and cross-sectional area As, and the outer one of 2024-T4 aluminum alloy and of area Aa, are compressed between heavy, flat end plates, as shown below. Determine the load-deflection curve of the assembly as it is compressed into the plastic region by an axial displacement. Assume that the end plates are so stiff that both tubes are shortened by exactly the same amount. Figure 125: Schematic Material Properties Geometric Properties Loading VMSIM047_CR_steel: = 10 in 1st Load step: δ = Steel: -0.032 in Es = 26,875,000 psi 2nd Load step: δ = σ(yp)s = 86,000 psi Inside radius = -0.05 in 1.9781692 in Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 191 VMMECH047 Material Properties Geometric Properties Loading VMSIM047_T4_aluminum alloy: Wall thickness = 0.5 3rd Load step: δ = in -0.10 in Ea = 11,000,000 psi Aluminum: σ(yp)a = 55,000 psi ν = 0.3 Inside radius = 3.5697185 in Wall thickness = 0.5 in Analysis Because the geometry and loading are symmetric about the y-axis, the above problem can be analyzed as an axisymmetric problem. Results Comparison Results Target Mechanical Error (%) 6 Load, lb for Deflection @ 0.032 in 1.0244 x 10 1033700 0.907 6 Load, lb for Deflection @ 0.05 in 1.262 x 10 1262700 0.056 6 Load, lb for Deflection @ 0.1 in 1.262 x 10 1267200 0.412 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 192 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH048: Bending of a Tee-Shaped Beam Overview Reference: S. H. Crandall, N. C. Dahl, An Introduction to the Mechanics of Solids, McGraw-Hill Book Co., Inc., New York, NY, 1959, pg. 294, ex. 7.2. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Beam Type(s): Test Case Find the maximum tensile and compressive bending stresses in an unsymmetrical T beam subjected to uniform bending Mz, with dimensions and geometric properties as shown below. Figure 126: Schematic Material Properties Geometric Properties Loading E = 3e7 psi b = 1.5 in Mz = 100,000 lbf-in h = 8 in (Z direction) y = 6 in Area = 60 in2 Iz = 2000 in4 Results Comparison Results Target Mechanical Error (%) StressBEND, Bottom (psi) 300 300 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 193 VMMECH048 Results Target Mechanical Error (%) StressBEND, Top (psi) -700 -700 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 194 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH049: Combined Bending and Torsion of Beam Overview Reference: S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 299, problem 2. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Beam Type(s): Test Case A vertical bar of length and radius r is subjected to the action of a horizontal force F acting at a distance d from the axis of the bar. Determine the maximum principal stress σmax. Figure 127: Problem Sketch Figure 128: Schematic Material Properties Geometric Properties Loading E = 3e7 psi = 25 ft F = 250 lb (Y ν = 0.3 r = 2.33508 in direction) d = 3 ft M = 9000 lbf-in (Z direction) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 195 VMMECH049 Results Comparison Results Target Mechanical Error (%) Principal stressmax (psi) 7527 7515.5 -0.153 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 196 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH050: Cylindrical Shell Under Pressure Overview Reference: S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 45, article 11. A. C. Ugural, S. K. Fenster, Advanced Strength and Applied Elasticity, Elsevier, 1981. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Axisymmetric Shell element Type(s): Test Case A long cylindrical pressure vessel of mean diameter d and wall thickness t has closed ends and is sub- jected to an internal pressure P. Determine the axial stress σy and the hoop stress σz in the vessel at the mid-thickness of the wall. An axial force of 5654866.8 lb ((Pπd2)/4) is applied to simulate the closed-end effect. Figure 129: Schematic Material Properties Geometric Properties Loading E = 3e7 psi t = 1 in P = 500 psi (radial ν = 0.3 d = 120 in direction) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 197 VMMECH050 Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model for ANSYS Mechanical: Figure 130: Schematic for ANSYS Mechanical Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model for ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 198 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH050 Figure 131: Schematic for ANSYS AIM Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Stressy (psi) 15000 15000 0 Stressz (psi) 30000 30002 0.007 Results Comparison for ANSYS AIM Results Target AIM Error (%) Stressy 15000 15000 0.0 (psi) Stressz 30000 30002 0.00650 (psi) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 199 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 200 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH051: Bending of a Circular Plate Using Axisymmetric Elements Overview Reference: S. Timoshenko, Strength of Material, Part II, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1956, pp. 96, 97, and 103. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Axisymmetric Shell element Type(s): Test Case A flat circular plate of radius r and thickness t is subject to various edge constraints and surface loadings. Determine the deflection δ at the middle and the maximum stress σmax for each case. Case 1: Uniform loading P, clamped edge Case 2: Concentrated center loading F, clamped edge Figure 132: Schematic Material Properties Geometric Properties Loading E = 3e7 psi r = 40 in Case 1: ν = 0.3 t = 1 in P = 6 psi Case 2: F = -7539.82 lb (y direction) Analysis Assumptions and Modeling Notes for ANSYS Mechanical Because the geometry and loading are symmetric about the y-axis, the above problem can be analyzed as an axisymmetric problem. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 201 VMMECH051 Figure 133: Schematic for ANSYS Mechanical Case 1: Case 2: Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model for ANSYS AIM: Figure 134: Schematic for ANSYS AIM Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 202 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH051 Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Case 1: Deflection (in) -0.08736 -0.087114 -0.282 Stressmax (psi) 7200 7212.8 0.178 Case 2: Deflection (in) -0.08736 -0.088025 0.761 Stressmax (psi) 3600 3607.9 0.219 Results Comparison for ANSYS AIM Results Target AIM Error (%) Case Deflection -0.08736 -0.08711 -0.281 1: (in) 7200 7212.8 0.178 Stressmax (psi) Case Deflection -0.08736 -0.088802 0.761 2: (in) 3600 3608 0.219 Stressmax (psi) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 203 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 204 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH052: Velocity of Pistons for Trunnion Mechanism Overview Reference: Any Basic Kinematics book Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case The Trunnion mechanism has the following data (all distances are center-to-center distances): • Crank radius OA = 100 mm and is oriented at 30 deg to Global Y Axis • AB = 400 mm • AC = 150 mm • CE = 350 mm • EF = 300 mm • Constant Angular Velocity at Crank = 12.57 rad/s • Center of Trunnion is at distance of 200 mm from line of stroke of Piston B horizontally and 300 mm vertical from Center of Crank • Find the Velocity of Piston (F) at the 180 deg from Initial Position • Find the Velocity of Piston (B) at the 180 deg from Initial Position Figure 135: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 205 VMMECH052 Material Properties Geometric Properties Loading E = 2e11 Pa AB = 400 mm Constant angular ν = 0.3 AC = 150 mm velocity at crank = CE = 350 mm 12.57 rad/s EF = 300 mm Analysis Analysis done using graphical solution. Consider the Space Diagram, Velocity Diagram at the 180° from Initial Position. Figure 136: Schematic Results Comparison Results Target Mechanical Error (%) Velocity of Piston (F) mm/s 501.8 497.04 -0.949 Velocity of Piston (B) mm/s 955 959.72 0.494 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 206 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH053: Simple Pendulum With SHM Motion Overview Reference: Any Basic Kinematics book Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case A simple pendulum as shown in Figure 137: Schematic (p. 207) has a SHM motion about its hinged point given by the following equation: θ = 1.571*sin (0.5235*t) rad The hinge point coordinates are: 1. Hinge point = (0, 0, -35.56) mm Find the relative angular acceleration of pendulum after t = 3s. Figure 137: Schematic Material Properties Geometric Properties Loading E = 2000000 MPa Hinge point = (0, 0, Rotation θ = ν = 0.3 -35.56) mm 1.571*sin (0.5235*t) rad Analysis The pendulum is having SHM motion in X-Z plane about the hinge. Angular acceleration of pendulum: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 207 VMMECH053 Results Comparison Results Target Mechanical Error (%) Relative angular acceleration -0.433 -0.43054 -0.568 of pendulum after t = 3s (rad/s2) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 208 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH054: Spinning Single Pendulum Overview Reference: Any Basic Kinematics book Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case A uniform bar A is connected to a vertical shaft by a revolute joint. The vertical shaft is rotating around its vertical axis at a constant velocity Ω. A point mass M is attached at the tip of the bar in the figure below. The length of bar A is L. Its mass is m, its rotational inertia to its principal axis are Jx, Jy, Jz. The angle of the bar A to the vertical axis is denoted as . The motion equation has been established as follows. The problem is solved for during the first second of motion. The WB/Mechanical results are compared to a fourth order Runge-Kutta solution. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 209 VMMECH054 Figure 138: Schematic Material Properties Geometric Properties Loading = tan-1(1,2) L= 2.2361 m Ω = 17.1522 =0 m = 551.45 kg M = 100.0 kg Jx = 229.97 kg-m2 Jy = 2.7293 kg-m2 Jz = 229.97 kg-m2 Results Comparison Results Target Mechanical Error (%) at 0.5 sec -1.3233 -1.3233 0.0 at 0.5 sec 116.1368 116.1368 0.0 at 1.0 sec -2.6755 -2.6755 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 210 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH054 Results Target Mechanical Error (%) at 1.0 sec 119.8471 119.8471 0.0 Figure 139: Plot of from 0 to 1 sec Figure 140: Plot of from 0 to 1 sec Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 211 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 212 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH055: Projector Mechanism - Finding the Acceleration of a Point Overview Reference: Any Basic Kinematics book Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case The mechanism shown in figure is used to pull a movie through a projector. The mechanism is driven by the drive wheel rotating at a constant -58.643 rad/s. The link lengths of all the links are constant as given below. • Link AB length r1 = 18mm • Link BC length r2 = 48mm • Length BX = x = 45 mm and CX = y = 28 mm The horizontal distance between A and C is length=34 mm. Determine the acceleration of point C with a change of angle of link AB (θ 1) from 0 to 60° in counter clockwise direction. Figure 141: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 213 VMMECH055 Material Properties Geometric Properties Loading E = 2e11 Pa r1 = 18 mm Constant rotational ν = 0.3 r2 = 48 mm velocity = -58.643 x = 45 mm rad/s y = 28 mm Analysis Linear acceleration of point C is given by Results Comparison Results Target Mechanical Error (%) Relative acceleration (θ 1 = -12.06 -12.043 -0.141 10) mm/s2 Relative acceleration (θ 1 = 1.317 1.3168 -0.015 30) mm/s2 Relative acceleration (θ 1 = 6.739 6.7386 -0.006 60) mm/s2 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 214 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH056: Coriolis Component of Acceleration (Rotary Engine Problem) Overview Reference: Any Basic Kinematics book Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case Kinematics diagram of one of the cylinders of a rotary engine is shown below. OA is 50mm long and fixed at point o. The length of the connecting rod AB is 125mm. The line of stroke OB is inclined at 50° to the vertical. The cylinders are rotating at a uniform speed of 300 rpm in a clockwise direction, about the fixed center O. Find Angular acceleration of the connecting rod. Figure 142: Schematic Material Properties Geometric Properties Loading E = 2e11 Pa Connecting rod AB is Constant rotational ν = 0.3 125mm Crank OA is velocity = 300 rpm 50mm long OB is inclined at 50° to the vertical. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 215 VMMECH056 Analysis Angular acceleration of the connecting rod is given by: Results Comparison Results Target Mechanical Error (%) Angular acceleration 294.52 294.53 0 (radian/s2) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 216 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH057: Calculation of Velocity of Slider and Force by Collar Overview Reference: Beer-Johnston ‘Vector Mechanics for Engineers’ Statics & Dynamics (In SI Units), 7th Edition, TATA McGRAW HILL Edition 2004, Problem 13.73, Page No: 793 Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case A 1.2 Kg collar is attached to a spring and slides without friction along a circular rod in a vertical plane. The spring has an undeformed length of 105 mm and a constant K = 300 N/m. Knowing that the collar is at rest at "C" and is given a slight push to get it moving. Length OP = 75 mm. Length OB = 180 mm. Determine the force exerted by the rod on the collar as it passes through point "A" and "B". Figure 143: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 217 VMMECH057 Material Properties Geometric Properties Loading E = 2e11 Pa Spring: Gravitational ν = 0.3 acceleration = Undeformed length = -9.8066 m/s2 (Y 105 mm Direction) Stiffness K = 300 N/m Results Comparison Results Target Mechanical Error (%) At point A (N) 14.88 14.992 0.753 At point B (N) -23.6 -23.667 0.3 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 218 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH058: Reverse Four Bar Linkage Mechanism Overview Reference: Results are simulated using MATLAB Solver(s): ANSYS Mechanical Analysis Rigid Dynamic Analysis Type(s): Element Multipoint Constraint Element Type(s): Test Case The figure (below) shows a reverse four bar linkage consisting of uniform rigid links PQ, QR, and RS and ground PS. Link PQ is connected with revolute joints to links QR and PS at points Q and P, respectively. Link RS is connected with revolute joints to links QR and PS at points R and S, respectively. The link lengths of all the links are constant as given below. • Fixed Link PS length r1 = 0.5 m • Crank Link PQ length r2 = 0.15 m • Link QR length r3 = 0.4 m • Link RS length r4 = 0.45 m • Gravity g = 9.81 m/sec2 Determine the angular accelerations, angular velocity and rotation of link RS at joint R. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 219 VMMECH058 Figure 144: Schematic Material Properties Geometric Properties Loading E = 2 x 1011 Pa Link PS length r1 = 0.5 Gravitational ν = 0.3 m acceleration = Link PQ length r2 = -9.8066 m/s2 (Y 0.15 m Direction) Link QR length r3 = 0.4 m Link RS length r4 = 0.45 m Analysis Results are obtained using MATLAB. Results Comparison Results Target Mechanical Error (%) 2 Angular Acceleration (rad/s ) 39.6 39.336 -0.667 Angular Velocity (rad/sec) -5.16 -5.1247 -0.7 Rotation (rad) -0.36 -0.36255 0.7 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 220 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH059: Bending of a Solid Beam (Plane Elements) Overview Reference: R. J. Roark, Formulas for Stress and Strain, 4th Edition, McGraw-Hill Book Co., Inc., New York, NY, 1965, pp. 104, 106. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element 2-D Plane Stress Shell element Type(s): Test Case A beam of length and height h is built-in at one end and loaded at the free end with: • a moment M • a shear force F For each case, determine the deflection δ at the free end and the bending stress σBend at a distance d from the wall at the outside fiber. Figure 145: Schematic Case 1: Case 2: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 221 VMMECH059 Material Properties Geometric Properties Loading E = 30 x 106 psi = 10 in Case 1: ν = 0.3 h = 2 in d = 1 in M = 2000 ibf-in (Z direction) Case 2: F = 300 lb (Y direction) Analysis Since the loading is uniform and in one plane, the above problem can be analyzed as a plane stress problem. Results Comparison Results Target Mechanical Error (%) Case 1: Deflection (in) 0.00500 0.00500 0 StressBend (psi) -3000 -3000 0 Case 2: Deflection (in) 0.00500 0.0051233 2.466 StressBend (psi) -4050 -4051.5 0.037 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 222 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH060: Crank Slot Joint Simulation with Flexible Dynamic Analysis Overview Reference: Mechanical APDL Multibody Analysis Solver(s): ANSYS Mechanical Analysis Flexible Dynamic Analysis Type(s): Element Solid and Multipoint Constraint Element Type(s): Test Case The figure shows crank slot model consists of a base and two rods. The two rods are attached to each other and the base with three bolts. The base of the model is fixed to the ground via a fixed joint and Bolt3 connected with slot joint to base. Define Rod1 and Rod2 as a flexible body and run the crank slot analysis using a Flexible Dynamic Analysis. Determine the Equivalent (von Mises) Stress for both flexible rods. Figure 146: Schematic Material Properties Geometric Properties Loading E = 2 x 105 MPa Rod1 length = 75 mm Constant angular ν = 0.3 Rod2 length = 115 mm acceleration at base to Bolt1 = 25 rad/s2 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 223 VMMECH060 Analysis Figure 147: Contour Plot Figure 148: Equivalent (von Mises) Stress Figure 149: Total Force at Base to Bolt1 Results Comparison Results Target Mechanical Error (%) Equivalent (von Mises) Stress 0.398 0.40591 1.987 (MPa) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 224 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH060 Results Target Mechanical Error (%) Force @ Bolt1 (N) 7.67 7.6813 0.147 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 225 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 226 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH061: Out-of-Plane Bending of a Curved Bar Overview Reference: S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 412, eq. 241. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Beam Type(s): Test Case A portion of a horizontal circular ring, built-in at A, is loaded by a vertical (Z) load F applied at the end B. The ring has a solid circular cross-section of diameter d. Determine the deflection δ at end B and the maximum bending stress σBend. Figure 150: Schematic Material Properties Geometric Properties Loading E = 30 x 106 psi r = 100 in F = -50 lb (Z ν = 0.3 d = 2 in direction) θ = 90° Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 227 VMMECH061 Results Comparison Results Target Mechanical Error (%) Deflection (in) -2.648 -2.655 0.264 StressBend (psi) 6366.0 6399.2 0.522 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 228 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH062: Stresses in a Long Cylinder Overview Reference: S. Timoshenko, Strength of Material, Part II, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1956, pg. 213, problem 1 and pg. 213, article 42. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Axisymmetric Shell Type(s): Test Case A long thick-walled cylinder is initially subjected to an internal pressure p. Determine the radial displace- ment δr at the inner surface, the radial stress σr, and tangential stress σt, at the inner and outer surfaces and at the middle wall thickness. Internal pressure is then removed and the cylinder is subjected to a rotation ω about its center line. Determine the radial σr and tangential σt stresses at the inner wall and at an interior point located at r = Xi. Figure 151: Schematic Material Properties Geometric Properties Loading E = 30 x 106 psi a = 4 in Case 1: ν = 0.3 b = 8 in Xi = 5.43 in Pressure = 30,000 ρ = 0.281826 lbm/in3 psi (radial direction) Case 2: Rotational velocity = 1000 rad/s (Y direction) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 229 VMMECH062 Analysis Assumptions and Modeling Notes for ANSYS Mechanical Because the geometry and loading are symmetric about the y-axis, the above problem can be analyzed as an axisymmetric problem in ANSYS Mechanical. Figure 152: Schematic for ANSYS Mechanical Case 1: Case 2: Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 230 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH062 Figure 153: Schematic for ANSYS AIM Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Case 1: Displacementr, in (r 0.0078666 0.0076267 -3.05 = 4 in) -30000. -29988 -0.04 Stressr, psi (r = 4 in) -7778. -7775.3 -0.035 Stressr, psi (r = 6 in) 0 0.79611 -- 50000. 49988 -0.024 Stresst, psi (r = 8 in) 27778. 27775 -0.011 Stresst, psi (r = 4 in) 20000. 19999 -0.005 Stresst, psi (r = 6 in) Stresst, psi (r = 8 in) Case 2: Stressr, psi (r = 4 in) 0 6.5483 -- Stresst, psi (r = 4 in) 40588. 41672 2.671 Stressr, psi (r = 5.43 4753. 4933.7 3.802 29436. 29719 0.961 in) Stresst, psi (r = 5.43 in) Results Comparison for ANSYS AIM Results Target AIM Error (%) Case 1: Displacementr, 0.0078666 0.0078692 0.0331 in (r = 4 -30000 -29997 -0.01 in) 0 0.1972 -- Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 231 VMMECH062 Results Target AIM Error (%) Stressr, psi 50000 49996.9 -0.0062 (r = 4 in) 20000 19999.8 -0.001 Stresst, psi (r = 8 in) Stressr, psi (r = 4 in) Stresst, psi (r = 8 in) Case 2: Stressr, psi 0 1.666 -- (r = 4 in) 40588 41710 2.764 Stresst, psi (r = 4 in) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 232 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH063: Large Deflection of a Cantilever Overview Reference: K. J. Bathe, E. N. Dvorkin, "A Formulation of General Shell Elements - The Use of Mixed Interpolation of Tensorial Components”, International Journal for Numerical Methods in Engineering, Vol. 22 No. 3, 1986, pg. 720. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Shell Type(s): Test Case A cantilever plate of length , width b, and thickness t is fixed at one end and subjected to a pure bending moment M at the free end. Determine the true (large deflection) free-end displacements and the top surface stress at the fixed end using shell elements. Figure 154: Schematic Material Properties Geometric Properties Loading E = 1800 N/mm2 = 12 mm M = 15.708 N-mm ν = 0.0 b = 1 mm (Y direction) t = 1 mm Analysis Large deformation is used to simulate the problem. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 233 VMMECH063 Results Comparison Results Target Mechanical Error (%) Directional Deformation -2.9 -2.9278 1.055 X-direction (mm) Directional Deformation -6.5 -6.6008 1.551 Z-direction (mm) Normal Stress X-direction 94.25 94.131 -0.126 (N/mm2) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 234 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH064: Small Deflection of a Belleville Spring Overview Reference: S. Timoshenko, Strength of Material, Part II, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1956, pg. 143, problem 2. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Shell Type(s): Test Case The conical ring shown below represents an element of a Belleville spring. Determine the deflection y produced by a load F per unit length on the inner edge of the ring. Figure 155: Schematic Material Properties Geometric Properties Loading E = 30 x 106 psi a = 1 in Line pressure = ν = 0.0 b = 1.5 in -100 lb/in (Y t = 0.2 in direction) β = 7° = 0.12217 rad Results Comparison Results Target Mechanical Error (%) Directional Deformation -0.0028205 -0.0029262 3.747 Y-direction (in) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 235 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 236 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH065: Thermal Expansion to Close a Gap at a Rigid Surface Overview Reference: C. O. Harris, Introduction to Stress Analysis, The Macmillan Co., New York, NY, 1959, pg. 58, problem 8. Solver(s): ANSYS Mechanical Analysis Static Thermal Stress Analysis Type(s): Element Solid and Shell Type(s): Test Case An aluminum-alloy bar is initially at a temperature of 70 °F. Calculate the stresses and the thermal strain in the bar after it has been heated to 170 °F. The supports are assumed to be rigid. Use a global mesh size of 0.25 in. Figure 156: Schematic Material Properties Geometric Properties Loading 6 E = 10.5 x 10 psi = 3 in. Δt = 170 °F - 70 °F Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 237 VMMECH065 Material Properties Geometric Properties Loading α = 1.25 x 10-5/°F δ = 0.002 in. ν = 0.0 Results Comparison Results Target Mechanical Error (%) Normal Stress Y (psi) -6125 -6122.4 -0.042 -3 -3 Thermal Strain Y (in/in) 1.25 x10 1.25 x 10 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 238 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH066: Bending of a Tapered Plate Overview Reference: C. O. Harris, Introduction to Stress Analysis, The Macmillan Co., New York, NY, 1959, pg. 114, problem 61. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Shell Type(s): Test Case A tapered cantilever plate of rectangular cross-section is subjected to a load F at its tip. Find the max- imum deflection δ and the maximum principal stress σ1 in the plate. Use a global mesh size of 0.75 in. Figure 157: Schematic Material Properties Geometric Properties Loading E = 30 x 106 psi L = 20 in F = 10 lbf ν = 0.0 d = 3 in t = 0.5 in Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 239 VMMECH066 Results Comparison Results Target Mechanical Error (%) Maximum Principal Stress 1600 1614.5 0.906 (psi) Directional Deformation Z -0.042667 -0.04275 0.195 (in) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 240 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH067: Elongation of a Solid Tapered Bar Overview Reference: C. O. Harris, Introduction to Stress Analysis, The Macmillan Co., New York, NY, 1959, pg. 237, problem 4. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Solid Type(s): Test Case A tapered aluminum alloy bar of square cross-section and length L is suspended from a ceiling. An axial load F is applied to the free end of the bar. Determine the maximum axial deflection δ in the bar and the axial stress σy at mid-length (Y = L/2). Use a global mesh size of 0.5 in. Figure 158: Problem Sketch Material Properties Geometric Properties Loading E = 10.4 x 106 psi L = 10 in F = 10000 lbf ν = 0.3 d = 2 in Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 241 VMMECH067 Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 159: ANSYS Mechanical Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Directional Deformation Y 0.0048077 0.0048215 0.287 (in) Normal Stress Y at L/2 (psi) 4444 4463 0.428 Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 242 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH067 Figure 160: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Directional Deformation Y 0.0048077 0.0048215 0.288 (in) Normal Stress Y at L/2 (psi) 4444 4463 0.428 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 243 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 244 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH068: Plastic Loading of a Thick Walled Cylinder Overview Reference: S. Timoshenko, Strength of Material, Part II, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1956, pg. 388, article 70. Solver(s): ANSYS Mechanical Analysis Static, Plastic Analysis (Plane Strain) Type(s): Element 2-D Structural Solid Type(s): Test Case A long thick-walled cylinder is subjected to an internal pressure p (with no end cap load). Determine the radial stress, σr, and the tangential (hoop) stress, σt, at locations near the inner and outer surfaces of the cylinder for a pressure, pel, just below the yield strength of the material, a fully elastic material condition. Determine the effective (von Mises) stress, σeff, at the same locations for a pressure, pult, which brings the entire cylinder wall into a state of plastic flow. Use a global mesh size of 0.4 in along with a mapped face meshing. Figure 161: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 245 VMMECH068 Material Properties Geometric Properties Loading E = 30 x 106 psi a = 4 in pel = 12,990 psi σyp = 30,000 psi b = 8 in pult = 24,011 psi ν = 0.3 Analysis This problem is modeled as a plane strain problem with only a quarter of the cross-section as shown in the above figures. Symmetry conditions are used on the edges perpendicular to X and Y axes. Load is applied in two steps as shown in the above table. The stresses are calculated at a distance of r = 4.4 in and 7.6 in, w.r.t a cylindrical coordinate system whose origin is same as that of the global coordinate system. Results Comparison Results Target Mechanical Error (%) Fully Elastic Stressr, psi (X = 4.4 in) -9984 -9948.8 -0.353 Stresst, psi (X = 4.4 in) 18645 18609 -0.193 Stressr, psi (X = 7.6 in) -468 -469.1 0.235 Stresst, psi (X = 7.6 in) 9128 9129.1 0.012 Fully Plastic Stresseff, psi (X = 4.4 in) 30000 30000 0 Stresseff, psi (X = 7.6 in) 30000 30000 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 246 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH069: Barrel Vault Roof Under Self Weight Overview Reference: R. D. Cook, Concepts and Applications of Finite Element Analysis, 2nd Edition, John Wiley and Sons, Inc., New York, NY, 1981, pp. 284-287. Solver(s): ANSYS Mechanical Analysis Static Analysis Type(s): Element Shell Type(s): Test Case A cylindrical shell roof of density ρ is subjected to a loading of its own weight. The roof is supported by walls at each end and is free along the sides. Find the x and y displacements at point A and the top and bottom stresses at points A and B. Express stresses in the cylindrical coordinate system. Use a global mesh size of 4 m. Figure 162: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 247 VMMECH069 Material Properties Geometric Properties Loading E = 4.32 x 108 N/m2 t = 0.25 m g = 9.8 m/s2 ν = 0.3 r = 25 m = 50 m ρ = 36.7347 kg/m3 Θ = 40° Analysis A one-fourth symmetry model is used. Displacements, UX and UY, and the longitudinal rotation, ROTZ, are constrained at the roof end to model the support wall. Results Comparison Results Target Mechanical Error (%) Directional Deformation Y -0.3019 -0.30908 2.378 @ A, m Directional Deformation X -0.1593 -0.1627 2.134 @ A, m Stressz, Top @ A, Pa 215570 223680 3.762 Stressz, Bottom @ A, Pa 340700 350100 2.759 Stressangle, Top @ B, Pa 191230 184290 -3.629 Stressangle, Bottom @ B, Pa -218740 -211000 -3.538 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 248 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH070: Hyperelastic Thick Cylinder Under Internal Pressure Overview Reference: J. T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill Book Co., Inc., New York, NY, 1972, pp. 325-331. Solver(s): ANSYS Mechanical Analysis Static, Large Deflection Analysis Type(s): Element 2-D Structural Solid Elements Type(s): Test Case An infinitely long cylinder is made of Mooney-Rivlin type material. An internal pressure of Pi is applied. Find the radial displacement at the inner radius and the radial stress at radius R = 8.16 in. Use a global mesh size of 1 in along with a mapped face meshing. Figure 163: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 249 VMMECH070 Material Properties Geometric Properties Loading Mooney-Rivlin material ri = 7.0 in Pi = 150 psi coefficients: ro = 18.625 in C10 = 80 psi C01 = 20 psi D1 = 0 /psi Analysis Because of the loading conditions and the infinite length, this problem is solved as a plane strain problem. A one-fourth symmetry model is used. The total pressure is applied in two load increments 90 and 150 psi. Stress and Deformation are expressed in cylindrical coordinate system. Results Comparison Results Target Mechanical Error (%) Deformation at inner radius 7.18 7.1819 0.026 in radial direction, in Radial Stress at r = 8.16 in, -122 -122 0 psi Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 250 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH071: Centerline Temperature of a Heat Generating Wire Overview Reference: W. M. Rohsenow, H. Y. Choi, Heat, Mass and Momentum Transfer, 2nd Printing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963, pg. 106, ex. 6.5. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Thermal Analysis Type(s): Element 2-D Thermal Solid Elements Type(s): Test Case Determine the centerline temperature TcL and the surface temperature Ts of a bare steel wire generating heat at the rate Q. The surface convection coefficient between the wire and the air (at temperature Ta) is h. Also, determine the heat dissipation rate q. Use a global mesh size of 0.02 ft along with a mapped face meshing. Figure 164: Schematic Material Properties Geometric Properties Loading k = 3.6111 x 10-3 ro = 0.03125 ft h = 1.3889 x 10-3 Btu/s-ft-°F Btu/s-ft2-°F Ta = 70°F Q = 30.92 Btu/s-ft3 Analysis Assumptions and Modeling Notes for ANSYS Mechanical Because of the symmetry in loading conditions and in the geometry, this problem is solved as an axisymmetric problem in ANSYS Mechanical. The solution is based on a wire 1 foot long. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 251 VMMECH071 Figure 165: Schematic for ANSYS Mechanical Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 252 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH071 Figure 166: Schematic for ANSYS AIM Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Centerline Temperature, °F 419.9 419.94 0.01 Surface Temperature, °F 417.9 417.85 0.012 Heat dissipation rate, BTU/s -0.094861 -0.094861 0.00 Results Comparison for ANSYS AIM Results Target AIM Error (%) Maximum Centerline Temperature, 419.9 419.8 -0.0225 °F Maximum Surface Temperature, °F 417.9 417.72 -0.0436 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 253 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 254 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH072: Thermal Stresses in a Long Cylinder Overview Reference: S. Timoshenko, Strength of Material, Part II, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co, Inc., New York, NY, 1956, pg. 234, problem 1. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Thermal Stress Analysis Type(s): Element 2-D Thermal Solid Elements Type(s): Test Case A long thick-walled cylinder is maintained at a temperature Ti on the inner surface and To on the outer surface. Determine the temperature distribution through the wall thickness. Also determine the axial stress σa and the tangential (hoop) stress σt at the inner and outer surfaces Edge sizing is used for all edges and edge behavior is defined as hard. Figure 167: Schematic Material Properties Geometric Properties Loading E = 30 x 106 psi a = 0.1875 in Ti = -1 °F -5 b = 0.625 in To = 0 °F α = 1.435 x 10 /°F ν = 0.3 k = 8.333 x 10-4 Btu/s-in-°F Analysis Assumptions and Modeling Notes for ANSYS Mechanical Because of the symmetry in loading conditions and in the geometry, this problem is solved as an axisymmetric problem in ANSYS Mechanical. The axial length is arbitrary and it is taken has 0.1 in. Nodal coupling is used in the static stress analysis. Model is used for the thermal and stress solutions. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 255 VMMECH072 Figure 168: Schematic in ANSYS Mechanical Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 256 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH072 Figure 169: Schematic in ANSYS AIM Results Comparison for ANSYS Mechanical Thermal Analysis Target Mechanical Error (%) T,°F (at X = 0.1875 in) -1 -1 0 T,°F (at X = 0.2788 in) -0.67037 -0.67054 0.025 T,°F (at X = 0.625 in) 0 0 0 Static Analysis Target Mechanical Error (%) Stressa, psi (at X = 0.1875 in) 420.42 416.92 -0.833 Stresst, psi (at X = 0.1875 in) 420.42 406.85 -3.227 Stressa, psi (at X = 0.625 in) -194.58 -195.06 0.247 Stresst, psi (at X = 0.625 in) -194.58 -195.01 0.221 Results Comparison for ANSYS AIM Thermal Analysis Target AIM Error (%) T, °F (at X = 0.1875 in) -1 -1 0 T, °F (at X = 0.625 in) 0 0 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 257 VMMECH072 Static Analysis Target AIM Error (%) Stressa, psi (at X = 420.42 421.97 0.368 0.1875 in) Stresst, psi (at X = 420.42 419.56 -0.204 0.1875 in) Stressa, psi (at X = 0.625 -194.58 -194.46 -0.059 in) Stresst, psi (at X = 0.625 -194.58 -194.58 0.205 in) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 258 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH073: Modal Analysis of a Cyclic Symmetric Annular Plate Overview Reference: R. D. Blevins, Formulas for Natural Frequency and Mode Shape, New York, NY, VanNostrand Reinhold Publishing Inc., 1979, PP. 246-247, 286-287. Solver(s): ANSYS AIM ANSYS Mechanical Analysis Mode-Frequency Analysis Type(s): Element Solid Type(s): Test Case The fundamental natural frequency of an annular plate is determined using a mode-frequency analysis. The lower bound is calculated from the natural frequency of the annular plates that are free on the inner radius and fixed on the outer. The bounds for the plate frequency are compared to the theoretical results. Figure 170: Problem Sketch Analysis According to Blevins, the lower bound for the fundamental natural frequency of the annular plate is found using the formula presented in Table 11-2 of the reference: (2) where: λ2 = 4.80 Material Properties Geometric Properties Loading 5 2 Outside Radius (a) = 50 E = 7.1 x 10 kg/cm ν = 0.3 cm -9 Inside Radius (b) = 18.5 ρ = 2.79 x 10 kg/cm2 cm γ = 1.415 x 10-6 Thickness (h) = 0.5 cm kg-sec2/cm3 Sector Angle = 30° Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 259 VMMECH073 Analysis Assumptions and Modeling Notes for ANSYS Mechanical The following figure shows the finite element model in ANSYS Mechanical: Figure 171: ANSYS Mechanical Schematic Results Comparison for ANSYS Mechanical Results Target Mechanical Error (%) Frequency (Hz) 23.38 23.074 -1.306 Analysis Assumptions and Modeling Notes for ANSYS AIM The following figure shows the finite element model in ANSYS AIM: Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 260 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH073 Figure 172: ANSYS AIM Schematic Results Comparison for ANSYS AIM Results Target AIM Error (%) Frequency (Hz) 23.38 23.080 1.283 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 261 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 262 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH074: Tension/Compression Only Springs Overview Reference: Rao, Singiresu S. Mechanical Vibrations. 4th ed. Singapore: Prentice Hall, 2004. 20. Solver(s): ANSYS Mechanical Analysis Rigid Body Dynamic Spring Analysis Type(s): Element Solid Type(s): Test Case This test calculates the elastic forces of both tension and compression only springs. A compression only spring uses a negative (compressive) displacement. A tension only spring uses a positive (tensile) dis- placement. Both spring types are analyzed in tension and compression loading. The detection of the spring state being in tension or compression is determined by the non-linear solver. Figure 173: Schematic Material Properties Geometric Properties Loading k = 1.0e7 N/m Lo = 1 m x1 = 0.5 m x2 = -0.5 m m = 7850 kg Analysis Assumptions and Modeling Notes Hooke’s Law: Elastic Force = Spring Constant * Displacement F = k*x Spring 1: Compression Only spring Spring 2: Tension Only spring Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 263 VMMECH074 Results Comparison Tensile Displacement (x1) Results Target Mechanical Error (%) Elastic Force (N) Spring 1 0 0 0 Elastic Force (N) Spring 2 5.0e6 5.0e6 0 Compressive Displacement (x2) Results Target Mechanical Error (%) Elastic Force (N) Spring 1 -5.0e6 -5.0e6 0 Elastic Force (N) Spring 2 0 0 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 264 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH075: Harmonic Response of Two-Story Building under Transverse Loading Overview Reference: W. T. Thomson, Theory of Vibration with Applications, 3rd Edition, 1999, Example 6.4-1, pg. 166 Solver(s): ANSYS Mechanical Analysis Harmonic Analysis Type(s): Element Solid Type(s): Test Case A two-story building has two columns (2K and K) constituting stiffness elements and two slabs (2M and M) constituting mass elements. Find the y directional deformation frequency response of the system at 70 Hz on each of the vertices for the frequency range of 0 to 500 Hz using mode-superposition as the solution method. Figure 174: Schematic Material Properties Material E (Pa) ν ρ (kg/m3) 18 Block 2 2 x 10 0.3 7850 10 Shaft 2 4.5 x 10 0.35 1 x 10-8 Block 1 2 x 1018 0.3 15700 10 Shaft 1 9 x 10 0.35 1 x 10-8 Geometric Properties Loading Block 1 and 2: Force = -1 x 105 N (y direction) 40 mm x 40 mm x 40 mm Shaft 1 and 2: 20 mm x 20 mm x 200 mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 265 VMMECH075 Analysis Assumptions and Modeling Notes The material of the columns is assigned negligible density to make them as massless springs. The slabs are allowed to move only in the y-direction by applying frictionless supports on all the faces of the slabs in the y direction. The end face of the column (2K) is fixed and a harmonic force is applied on the face of the slab (M) as shown in Figure 174: Schematic (p. 265). Set the solution intervals to 50. Add the frictionless support and fixed support in a modal system, and then link the modal system to a harmonic response system. Note There are frictionless supports on 8 faces of the geometry shown. Results Comparison Results Target Mechanical Error (%) Maximum Amplitude for 0.20853 0.21214 1.731 Vertex A (m) Maximum Amplitude for 0.074902 0.075962 1.415 Vertex B (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 266 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH076: Elongation of a Tapered Shell With Variable Thickness Overview Reference: C. O. Harris, Introduction to Stress Analysis, The Macmillan Co., New York, NY, 1959, pg. 237, problem 4. Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Shell Type(s): Test Case A tapered aluminum alloy plate of length L with varying thickness across length is suspended from a ceiling. An axial load F is applied to the free end of the plate. Determine the maximum axial deflection δ in the plate and the axial stress σy at mid-length (Y = L/2). Use a global mesh size of 0.5 in with mapped face meshing. Figure 175: Schematic Material Properties Geometric Properties Loading 6 Tapered plate: F = 10000 lbf E = 10.4 x 10 psi ν = 0.3 L = 10 in Base width = 2 in Top width = 1 in Thickness varying from 2 in to 1 in from base to top. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 267 VMMECH076 Results Comparison Results Target Mechanical Error (%) Directional Deformation Y 0.0048077 0.0048137 -0.1248 (in) Normal Stress Y at L/2 (psi) 4444 4454.6 -0.2385 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 268 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH077: Heat Transfer in a Bar with Variable Sheet Thickness Overview Reference: For basic equation: Frank P. Incropera and David P. DeWitt, Heat and Mass Transfer, John Wiley & Sons, Inc, 2002, 5th Edition pg. 5. Solver(s): ANSYS Mechanical Analysis Static Thermal Analysis Type(s): Element Shell Type(s): Test Case A 10 x 50 mm plate with a thickness varying from 1 mm to 4 mm is maintained at temperatures of 100 °C and 200 °C as shown below. Find the following: • Temperatures at mid of the surface. • Heat flow reactions on end edges. Figure 176: Schematic Material Properties Geometric Properties Loading E = 2.0 x 1011 Pa Plate Dimensions : 10 Temperature (T1) v=0 X 50 mm. on edge (@ 1mm Thickness Variation : 1 thickness) = 100 °C α = 1.2 x 10-5 1/°C mm to 4 mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 269 VMMECH077 Material Properties Geometric Properties Loading k = 60.5 W/m°C Temperature (T2)on edge (@ 4mm thickness) = 200 °C Analysis Heat flow due to conduction is given by: (3) The area for conduction varies from A1 to A2. The area Ay at any distance y is given as: (4) Inserting Equation 4 (p. 270) in equation Equation 3 (p. 270) and integrating the equation from 0 to L, (5) Temperature at any point y is given as: (6) Results Comparison Results Target Mechanical Error (%) Heat reaction at T1 (W) 2.618 2.6188 0.031 Heat reaction at T2 (W) -2.618 -2.6188 0.031 Temperature at mid of 166.083 166.09 0.004 surface (°C) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 270 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH078: Gasket Material Under Uniaxial Compression Loading-3-D Analysis Overview Reference: Any Nonlinear Material Verification Text Solver(s): ANSYS Mechanical Analysis Static Analysis (ANTYPE=0) Type(s): Element 3-D Structural Solid Elements Type(s): 3-D Gasket Elements Test Case A thin interface layer of thickness t is defined between two blocks of length and width l placed on top of each other. The blocks are constrained on the left and bottom and back faces. The blocks are loaded with pressure P on the top face. Determine the pressure-closure response for gasket elements. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 271 VMMECH078 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 272 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH078 Material Properties Geometric Properties Loading E = 1.04728 x 106 Pa L=1m P1 = 44006400 Pa ν = 0.21 T = 0.02 m P2 = 157147000 Pa Analysis A 3-D analysis is performed first using a mesh of 4 x 4 gasket elements. In order to simulate the loading- unloading behavior of gasket material, the model is first loaded with a pressure P1 and unloaded and then loaded with a pressure P2 and unloaded. The pressure-closure responses simulated are compared to the material definition. Because of convergence issues, the model could not be unloaded to 0 Pa and was instead unloaded to 100 Pa. Results Comparison Target Mechanical Error (%) Gasket Pressure and Closure at End of 1st Loading: GK-PRES -4.4006 x -4.4006 x 0 107 107 GK-CLOS -4.064 x 104 -4.064 x 104 0 Gasket Pressure and Closure at End of 2nd Loading: GK-PRES -1.5715 x -1.5715 x 0 108 108 GK-CLOS -6.8327 x -6.8327 x 0 104 104 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 273 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 274 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH079: Natural Frequency of a Motor-Generator Overview Reference: W. T. Thomson. “Vibration Theory and Applications”. 2nd Printing, Prentice-Hall, Inc., Englewood Cliffs, NJ. pg. 10, ex. 1.3-3. 1965. Solver(s): ANSYS Mechanical Analysis Mode-Frequency Analysis Type(s): Element Pipe Element Type(s): Test Case A small generator of mass m is driven by a main engine through a solid steel shaft of diameter d. If the polar moment of inertia of the generator rotor is J, determine the natural frequency f in torsion. Assume that the engine is large compared to the rotor so that the engine end of the shaft may be assumed to be fixed. Neglect the mass of the shaft also. Figure 177: Schematic Material Properties Geometric Properties Loading E = 31.2 x 106 psi d = .375 in = 8.00 in m = 1 lb-sec2/in J = .031 lb-in-sec2 Results Comparison Results Target Mechanical Error (%) Lower Order F, Hz 48.781 48.781 0 Higher Order F, Hz 48.781 48.781 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 275 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 276 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH080: Transient Response of a Spring-Mass System Overview Reference: R. K. Vierck. “Vibration Analysis”. 2nd Edition. Harper & Row Publishers, New York, NY, 1979. sec. 5-8. Solver(s): ANSYS Mechanical Analysis Transient Dynamic Mode-Superposition Analysis Type(s): Element Type(s): Test Case A system containing two masses, m1 and m2, and two springs of stiffness k1 and k2 is subjected to a pulse load F(t) on mass 1. Determine the displacement response of the system for the load history shown. Figure 178: Schematic Material Geometric Loading Properties Properties k1 = 6 N/m F0 = 50 k2 = 16 N/m N m1 = 2 Kg Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 277 VMMECH080 Material Geometric Loading Properties Properties m2 = 2 Kg td = 1.8 sec Results Comparison Results Target Mechanical Error (%) Y1 , m (@ t = 1.3s) 14.48 14.432 -0.3314 Y2 , m (@ t = 1.3s) 3.99 3.9621 -0.699 Y1 , m (@ t = 2.4s) 18.32 18.589 1.468 Y2 , m (@ t = 2.4s) 6.14 6.2443 2.670 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 278 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH081: Statically Indeterminate Reaction Force Analysis Overview Reference: P.Bezler, M. Hartzman, and M. Reich. Dynamic Analysis of Uniform Support Motion Response Spectrum Method, (NUREG/CR-1677), Brookhaven National Laboratory, August 1980. Problem 2. Pages 48-80. Solver(s): ANSYS Mechanical Analysis Modal analysis Type(s): Spectral analysis Element Elastic straight pipe elements Type(s): Structural Mass element Test Case This benchmark problem contains three-dimensional multi-branched piping systems. The total mass of the system is represented by structural mass elements specified at individual nodes. Modal and response spectrum analyses are performed on the piping model. Frequencies obtained from modal solve and the nodal/element solution obtained from spectrum solve are compared against reference results. The NUREG intermodal/interspatial results are used for comparison. This benchmark problem is also presented in VM-NR1677-01-2-a in the Mechanical APDL Verification Manual. Figure 179: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 279 VMMECH081 Material Properties Geometric Properties Loading Pipe Elements: Straight Pipe: Acceleration response spectrum curve E = 27.8999 x 106 psi. Outer defined Nu = 0.3 Diameter by SV and FREQ commands. Density = 2.587991718 x = 2.375 in Wall 10-10 lb-sec2/in4 Thickness = 0.154 in Mass Elements (lb-sec2/in): (Mass is isotropic) Mass @ node 1: M = 0.447000518 x 10-1 Mass @ node 2: M = 0.447000518 x 10-1 Mass @ node 3: M = 0.447000518 x 10-1 Mass @ node 4: M = 0.447000518 x 10-1 Mass @ node 5: M = 0.432699275 x 10-1 Mass @ node 6: M = 0.893995859 x 10-2 Mass @ node 7: M = 0.432699275 x 10-1 Mass @ node 8: M = 0.893995859 x 10-2 Mass @ node 9: M = 0.893995859 x 10-2 Mass @ node 10: M = 0.432699275 x 10-1 Mass @ node 11: M = 0.893995859 x 10-2 Mass @ node 12: M = 0.432699275 x 10-1 Mass @ node 13: M = 0.893995859 x 10-2 Mass @ node 14: M = 0.893995859 x 10-2 Results Comparison Results Target Mechanical Error (%) 1 8.712 8.7121 0.001 2 8.806 8.8091 0.035 3 17.510 17.509 -0.006 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 280 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH081 Results Target Mechanical Error (%) 4 40.370 40.368 -0.005 5 41.630 41.642 0.029 Results Node Target Mechanical Error (%) UX at node6 0.46186 0.46186 0.00 UY at node8 0.0025747 0.0025747 0.00 UZ at node8 0.446591 0.44949 0.65 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 281 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 282 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH082: Fracture Mechanics Stress for a Crack in a Plate Overview Reference: W.F.Brown, Jr., J.E.Srawley, Plane strain crack toughness testing of high strength metallic materials, ASTM STP-410, (1966). Solver(s): ANSYS Mechanical Analysis Static Structural Analysis Type(s): Element Solid Type(s): Test Case A long plate with a center crack is subjected to an end tensile stress 0 as shown in problem sketch. Symmetry boundary conditions are considered and the fracture mechanics stress intensity factor KI is determined. This problem is also presented in VM256 in the Mechanical APDL Verification Manual. Figure 180: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 283 VMMECH082 Material Geometric Loading Properties Properties E = 30 x 106 a = 1 in σ0 = psi 0.5641895 b = 5 in psi ν = 0.3 h = 5 in t = 0.25 in  Results Comparison Results Target Mechanical Error (%) Stress Intensity KI 1.0249 1.05 2.449 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 284 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH083: Transient Response to a Step Excitation Overview Reference: W. T. Thomson, Vibration Theory and Applications, 2nd Printing, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965, pg. 102, article 4.3. Solver(s): ANSYS Mechanical Analysis Mode-Superposition Transient Dynamic Analysis Type(s): Element Type(s): Test Case A spring-mass-damping system that is initially at rest is subjected to a step force change F acting on the mass. Determine the displacement u at time t for damping ratio, ξ = 0.5. This problem is also presented in VM75 in the Mechanical APDL Verification Manual. Figure 181: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 285 VMMECH083 Material Loading Properties m = 0.5 F= 2 200 lb-sec /in lb k = 200 lb/in Analysis Assumptions and Modeling Notes The damping coefficient c is calculated as lb-sec/in for . Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 286 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH083 Results Comparison Results Target Mechanical Error (%) Total Def Max (ξ = 0.5) Time 1.1531 1.1544 0.113 = 0.20 sec Figure 182: Maximum Deformation vs. Time (damped) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 287 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 288 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH084: Mullins Effect on a Rubber Tube Model Subjected to Tension Loading Overview Reference: .W.Ogden, et al., “A Pseudo-elastic Model for the Mullins Effect in Filled Rubber", Royal Society of London Proceedings Series A., (1989), pg: 2861-2877. Solver(s): ANSYS Mechanical Analysis Static Analysis Type(s): Element Solid Type(s): Test Case An axisymmetric rubber plate made of Neo-Hookean material is modeled with radius R and height H. The model is subjected to cyclic displacement loading on the top surface. The axial stress obtained at different load steps is compared against the reference solution. This problem is also presented in VM268 in the Mechanical APDL Verification Manual. Figure 183: Schematic Material Properties Geometric Loading Properties Neo-Hookean Constants: R = 0.5m One cycle of loading µ = 8 MPa H = 1m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 289 VMMECH084 Material Properties Geometric Loading Properties Step 1: λ = 1.5 Ogden-Roxburgh Mullins Constants: Step 2: λ = 2.0 r = 2.104 Step 3: λ = 3.0 m = 30.45 β =0.2 Step 4: λ = 2.0 Step 5: λ = 1.5 Step 6: λ = 1.0 Results Comparison Results Axial Stress (Pa) Stretch λ Target Mechanical Error (%) 1.5 12.666 12.667 0.008 2.0 28.000 28.000 0.0 3.0 69.333 69.333 0.0 2.0 20.819 20.823 0.019 1.5 8.660 8.6704 0.12 1.0 0.000 0.0 0.0 Figure 184: Variation of Axial Stress Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 290 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH085: Bending of a Composite Beam Overview Reference: R. J. Roark, W. C. Young, Formulas for Stress and Strain, McGraw-Hill Book Co., Inc., New York, NY, 1975, pg. 112-114, article 7.2. Solver(s): ANSYS Mechanical Analysis Static Analysis Type(s): Element Solid Type(s): Test Case A beam of length and width w made up of two layers of different materials is subjected to a uniform rise in temperature from Tref to To, and a bending moment My at the free-end. Ei and α i correspond to the Young's modulus and thermal coefficient of expansion for layer i, respectively. Determine the free-end displacement δ (in the Z-direction) and the X-direction stresses at the top and bottom surfaces of the layered beam. This problem is also presented in VM144 in the Mechanical APDL Verification Manual. Figure 185: Schematic Material Properties Geometric Loading Properties MAT1: = 8 in To = 100°F E1 = 1.2 w = 0.5 in x 106 psi Tref = t1 = 0.2 in α1= 0°F 1.8 x t2 = 0.1 in My = 10-4 10.0 in/in/°F in-lb Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 291 VMMECH085 Material Properties Geometric Loading Properties MAT2: E2 = 0.4 x 106 psi α2= 0.6 x 10-4 in/in/°F Results Comparison Results Target Mechanical Error (%) Displacement, in -0.832 -0.832 0.0 StressxTOP , psi 1730.6 1730.6 0.0 StressxBOT , psi 2257.6 2257.5 0.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 292 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH086: Stress Concentration at a Hole in a Plate Overview Reference: R. J. Roark, Formulas for Stress and Strain, 4th Edition, McGraw-Hill Book Co., Inc., New York, NY, 1965, pg. 384 Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural, Submodeling (2D-2D) Element Type(s): Solid Test Case Determine the maximum stress at a circular hole cut into a rectangular plate loaded with uniform tension P. This problem is also presented in VM142 in the Mechanical APDL Verification Manual. Figure 186: Plate Problem Sketch Material Properties Geometric Properties Loading E = 30 x 106 psi L = 12 in P = 1000 psi υ = 0.3 d = 1 in t = 1 in Analysis Assumptions and Modeling Notes Due to symmetry, only a quarter sector of the plate is modeled. The reference result is from an infinitely long plate. Using a transferred load from the coarse model, the submodel result closely approximates the fine model. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 293 VMMECH086 Results Comparison 2D-2D Results Results Target Mechanical Error (%) Fine Model Equivalent Stress - Max 3018 2871.9 -4.841 Coarse Model Equivalent Stress - Max 3018 3008.6 -3.114 Submodel Equivalent Stress - Max 3018 3023.6 -0.8556 Figure 187: 2D-2D Fine Model Equivalent Stress Figure 188: 2D-2D Coarse Model Equivalent Stress Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 294 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH086 Figure 189: 2D-2D Submodel Equivalent Stress Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 295 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 296 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH087: Campbell Diagrams and Critical Speeds Using Symmetric Orthotropic Bearings Overview Reference: Nelson, H.D., McVaugh, J.M., “The Dynamics of Rotor-Bearing Systems Using Finite Elements”, Journal of Engineering for Industry, Vol 98, pp. 593-600, 1976 Solver(s): ANSYS Mechanical Analysis Type(s): Modal Analysis Element Type(s): Line Body Point Mass Bearing Connection Test Case A rotor-bearing system is analyzed to determine the forward and backward whirl speeds. The distributed rotor is modeled as a configuration of six elements, with each element composed of subelements. See Table 1: Geometric Data of Rotor-Bearing Elements (p. 297) for a list of the geometric data of the indi- vidual elements. Two symmetric orthotropic bearings are located at positions four and six. A modal analysis is performed on the rotor-bearing system with multiple load steps to determine the whirl speeds and Campbell values for the system. This problem is also presented in VM254 in the Mechanical APDL Verification Manual. Figure 190: Rotor-Bearing Configuration Table 1: Geometric Data of Rotor-Bearing Elements Element Number Subelement Axial Distance to Inner Diameter Outer number Subelement (cm) Diameter (cm) 1 1 0.00 1.02 2 1.27 2.04 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 297 VMMECH087 Element Number Subelement Axial Distance to Inner Diameter Outer number Subelement (cm) Diameter (cm) 2 1 5.08 1.52 2 7.62 4.06 3 1 8.89 4.06 2 10.16 6.60 3 10.67 1.52 6.60 4 11.43 1.78 5.08 5 12.70 5.08 6 13.46 2.54 4 1 16.51 2.54 2 19.05 3.04 5 1 22.86 3.04 2 26.67 2.54 6 1 28.70 2.54 2 30.48 7.62 3 31.50 4.06 4 34.54 1.52 4.06 Material Properties Geometric Properties Loading Shaft Refer to Rotational Velocity Table 1: Geometric Data E11 = 2.078 x 1011 Pa of Rotor-Bearing Spin (1) = Elements (p. 297) 1000 RPM G12 = 1.0 x 1014 Pa Spin (2) = Density = 7806 kg/m3 20000 RPM Spin (3) = Mass Element 40000 RPM Spin (4) = Mass = 1.401 kg 60000 RPM Polar inertia = 0.002 kg⋅m2 Spin (5) = Diametral inertia = 0.00136 kg⋅m2 80000 RPM Spin (6) = Bearing Element 100000 RPM Spring coefficients K11 = K22 = 3.503 x 107 N/m K12 = K21 = -8.756 x 106 N/m Analysis Assumptions and Modeling Notes A modal analysis is performed on the rotor-bearing system with QR Damp methods using pipe elements (PIPE288) to determine the whirl speeds and Campbell values. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 298 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH087 A point mass is used to model the rigid disk (concentrated mass). Two symmetric orthotropic bearings are used to assemble the rotor system. No shear effect is included in the rotor-bearing system. The displacement and rotation along and around the X-axis is constrained so that the rotor-bearing system does not have any torsion or traction related displacements. Backward and forward whirl speeds for slope = 1 @ 100000 RPM are determined from the modal ana- lysis. Results Comparison Target Mechanical Error (%) Backward and forward whirl speeds for slope = 1 @ 100000 RPM RPM = Hz * 60 PIPE288 Mode 1 (BW) 10747 10803 0.521 Mode 2 (FW) 19665 19567.8 -0.494 Mode 3 (BW) 39077 39663.6 1.501 Mode 4 (FW) 47549 48202.2 1.014 Figure 191: Campbell Diagram for Rotor-Bearing System Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 299 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 300 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH088: Harmonic Response of a Guitar String Overview Reference: Blevins, R.D., Formulas for Natural Frequency and Mode Shape, Nostrand Reinhold Co., New York, NY, 1979, pg. 90, tab. 7-1 Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Linear Perturbed Modal Linear Perturbed Harmonic Element Type(s): Beam Test Case A uniform stainless steel guitar string of length l and diameter d is stretched between two rigid supports by a tensioning force F1, which is required to tune the string to the E note of a C scale. The string is then struck near the quarter point with a force F2. Determine the fundamental frequency, f1. Also, show that only the odd-numbered frequencies produce a response at the midpoint of the string for this ex- citation. This problem is also presented in VM76 in the Mechanical APDL Verification Manual. Material Properties Geometric Properties Loading E = 190 x 109 Pa l = 710 mm F1 = 84 N c = 165 mm F2 = 1 N ρ = 7920 kg/m3 d = 0.254 mm Analysis Assumptions and Modeling Notes Enough elements are selected so that the model can be used to adequately characterize the string dy- namics. The stress stiffening capability of the elements is used. Linear perturbed harmonic analysis de- termines the displacement response to the lateral force F2. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 301 VMMECH088 Figure 192: Guitar String Problem Results Comparison Target Mechanical Error (%) Modal f, Hz 322.2 322.621 1.001 Frequency f1, (322.2 Hz) Response Response, 320 < f < - Response 328 f2, (644.4 Hz) No Response No Response - f3, (966.6 Hz) Response Response, 966 < f < - 974 f4, (1288.8 Hz) No Response No Response - f5, (1611.0 Hz) Response Response, 1611 < f < - 1619 f6, (1933.2 Hz) No Response No Response - Figure 193: String Midpoint Displacement Amplitude Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 302 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH089: Delamination Analysis of a Double Cantilever Beam Using Contact-Based Debonding Overview Reference: Alfano, G., Crisfield, M.A., “Finite Element Interface Models for the Delamination Analysis of La Mechanical and Computation Issues”, International Journal for Numerical Methods in Engineerin 1701-1736, 2001 Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): Solid Test Case A double cantilever beam of length l, width w, and height h with an initial crack of length a at the free end is subjected to a maximum vertical displacement Umax at the top and bottom free end nodes. De- termine the vertical reaction at point P, based on the vertical displacement using the contact-based debonding capability. This problem is also presented in VM255 in the Mechanical APDL Verification Manual. Figure 194: Double Cantilever Beam Sketch Material Properties Geometric Properties Loading Composite l = 100 mm Umax = 10 mm a = 30 mm E11 = 135.3 GPa h = 3 mm E22 = 9.0 GPa w = 20 mm E33 = 9.0 GPa G12 = 5.2 GPa ν12 = 0.24 ν13 = 0.24 ν23 = 0.46 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 303 VMMECH089 Material Properties Geometric Properties Loading Interface C1 = 1.7 MPa C2 = 0.28 N/mm C5 = 1.0 x 10-5 Analysis Assumptions and Modeling Notes A double cantilever beam is analyzed under displacement control using 2-D plane strain formulation elements. An imposed displacement of Uy = 10 mm acts at the top and bottom free vertex. Contact debonding is inserted at the interface. Defined fracture-energy based debonding material is used to define the material for contact debonding. Equivalent separation-distance based debonding material is also used for the contact debonding object. Based on the interface material parameters used, results obtained using Mechanical are compared to results shown in Figure 15(a) of the reference material. Results Comparison Target Mechanical Error (%) Max RFORCE and corresponding displacement using debonding RFORCE FY (N) 50.619 50.619 0.999 DISP UY (mm) 1.50 1.50 1.000 RFORCE and corresponding displacement U = 10.0 using debonding RFORCE FY (N) 24.543 24.543 1.000 DISP UY (mm) 10.00 10.00 1.000 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 304 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH090: Delamination Analysis of a Double Cantilever Beam Using Interface Delamination Overview Reference: Alfano, G., Crisfield, M.A., “Finite Element Interface Models for the Delamination Analysis of La Engineering, Vol 50, pp. 1701-1736, 2001 Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): Solid Test Case A double cantilever beam of length l, width w, and height h with an initial crack of length a at the free end is subjected to a maximum vertical displacement Umax at the top and bottom free end nodes. De- termine the vertical reaction at point P based on the vertical displacement for the interface model. This problem is also presented in VM248 in the Mechanical APDL Verification Manual. Figure 195: Double Cantilever Beam Sketch Material Properties Geometric Properties Loading Composite l = 100 mm Umax = 10 mm a = 30 mm E11 = 135.3 GPa h = 3 mm E22 = 9.0 GPa w = 20 mm E33 = 9.0 GPa G12 = 5.2 GPa ν12 = 0.24 ν13 = 0.24 ν23 = 0.46 Interface C1 (maximum stress) = 25 MPa Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 305 VMMECH090 Material Properties Geometric Properties Loading C2 (normal separation) = 0.004 mm C3 (shear separation) = 1000 mm Analysis Assumptions and Modeling Notes A double cantilever beam is analyzed under displacement load using interface elements for delamination and 2-D plane strain formulation elements. An imposed displacement of Uy = 10 mm acts at the top and bottom free vertex. An Interface Delamination object is inserted to model delamination. Equivalent material constants are used for the interface material, as Mechanical uses the exponential form of the cohesive zone model and the reference uses a bilinear constitutive model. Results Comparison Lower Order Results Target Mechanical Error (%) Max RFORCE and corresponding DISP: RFORCE FY 60.00 60.069 1.001 (N) DISP UY 1.00 1.000 1.000 (mm) End RFORCE and corresponding DISP RFORCE FY 24.00 23.966 0.999 (N) DISP UY 10.00 10.00 1.00 (mm) Higher Order Results Target Mechanical Ratio Max RFORCE and corresponding DISP RFORCE FY 60.00 60.063 1.001 (N) DISP UY 1.00 1.000 1.000 (mm) End RFORCE and corresponding DISP RFORCE FY 24.00 23.966 0.999 (N) DISP UY 10.00 10.00 1.00 (mm) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 306 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH091: Unbalanced Harmonic Response of a Shaft Carrying Single Rotor with Damping Overview Reference: Any basic vibration analysis book. Solver(s): ANSYS Mechanical Analysis Type(s): Harmonic Element Type(s): Beam Test Case A disc with mass of 5 kg is mounted midway on a simply supported shaft of diameter 0.01 m and length 0.5 m. All bodies are cylindrical. The center of gravity of the disc is 5 x 10-3 m away from the geometric center. The equivalent viscous damping at the center of the disc is 40 N⋅sec/m. The shaft rotates axially at 740 RPM. Find the frequency response in the Y direction at the midpoint of the shaft. Figure 196: Schematic Material Properties Geometric Properties Loading E = 2 x 1011 Pa Length of shaft, Point Mass at -14 3 L = 0.5 m midpoint of shaft, ρ = 1 x 10 kg/m Diameter of m = 5 kg shaft, D = 0.01 m Rotating radius (eccentricity), e = 5 x 10-3 m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 307 VMMECH091 Material Properties Geometric Properties Loading Rotating force, (m x e) = 2.5 x 10-2 kg⋅m Analysis Assumptions and Modeling Notes Static deflection of shaft, Weight of shaft, Stiffness of shaft, Critical speed, Critical damping coefficient, Damping ratio, The amplitude of vibration is therefore: where = eccentricity = speed of shaft = damping coefficient = mass of shaft Results Comparison Results Target Mechanical Error (%) Frequency response (Y) at the midpoint of the shaft @ 740 RPM (12.333 0.018024 0.018039 -0.083 Hz) (m) Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 308 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH091 Figure 197: Bode Plot Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 309 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 310 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH092: Convection Treatment Problem for a Hollow Cylinder with Fluid Flow Overview Reference: Arpaci, V.S., Selamet, A., and Kao, S.H., “Introduction to Heat Transfer”, 2000, pp. 90-100 Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): Coupled Thermal Pipe Thermal Surface Thermal Solid Test Case A hollow cylinder is modeled with an inner radius of 0.01105 m, an outer radius of 0.02 m, and a length of 0.1 m. Fluid is made to flow through the cylinder to simulate the convection problem. Surface effect elements with film coefficients are used in between the fluid and cylinder to include the convection loads. The inlet temperature of the fluid, mass flow rate of the fluid, and the bulk temperature at the outer cylinder surface are defined. A static analysis is performed on the model to determine the nodal temperature of the fluid elements. This problem is also presented in VM271 in the Mechanical APDL Verification Manual. Figure 198: Schematic Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 311 VMMECH092 Material Properties Geometric Properties Loading Fluid: Inner radius, r1 = Inlet temperature Specific heat = 0.5474 J/(kg⋅C) 0.01105 m of fluid, Tinlet = Thermal conductivity = 1.0 x 10-16 Outer radius, r2 = 700 °C W/(m⋅C) 0.02 m Temperature at Cylinder: Length, l = 0.1 m the outer cylinder Thermal conductivity = 1000 W/(m⋅C) surface, Tbulk = 2000 °C Film coefficients for surface element = 300 W/m2⋅C Mass flow rate for fluid = 7.2 kg/s Analysis Assumptions and Modeling Notes The line body model type is set to thermal fluid and the discretization type is set to upwind/exponential in a 3-D steady state thermal analysis. The existing convection is scoped to the inner face of the cylinder and is modified to consider fluid flow with fluid flow edge(s) of line bodies. The mass flow rate is applied using line body edge. The problem is solved using MKS as the unit system. Results are evaluated at different locations on a line body using nodal named selections. Results Comparison Result Target Mechanical Error (%) Temperature (°C) @ Y = 0.0 700 700 1.000 Temperature (°C) @ Y = 3.33 x 913.537 916.91 1.003 10-2 Temperature (°C) @ Y = 6.66 x 1091.999 1096.9 1.004 10-2 Temperature (°C) @ Y = 1.0 1241.147 1239.4 0.998 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 312 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH093: C*-Integral Calculation for a Single-Edge Cracked Plate Using Pre-Meshed Crack Overview Reference: Kanninen, M.F., Popelar, C.H., Advanced Fracture Mechanics, Oxford University Press, p. 550, 198 Solver(s): ANSYS Mechanical Analysis Type(s): Static Analysis Element Type(s): 2-D Structural Solid Element Test Case A rectangular plate with an edge crack is subjected to a tensile load . The load is applied instantly and held for 1000 hours. The C* integral is then determined using Pre-Meshed Crack and compared against the reference solution. Figure 199: Two-Dimensional Fraction Problem Sketch Material Properties Geometric Properties Loading E=200000 MPa L = 100 mm = 150 MPa = 0.3 a = 2.5 mm Parameters for strain hardening b = 20 mm creep c = 17.5 mm C1 = 5 x 10-12 C2 = 3 C3 = 0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 313 VMMECH093 Material Properties Geometric Properties Loading C4 = 0 Analysis Assumptions and Modeling Notes The problem is solved using 2-D structural solid element with plane strain element behavior. One half of the plate is modeled and symmetric boundary conditions are considered. The crack tip nodes and the number of paths surrounding the crack tip nodes are defined using Pre-Meshed Crack. The compu- tation is completed in two steps. In the first step, the instal loading is applied on the top edge. In the second tep, the load is held for 1000 hours. The C* integral is computed for the crack tip nodes by av- eraging the C* integral values from contour 2 to contour 9. The reference value for the C* integral is determined using the following equation: where A = 5 x 10-12 a = 2.5 c = 17.5 C1 = 17.5 h1 = 8.57 (according to the reference material) P = 3000 Results Comparison Result Target Mechanical Error (%) C*-Integral 0.032 0.03312525 -3.5164 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 314 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH094: Residual Vector in Stand-Alone and Linked Mode-Superposition Harmonic Analysis Overview Reference: Dickens, J.M., Nakagawa, J.M., Wittbrodt, M.J., "A Critique of Mode Acceleration and Modal Tru 1997 Solver(s): ANSYS Mechanical Analysis Type(s): Modal Analysis Harmonic Analysis Element Type(s): Spring-Damper Structural Mass Test Case A mode-superposition harmonic analysis (stand-alone and linked) with an excitation frequency range of 3-70 Hz and a force load along the X-direction is performed on a spring-mass model, extracting one mode and residual vector. The spring-mass model is represented using Springs and Point Masses. Refer to Mechanical APDL VM149 for more details. Figure 200: Spring-Mass Model Material Properties Geometric Properties Loading String stiffness (K) = 1000 N/m Total length = 5 m Force along Mass 1 = 1.0 kg X-direction at Mass 2 = 0.5 kg node 4 = 1 N Damping ratio = 2% Excitation frequency: 3-70 Hz Analysis Assumptions and Modeling Notes The spring-mass model is represented using 2-D Spring-Damper and 2-D Structural Mass elements without rotary inertia. The X and Y-axes of the spring model are inverted from the reference model to model it along the global X-axis. The model is fixed at both ends, and the displacement along the Y- direction is constrained on all nodes. In order to obtain four distinct modes, the mass at node 5 is set to half the value of the other three masses defined at nodes 2, 3, and 4. A modal analysis is performed Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 315 VMMECH094 first using the Block Lanczos eigensolver. A mode-superposition harmonic analysis is then performed with an excitation frequency range of 3-70 Hz and a force load along the X-direction at node 4. A constant damping ratio of 0.02 (2%) is defined in the analysis. Results Comparison Result Target Mechanical Error (%) Standalone Results UX_MAX (m) 0.00226 0.0022611 0.0487 F_MAX (N) 10.05002 10.05007 0.00050 Frequency @ UX_MAX 10.102 10.102 0 (Hz) F_MAX2 (N) 4.72924 4.72928 0.00086 Linked Results UX_MAX (m) 0.00226 0.0022611 0.0487 F_MAX (N) 10.05002 10.05007 0.00050 Frequency @ UX_MAX 10.102 10.102 0 (Hz) F_MAX2 (N) 4.72924 4.72928 0.00086 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 316 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH095: 2-D Double Cantilever Beam Problem Overview Reference: Madell, J.F., et al., "Prediction of Delamination in Wind Turbine Blade Structural Details", Journa Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): 2-D Structural Elements Test Case A double cantilever beam is modeled with composite material. One end of the beam is fixed, and the other end is loaded with two external forces of the same magnitude of the same magnitude but opposite directions as shown in the problem schematic. G computation for the cracked tip is conducted for VCCT using a pre-meshed crack object and compared with Equation 1 in the reference. Figure 201: Problem Schematic Material Properties Geometric Properties Loading E = 210 GPa L = 100 P = 10 N = 0.3 a = 60 mm h = 5 mm b = 1 mm Analysis Assumptions and Modeling Notes The problem is solved using 2-D element with plain strain element behavior. A pre-meshed crack object is defined. The plate is subjected to vertical loading the middle. G values are computed for the crack tip node. Results Comparison Result Target Mechanical Error (%) G 0.201 0.193 -4.5 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 317 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 318 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH096: 2-D Fracture Problem Under Thermal Loading Overview Reference: Wilson, W.K., et al., "The Use of the J-Integral in Thermal Stress Crack Problems", International J Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): 2-D Structural Elements Test Case An edge-cracked strip is modeled with its ends constrained. The strip is subject to a linear temperature gradient through its thickness, starting at zero at the mid-thickness and reaching final value T0 at the right edge. Stress intensity factor for the cracked strip is calculated and compared against the reference value. Figure 202: Problem Schematic L W T0 0 -T0 Temperature Distribution Material Properties Geometric Properties Loading E = 1 x 105 Pa Crack length = 1 T0 = 10 ° = 0.3 mm L = 4 mm Thermal expansion, = 1 x 10-4 W = 2 mm Analysis Assumptions and Modeling Notes The problem is solved using 2-D structural elements with plain strain element behavior. Results Comparison Result Target Mechanical Error (%) Stress intensity, KI 126.604 128.633 1.602 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 319 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 320 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH097: Inclined Crack in 2-D Plate Under Uniform Tension Loading Overview Reference: Anderson, T.L., "Fracture Mechanics: Fundamentals and Applications", CRC Press, Boca Raton, F Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): 2-D Structural Elements Test Case A 2-D plate with length L is subjected to uniform tension loading. An inclined crack of length 2a is modeled with an angle of between the crack surface and loading direction. Stress intensity factor is calculated and compared against analytical value. Figure 203: Problem Sketch 2a L L Material Properties Geometric Properties Loading E = 210 GPa L = 0.3 m = 10 MPa = 0.3 Crack length, 2a = 0.09 m = 30° Analysis Assumptions and Modeling Notes The problem is solved using 2-D elements with plain strain element behavior. The plate is constrained along the X-direction at X = 0 and along the Y-direction at Y = 0. Stress intensity factors K1 and K2 are computed using a Pre-Meshed Crack object and compared against the analytical value. Results Comparison Results Target Mechanical Error (%) KI_Right tip (Mode1) 2819957 2839875 0.706% KII_Right tip (Mode2) 1628103 1647625 1.199% Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 321 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 322 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH098: 2-D End Notched Flexure Problem Overview Reference: Mandell, J.F., et al., "Prediction of Delamination in Wind Turbine Blade Structural Details", Journ Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): 2-D Structural Elements Test Case A beam is clamped at one end and contains a delamination of length a at the other end. A load P is applied in the middle to cause crack growth. VCCT G result is computed using a Pre-Meshed Crack object and compared against the analytical solution (equation 2 in reference). Figure 204: Problem Sketch P a 2L P/2 P/2 Material Properties Geometric Properties Loading E = 210000 MPa L = 30 mm P = 10 N = 0.3 a = 10 mm Analysis Assumptions and Modeling Notes The problem is solved using 2-D elements with plain strain element behavior. Results Comparison Result Target Mechanical Error (%) G 0.11338 0.11702 3.274 % Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 323 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 324 of ANSYS, Inc. and its subsidiaries and affiliates. VMMECH099: Mode I Crack Growth Analysis of DCB Using Interface Delamination Overview Reference: Krueger, R., "Application of Benchmark Examples to Assess the Single and Mixed-Mode Static Solver(s): ANSYS Mechanical Analysis Type(s): Static Structural Element Type(s): 2-D Structural Elements Test Case A double cantilever beam is fixed at one end and contains a crack of length a0 at the other end. Equal and opposite displacements are applied to top and bottom vertices to cause crack growth. Critical load and corresponding displacement values for Mode I failure are computed using the VCCT method of interface delamination and compared against results shown in Figure 14 of the reference material. Figure 205: Problem Schematic Material Properties Geometric Properties Loading T300/1076 Unidirectional B = 25 mm d = 1.96 mm Graphite/Epoxy Prepreg 2h = 3 mm 2L = 150 mm E11 = 139 GPa a0 = 20 mm E22 = 10.16 GPa E33 = 10.16 GPa = 0.3 = 0.3 = 0.436 GIc = 0.170 kJ/m2 GIIc = 0.494 kJ/m2 n (material constant) = 1.62 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 325 VMMECH099 Analysis Assumptions and Modeling Notes The problem is solved using 2-D elements with plain strain element behavior. This problem is solved using two methods of VCCT-based interface delamination, node matching, and matched meshing. Results Comparison Result Target Mechanical Error (%) VCCT (G1) Node 0.17 0.17061 0.359 Matching VCCT (G1) Matched 0.17 0.17061 0.359 Meshing Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 326 of ANSYS, Inc. and its subsidiaries and affiliates. Part IV: Design Exploration Descriptions VMDX001: Optimization of L-Shaped Cantilever Beam Under Axial Load Overview Reference: From the Basic Principle Analysis Goal Driven Optimization Type(s): Element 3-D Solid Type(s): Test Case An L-shaped beam with dimensions 30 x 25 mm with 4 mm as the rib thickness and 300 mm in length has the surface fixed at one end. A force of 10,000 N is then applied to the opposite end of the beam. Input Parameters: Width, Height, and Length (CAD Geometry) Response Parameters: Volume, Stress, and Deflection Figure 206: Schematic Material Properties Geometric Loading E = 2e11 Pa Properties Fixed Support ν=0 Width = 25 mm Force F = ρ = 7850 kg/m3 Height = 30 10000 N (Z mm direction) Rib Thickness = 4 mm Length = 300 mm Parameter Type Limits Desired Value Importance Width Input 20 mm W 30 No Preference High mm Height Input 25 mm H 35 No Preference High mm Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 329 VMDX001 Parameter Type Limits Desired Value Importance Length Input 250 mm L 350 No Preference High mm Volume Output n/a Minimum Possible Low Stress Output n/a Minimum Possible High Deflection Output n/a Minimum Possible High Analysis Beam volume: Maximum axial deformation under load F: Normal stress along Z-direction: Combined objective function becomes: Minimizing ϕ we get dimensions as: L = Length = 0.250 m W = Width = 0.030 m H = Height = 0.035 m Results Comparison Results Target DesignXplorer Error (%) 3 Volume (V) 6.9e-5 m 6.9E-05 m3 0.0 Deformation (D) 4.5290e-5 m 4.5339E-05 0.10862 m Stress (σ) 3.62319e7 3.623065E-07 0.00046 Pa Pa Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 330 of ANSYS, Inc. and its subsidiaries and affiliates. VMDX002: Optimization of Bar with Temperature-Dependent Conductivity Overview Reference: From the Basic Principle Analysis Goal Driven Optimization Type(s): Element 3-D Solid Type(s): Test Case A long bar 2 X 2 X 20 m is made up of material having thermal conductivity linearly varying with the temperature K = k0*(1 + a*T) W/m-°C, k0 = 0.038, a = 0.00582. The bar is constrained on all faces by frictionless support. A temperature of 100°C is applied at one end of the bar. The reference temperature is 5°C. At the other end, a constant convection coefficient of 0.005 W/m2°C is applied. The ambient temperature is 5°C. Input Parameters: Convection coefficient, coefficient of thermal expansion and length Response Parameters: Temperature (scoped on end face), thermal strain Figure 207: Schematic Material Properties Geometric Loading E = 2e11 Pa Properties Frictionless Support (on ν=0 Breadth B = 2 all faces) α = 1.5E-05/°C m Reference temperature K = k0*(1 + a*T) Width W = 2 m = 5°C W/m-°C Length L = 20 Temperature on end k0 = 0.038 m face T = 100°C a = 0.00582 Convection on other end face Convection coefficient h = 5e-3 W/m2°C Ambient temperature Ta = 5°C Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 331 VMDX002 Parameter Type Limits Desired Value Importance Length (l) Input 15 m l 25 m No Preference Low 2 Convection Input 0.004 W/m °C h No Preference Low coefficient 0.006 W/m2°C (h) Coefficient Input 1.4e-5/°C α No Preference Low of 1.6e-5/°C temperature expansion (α) Temperature Output n/a Minimum Possible High (T) Thermal Output n/a Minimum Possible High strain (ε) Analysis Temperature: Thermal strain: Combined objective function becomes, Minimizing ϕ we get input parameters as: l = beam length = 25 m h = convection coefficient = 0.006 W/m2°C α = coefficient of thermal expansion = 1.4e-5/°C Results Comparison Results Target DesignXplorer Error (%) Length (l) 25 m 25 m 0 Convection coefficient (h) 0.006 0.006 0 W/m2°C W/m2°C Coefficient of thermal 1.4e-5/°C 1.4e-5/°C 0 expansion (α) Temperature (T) 29.6528°C 29.553°C -0.3278 Thermal strain (ε) 3.4514e-4 3.437e-4 -0.4115 m/m m/m Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 332 of ANSYS, Inc. and its subsidiaries and affiliates. VMDX003: Optimization of Water Tank Column for Mass and Natural Frequency Overview Reference: S. S. Rao, Optimization Theory and Application Second edition, example 1.10, page 28-30 Analysis Goal Driven Optimization with APDL Type(s): Element 3-D Solid Type(s): Test Case A uniform column of rectangular cross section b and d m is to be constructed for supporting a water tank of mass M. It is required to: 1. minimize the mass of the column for economy 2. maximize the natural frequency of transverse vibration of the system for avoiding possible resonance due to wind. Design the column to avoid failure due to direct compression (should be less than maximum permissible compressive stress) and buckling (should be greater than direct compressive stress). Assume the max- imum permissible compressive stress as σmax. The design vector is defined as: where: b = width of cross-section of column d = depth of cross-section of column Input Parameters: Width and Height Response Parameters: Mass, Natural Frequency, Direct Stress, Buckling Stress Material Geometric Properties Loading Properties E = 3e10 Pa Width, b = 0.4 m Mass of water tank M = 1000000 Kg ρ = 2300 Depth, d =1.2 m 3 Acceleration due to gravity = 9.81 Kg/m Length, I = 20 m m/s2 σ max = 4.1e7 Pa Sample Size: 10000 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 333 VMDX003 Results Target DesignXplorer Error (%) Width b 0.36102 m 0.36102 m 0.000 Depth d 1.3181 m 1.318137 m 0.002 Mass of column M 21890 kg 21889.77 kg –0.001 Natural frequency w 0.87834 0.87816 -0.020 rad/sec rad/sec Direct stress 2.0386e7 Pa 2.0383e7 Pa -0.015 Buckling stress 6.1526e6 Pa 6.15174e6 -0.013 Pa Analysis Minimize: Maximize: Subject to constraints: Required objective is obtained by having: b = 0.36102 m d = 1.3181 m M = (minimum) = 21890 kg W = (maximum) = 0.87834 rad/sec Direct stress = 2.0386e7 Pa Buckling stress = 6.1526e6 Pa Results Comparison Results Target DesignXplorer Error (%) Width b 0.36102 m 0.36102 m 0.000 Depth d 1.3181 m 1.318137 m 0.002807 Mass of column M 21890 kg 21890.1957 -0.00089 kg Natural frequency w 0.87834 0.87816 -0.02074 rad/sec rad/sec Direct stress 2.0386e7 Pa 2.0383e7 Pa -0.01277 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 334 of ANSYS, Inc. and its subsidiaries and affiliates. VMDX003 Results Target DesignXplorer Error (%) Buckling stress 6.1526e6 Pa 6.15174e6 -0.0139 Pa Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 335 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 336 of ANSYS, Inc. and its subsidiaries and affiliates. VMDX004: Optimization of Frequency for a Plate With Simple Support at all Vertices Overview Reference: Blevins, Formula for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company Inc., 1979, pg. 269-271 Analysis Goal Driven Optimization Type(s): Element 3-D Shell Type(s): Test Case A square plate of side 250 mm and thickness 5 mm is simply supported on all its vertices. Input Parameters: Young's modulus, Poisson's ratio and density Response Parameters: First natural frequency Figure 208: Schematic Material Properties Geometric Loading E = 2e5 MPa Properties All vertices are ν = 0.3 Length a = 250 simply ρ = 7.850 e-6 mm supported kg/mm3 Width b = 250 mm Thickness h = 5 mm Parameter Type Constraints Desired Value Importance Young's Modulus E Input 1.8e11 Pa E No Preference Low 2.2e11Pa Poisson's Ratio μ Input 0.27 μ 0.30 No Preference Low Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 337 VMDX004 Parameter Type Constraints Desired Value Importance 3 Density ρ Input 7065 kg/m ρ No Preference Low 8635 kg/m3 First Natural Output N/a Minimum Possible High Frequency w Analysis First Natural Frequency: Objective function becomes: Minimizing ϕ we get dimensions as: Young's Modulus E = 1.8e11 Pa Poisson's Ratio μ = 0.27 Density ρ = 8635 kg/m3 First Natural Frequency w = 124.0913 rad/s Results Comparison Results Target DesignXplorer Error (%) Young's Modulus E 1.8e11 Pa 1.8e11 Pa 0.00 Poisson's Ratio μ 0.27 0.27 0.00 3 3 Density ρ 8635 kg/m 8635 kg/m 0.00 First Natural Frequency w 124.0913 123.36 rad/s -0.5894 rad/s Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 338 of ANSYS, Inc. and its subsidiaries and affiliates. VMDX005: Optimization of Buckling Load Multiplier With CAD Parameters and Young's Modulus Overview Reference: Timoshenko, Strength of Materials, Part 2 (Advanced theory and problems), pg. 167–168 Analysis Goal Driven Optimization Type(s): Element 3-D Solid Type(s): Test Case The cantilever bar of length 25 feet is loaded by uniformly distributed axial force p = 11 lbf on one of the vertical face of the bar in negative Z-direction. The bar has a cross-sectional area A is 0.0625 ft2. Input Parameters: Side of Square C/S , Length of Cantilever Bar and Young's Modulus Response Parameters: Load Multiplier of the First Buckling Mode Optimization Method: Genetic Algorithm Sample Size: 200 Figure 209: Schematic Material Properties Geometric Loading E = 4.1771e 9 psf Properties Fixed support ν = 0.3 Cross-section of on one face, ρ = 490.45 lbm/ft3 square = 0.25 Force = 11 lbf ft. x 0.25 ft. (Negative Length of bar = Z-direction) on 25 ft. top face Parameter Type Constraints Desired Value Importance Cross-section side Input 0.225 ft. a 0.275 No Preference N/A ft. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 339 VMDX005 Parameter Type Constraints Desired Value Importance Length Input 22.5 ft. l 27.5 ft. No Preference N/A Young's Modulus Input 3.7594e9 psf E No Preference N/A 4.5948e9 psf First buckling mode Output N/A Maximum Possible N/A load multiplier Analysis Assuming that under the action of uniform axial load a slight lateral bucking occurs. The expression for deflection is: The critical load is given by, where: q = force per unit length The first critical buckling load is: The load multiplier is given by the ratio of critical load to applied load . The first buckling multiplier is: Combined objective function becomes: Minimizing ϕ we get dimensions as: Cross-section side a = 0.275 ft. Length l = 22.5 ft. Young's Modulus E = 4.5948e9 psf Buckling load multiplier = 3083.32 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 340 of ANSYS, Inc. and its subsidiaries and affiliates. VMDX005 Results Comparison Results Target DesignXplorer Error (%) First buckling mode load 3083.32 3037.1 -1.499 multiplier Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 341 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 342 of ANSYS, Inc. and its subsidiaries and affiliates. Part V: Explicit Dynamics Descriptions EXDVM2: Propagation of Shock and Rarefaction Waves in a Shock Tube Overview Reference: Harlow, Francis H. et al., Fluid Dynamics – A LASL Monograph, LA-4700, June 1971. Analysis Autodyn 3D Type(s): Elements: Multi-material Euler Quad Euler Ideal Gas Quad Boundary Default Wall Conditions: Structural No Interactions: Fluid-Structure No Interactions: Bonds: No Materials: Ideal Gas Test Case A 1000 mm (1000 element) shock tube is divided into two sections be a diaphragm at its midpoint. Initially, gas is at rest on both sides. To the left of the diaphragm, the gas is initially at a higher density and pressure than on the right . Both sides of the diaphragm have the same internal energy . At t = 0, the diaphragm is removed and the gas in the tube is observed. A shock is observed moving from left to right, as well as a a contact discontinuity moving from the left to the right, and a rarefaction wave moving from the right to the left. There is no significant length to the system. The appearance of the system at a later time is a magnific- ation of an earlier appearance (similarity solution). Material Properties Geometric Properties Loading Heat capacity ratio, Length of shock tube = 1000 Density of gas on left of diaphragm, mm g/mL Density of gas on right of column, g/mL Pressure of gas on left of diaphragm, MPa Pressure of gas on right of column, MPa Internal energy, J/kg Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 345 EXDVM2 Analysis Assumptions and Modeling Notes Two identical 1000 mm shock tubes are modeled using 1000 elements for each tube. One tube uses the Multi-Material Euler solver and the other uses the Ideal Gas (Blast) solver. Because of a limitation on creating Euler inflow boundary conditions, two interacting parts are modeled, one with Euler inflow properties and the other with Ideal Gas properties. The calculation is run to t = 0.8 ms, the time at which the shock front nears the right boundary. Results Comparison Applying the theoretical solution to the initial conditions used for the simulation, the following values are obtained: Final pressure in tube, MPa Density of gas on left of diaphragm, g/mL Density of gas on right of diaphragm, g/mL Continuous pressure, MPa Comparing 28 profiles of pressure and density along the length of the shock tube at t = 0.8 ms show that these theoretical values are closely matched by the simulation. The results using the Ideal Gas (Blast) solver show steeper rise times for the shock and density discontinuity, reflecting the more accurate 2nd order solution calculated by this solver. Figure 210: Pressure Along the Shock Tube at t = 0.8 ms Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 346 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM2 Figure 211: Density of Gas Along the Shock Tube at t = 0.8 ms Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 347 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 348 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM3: Flow of Gas Past an Infinite Two-Dimensional Wedge Overview Reference: Harlow, Francis H. et al., Fluid Dynamics – A LASL Monograph, LA-4700, June 1971. Analysis Autodyn 2D, Planar Symmetry Type(s): Elements: Multi-material Euler Quad Euler Ideal Gas Quad Boundary Euler Inflow, Euler Outflow Conditions: Structural No Interactions: Fluid-Structure Yes, fully coupled Interactions: Bonds: No Materials: Ideal Gas Test Case Perform a dynamic flow analysis of a Mach 2 gas flowing past a 2-D wedge until a steady-state config- uration is obtained. An attached shock is formed, the angle of which is determined by the flow conditions. Since the appearance of the configuration is independent of magnification, there is no significant length to the system. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 349 EXDVM3 Figure 212: Model of Ideal Gas Flowing Past a Two-Dimensional Wedge Material Properties Geometric Loading Properties Heat capacity ratio, Angle of wedge = 25 ° Density of gas, g/mL Pressure of gas, MPa Internal energy, J/kg Flow velocity, m/s Analysis Assumptions and Modeling Notes Two 200 x 200 mm square regions are modeled using two 200 x 200 element Euler Parts. One part uses the Euler Multi-Material solver, the other part uses the Ideal Gas (Blast) solver. Single element fixed Lagrange parts are superimposed on top of each Euler part to impose a 25 degree wedge on the flow. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 350 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM3 An inflow boundary condition is applied to the left boundary and an outflow boundary condition is applied to the right boundary of the Euler parts. The default wall boundary condition is maintained elsewhere. The calculation is run for 3000 cycles, by which time a steady-state condition is achieved. Results Comparison The sound speed of the gas is given by m/s. The incoming flow Mach number is therefore . According to the reference document, the angle of the detached shock is approx- imately 49 °. Figure 213: Pressure of the Gas in Steady-State Condition Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 351 EXDVM3 Figure 214: Density of the Gas in Steady-State Condition Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 352 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM4: Regular and Mach Deflections Off a Two-Dimensional Wedge Overview Reference: Harlow, Francis H. et al., Fluid Dynamics – A LASL Monograph, LA-4700, June 1971. Analysis Autodyn 2D, Planar Symmetry Type(s): Elements: Multi-material Euler Quad Euler Ideal Gas Quad Boundary Euler Inflow, Euler Outflow Conditions: Structural No Interactions: Fluid-Structure Yes Interactions: Bonds: No Materials: Ideal Gas Test Case Perform a dynamic flow analysis of a Mach 1.1 shock reflection off a 45 ° wedge, and a Mach 2 shock reflection off a 25 ° wedge. The first analysis should produce a regular reflection, and the second a Mach reflection. Since the appearance of the configuration is independent of magnification, there is no significant length to the system. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 353 EXDVM4 Figure 215: Mach 1.1 and Mach 2.0 Problem Sketches for the Ideal Gas Solver and Multi-Material Euler Solver Material Properties Geometric Loading Properties Initial Conditions Mach 1.1 Flow Angle of wedge Heat capacity ratio, = 45 ° Density of gas, Mach 2.0 Flow g/mL Angle of wedge Pressure of gas, = 25 ° MPa Internal energy, J/kg Mach 1.1 Flow Velocity of gas, m/s Pressure of gas, MPa Density of gas, g/mL Internal energy, J/kg Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 354 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM4 Material Properties Geometric Loading Properties Mach 2.0 Flow Velocity of gas, m/s Pressure of gas, MPa Density of gas, g/mL Internal energy, J/kg Analysis Assumptions and Modeling Notes Four Euler parts are creating, two using the Euler Multi-Material solver and two using the Ideal Gas (Blast) solver. Single element fixed Lagrange parts are superimposed on top of each Euler part to impose a wedge on the flow. Two of the wedges have a 45 degree angle, and two have a 25 degree angle. An inflow boundary condition is applied to the left boundary of each Euler part. Mach 1.1 flow conditions are used for the parts containing the 45 ° wedge, and Mach 2.0 flow conditions are used for the parts containing the 25 ° wedge. The calculation is run for 0.85 ms, by which time the shock has progressed almost to the right edge of the Euler parts. Results Comparison Results shown below are from the end of the simulation (t = 0.85 ms). Both plots are from the same result, but have the pressure scales adjusted to show good resolution of the desired reflection. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 355 EXDVM4 Figure 216: Regular Reflection Off the 45 ° Wedge Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 356 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM4 Figure 217: Mach Reflection Off the 25 ° Wedge Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 357 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 358 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM6: 3-D Taylor Cylinder Impact Overview Reference: No theoretical solution, experimental results and code-comparisons are available: Taylor, G.I., “The Use of Flat Ended Projectiles for Determining Yield Stress, Part I: Theoretical Considerations,” Proceedings of the Royal Society (London), Vol. 194, pp. 289-299, 1948 Lacy, J.M., Novascone, S.R., Richins, W.D., and Larson, T.K., “A Method for Selecting Software for Dynamic Event Analysis II: The Taylor Anvil and Dynamic Brazilian Tests,” Proceedings of the 16th International Conference on Nuclear Engineering, INL/CON-08-13727, Idaho National Laboratory, 2008 Analysis Autodyn 3D Type(s): Elements: 8-Node Linear Interpolated Reduced Integration Hex Boundary Fixed Constraint Conditions: Structural Yes, trajectory contact Interactions: Fluid-Structure No Interactions: Bonds: No Materials: Copper Test Case The Taylor cylinder impact test uses a right circular cylinder of a test material which impacts a theoret- ically rigid target. In this test, an OHFC copper cylinder, 0.762 cm in diameter and 2.54 cm in length, impacts a rigid plate at 19000 cm/s. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 359 EXDVM6 Figure 218: Finite Element Model of Copper Taylor Cylinder with 1/4 Symmetry Material Properties Geometric Properties Loading Cylinder material = Diameter of cylinder Impact velocity = 190 copper = 0.762 cm m/s Length of cylinder = 2.54 cm Analysis Assumptions and Modeling Notes Two parts are created to model the copper cylinder and the rigid plate. Quarter symmetry is used to reduce simulation time. Material data for copper is obtained from the Explicit Materials data source in Engineering Data. This data is the same as the material data used in the code comparison reference. A 0.2 mm element size is used to mesh the cylinder. The rigid plate is modeled with a single element. Trajectory contact is used to compute the impact of the cylinder on the plate. The initial velocity of the cylinder is 190 m/s and the simulation is run for 8 x 10-5 s. Results Comparison The final cylinder profile is similar to the profile shown for the Autodyn results shown in the code comparison reference. The cylinder radius in the impact plane agrees well with the experimentally ob- tained values, as well as with other simulation programs. The final cylindar length in this simulation is greater than the experimental value, but agrees well with the length computed by the other simulation programs using the same material model for copper. The plot below shows the final share of the de- formed copper cylinder at the end of the simulation, compared to the original shape (semi-transparent). Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 360 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM6 Figure 219: Final Deformed Shape of the Taylor Cylinder Figure 220: Profile of Final Deformed Shape of Taylor Cylinder With Scale Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 361 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 362 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM7: 2-D Taylor Cylinder Impact Overview Reference: No theoretical solution, experimental results and code-comparisons are available: Taylor, G.I., “The Use of Flat Ended Projectiles for Determining Yield Stress, Part I: Theoretical Considerations,” Proceedings of the Royal Society (London), Vol. 194, pp. 289-299, 1948 Lacy, J.M., Novascone, S.R., Richins, W.D., and Larson, T.K., “A Method for Selecting Software for Dynamic Event Analysis II: The Taylor Anvil and Dynamic Brazilian Tests,” Proceedings of the 16th International Conference on Nuclear Engineering, INL/CON-08-13727, Idaho National Laboratory, 2008 Analysis Autodyn 2D Type(s): Elements: 4-Node Hex Boundary Fixed Constraint Conditions: Structural Yes, proximity-based contact Interactions: Fluid-Structure No Interactions: Bonds: No Materials: Copper Test Case The Taylor cylinder impact test uses a right circular cylinder of a test material which impacts a theoret- ically rigid target. In this test, an OHFC copper cylinder, 0.762 cm in diameter and 2.54 cm in length, impacts a rigid plate at 19000 cm/s. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 363 EXDVM7 Figure 221: Finite Element Model of a Copper Taylor Cylinder with 2-D Axial Symmetry Material Properties Geometric Properties Loading Cylinder material = Diameter of cylinder Impact velocity = 190 copper = 0.762 cm m/s Length of cylinder = 2.54 cm Analysis Assumptions and Modeling Notes Two parts are created to model the copper cylinder and the rigid plate. 2-D axial symmetry is used to reduce simulation time. Material data for copper is obtained from the Explicit Materials data source in Engineering Data. This data is the same as the material data used in the code comparison reference. A ~0.15 mm element size is used to mesh the cylinder. The rigid plate is modeled with a single element. Proximity-based contact is used to compute the impact of the cylinder on the plate. The initial velocity of the cylinder is 190 m/s and the simulation is run for 8 x 10-5 s. Results Comparison The final cylinder profile is similar to the profile shown for the Autodyn results shown in the code comparison reference. The cylinder radius in the impact plane agrees well with the experimentally ob- tained values, as well as with other simulation programs. The final cylindar length in this simulation is greater than the experimental value, but agrees well with the length computed by the other simulation programs using the same material model for copper. The plot below shows the final share of the de- formed copper cylinder (rotated through 360 °) at the end of the simulation. Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 364 of ANSYS, Inc. and its subsidiaries and affiliates. EXDVM7 Figure 222: Final Deformed Shape of the Taylor Cylinder (Rotated Through 360 °) Figure 223: Profile of Final Deformed Shape of Taylor Cylinder With Scale Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 365 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 366 of ANSYS, Inc. and its subsidiaries and affiliates. Part VI: Aqwa Descriptions AQVM1: Hydrostatic Evaluation of a Floating Inverted Pyramid Overview Reference: Floating Structures: a guide for the design and analysis, Ed. N.D.P. Barltrop, Oilfield Publications Ltd., 1998 Analysis Hydrostatic Type(s): Test Case An inverted pyramid is floating in static equilibrium in a large body of water with a density of 1025 kg/m3. Calculate the hydrostatic properties for this model. Figure 224: Problem Sketch of a Floating Inverted Pyramid Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 369 AQVM1 Figure 225: Finite Element Mesh of a Floating Inverted Pyramid Material Properties Geometric Properties Loading Density of water = 1025 Center of gravity at Z = Acceleration due to kg/m3 0 gravity = 9.8 m/s2 Results Comparison Target Aqwa Ratio 3 Waterplane Aw 2000 2.000 x 10 1.0 area Displaced ∇ 13333.333 1.33333 x 104 1.0 volume Center of Bx 25.0 25.000 1.0 buoyancy By 10.0 10.000 1.0 Bz -5.0 -5.000 1.0 Center of LCF 33.333 33.333 1.0 floatation TCF 13.333 13.333 1.0 6 6 Waterplane 1.1613 x 10 1.161 x 10 1.0 inertia 1.2757 x 105 1.276 x 105 1.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 370 of ANSYS, Inc. and its subsidiaries and affiliates. AQVM1 Target Aqwa Ratio 12.7317 12.732 1.0 Metacenters BMX 9.5677 9.568 1.0 BMY 87.0975 87.099 1.0 Metacentric GMX 4.5677 4.568 1.0 heights GMY 82.0975 82.099 1.0 7 7 Restoring Roll 1.0677 x 10 1.068 x 10 1.0 moment MCT 1.9191 x 10 8 1.919 x 10 8 1.0 7 7 Stiffness K33 2.0090 x 10 2.009 x 10 1.0 matrix K34, K43 6.6966 x 10 7 6.697 x 10 7 1.0 8 8 K35, K53 -1.6742 x 10 -1.674 x 10 1.0 9 9 K44 1.3393 x 10 1.339 x 10 1.0 9 9 K45 1.6742 x 10 1.674 x 10 1.0 10 10 K55 1.1887 x 10 1.189 x 10 1.0 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 371 Release 18.1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information 372 of ANSYS, Inc. and its subsidiaries and affiliates.