NX Nastran Verification Manual

NX Nastran Verification Manual

Proprietary & Restricted Rights Notice

© 2007 UGS Corp. All Rights Reserved. This software and related documentation are proprietaryto UGS Corp.

NASTRAN is a registered trademark of the National Aeronautics and Space Administration. NXNastran is an enhanced proprietary version developed and maintained by UGS Corp.

MSC is a registered trademark of MSC.Software Corporation. MSC.Nastran and MSC.Patranare trademarks of MSC.Software Corporation.

All other trademarks are the property of their respective owners.

2 NX Nastran Verification Manual

Contents

Part I: Introduction

Overview of the Verification Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Running the Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

Part II: Linear Statics Verification Using Theoretical Solutions

Overview of Linear Statics Verification Using Theoretical Solutions . . . . . . . . . . . 3-1

Understanding the Test Case Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Understanding Comparisons with Theoretical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3-2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

Point Load on a Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1Axial Distributed Load on a Linear Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3Distributed Loads on a Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5Moment Load on a Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7Edge Pressure on Beam Element - Torque Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9Thermal Strain, Displacement, and Stress on Heated Beam . . . . . . . . . . . . . . . . . . . . . . 4-11Uniformly Distributed Load on Linear Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13Membrane Loads on a Linear Quadrilateral Thin Shell Element . . . . . . . . . . . . . . . . . . . 4-15Axial Loading on Rod Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17Stress on a Beam as It Expands and Closes a Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19Thin Wall Cylinder in Pure Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21Thin Shell Beam Wall in Pure Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23Strain Energy of a Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

Part III: Linear Statics Verification Using Standard NAFEMS Benchmarks

Overview of Linear Statics Verification Using Standard NAFEMS Benchmarks . . . 5-1

Understanding the Test Case Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

Elliptic Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Cylindrical Shell Patch Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6Hemisphere-Point Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10Z-Section Cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13Skew Plate Normal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15Axisymmetric Cylinder/Sphere — Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17Axisymmetric Shell Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21Thick Plate Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-25Solid Cylinder/Taper/Sphere — Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-30

NX Nastran Verification Manual 3

Contents

Part IV: Normal Mode Dynamics Verification

Overview of Normal Mode Dynamics Verification Using Theoretical Solutions . . . 7-1

Understanding the Test Case Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Understanding Comparisons with Theoretical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

Natural Frequency of Circular Ring with Axisymmetric Model . . . . . . . . . . . . . . . . . . . . . 8-1Undamped Free Vibration — Single Degree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3Two Degrees of Freedom Undamped Free Vibration — Principle Modes . . . . . . . . . . . . . . . 8-5Three Degrees of Freedom Torsional System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7Two Degrees of Freedom Vehicle Suspension System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8Two Degrees of Freedom Vehicle Suspension System . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-10Cantilever Beam Undamped Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12Natural Frequency of a Cantilevered Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

Part V: Normal Mode Dynamics Verification Using Standard NAFEMS Benchmarks

Overview of Normal Mode Dynamics Verification Using Standard NAFEMSBenchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

Understanding the Test Case Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-1

Beam Element Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

Pin-ended Cross — In-plane Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1Pin-ended Double Cross - In-plane Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4Free Square Frame - In-plane Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-7Cantilever with Off-center Point Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-9Deep Simply-Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-11Circular Ring — In-plane and Out-of-plane Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 10-14Cantilevered Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-16

Shell Element Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

Thin Square Cantilevered Plate — Symmetric Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1Thin Square Cantilevered Plate — Anti-symmetric Modes . . . . . . . . . . . . . . . . . . . . . . . 11-4Free Thin Square Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7Simply Supported Thin Square Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-10Simply Supported Thin Annular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13Clamped Thin Rhombic Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-16Cantilevered Thin Square Plate with Distorted Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 11-19Simply Supported Thick Square Plate, Test A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-24Simply Supported Thick Square Plate, Test B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-27Clamped Thick Rhombic Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-30Simply Supported Thick Annular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-33Cantilevered Square Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-36Cantilevered Tapered Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-39Free Annular Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-42Cantilevered Thin Square Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-45


Contents

Axisymmetric Solid and Solid Element Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . 12-1

Free Cylinder — Axisymmetric Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1Thick Hollow Sphere — Uniform Radial Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4Simply Supported Annular Plate — Axisymmetric Vibration . . . . . . . . . . . . . . . . . . . . . . 12-7Deep Simply Supported "Solid" Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-10Simply Supported "Solid" Square Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-13Simply Supported "Solid" Annular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-16Cantilevered Solid Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-19

Part VI: Verification Test Cases from the Societe Francaise des Mecaniciens

Overview of Verification Test Cases Provided by the Societe Francaise desMecaniciens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-1

Understanding the Test Case Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-1Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-2

Mechanical Structures — Linear Statics Analysis with Beam or Rod Elements . . 14-1

Short Beam on Two Articulated Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-1Clamped Beams Linked by a Rigid Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-3Transverse Bending of a Curved Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-5Plane Bending Load on a Thin Arch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-8Grid Point Load on an Articulated CONROD Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-11Articulated Plane Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-14Beam on an Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-18

Mechanical Structures — Linear Statics Analysis with Shell Elements . . . . . . . . . 15-1

Plane Shear and Bending Load on a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-1Infinite Plate with a Circular Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-4Uniformly Distributed Load on a Circular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-7Torque Loading on a Square Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-10Cylindrical Shell with Internal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-13Uniform Axial Load on a Thin Wall Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-17Hydrostatic Pressure on a Thin Wall Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-21Gravity Loading on a Thin Wall Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-25Pinched Cylindrical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-29Spherical Shell with a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-32Bending Load on a Cylindrical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-35Uniformly Distributed Load on a Simply-Supported Rectangular Plate . . . . . . . . . . . . . 15-38Uniformly Distributed Load on a Simply-Supported Rhomboid Plate . . . . . . . . . . . . . . . 15-42Shear Loading on a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-45

Mechanical Structures — Linear Statics Analysis with Solid Elements . . . . . . . . . 16-1

Solid Cylinder in Pure Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-1Internal Pressure on a Thick-Walled Spherical Container . . . . . . . . . . . . . . . . . . . . . . . . 16-7Internal Pressure on a Thick-Walled Infinite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 16-11Prismatic Rod in Pure Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-15Thick Plate Clamped at Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-19

Mechanical Structures — Normal Mode Dynamics Analysis . . . . . . . . . . . . . . . . . 17-1

Lumped Mass-Spring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-1Short Beam on Simple Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-4


Contents

Axial Loading on a Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-8Cantilever Beam with a Variable Rectangular Section . . . . . . . . . . . . . . . . . . . . . . . . . 17-10Thin Circular Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-13Thin Circular Ring Clamped at Two Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-16Vibration Modes of a Thin Pipe Elbow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-19Cantilever Beam with Eccentric Lumped Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-23Thin Square Plate (Clamped or Free) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-26Simply-Supported Rectangular Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-29Thin Ring Plate Clamped on a Hub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-32Vane of a Compressor - Clamped-free Thin Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-35Bending of a Symmetric Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-38Hovgaard’s Problem — Pipes with Flexible Elbows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-41Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-44

Mechanical Structures — Normal Mode Dynamics Analysis and ModelResponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-1

Transient Response of a Spring-Mass System with Acceleration Loading . . . . . . . . . . . . . 18-1Transient Response of a Clamped-free Post . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-5

Stationary Thermal Tests — Heat Transfer Analysis . . . . . . . . . . . . . . . . . . . . . . . 19-1

Hollow Cylinder - Fixed Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-2Hollow Cylinder - Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-5Cylindrical Rod - Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-7Hollow Cylinder with Two Materials - Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-10Wall-Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-13Wall-Fixed Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-16L-Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-18Orthotropic Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-21Hollow Sphere - Fixed Temperatures, Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-25Hollow Sphere with Two Materials - Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-28

Thermo-mechanical Tests — Linear Statics Analysis . . . . . . . . . . . . . . . . . . . . . . . 20-1

Orthotropic Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1Thermal Gradient on a Thin Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-5Simply-Supported Arch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-8

Part VII: Material Nonlinear (Plasticity) Verification Using Standard NAFEMSBenchmarks

Overview of the Material Nonlinear (Plasticity) Verification Using NAFEMS TestCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1

Understanding the Verification Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1

Plane Strain Elements - Perfect Plasticity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1Plane Strain Elements - Isotropic Hardening Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-6Plane Stress Elements - Perfect Plasticity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-11Plane Stress Elements - Isotropic Hardening Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-16Solid Element - Perfect Plasticity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-21Solid Element - Isotropic Hardening Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-27


Contents

Part VIII: Geometric Nonlinear Verification Using Standard NAFEMS Benchmarks

Overview of the Geometric Nonlinear Verification Using NAFEMS Test Cases . . . 23-1

Understanding the Verification Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-1Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-1

Straight Cantilever with End Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-1Straight Cantilever with Axial End Point Load - Brick Elements . . . . . . . . . . . . . . . . . . . 24-6Straight Cantilever with Axial End Point Load - BEAM Elements . . . . . . . . . . . . . . . . . 24-10Lee’s Frame Buckling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-15Large Displacement Elastic Response of a Hinged Spherical Shell Under Uniform PressureLoading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-18Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-20Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-20


Part

I Introduction

Overview of the Verification Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Running the Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1


Chapter

1 Overview of the VerificationManual

This guide contains verification test cases for NX Nastran. These test cases verify the functionof the different NX Nastran analysis types using theoretical and benchmark solutions fromwell-known engineering test cases. Each test case contains test case data and information, suchas element type and material properties, results, and references.

The guide contains test cases for:

• Linear Statics verification using theoretical solutions

• Linear Statics verification using standard NAFEMS benchmarks

• Normal Mode Dynamics verification using theoretical solutions

• Normal Mode Dynamics verification using standard NAFEMS benchmarks

• Verification Test Cases from the Societe Francaise des Mecaniciens

• Material Nonlinear (Plasticity) verification using standard NAFEMS benchmarks (NXNastran only)

• Geometric Nonlinear verification using standard NAFEMS benchmarks

NX Nastran Verification Manual 1-1

Chapter

2 Running the Test Cases

All verification test cases are available as *.dat files and are included in the NX Nastraninstallation in the directory path install_dir/NXr/nast/demo.

The test cases are relatively simple, and most have closed-form theoretical solutions. Differencesbetween finite element and theoretical solutions are in most cases negligible. Some tests wouldrequire an infinite number of elements to achieve an exact solution. Elements are chosen toachieve reasonable engineering accuracy with reasonable computing times.

Actual results from NX Nastran may vary insignificantly from the results presented inthis document. This variation is generally due to different methods of performing realnumber algorithms on different systems.


Part

II Linear Statics VerificationUsing Theoretical Solutions

Overview of Linear Statics Verification Using Theoretical Solutions . . . . . . . . . . . . . . . . . 3-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1


Chapter

3 Overview of Linear StaticsVerification Using TheoreticalSolutions

The purpose of these linear statics test cases is to verify the function of the NX Nastran softwareusing theoretical solutions. The test cases are relatively simple in form and most of them haveclosed-form theoretical solutions.

The theoretical solutions shown in these examples are from well-known engineering texts.For each test case, a specific reference is cited. All theoretical reference texts are listed at theend of this topic.

The finite element method is very flexible in the types of physical problems represented. Theverification tests provided are not exhaustive in exploring all possible problems, but representcommon types of applications.

This overview provides information on the following:

• Understanding the test case format

• Understanding comparisons with theoretical solutions

• References

3.1 Understanding the Test Case FormatEach test case is structured with the following information.

• Test case data and information:

– Physical and material properties

– Finite element modeling (modeling procedure or hints)

– Units

– Solution type

– Element type

– Boundary conditions (loads, restraints)

• Results


Chapter 3 Overview of Linear Statics Verification Using Theoretical Solutions

• References (text from which a closed-form or theoretical solution was taken)

In addition to these example problems, test cases from NAFEMS (National Agency for FiniteElement Methods and Standards, National Engineering Laboratory, Glasgow, U.K.) have beenexecuted. Results for these test cases can be found in the next section, Linear Statics AnalysisVerification Using NAFEMS Standard Benchmarks.

3.2 Understanding Comparisons with Theoretical SolutionsWhile differences in finite element and theoretical results are, in most cases, negligible, sometests would require an infinite number of elements to achieve the exact solution. Elements arechosen to achieve reasonable engineering accuracy with reasonable computing times.

Results reported here are results which you can compare to the referenced theoretical solution.Other results available from the analyses are not reported here. Results for both theoretical andfinite element solutions are carried out with the same significant digits of accuracy.

The closed-form theoretical solution may have restrictions, such as rigid connections, that donot exist in the real world. These limiting restrictions are not necessary for the finite elementmodel, but are used for comparison purposes. Verification to real world problems is more difficultbut should be done when possible.

The actual results from the NX Nastran software may vary insignificantly from the resultspresented in this document. This variation is due to different methods of performing realnumerical arithmetic on different systems. In addition, it is due to changes in elementformulations which have been made to improve results under certain circumstances.

3.3 ReferencesThe following references have been used in the Linear Statics Analysis verification problemspresented:

1. Beer and Johnston. Mechanics of Materials. New York: McGraw-Hill, Inc., 1992.

2. Harris, C. O. Introduction to Stress Analysis. New York Macmillan1959.

3. Roark, R. and Young, W. Formulas for Stress and Strain, 5th Edition. New York:McGraw-Hill Book Company, 1975.

4. Shigley, J. and Mitchel L. Mechanical Engineering Design, 4th Edition. New York:McGraw-Hill Book Company, 1983.

5. Timoshenko, S. Strength of Materials, Part I, Elementary Theory and Problems. New YorK:Van Norstrand Reinhold Company, 1955.

3-2 NX Nastran Verification Manual

Chapter

4 Test Cases

4.1 Point Load on a Cantilever BeamDetermine the deflection of a beam at the free end. Determine the stress at the midpoint of the

beam.

Test Case Data and Information

Input Files

mstvl001.dat

Units

Inch

Model Geometry

Length = 480 in.


Chapter 4 Test Cases

Cross Sectional Properties

• Area = 30 x 30 in.

• Iy = Iz = 67500 in.4

Material Properties

• E = 30E06 psi

Finite Element Modeling

Create four successive linear beam (CBAR) elements along the X axis.

Boundary Conditions

• Restraints

– Restrain the left end of the beam in all six degrees.

• Loads

– Set grid force to 50,000 lb in. the -Y direction.

Solution Type

SOL 101 — Linear Statics

Results

Result Bench Value NX Nastran

Von Mises Stress, grid point 1 (psi) 5333. 5333.

Y Deflection, grid point 5 (in) 0.9102 0.9130

References

Beer and Johnston. Mechanics of Materials. New York: McGraw-Hill, Inc., 1992. p. 716.


Test Cases

4.2 Axial Distributed Load on a Linear BeamDetermine the stress, elongation and resultant force due to an axial loading along a linear beam

element.


Input Filesmstvl002.dat

UnitsInch

Model Geometry• Length = 300 in.

Cross Sectional Properties• Area = 9 in.2

• square cross section (3 in. x 3 in.)

• I = 6.75 in.4

Material Properties• E = 30E+6 psi

Finite Element ModelingCreate 30 beam element along the X axis, each 10 inches long.

Boundary Conditions• Restraints

– Restrain one end of the beam in all six degrees.

• Loads

– Set the axial distributed load (force per unit length) to 1000 lb/in. for the 10-inch longelement furthest from the restrained end in the X direction.



The boundary conditions are shown in the following figure:

Solution Type


Results


Von Mises Stress, grid point 1 (psi) 1111. 1111.

Deflection in X, grid point 2 (in) 0.01111 0.01093

Reaction in X, grid point 1 (lb) –1.000E4 –1.000E4

References

Beer and Johnston. Mechanics of Materials.. New York: McGraw-Hill, Inc., 1992. p. 76.


Test Cases

4.3 Distributed Loads on a Cantilever BeamDetermine the deflection of a beam at the free end. Determine the stress at the midpoint of thebeam and the reaction force at the restrained end.



UnitsInch

Model GeometryLength = 480 in.


• Square cross section (30 in. x 30 in.)

• Iy = Iz = 67500 in.4

Material Properties• E = 30E06 psi

Finite Element ModelingCreate eight successive linear beam (CBAR) elements along the X axis.





• Loads

– Define a distributed load of 250 lb/in. in the –Y direction.

Solution Type


Results


X Stress at grid point 1 (psi) 6,400. 6,383.

Deflection Magnitude at grid point 5 (in) 0.8190 0.8225 *

Reaction Force Magnitude at grid point 1 (lb) 1.200E5 1.200E5

* Includes shear deformation which is neglected in theoretical value.

References



Test Cases

4.4 Moment Load on a Cantilever BeamDetermine the deflection of a beam at the free end. Determine the bending stress of the beamand the reaction force at the restrained end.



UnitsInch



• Iy = Iz = 67500 in.4

• Square cross section 30” x 30” inches

Material Properties• E = 30 E+06 psi

Finite Element ModelingCreate eight successive linear beam (CBAR) elements along the X axis.





• Loads

– Set the Z-moment of the end grid point to 2.5E06 in.-lb.

Solution Type


Results


Von Mises Stress at grid point 1 (psi) 555.6 555.6

Deflection Magnitude at grid point 5 (in) 0.1422 0.1422

Reaction Force Z Direction at grid point 1 (lb) 2.500E6 2.499E6

* Includes shear deformation which is neglected in theoretical value.

References



Test Cases

4.5 Edge Pressure on Beam Element - Torque LoadingDetermine the stress, elongation and resultant force due to a torque applied to a hollow cylinderat the free end.



UnitsSI - meter

Model GeometryLength = 1.5 m

Cross Sectional Properties• Radius1 = 0.02 m

• Radius2 = .03 m

Material Properties• E = 208.6 GPa

Finite Element Modeling• Create a CBAR element along the X axis.



• To find the maximum shearing stress, set the effective radius in torsion to 0.03 m.

• The minimum shearing stress is located at a radius equal to 0.02 m.

Boundary Conditions

• Restraints


• Loads

– Apply an edge torque equal to 4.08 kN-m along the 10 cm linear beam (CBAR) elementfurthest from the restrained end.

Solution Type


Results


Max Torsional Shear Stress (MPa) 120.0 120.0

Min Torsional Shear Stress (MPa) 80.00 80.00

Post Processing

To obtain the minimum and maximum shear stress values, a post processor which supportscontour plots of the torsional shear stress on the cross section using the linear beam (CBAR)element forces must be used. The cross section location can be anywhere except the free end ofthe beam.

References



Test Cases

4.6 Thermal Strain, Displacement, and Stress on Heated BeamA beam originally 1 meter long and at –50° C is heated to 25° C. First, determine the displacementand thermal strain on a cantilever beam. Fix the beam at the free end and then determine thedisplacement, reaction forces, and stresses along the beam. Next, fix the beam at both ends.


Input Files

mstvl007.dat

Units

SI - meter

Model Geometry

Length = 1 m


• Area = 0.01 m2

Material Properties

• E = 2.068E11 Pa

• Coefficient of thermal expansion = 1.2E–05

• v = 0.3




• Create 10 linear beam (CBAR) elements on the X axis and restrain the end grids in alldirections.

• Apply a temperature on all grid points.

Boundary Conditions

• Restraints

– Case 1: Restrain one end of the beam in all six directions.

– Case 2: Restrain both ends of the beam in all six directions.

• Loads

– Set grid temperatures to 25°C. Set the reference temperature to –50°C.

Solution Type


Results

Case 1


X Displacement at grid 11 (m) .0009000 .0009000

Axial Thermal Strain .0009000 .0009000

Case 2


X Displacement (m) 0 0

Axial Stress (Pa) 1.860E8 1.861E8

X Reaction Force (N) 1.860E6 1.861E6

References



Test Cases

4.7 Uniformly Distributed Load on Linear BeamA beam 40 feet long is restrained and loaded as shown with a distributed load of –833 lbs. perfoot. Determine the bending stress and the deflection at the middle of the beam.


Input Files

mstvl008.dat

Units

Inch

Model Geometry

Length = 480 in.




• Rectangular cross section (1.17 in. x 43.24 in.)

• Iz = 7892 in.4

Material Properties

• E = 30E06 psi


Create 4 successive linear beam (CBAR) elements that are each 10 feet long.

Boundary Conditions

• Restraints

– Restrain the second and the fourth grids in five degrees of freedom. Do not restrainrotation about Z.

• Loads

– Define a distributed load (force per unit length) of –833 lb/foot (global negative Ydirection) on the end elements.

Solution Type


Results


Y Displacement at grid 3 (in.) 0.1820 0.182

Max bending stress (psi) 1.644E4 1.644E4

References



Test Cases

4.8 Membrane Loads on a Linear Quadrilateral Thin Shell ElementA circle is scribed on an unstressed aluminum plate. Forces acting in the plane of the plate causenormal stresses. Determine the change in the length of diameter AB and of diameter CD.


Element Typescquad4


UnitsInch


• Diameter = 9 in.



• Thickness = 3/4 in.

Material Properties

• E = 10E06 psi

• Poisson’s ratio = 1/3

• F(x)/L = 9,000 lb/in.

• F(z)/L = 15,000 lb/in.


Create 1/4 of the model and apply symmetry boundary conditions. Then multiply the answer by2 for correct results. Remember to account for the ratio of the circle diameter to plate length.

Boundary Conditions

• Restraints


• Loads

– Set the edge pressure to 9,000 lb/in. in the X direction and 15,000 lb/in. in the Z direction.

Solution Type


Results


X Diameter Change (in.) –4.800E3 –4.800E3

Z Diameter Change (in.) –14.40E3 –14.40E3

Post Processing

Deflection

• (dx at grid point 7 – dx at grid point 10) x 2 = (0.004 – 0.0016) x 2 = 0.0048

• (dz at grid point 7 – dz at grid point 24) x 2 = (0.012 –0.0048) x 2 = 0.0144

References



Test Cases

4.9 Axial Loading on Rod ElementDetermine the stress, elongation, and strain due to an axial load on a rod element.


Input Files

mstvl011.dat

Units

SI - meters

Model Geometry

Length = 10 m


• Area = 0.01 m2

Material Properties

• E = 200.0 GPa


Create a rod (CROD) element along the X axis.

Boundary Conditions

• Restraints

– Restrain an end of the rod in the 3 translational degrees.

• Loads

– Apply a grid point force in the positive X-direction of 500 kN.

Solution Type


Results


Axial Stress (MPa) 50.00 50.00

Axial Strain 0.0002500 0.0002500

Elongation (mm) 2.500 2.500



References



Test Cases

4.10 Stress on a Beam as It Expands and Closes a GapDetermine the stress on a beam as it expands thermally and closes a 0.002 inch gap. It isinitially at 70 °F and is heated to 170 °F.



UnitsInch


Material Properties• E = 1.05E07 psi

• Coefficient of thermal expansion = 1.25E–05 in./(in.–°F)

Finite Element Modeling• Create a single linear beam (CBAR) element on the X axis.



• Create an MPC to define the closing of the gap.

Boundary Conditions

• Restraints

– Restrain the free end of the beam in all six degrees.

• Loads

– Set grid temperature to 170 °F.

– Set the reference temperature to 70 °F.

Solution Type


Results


Axial Stress (psi) –6.125E3 –6.125E3

References

Harris, C. O. Introduction to Stress Analysis 1959. p. 58.


Test Cases

4.11 Thin Wall Cylinder in Pure TensionDetermine the stress and deflection of a thin wall cylinder with a uniform axial load.


Input Files

mstvl014.dat

UnitsInch

Model Geometry• R = 0.5 in.

• Thickness = 0.01 in.

• y = 1.0 in.



Material Properties

• E = 10,000 psi

• ν = 0.3


Create 1/4 model of the cylinder with thin shell linear quadrilateral (CQUAD4) elements andsymmetry boundary conditions.

Boundary Conditions

• Restraints

– Restrain edges of symmetry, in translation, in hoop direction, and rotation about Z axis.

– Restrain one end in Y direction.

Loads

– Apply membrane edge pressure of p / (pi)D = 3.1831 where p = 10 psi

Solution Type


Results


Axial (Z) Stress (psi) 1.000E3 1.000E3

Axial (Z) Deflection (in.) 1.000 1.000

Radial Deflection (in.) –0.01500 –0.01500

References

Roark, R. and Young, W. Formulas for Stress and Strain, 6th Edition. New York: McGraw-HillBook Company, 1989. p. 518, Case 1a.


Test Cases

4.12 Thin Shell Beam Wall in Pure BendingDetermine the maximum stress, maximum deflection, and strain energy of a thin shell beamwall with a uniform bending load.


Input Files

mstvl015.dat

Units

Inch

Model Geometry

• Length = 30 in.

• Width = 5 in.

• Thickness = 0.1 in.



Material Properties

• E = 30E06 psi

• ν = 0.03


Create a 30 in. x 5 in. plate with thing shell (CQUAD4) elements.

Boundary Conditions

• Restraints

– Restrain at one of the ends in all directions.

• Loads

– Apply edge pressure of p/w = 1.2 lbs/in. where p = 6.0 lb.

Solution Type


Results


Max Z Deflection (in.) 4.320 4.264

Max Z Stress (psi) 2.160E4 1.980E4

Total Strain Energy (lb in.) 12.96 12.79

References

Shigley, J. and Mitchel L. Mechanical Engineering Design, 4th Edition. New York: McGraw-Hill,Inc., 1983. pp. 134, 804.


Test Cases

4.13 Strain Energy of a TrussDetermine the strain energy of a truss. The cross-sectional area of the diagonal members is twicethe cross-sectional area of the horizontal and vertical members.



UnitsInch


Cross Sectional Properties• Cross-sectional area (A) = 0.01 in.2

Material Properties• E = 30E06 psi

Finite Element ModelingCreate truss shown using rod (CROD) elements.




– Restrain far left grid in directions: X, Y, Z, RX, RY.

– Restrain far right grid in directions: Y, Z, RX, RY.

• Loads

– Apply grid force in Y direction on lower center grid; F= 300 lb.

Solution Type


Results


Total Strain Energy (lb in.) 5.846 5.846

References



Part

III Linear Statics VerificationUsing Standard NAFEMSBenchmarks

Overview of Linear Statics Verification Using Standard NAFEMS Benchmarks . . . . . . . . . 5-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1


Chapter

5 Overview of Linear StaticsVerification Using StandardNAFEMS Benchmarks

The purpose of these linear statics test cases is to verify the function of NX Nastran usingstandard benchmarks published by NAFEMS (National Agency for Finite Element Methods andStandards, National Engineering Laboratory, Glasgow, U.K.).

These standard benchmark tests were created by NAFEMS to stretch the limits of the finiteelements in commercial software. All results obtained using NX Nastran compare favorably withother commercial finite element analysis software.

5.1 Understanding the Test Case FormatEach test case is structured with the following information:

• Test case data and information



– Units

– Finite element modeling information

– Boundary conditions (loads and restraints)

– Solution type

• Results

• Reference

5.2 ReferenceThe following reference has been used in these test cases:

NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks. Glasgow:NAFEMS, Rev. 3, 1990.


Chapter

6 Test Cases

6.1 Elliptic MembraneThis test is a linear elastic analysis of an elliptic membrane (shown below) using coarse and finemeshes of plane stress elements and thin shell elements. It provides the input data and resultsfor NAFEMS Standard Benchmark Test LE1.

Ellipses:


Input Files

le101.dat (plane stress quadrilateral)

le102.dat (plane stress triangle)

le103.dat (thin shell)



Physical and Material Properties

• Thickness = 0.1 m

• Isotropic material

• E = 210E3 MPa

• v = 0.3

Units

SI


• Plane stress (only MID1 defined on PSHELL) linear (CQUAD4) and parabolic (CQUAD8)quadrilaterals — coarse and fine mesh.

• Plane stress (only MID1 defined on PSHELL) linear (CTRI3) and parabolic (CTRI6) triangles— coarse and fine mesh.

• Thin shell (MID1, MID2 and MID3 defined on PSHELL) linear (CQUAD4) and parabolic(CQUAD8) quadrilaterals — coarse and fine mesh.

• The fine mesh is created by approximately halving the coarse mesh.


Test Cases

Boundary Conditions

• Uniform outward pressure at outer edge BC = 10 MPa

• Inner curved edge AD unloaded

• X displacement (edge AB) = 0



• Y displacement (edge CD) = 0

Solution Type


Results

Output — tangential edge stress at D (stress in Y direction)

Plane Stress Elements

Test case Grid point # Bench Value NX Nastran

Linear quad — coarse mesh 4.000 92.70 62.10

Linear quad — fine mesh 204.0 92.70 79.60

Parabolic quad — coarse mesh 104.0 92.70 84.00

Parabolic quad — fine mesh 304.0 92.70 88.70

Linear triangle — coarse mesh 4.000 92.70 52.90

Linear triangle — fine mesh 204.0 92.70 70.90

Parabolic triangle — coarsemesh

104.0 92.70 76.80

Parabolic triangle — fine mesh 304.0 92.70 93.60

Thin Shell Elements


Linear quad — coarse mesh 4.000 92.70 62.10


Test Cases


Linear quad — fine mesh 204.0 92.70 79.60

Parabolic quad — coarse mesh 104.0 92.70 84.00

Parabolic quad — fine mesh 304.0 92.70 88.70

References

NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, TestNo. LE1. Glasgow: NAFEMS, Rev. 3, 1990.



6.2 Cylindrical Shell Patch TestThis test is a linear elastic analysis of a cylindrical shell (shown below) using thin shell elementsand two different loadings. It provides the input data and results for NAFEMS StandardBenchmark Test LE2.


Input Files

• le201a.dat (linear shell, case 1)

• le201b.dat (parabolic shell, case 1)

• le202a.dat (linear shell, case 2)

• le202b.dat (parabolic shell, case 2)




• E = 210E3 MPa

• v = 0.3


Test Cases

Units

SI


• Thin shell linear (CQUAD4) and parabolic (CQUAD8) quadrilaterals

Boundary Conditions

• Translations and rotations (edge AB) = 0

• Z translations and normal rotations (edge AD and edge BC) = 0

Case 1 loading:

• Uniform normal edge moment on DC = 1.0 kNm/m

Case 2 loading:

• Uniform outward normal pressure at mid-surface ABCD = 0.6 MPa

• Tangential outward normal pressure on edge DC = 60.0 MPa



Solution Type


Results

Output — outer (convex) surface tangential stress at point E (grid point 2):

Test case Bench Value NX Nastran

Linear quad — case 1 60.00 51.80

Linear quad — case 2 60.00 54.00*

Parabolic quad — case 1 60.00 51.10

Parabolic quad — case 2 60.00 55.10

* Since the shapes of the shells are an approximation to a cylindrical surface, an edge loadwill not be in the correct direction. To get this result, the edge load must be input as gridforces in the tangential direction.

Post Processing

• Stress component: Y


Test Cases

• Results obtained on the element top surface in cylindrical coordinate system

References




6.3 Hemisphere-Point LoadsThis test is a linear elastic analysis of hemisphere point loads (shown below) using coarse andfine meshes of thin shell elements. It provides the input data and results for NAFEMS StandardBenchmark Test LE3.


Input Files

• le301.dat (linear quad, coarse mesh)

• le302.dat (linear quad, fine mesh)

• le303.dat (parabolic quad, coarse mesh)

• le304.dat (parabolic quad, fine mesh)




• E = 68.25 × 103 MPa

• v = 0.3


Test Cases

Units

SI


• Thin shell linear (CQUAD4) and parabolic (CQUAD8) quadrilaterals — coarse and fine mesh

• Equally spaced grid points on AC, CE, EA

• Point G at X = Y = Z = 10 /( 31/2) grid point 7

Boundary Conditions

• Edge AE symmetry about XZ plane (y = rotation x = rotation z = 0)

• Edge CE symmetry about YZ plane (x = rotation y = rotation z = 0)

• Point E (x = y = z = 0)

• All other displacements on edge AC are free.

• Concentrated radial load outward at A = 2KN

• Concentrated radial load inward at C = 2KN



Solution Type


Results

Output — X displacement at point A

Mesh Test Case Bench Value NX Nastran

linear quad — coarse mesh le301 0.1850 0.1890

linear quad — fine mesh le302 0.1850 0.1870

parabolic quad — coarsemesh

le303 0.1850 0.1420

parabolic quad — fine mesh le304 0.1850 0.1710

References



Test Cases

6.4 Z-Section CantileverThis test is a linear elastic analysis of a Z-section cantilever (shown below) using thin shellelements. It provides the input data and results for NAFEMS Standard Benchmark Test LE5.


Input Files

• le501.dat (linear quadrilateral)

• le502.dat (parabolic quadrilateral)




• E = 210E3 MPa

• v = 0.3

Units

SI


• Thin shell linear (CQUAD4) and parabolic (CQUAD8) quadrilaterals

Boundary Conditions

• All displacements on edges B1, B2, B3 = 0

• Torque of 1.2MN applied at end C by two edge shears (at C1 & C3) of 0.6 MN



Solution Type


Results

Output — averaged axial stress at mid-surface, point A, grid point 30 (compression)


Linear quad - point A/grid point 30 –108.0 –111.0

Parabolic quad - point A/grid point 30 –108.0 –109.3

References

NAFEMS Finite Element Methods & Standards. The Standard NAFEMS Benchmarks, TestNo. LE5. Glasgow: NAFEMS, Rev. 3, 1990.


Test Cases

6.5 Skew Plate Normal PressureThis test is a linear elastic analysis of a plate (shown below) using thin shell elements. Itprovides the input data and results for NAFEMS Standard Benchmark Test LE6.


Input Files

• le601.dat (linear and parabolic quad)

• le602.dat (linear and parabolic triangle)




• E = 210E3 MPa

• v = 0.3

Units

SI


• Thin shelllinear (CQUAD4) and parabolic (CQUAD8) quadrilaterals — coarse and fine mesh

• Thin shell linear (CTRI3) and parabolic (CTRI6) triangles — coarse and fine mesh

Boundary Conditions

• Simple supports

• Z displacement = 0

• Normal pressure = –0.7KPa in the Z direction



Solution Type


Results

Output — maximum principal stress on the bottom surface at the plate center.

Case le601

Mesh Grid point # Bench Value NX Nastran

Linear quad coarse mesh 9.000 0.8020 0.3250

Linear quad fine mesh 18.00 0.8020 0.6830

Parabolic quad coarse mesh 43.00 0.8020 0.6250

Parabolic quad fine mesh 52.00 0.8020 0.7190

Case le602

Mresh Grid point # BenchValue

NX Nastran

Linear triangle coarse mesh 9.000 0.8020 0.3960

Linear triangle fine mesh 18.00 0.8020 0.7200

Parabolic triangle coarsemesh

43.00 0.8020 0.9260

Parabolic triangle fine mesh 52.00 0.8020 0.8570

References



Test Cases

6.6 Axisymmetric Cylinder/Sphere — PressureThis test is a linear elastic analysis of an axisymmetric cylinder (shown below) usingaxisymmetric shell elements. It provides the input data and results for NAFEMS StandardBenchmark Test LE7.


Input Files

• le701a.dat (coarse linear mesh)

• le701b.dat (fine linear mesh)




• E = 210E3 MPa

• v = 0.3

Units

SI




• Axisymmetric shell — coarse and fine mesh

• Elements uniformly spaced between points A, B, C, D, E, F

Boundary Conditions

• Point A — radial displacement and rotation = 0

• Point F — axial displacement (z direction) = 0

• Uniform internal pressure = 1 MPa


Test Cases

Solution Type


Results

Output — axial stress on outer surface at point D

2-noded axisymmetric shell

Mesh Bench Value NX Nastran

Coarse mesh 25.86 25.30

Fine mesh 25.86 25.65




Fine mesh 25.86 25.99

Post Processing


• σz at grid point 8







References



Test Cases

6.7 Axisymmetric Shell PressureThis test is a linear elastic analysis of axisymmetric shell pressure (shown below) usingaxisymmetric shell elements. It provides the input data and results for NAFEMS StandardBenchmark Test LE8.


Input Files

• le801.dat (linear)

• le801a.dat (parabolic)




• E = 210E3 MPa

• v = 0.3

Units

SI




• Axisymmetric shell — coarse and fine mesh

• Elements uniformly spaced between points A, B, C, D, E

Boundary Conditions

• Point E — radial displacement and rotation = 0

• Point A — Z-displacement = 0

• Uniform internal pressure = 1 MPa


Test Cases

Solution TypeLinear Statics

ResultsOutput — hoop stress on outer surface at point D




Fine mesh 94.55 83.14




Fine mesh 94.55 94.88

Post Processing


Fine Mesh

• Grid point 29

• Y stress



• Bottom surface

Coarse Mesh

• Grid point 73

• Y stress

• Bottom surface


Fine Mesh

• Grid point 25

• Y stress

• Bottom surface

Coarse Mesh

• Grid point 1

• Y stress

• Bottom surface

References



Test Cases

6.8 Thick Plate PressureThis article provides the input data and results for NAFEMS Standard Benchmark Test LE10.This test is a linear elastic analysis of a thick (shown below) using coarse and fine meshes ofsolid elements.

Ellipses:


Input Files

• le1001.dat (linear and parabolic brick)

• le1002.dat (linear and parabolic wedge)

• le1003.dat (linear and parabolic tetrahedron)



• E = 210E3 MPa

• v = 0.3

Units

SI


• Solid brick (CHEXA) linear and parabolic - coarse and fine mesh



• Solid wedge (CPENTA) linear and parabolic - coarse and fine mesh

• Solid tetrahedron (CTETRA) - linear and parabolic - coarse and fine mesh

Solid Brick


Test Cases

Solid Wedge

Solid Tetrahedron — fine mesh only

Boundary Conditions

• Uniform normal pressure on the upper surface of the plate = 1 MPa

• Inner curved edge AD unloaded

• X and Y displacements on faces DCD’C′ and ABA′B′ = 0

• X and Y displacements on face BCB′C′ are fixed



• Z displacements along mid-plane are fixed

Solution Type


Results

Output — direct stress at point Dσyy

Test Case 1e1001


Linear brick — coarse mesh 4.000 –5.500 –5.410

Linear brick — fine mesh 204.0 –5.500 –5.670

Parabolic brick — coarsemesh

104.0 –5.500 –6.130

Parabolic brick — fine mesh 304.0 –5.500 –6.040

Test Case 1e1002


Linear wedge — coarse mesh 4.000 –5.500 –5.940

Linear wedge — fine mesh 204.0 –5.500 –5.830

Parabolic wedge — coarse mesh 104.0 –5.500 –5.320

Parabolic wedge — fine mesh 304.0 –5.500 –6.010

Test Case 1e1003

Result Grid point # Bench Value NX Nastran

Linear tetra — fine mesh 40.00 –5.500 –2.410


Test Cases

Result Grid point # Bench Value NX Nastran

Parabolic tetra — fine mesh 171.0 –5.500 –5.280

References




6.9 Solid Cylinder/Taper/Sphere — TemperatureThis test is a linear elastic analysis of a solid cylinder with a temperature gradient (shownbelow) using coarse and fine meshes of solid elements. It provides the input data and results forNAFEMS Standard Benchmark Test LE11.


Input Files

• le1101a.dat (linear brick — coarse mesh)

• le1101b.dat (linear brick — fine mesh)

• le1102a.dat (parabolic brick — coarse mesh)

• le1102b.dat (parabolic brick — fine mesh)

• le1103a.dat (linear wedge — coarse mesh)

• le1103b.dat (linear wedge — fine mesh)

• le1104a.dat (parabolic wedge — coarse mesh)

• le1104b.dat (parabolic wedge — fine mesh)

• le1105a.dat (linear tetra — coarse mesh)


Test Cases

• le1105b.dat (linear tetra — fine mesh)

• le1106a.dat (parabolic tetra — coarse mesh)

• le1106b.dat (parabolic tetra — fine mesh)



• E = 210E3 MPa

• v = 0.3

• a = 2.3E–4 °C

Units

SI


• Solid brick (CHEXA) linear and parabolic — coarse and fine mesh

• Solid wedge (CPENTA) linear (6 grid point) and parabolic (15 grid point) — coarse andfine mesh

• Solid tetrahedron (CTETRA) linear and parabolic — coarse and fine mesh



Solid Brick

Solid Tetrahedron


Test Cases

Boundary Conditions

• Linear temperature gradient in the radial and axial direction

T° C = (X2 + Y2)1/2 + Z

• X, Y, and Z displacements = 0

• X and Y displacements on face BCB′C′ are fixed

• Z displacements on XY-plane face and HIH′I′ face = 0

Solution Type


Results

Output - direct stress σyy at point A

File Name Result Grid point at Point A BenchValue

NX Nastran

le1101a Linear brick — coarsemesh

30.00 –105.0 –88.29

le1101b Linear brick — finemesh

71.00 –105.0 –93.68

le1102a Parabolic brick —coarse mesh

67.00 –105.0 –100.4

le1102b Parabolic brick — finemesh

159.0 –105.0 –111.2

le1103a Linear wedge — coarsemesh

33.00 –105.0 –10.00



File Name Result Grid point at Point A BenchValue

NX Nastran

le1103b Linear wedge — finemesh

74.00 –105.0 –48.30

le1104a Parabolic wedge —coarse mesh

71.00 –105.0 –87.20

le1104b Parabolic wedge — finemesh

187.0 –105.0 –96.20

le1105a Linear tetra — coarsemesh

8.000 –105.0 –31.40

le1105b Linear tetra — finemesh

8.000 –105.0 –65.20

le1106a Parabolic tetra —coarse mesh

8.000 –105.0 –89.60

le1106b Parabolic tetra — finemesh

8.000 –105.0 –97.30

References



Part

IV Normal Mode DynamicsVerification

Overview of Normal Mode Dynamics Verification Using Theoretical Solutions . . . . . . . . . . 7-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1


Chapter

7 Overview of Normal ModeDynamics Verification UsingTheoretical Solutions

The purpose of these normal mode dynamics test cases is to verify the function of NX Nastranusing theoretical solutions. The test cases are relatively simple in form and most of them haveclosed-form theoretical solutions.

The theoretical solutions shown in these examples are from well known engineering texts.For each test case, a specific reference is cited. All theoretical reference texts are listed at theend of this topic.

The finite element method is very flexible in the types of physical problems represented. Theverification tests provided are not exhaustive in exploring all possible problems, but representcommon types of applications.

This overview provides information on the following:

• Understanding the test case format

• Understanding comparisons with theoretical solutions

• References





– Units

– Solution type

– Boundary conditions (loads and restraints/constraints)

• Results

• Reference


Chapter 7 Overview of Normal Mode Dynamics Verification Using TheoreticalSolutions

7.2 Understanding Comparisons with Theoretical SolutionsWhile differences in finite element and theoretical results are, in most cases, negligible, sometests would require an infinite number of elements to achieve the exact solution. Elements arechosen to achieve reasonable engineering accuracy with reasonable computing times.

Results reported here are results which you can compare to the referenced theoretical solution.Other results available from the analyses are not reported here. Results for both theoretical andfinite element solutions are carried out with the same significant digits of accuracy.

The closed-form theoretical solution may have restrictions, such as rigid connections, that donot exist in the real world. These limiting restrictions are not necessary for the finite elementmodel, but are used for comparison purposes. Verification to real world problems is more difficultbut should be done when possible.

The actual results from NX Nastran may vary insignificantly from the results presented in thisdocument. This variation is due to different methods of performing real numerical arithmetic ondifferent systems. In addition, it is due to changes in element formulations which have beenmade to improve results under certain circumstances.

7.3 ReferenceThe following references have been used in the normal mode dynamics analysis verificationproblems presented:

1. Blevins, R. Formulas For Natural Frequency and Mode Shape, 1st Edition. New York: VanNorstrand Reinhold Company, 1979.

2. Timoshenko and Young. Vibration Problems in Engineering. New York: Van NorstrandReinhold Company, 1955.

3. Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, Theory and Applications. Boston:Allyn and Bacon, Inc., 1978.

4. Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, 2nd Edition. Boston: Allyn andBacon, Inc., 1978.


Chapter

8 Test Cases

8.1 Natural Frequency of Circular Ring with Axisymmetric ModelDetermine the frequency of radial vibration of an axisymmetric ring.


Input Filemstvn001.dat

UnitsInch

Model Geometry• Thickness = 0.05 in.

• Radius = 100 in.

Material Properties• Density = 0.00073 lb-sec2/in.4

• E = 30E6 psi

Finite Element ModelingCreate a linear axisymmetric thin shell element (CCONEAX) .05 inches long at a radius of100 inches from the center.



Solution Type

SOL 103 — Normal Mode Dynamics, Lanczos Method

Results


Frequency (Hz) 322.6 322.6

References

Timoshenko and Young. Vibration Problems in Engineering, p. 425. New York: Van NorstrandReinhold Company, 1955.


Test Cases

8.2 Undamped Free Vibration — Single Degree of FreedomDetermine the natural frequency of the system shown.


Input File

mstvn002.dat

Units

SI - meter

Model Geometry

• Length = 0.5 m

• a = 0.3 m

Physical Properties

• mass = 20 Kg

• k = 8 KN/m




• Create 5 rigid bar (RBAR) elements along the X axis. Each bar should be 0.1 m long.

• A lumped mass (CONM2) element is applied on the end grid point.

• A grid point-to-ground spring element (CELAS1) is applied 0.2 m from the lumped mass.

Boundary Conditions

• Restrain the first grid point to allow rotation only in the Z direction.

Solution Type


Results


Frequency (Hz) 1.910 1.910

References

Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, Theory and Applications, p. 75. Boston:Allyn and Bacon, Inc., 1978.


Test Cases

8.3 Two Degrees of Freedom Undamped Free Vibration — PrincipleModesDetermine the natural frequencies of a dynamic system with two degrees of freedom.


Input File

mstvn003.dat

Units

SI- meter



Element Types

• Translational springs (CELAS1)

• Lumped mass (CONM2)

Physical Properties

• Mass = 1 kg

• k = 1 N/m


• Create four grid points on the Y axis.

• Create three linear springs (CELAS1) with stiffness of 1 N/m and with a uniaxial stiffnessreference coordinate system.

• Create two lumped mass elements (CONM2) with a mass of 1 kg.

Boundary Conditions

• Restrain ends in all directions.

• Restrain other grid points in all directions but Y.

Solution Type

Normal Mode Dynamics - SOL 103, Lanczos method

Results


Frequency of Mode 1 (Hz) 0.1592 0.1592


References

Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, 2nd Edition, pp. 145-149. Boston:Allyn and Bacon, Inc., 1978.


Test Cases

8.4 Three Degrees of Freedom Torsional SystemDetermine the natural frequencies of a dynamic system with three degrees of freedom.


Input File

mstvn004.dat

Element Types

• Rotational springs (CELAS1)


Units

SI — meter

Physical Properties

• J = J1 = J2 = J3 = 0.1

• k = k1 = k2 = k3 = 1 N*m


• Create four grid points on the X axis.

• Create three linear torsional springs (CELAS1) with stiffness of 1 N*m and with a stiffnessreference coordinate system being uniaxial.

• Create three lumped mass elements (CONM2) with a mass coordinate system = 1 and withmass inertia system of: 0.1, 0.0, 0.0, 0.0, 0.0, 0.0.

Boundary Conditions

• Restrain one end in all directions.

• Restrain the other grid points in all directions but RX.



Solution Type

SOL 103 — Normal Mode Dynamics, Lanczos method

Results





References

Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, 2nd Edition, pp. 153-155. Boston:Allyn and Bacon, Inc., 1978.

8.5 Two Degrees of Freedom Vehicle Suspension SystemDetermine the natural frequencies of dynamic system with two degrees of freedom. Degrees offreedom are one translational and one rotational.


Input Files

mstvn005.dat


Test Cases

Units

SI - meter

Physical Properties

• Mass = 1800 kg

• K1 = 42000 N/m

• K2 = 48000 N/m


• Create a linear translation spring (CELAS1) with stiffness of K1

• Create a linear translation spring (CELAS1) with stiffness of K2

• Create a lumped mass element (CONM2) with a mass coordinate system = 1 and massinertia system of: 0.0, 0.0, 3528, 0.0, 0.0, 0.0.

• Create a three-noded rigid element (RBE2)

Boundary Conditions

• Nodal displacement restraints

– Restrain grid points 4 and 5 in all directions.

– Restrain the other grid points in all directions but Y and RZ.

Solution Type


Results



Frequency of Mode 2(Hz) 1.496 1.496

References

Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations. Boston: Allyn and Bacon, Inc., 1978.pp. 150-153.



8.6 Two Degrees of Freedom Vehicle Suspension SystemDetermine the natural frequencies of dynamic system with two degrees of freedom. Degrees offreedom are one translational and one rotational.


Input File

mstvn005.dat

Element Types

• Translational springs (CELAS1)


• Rigid (RBE2)

Units

SI — meter

Model Geometry

• Length1 = 1.6 m

• Length2 = 2.0 m

• r = 1.4 m (gyration radius; J = m*r*r)


Test Cases

Physical Properties• mass = 1800 kg

• K1 = 42000 N/m

• K2 = 48000 N/m

Finite Element Modeling• Create five grid points in the XY plane with the following coordinates:

– Grid point 1 = (0,0)

– Grid point 2 = (12,0)

– Grid point 3 = (–L1,0)

– Grid point 4 = (L2,–1)

– Grid point 5 = (–L1,–1)

• Create a linear translation spring (CELAS1) with stiffness of K1 between grid point 1 andgrid point 5.

• Create a linear translation spring (CELAS1) with stiffness of K2 between grid point 2 andgrid point 4.

• Create a lumped mass element (CONM2) with a mass coordinate system = 1 and massinertia system of: 0.0, 0.0, 3528, 0.0, 0.0, 0.0.

• Create a three-grid-point rigid element (RBE2) using grid point 1, grid point 2, and gridpoint 3.

Boundary Conditions• Restrain grid points 4 and 5 in all directions.

• Restrain the other grid points in all directions but Y and RZ.

Solution TypeSOL 103 — Normal Mode Dynamics, Lanczos Method

Results



Frequency of Mode 2(Hz) 1.496 1.496

ReferencesTse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, pp. 150-153. Boston: Allyn and Bacon,Inc., 1978.



8.7 Cantilever Beam Undamped Free VibrationsDetermine the natural frequencies of a cantilever beam.



Element TypeLinear beam (CBEAM)

UnitsInch


• Height = 2 in.

Physical and Material Properties• w = 1 lb/in.

• J = .10

• Poisson’s ratio = .3

Calculated Data• A = h2 = 4 in2

• I = h4/12 = 1.33333

• G = E/2 × 1/1 + nu = 11538461.54

• m = w/g = 2.59067375E–3

• Ip = Ixx + Iyy = 2.66666


Test Cases


• Create 11 grid points on X axis.

• Create 10 linear beams (CBEAM) between grid points.

Boundary Conditions

• Restrain one end grid point in all directions.

Solution Type


Results


Frequency of Modes 1 & 2 (TransverseVibration)

6.953 6.953


43.58 43.58

Frequency of Mode 5 (Torsional Vibration) 64.68 64.68


122.0 122.0

Frequency of Mode 8 (Torsional Vibration) 193.9 195.7


238.8 239.3

References

Blevins, R. Formulas For Natural Frequency and Mode Shape, 1st Edition, pp. 108,193. NewYork: Van Norstrand Reinhold Company, 1979.



8.8 Natural Frequency of a Cantilevered MassDetermine the natural frequencies of a dynamic system consisting of a massless beam and alumped mass at the end.



Element Types• Linear beam (CBAR)


UnitsInch


Physical and Material Properties• Mass = 0.5 lbm

• E = 30E6 psi

• Density = 1.0E–6

• I = 1.5 in.4

Finite Element Modeling• Create 2 grid points on the X axis with coordinates (0,0,0) and (30,0,0).

• Create a linear beam (CBAR) element between grid points with shear area ratio = 0.

• Create a lumped mass (CONM2) on one grid point with mass of 0.5 lbm.


Test Cases

Boundary Conditions

• Restrain wall end in all directions.

• Restrain mass end in directions of Z, RX, and RY.

Solution Type

SOL 103 — Normal Mode Dynamics, Lanczos method

Results


Mode 2 Frequency (Hz) 15.92 15.92

References

Tse, F., Morse, I., and Hinkle, R. Mechanical Vibrations, 2nd Edition, p. 72. Boston: Allynand Bacon, Inc., 1978.


Part

V Normal Mode DynamicsVerification Using StandardNAFEMS Benchmarks

Overview of Normal Mode Dynamics Verification Using Standard NAFEMS Benchmarks . . 9-1

Beam Element Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1

Shell Element Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

Axisymmetric Solid and Solid Element Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-1


Chapter

9 Overview of Normal ModeDynamics Verification UsingStandard NAFEMS Benchmarks

The purpose of these normal mode dynamics test cases is to verify the function of NX Nastranusing standard benchmarks published by NAFEMS (National Agency for Finite ElementMethods and Standards, National Engineering Laboratory, Glasgow, U.K.).

These standard benchmark tests were created by NAFEMS to stretch the limits of the finiteelements in commercial software. All results obtained using NX Nastran compare favorablywith other commercial finite element analysis software. Results of these test cases using othercommercial finite element analysis software programs are available from NAFEMS.



– Units



– Boundary conditions (loads and restraints/constraints)

– Solution type

• Results

• Reference


NAFEMS Finite Element Methods & Standards. Abbassian, F., Dawswell, D. J., and Knowles, N.C. Selected Benchmarks for Natural Frequency Analysis. Glasgow: NAFEMS, Nov., 1987.


Chapter

10 Beam Element Test Cases

10.1 Pin-ended Cross — In-plane VibrationThis test is a normal mode dynamic analysis of a pin-ended cross (shown below) using beamelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 1.

Attributes of this test are:

• Coupling between flexural and extensional behavior

• Repeated and close eigenvalues


Input Files

• nf001ac.dat (linear consistent)

• nf001al.dat (linear lumped)

Units

SI


Chapter 10 Beam Element Test Cases

Cross Sectional Properties• Area = .015625 m2

Shear ratio:

• Y = 0

• Z = 0

Material Properties• E = 200E09 N/m2

• ρ=8000 kg/m3

• ν = 0.29 (Poisson’s ratio)

• G = 8.01E10

Finite Element Modeling• Four linear beam (CBAR) elements per arm

Boundary Conditions• X = Y = 0 at A, B, C, D

• Z = Rx = Ry = 0 at all grid points


NX Nastran results were obtained in two different ways:

• Using lumped mass (lumped mass on, param coupmass = –1)


Beam Element Test Cases

• Using coupled mass (lumped mass off, param coupmass = 1)

Results

Mode ReferenceValue (Hz)

NAFEMSTarget Value(Hz)

NX Nastran Result(lumped mass) (Hz)

NX Nastran Result(coupled mass) (Hz)

1 11.34 11.34 11.33 11.34

2, 3 17.71 17.69 17.66 17.69

4 17.71 17.72 17.69 17.72

5 45.35 45.48 45.02 45.52

6, 7 57.39 57.36 56.06 57.43

8 57.39 57.68 56.34 57.75

References

NAFEMS Finite Element Methods & Standards. Abbassian, F., Dawswell, D. J., and Knowles,N. C.Selected Benchmarks for Natural Frequency Analysis Test No. 1. Glasgow: NAFEMS,Nov., 1987.



10.2 Pin-ended Double Cross - In-plane VibrationThis test is a normal mode dynamic analysis of a pin-ended double cross (shown below) usingbeam elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 2.





Input Files



Units

SI


Key-in section:

• Area = .015625 m2

Shear ratio:

• Y = 0

• Z = 0




• ρ = 8000 kg/m3

Finite Element Modeling• Four linear beam (CBAR) elements per arm

Boundary Conditions• X = Y = 0 at A, B, C, D, E, F, G, H

• Z = Rx= Ry = 0 at all grid points



• Using lumped mass (lumped mass toggle on, param coupmass = –1)

• Using coupled mass (lumped mass toggle off, param coupmass = 1)

Results


NAFEMS TargetValue (Hz)

NX NastranResult (lumpedmass) (Hz)

NX NastranResult (coupledmass) (Hz)

1 11.34 11.34 11.33 11.34

2, 3 17.71 17.69 17.66 17.69







4, 5, 6, 7,8 17.71 17.72 17.69 17.72

9 45.35 45.48 45.02 45.52

10, 11 57.39 57.36 56.06 57.43

12, 13, 14, 15,16

57.39 57.68 56.34 57.75

References

NAFEMS Finite Element Methods & Standards. Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis, Test No. 2. Glasgow: NAFEMS,Nov., 1987.



10.3 Free Square Frame - In-plane VibrationThis test is a normal mode dynamic analysis of a free square frame (shown below) using beamelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 3.



• Rigid body modes (3 modes)



Input Files



Units

SI


Shear ratio:

• Y = 1.0

• Z = 1.0

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3




• Four linear beam (CBAR) elements per arm

Boundary Conditions

• Rotations fixed, translations free

Solution Type



• Using lumped mass (lumped mass toggle on, param coupmass = –1)

• Using coupled mass (lumped mass toggle off, param coupmass = 1)

Results





4 3.261 3.262 3.259 3.259

5 5.668 5.665 5.660 5.663

6, 7 11.14 11.15 10.89 11.13

8 12.85 12.83 12.74 12.80

9 24.57 24.66 23.53 24.64

10, 11 28.70 28.81 28.13 28.73

References




10.4 Cantilever with Off-center Point MassesThis test is a normal mode dynamic analysis of a cantilever with off-center point masses (shownbelow) using beam elements. This document provides the input data and results for NAFEMSSelected Benchmarks for Natural Frequency Analysis,Test 4.


• Coupling between torsional and flexural behavior

• Inertial axis non-coincident with flexibility axis

• Discrete lumped mass, rigid links

• Close eigenvalues


Input Files• nf004a.dat

UnitsSI

Cross Sectional PropertiesShear ratio:

• Y = 1.128

• Z = 1.128

Material Properties• E = 200E09 N/m 2

• ρ = 8000 kg/m3

• ν = 0.3

Finite Element Modeling• Five linear beam (CBAR) elements along cantilever



Boundary Conditions

• X = Y = Z = Rx = Ry = Rz = 0 at A

Solution Type

SOL 103 — Normal Mode Dynamics, Lanczos (Parameter COUPMASS = –1)

Results

Mode Reference Value(Hz)


NX Nastran Result(Hz)

1 1.723 1.723 1.714

2 1.727 1.727 1.720

3 7.413 7.413 7.554

4 9.972 9.972 9.954

5 18.16 18.16 17.68

6 26.96 26.97 26.78

References




10.5 Deep Simply-Supported BeamThis test is a normal mode dynamic analysis of a deep simply supported beam (shown below).This document provides the input data and results for NAFEMS Selected Benchmarks forNatural Frequency Analysis, Test 5.


• Shear deformation and rotary inertial (Timoshenko beam)

• Possibility of missing extensional modes when using iteration solution methods

• Repeated eigenvalues


Input Files

• nf005ac.dat (linear consistent, param coupmass = 1)

• nf005al.dat (linear lumped, param coupmass = –1)

Units

SI


Shear ratio:

• Y = 1.176923

• Z = 1.176923

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




• Five linear beam elements (CBEAM)

Boundary Conditions

• X = Y = Z = Rx =0 at A

• Y = Z = 0 at B

Solution Type



• Using lumped mass (lumped mass on, param coupmass = –1)

• Using coupled mass (lumped mass off, param coupmass = 1)

Results





1, 2 (flexural) 42.65 42.57 43.15 43.26

3 (torsional) 77.54 77.84 77.20 77.84

4 (extensional) 125.0 125.5 124.5 125.5

5, 6 (flexural) 148.3 145.5 149.9 154.9

7 (torsional) 233.1 241.2 224.1 241.2

8, 9 (flexural) 284.6 267.0 271.0 306.7

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis, Test No. 5. Glasgow: NAFEMS,Nov., 1987.



10.6 Circular Ring — In-plane and Out-of-plane VibrationThis test is a normal mode dynamic analysis of a circular ring (shown below) using beamelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 6.


• Rigid body modes (six modes)



Input Files• nf006ac.dat (param coupmass = 1)

• nf006al.dat (param coupmass = –1)

UnitsSI

Cross Sectional PropertiesShear ratio:

• Y = 1.128205

• Z = 1.128205


• ρ = 8000 kg/m3

• ν = 0.3




• 20 linear beam (CBAR) elements

Boundary Conditions

• X = Y = Z = Rx = Ry = Rz active

• Model is unsupported.

Solution Type


NX Nastran results were obtained two different ways:

• Using coupled mass (param coupmass = –1)

• Using lumped mass (param coupmass = 1)

Results





7, 8 (out ofplane)

51.85 52.29 51.63 52.38

9, 10 (in plane) 53.38 53.97 54.05 53.80

11, 12 (out ofplane)

148.8 149.7 146.9 149.7

13, 14 (in plane) 151.0 152.4 152.2 151.5

15 (out of plane) 287.0 288.3 280.4 287.4

16 (in plane) 289.5 288.3 289.2 289.1

References




10.7 Cantilevered BeamThis test is a normal mode dynamic analysis of a cantilevered beam (shown below). Thisdocument provides the input data and results for NAFEMS Selected Benchmarks for NaturalFrequency Analysis, Test 71.


• Ill-conditioned stiffness matrix


Input Files

• nf071a.dat (Test 1)

• nf071b.dat (Test 2)

• nf071c.dat (Test 3)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ=8000 kg/m3


Three tests — all use linear beam (CBAR) elements

• Test 1: a = b

• Test 2: a = 10b

• Test3: a = 100b



Boundary Conditions

• X = Y = Rz = 0 at A

• Z = 0 at all grid points

• Rx = Ry = 0 at all grid points

Solution Type


Beams always use a coupled mass formulation (param coupmass = 1).

Results

Mode Reference Value(Hz)

Mesh NX Nastran Result (Hz)

1 1.010 a = b

a = 10b

a = 100b

1.010

1.010

1.010

2 6.327 a = b

a = 10b

a = 100b

6.324

6.327

6.330

3 17.72 a = b

a = 10b

a = 100b

17.70

17.80

17.83

4 34.72 a = b

a = 10b

a = 100b

34.70

34.87

35.07

5 57.39 a = b

a = 10b

a = 100b

57.48

60.64

64.83

6 85.73 a = b

a = 10b

a = 100b

86.24

101.9

104.7



References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis Test No. 71. Glasgow: NAFEMS,Nov., 1987.


Chapter

11 Shell Element Test Cases

11.1 Thin Square Cantilevered Plate — Symmetric ModesThis test is a normal mode dynamic analysis of a thin, square, cantilevered plate meshed withNX Nastran shell elements. This document provides the input data and results for NAFEMSSelected Benchmarks for Natural Frequency Analysis, Test 11a.


• Symmetric modes, symmetric boundary conditions along the cutting plane


Input Files

• nf011a_l.dat (4-noded quadrilateral, lumped mass)

• nf011a_c.dat (4-noded quadrilateral, coupled mass)

• nf011ha_l.dat (8-noded quadrilateral, lumped mass)

• nf011ha_c.dat (8-noded quadrilateral, coupled mass)

Units

SI


Chapter 11 Shell Element Test Cases


• ρ = 8000 kg/m3

• ν = 0.3

Finite Element ModelingTwo tests:

• 32 linear quadrilateral thin shell (CQUAD4) elements — thickness = 0.05m

• 8 parabolic quadrilateral thin shell (CQUAD8) elements — thickness = 0.05m

Boundary Conditions• X = Y = Rz = 0 at all grid points

• Z = Ry = Rx = 0 along Y-axis

• Rx = 0 along Y = 5m


Results were obtained in two different ways:

• Using lumped mass (param coupmass = –1)


Shell Element Test Cases

• Using coupled mass (param coupmass = 1)

Results


Mesh NX NastranResult (lumpedmass)(Hz)

NX NastranResult (coupledmass(Hz)

1 0.4210 Linear

Parabolic

0.4150

0.4150

0.4180

0.4180

2 2.582 Linear

Parabolic

2.490

2.478

2.604

2.567

3 3.306 Linear

Parabolic

3.115

3.134

3.314

3.271

4 6.555 Linear

Parabolic

6.044

6.163

6.538

6.539

5 7.381 Linear

Parabolic

7.094

7.099

7.808

7.495

6 11.40 Linear

Parabolic

10.57

10.99

12.34

12.08

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis, Test No. 11a. Glasgow: NAFEMS,Nov., 1987.



11.2 Thin Square Cantilevered Plate — Anti-symmetric ModesThis test is a normal mode dynamic analysis of a thin, square, cantilevered plate meshed withshell elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 11b.


• Anti-symmetric modes


Input Files

• nf011b.dat (linear (4-noded) quadrilateral)

• nf011hb.dat (parabolic (8-noded) quadrilateral)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3


Two tests:



Mesh only half the plate (10m × 5m).



Boundary Conditions

• X = Y = Rz = 0 at all grid points

• Z = Ry = Rx = 0 along Y-axis

• Rx = 0 along Y = 5m

Solution Type



• Using lumped mass (param coupmass = —1)


Results


Mesh NAFEMSTarget Value(Hz)

NX NastranResult(lumpedmass)(Hz)

NX NastranResult(coupledmass(Hz)

1 1.029 Linear

Parabolic

1.019

1.018

1.000

1.005

1.020

1.022





NX NastranResult(lumpedmass)(Hz)


2 3.753 Linear

Parabolic

3.839

3.710

3.570

3.597

3.767

3.725

3 7.730 Linear

Parabolic

8.313

7.768

7.091

7.026

8.113

7.786

4 8.561 Linear

Parabolic

9.424

8.483

8.047

8.133

9.025

8.690

5 not available Linear

Parabolic

11.73

11.19

9.940

10.15

11.69

11.19

6 not available Linear

Parabolic

17.82

15.76

14.22

14.21

17.44

16.78

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis Test No. 11b. Glasgow: NAFEMS,Nov., 1987.



11.3 Free Thin Square PlateThis test is a normal mode dynamic analysis of a free thin square plate meshed with shellelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 12.


• Rigid body modes (three modes)



Input Files• nf012l_l.dat (linear (4-noded) quadrilateral, lumped mass)

• nf012l_c.dat (linear (4-noded) quadrilateral, coupled mass)

• nf012h_l.dat (parabolic (8-noded) quadrilateral, lumped mass)

• nf012h_c.dat (parabolic (8-noded) quadrilateral, coupled mass)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3






Boundary Conditions


Solution Type


Results were obtained in two different ways:



Results



NX NastranResult(lumpedmass) (Hz)

NX NastranResult(coupledmass) (Hz)

4 1.622 Linear

Parabolic

1.632

1.532

1.578

1.584

1.624

1.619

5 2.360 Linear

Parabolic

2.402

2.356

2.241

2.233

2.389

2.363

6 2.922 Linear

Parabolic

3.006

2.861

2.804

2.808

2.979

2.929

7, 8 4.233 Linear

Parabolic

4.251

4.122

3.931

3.944

4.237

4.158

9 7.416 Linear

Parabolic

7.859

7.363

6.822

6.813

7.790

7.477

10 Notavailable

Linear

Parabolic

8.027

7.392

6.822

6.813

7.790

7.477



References




11.4 Simply Supported Thin Square PlateThis test is a normal mode dynamic analysis of a free thin square plate meshed with shellelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 13.


• Well established



Input Files• nf013l_l.dat (linear quadrilateral, lumped mass)

• nf013l_c.dat (linear quadrilateral, coupled mass)

• nf013h_l.dat (parabolic quadrilateral, lumped mass)

• nf013h_c.dat (parabolic quadrilateral, coupled mass)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3






Boundary Conditions


• Z = Rx = 0 along edges X = 0 and X = 10m

• Z = Ry = 0 along edges Y = 0 and Y = 10m

Solution Type





Results


Mesh NX NastranResult (lumpedmass) (Hz)


1 2.377 Linear

Parabolic

2.332

2.376

2.392

2.382

2, 3 5.942 Linear

Parabolic

5.797

5.938

6.181

6.026

4 9.507 Linear

Parabolic

8.963

9.747

9.933

10.22

5, 6 11.88 Linear

Parabolic

11.67

11.87

13.27

12.39

7, 8 15.45 Linear

Parabolic

14.45

16.56

17.07

18.17



References




11.5 Simply Supported Thin Annular PlateThis test is a normal mode dynamic analysis of a free thin square plate meshed with shellelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 14.


• Curved boundary (skewed coordinate system)



Input Files• nf014l_l.dat (linear quadrilateral, lumped mass)

• nf014l_c.dat (linear quadrilateral, coupled mass)

• nf014h_l.dat (parabolic quadrilateral, lumped mass)

• nf014h_c.dat (parabolic quadrilateral, coupled mass)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3




Two tests:

• 160 linear quadrilateral thin shell (CQUAD4) elements — thickness = 0.06 m

• 48 parabolic quadrilateral thin shell (CQUAD8) elements — thickness = 0.06 m

Boundary Conditions


• Z′ = Rx′ = 0 around the circumference

Solution Type





Results




1 1.870 Linear

Parabolic

1.859

1.840

1.877

1.873

2, 3 5.137 Linear

Parabolic

5.293

5.111

5.249

5.151

4, 5 9.673 Linear

Parabolic

10.03

9.673

9.983

9.713






6 14.85 Linear

Parabolic

14.37

13.95

15.41

14.92

7, 8 15.57 Linear

Parabolic

16.10

15.55

15.55

15.71

9 18.38 Linear

Parabolic

18.07

17.38

19.09

18.52

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis,, Test No. 13. Glasgow: NAFEMS,Nov., 1987.



11.6 Clamped Thin Rhombic PlateThis test is a normal mode dynamic analysis of a free thin square plate meshed with I-DEASshell elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 15.


• Distorted elements


Input Files• nf015l.dat linear (lumped)

• nf015ha.dat parabolic (lumped)

• nf015hb.dat parabolic (consistent)

• nf015hc.data linear (consistent)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3






Boundary Conditions• X = Y = Rz = 0 at all grid points

• Z′ = Rx′ = Ry′ = 0 along all four edges

Solution TypeSOL103 — Normal Mode Dynamics




Results





1 7.938 Linear

Parabolic

8.142

7.873

7.818

7.902

7.955

7.929

2 12.84 Linear

Parabolic

13.89

12.48

12.83

12.85

13.39

13.01

3 17.94 Linear

Parabolic

20.04

17.31

17.81

17.95

19.07

18.47

4 19.13 Linear

Parabolic

20.17

18.74

18.55

18.96

19.24

19.17







5 24.01 Linear

Parabolic

27.70

27.95

23.67

23.88

26.19

25.23

6 27.92 Linear

Parabolic

32.05

25.88

27.70

27.91

29.82

28.81

References




11.7 Cantilevered Thin Square Plate with Distorted MeshThis test is a normal mode dynamic analysis of a free thin square plate meshed with I-DEASshell elements. This document provides the input data and results for NAFEMSSelectedBenchmarks for Natural Frequency Analysis, Test 16.


• Distorted meshes


Input Files• nf016a1.dat: (16 parabolic quad, lumped mass)

• nf016a2.dat: (16 parabolic quad, coupled mass)

• nf016b1.dat: (16 parabolic quad, lumped mass)

• nf016b2.dat: (16 parabolic quad, coupled mass)

• nf016c1.dat: (4 parabolic quad, lumped mass)

• nf016c2.dat: (4 parabolic quad, coupled mass)

• nf016d1.dat: (4 parabolic quad, lumped mass)

• nf016d2.dat: (4 parabolic quad, coupled mass)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3




All tests — parabolic quadrilateral thin shell elements — thickness = 0.05m

Four tests:

• Test 1 — 16 elements

• Test 2 — 16 elements with specified grid points at the following XY coordinates:

Coordinates

Node X Y

1 4.000 4.000

2 2.250 2.250

3 4.750 2.500

4 7.250 2.750

5 7.500 7.250

6 5.250 7.250

7 5.250 7.250

8 2.250 7.250

9 2.500 4.750

• Test 3 — 4 elements



• Test 4 — 4 elements with a specified grid point at the following XY coordinate:

Coordinates

Node X Y

1 4.000 4.000

Boundary Conditions

• X = Y = Z = Ry = 0 along Y-axis

Solution Type

SOL103 — Normal Mode Dynamics






Results


Test NAFEMSTarget Value(Hz)



1 0.4210 1

2

3

4

0.4174

0.4174

0.4144

0.4145

0.4139

0.4135

0.4021

0.4000

0.4181

0.4182

0.4189

0.4192

2 1.029 1

2

3

4

1.020

1.020

0.9990

1.002

0.9985

0.9967

0.9347

0.9202

1.024

1.024

1.021

1.025

3 2.582 1

2

3

4

2.564

2.571

2.554

2.565

2.444

2.445

2.132

2.112

2.569

2.566

2.708

2.698

4 3.306 1

2

3

4

3.302

3.317

3.401

3.424

3.082

3.072

2.707

2.697

3.281

3.280

3.449

3.430

5 3.753 1

2

3

4

3.769

3.780

3.697

3.714

3.540

3.535

3.136

3.077

3.728

3.731

3.913

3.881

6 6.555 1

2

3

4

6.805

6.883

5.455

5.133

6.018

5.994

5.458

5.459

6.551

6.552

7.108

6.858



References




11.8 Simply Supported Thick Square Plate, Test AThis test is a normal mode dynamic analysis of a free thin square plate meshed with I-DEASshell elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 21a.




• Effect of secondary restraints


Input Files• nf021a.dat: linear (lumped mass)

• nf021ha.dat: parabolic (lumped mass)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3






Boundary Conditions

• Z = 0 along all four edges


• Rx = 0 along edges X = 0 and X = 10 m

• Ry = 0 along edges Y = 0 and Y = 10 m

Solution Type







Results





1 45.90 Linear

Parabolic

46.66

45.94

45.83

46.17

46.35

45.83

2, 3 109.4 Linear

Parabolic

115.8

110.4

110.6

110.3

114.1

109.4

4 167.9 Linear

Parabolic

177.5

170.4

164.8

167.3

174.3

169.8

5, 6 204.5 Linear

Parabolic

233.4

212.8

211.8

204.6

227.1

208.2

7, 8 256.5 Linear

Parabolic

283.6

270.0

250.5

249.3

276.9

268.4

9 336.6 Linear

Parabolic

371.1

344.8

313.1

311.4

364.3

319.4

10 336.6 Linear

Parabolic

371.1

344.8

338.4

347.6

385.8

319.4

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis, Test No. 21a. Glasgow: NAFEMS,Nov., 1987.



11.9 Simply Supported Thick Square Plate, Test BThis test is a normal mode dynamic analysis of a free thin square plate meshed with shellelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 21b.




• Effect of secondary restraints


Input Files• nf021b_c.dat (quadrilateral thin shell elements — coupled mass)

• nf021b_l.dat (quadrilateral thin shell elements — lumped mass)

• nf021hb_c.dat (parabolic thin shell elements — coupled mass)

• nf021hb_l.dat (parabolic thin shell elements — lumped mass)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3




• 64 linear quadrilateral thin shell elements — thickness = 1.0 m

• 16 parabolic quadrilateral thin shell elements — thickness = 1.0 m

Boundary Conditions

• Z = 0 along all four edges; X = Y = Rz = 0 at all grid points

Solution Type







Results





1 45.90 Linear

Parabolic

44.75

44.13

44.65

44.82

44.96

44.49

2, 3 109.4 Linear \

Parabolic

112.9

107.9

109.1

108.5

112.3

107.6

4 167.9 Linear

Parabolic

170.3

164.2

161.4

163.6

170.2

165.7

5, 6 204.5 Linear

Parabolic

230.2

20.07

210.5

203.1

225.4

206.5

7, 8 256.5 Linear

Parabolic

274.2

260.3

247.1

245.7

272.5

263.6

9 336.6 Linear

Parabolic

356.0

342.8

308.8

307.2

358.4

318.6

10 336.6 Linear

Parabolic

356.0

342.8

337.6

346.9

384.8

318.6

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis, Test No. 21b. Glasgow: NAFEMS,Nov., 1987.



11.10 Clamped Thick Rhombic PlateThis test is a normal mode dynamic analysis of a free thin square plate meshed with shellelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 22.


• Distorted elements


Input Files• nf022l_l.dat

• nf022l_c.dat

• nf022h_l.dat

• nf022h_c.dat

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3


• 100 linear quadrilateral thin shell (CQUAD4) elements - thickness = 1.0 m

• 36 parabolic quadrilateral thin shell (CQUAD8) elements - thickness = 1.0 m



Boundary Conditions


• Z′ = Rx′ = Ry′ = 0 along all four edges

Solution Type

SOL 103 – Normal Mode Dynamics


• Using lumped mass (parm coupmass = –1)


Results





1 134.0 Linear

Parabolic

137.8

133.9

131.2

134.9

134.3

135.2

2 201.4 Linear

Parabolic

218.5

203.3

200.4

204.4

211.9

206.3

3 265.8 Linear

Parabolic

295.4

271.4

262.0

270.3

286.6

276.4







4 282.7 Linear

Parabolic

296.8

283.7

273.6

286.9

287.0

289.1

5 334.5 Linear

Parabolic

383.6

346.4

327.0

337.5

373.3

353.8

6 Notavailable

Linear

Parabolic

426.6

386.6

372.2

384.7

410.6

394.0

References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C.,Selected Benchmarks for Natural Frequency Analysis, Test No. 22. Glasgow: NAFEMS,Nov., 1987.



11.11 Simply Supported Thick Annular PlateThis test is a normal mode dynamic analysis of a simply supported thick annular plate meshedwith shell elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 23.





Input Files

nf023l_l.dat

nf023l_c.dat

nf023h_l.dat

nf023h_c.dat

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




Two tests:



Boundary Conditions


• Z′ = Rx′ = 0 around the circumference

Solution Type

SOL 103 — Normal Mode Dynamics



• Using coupled mass (param coupmas = 1)



Results





1 18.58 Linear

Parabolic

18.82

18.59

18.40

18.53

18.64

18.65

2, 3 48.92 Linear

Parabolic

49.82

49.02

50.00

49.22

50.81

49.41

4, 5 92.59 Linear

Parabolic

96.06

92.90

93.09

93.41

96.00

93.73

6 140.2 Linear

Parabolic

148.3

140.9

134.6

140.2

147.0

143.1

7, 8 Notavailable

Linear

Parabolic

153.7

146.6

144.0

147.0

152.1

148.2

9 166.4 Linear

Parabolic

174.5

167.3

162.2

166.9

177.1

170.4

References




11.12 Cantilevered Square MembraneThis test is a normal mode dynamic analysis of a cantilevered square membrane meshed withshell elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 31.




Input Files

• nf031l.dat (linear quadrilateral, lumped mass)

• nf031a.dat (linear quadrilateral, coupled mass)

• nf031h.dat (parabolic quadrilateral, lumped mass)

• nf031j.dat (parabolic quadrilateral, coupled mass)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




Two tests:

• 64 linear quadrilateral thin shell (CQUAD4) elements - thickness = 0.05 m

• 16 parabolic quadrilateral thin shell (CQUAD8) elements - thickness = 0.05 m

Boundary Conditions

• X = Y = 0 along the Y axis


Solution Type







Results





1 52.40 Linear

Parabolic

52.91

52.64

52.48

52.30

52.78

52.60

2 125.7 Linear

Parabolic

126.1

125.9

125.6

125.7

126.1

125.9

3 140.8 Linear

Parabolic

143.2

141.5

139.6

139.5

142.9

141.4

4 222.5 Linear

Parabolic

228.9

224.6

215.1

214.4

227.5

224.3

5 241.4 Linear

Parabolic

247.9

243.3

240.1

242.3

247.4

242.9

6 255.7 Linear

Parabolic

260.6

256.8

252.4

254.6

259.8

256.6

References




11.13 Cantilevered Tapered MembraneThis test is a normal mode dynamic analysis of a cantilevered tapered membrane meshed withshell elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 32.


• Shear behavior

• Irregular mesh

• Symmetry


Input Files

• nf032l.dat (linear quadrilateral, lumped mass)




Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




Two tests:

• 128 linear quadrilateral thin shell (CQUADR) elements — thickness = 0.1 m


Boundary Conditions

• X = Y = 0 along the Y axis


Solution Type



• Using lumped mass (param coupmas = –1)




Results





1 44.62 Linear

Parabolic

44.91

44.64

44.66

44.54

44.78

44.63

2 130.0 Linear

Parabolic

132.1

130.1

130.3

129.7

131.8

130.1

3 162.7 Linear

Parabolic

162.8

162.7

162.6

162.7

162.8

162.7

4 246.1 Linear

Parabolic

253.0

246.6

246.1

245.1

252.3

246.4

5 379.9 Linear

Parabolic

393.3

382.0

377.9

377.9

393.2

381.4

6 391.4 Linear

Parabolic

396.3

391.6

389.7

390.9

395.0

391.5

References




11.14 Free Annular MembraneThis test is a normal mode dynamic analysis of a free annular membrane meshed with shellelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 33.





Input Files• nf033l.dat (linear quadrilateral, lumped mass)




UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3




Two tests:



Boundary Conditions


Solution Type







Results





4, 5 129.2 Linear

Parabolic

129.5

126.5

127.8

125.7

128.8

125.8

6 226.2 Linear

Parabolic

225.5

224.3

224.5

224.0

225.3

224.2

7, 8 234.7 Linear

Parabolic

234.9

233.0

229.9

230.8

234.9

233.0

9, 10 264.7 Linear

Parabolic

272.1

264.8

264.3

262.6

271.2

263.6

11, 12 336.6 Linear

Parabolic

340.3

335.7

329.0

331.5

339.9

335.7

13, 14 376.8 Linear

Parabolic

392.0

378.6

369.9

373.3

390.5

377.4

References




11.15 Cantilevered Thin Square PlateThis test is a normal mode dynamic analysis of a cantilevered thin square plate meshed withshell elements. This document provides the input data and results for NAFEMSSelectedBenchmarks for Natural Frequency Analysis, Test 73.


Input Files• nf073a.dat (Test 1)

• nf073b.dat (Test 2)

• nf073c.dat (Test 3)

• nf073d.dat (Test 4)

UnitsSI


• ρ = 8000 kg/m3

• ν = 0.3

Finite Element Modeling16 parabolic quadrilateral thin shell (CQUAD8) elements — thickness = 0.05 m



Boundary ConditionsX = Y = Z = Ry = 0 along the Y axis

Solution TypeSOL 103 — Normal Mode Dynamics






Results





1 0.4210 Test 1

Test 2

Test 3

Test 4

0.4174

0.4174

0.4175

0.4184

0.4154

0.4154

0.4154

0.4161

0.4183

0.4183

0.4184

0.4192

2 1.029 Test 1

Test 2

Test 3

Test 4

1.020

1.020

1.021

1.032

1.051

1.006

1.007

1.015

1.023

1.023

1.027

1.024

3 2.582 Test 1

Test 2

Test 3

Test 4

2.564

2.597

2.677

2.850

2.485

2.509

2.524

2.563

2.579

2.605

2.675

2.672

4 3.306 Test 1

Test 2

Test 3

Test 4

3.302

3.345

3.365

3.571

3.150

3.180

3.196

3.373

3.298

3.332

3.344

3.535

5 3.753 Test 1

Test 2

Test 3

Test 4

3.769

3.888

4.035

5.466

3.622

3.713

3.828

4.935

3.765

3.862

4.000

5.360

6 376.8 Test 1

Test 2

Test 3

Test 4

6.805

7.517

7.495

——

6.292

6.901

6.879

——

6.719

7.399

7.387

——

References



Chapter

12 Axisymmetric Solid and SolidElement Test Cases

12.1 Free Cylinder — Axisymmetric VibrationThis test is a normal mode dynamic analysis of a free cylinder meshed with axisymmetricelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 41.


• Rigid body modes (one mode)

• Coupling between axial, radial, and circumferential behavior

• Close eigenvalues


Input Files• nf041.dat (linear axisymmetric, lumped mass)

• nf041a.dat (linear axisymmetric, coupled mass)

• nf041h.dat (parabolic axisymmetric, lumped mass)

• nf041j.dat (parabolic axisymmetric, coupled mass)

UnitsSI



Chapter 12 Axisymmetric Solid and Solid Element Test Cases

• ρ = 8000 kg / m3

• ν = 0.3


• 16 axisymmetric solid linear quadrilateral (CQUADX) elements

• 8 axisymmetric solid parabolic quadrilateral (CQUADX) elements

Boundary ConditionsUnsupported






Axisymmetric Solid and Solid Element Test Cases

Results

Mode # ReferenceValue (Hz)

Mesh NAFEMSTargetValue (Hz)



2 243.5 Linear

Parabolic

244.0

243.5

243.1

243.4

243.9

243.5

3 377.4 Linear

Parabolic

379.4

377.5

372.1

376.4

378.4

377.4

4 394.1 Linear

Parabolic

395.4

394.3

385.8

392.4

394.4

394.2

5 397.7 Linear

Parabolic

401.4

397.9

386.9

392.8

398.5

397.9

6 405.3 Linear

Parabolic

421.9

406.4

391.7

397.2

415.4

406.0

The reference value refers to the accepted solution to the problem.

References




12.2 Thick Hollow Sphere — Uniform Radial VibrationThis test is a normal mode dynamic analysis of a thick hollow sphere meshed using axisymmetricelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 42.



• Constraint equations


Input Files

• nf042.dat (linear axisymmetric, lumped mass)

• nf042a.dat (linear axisymmetric, coupled mass)

• nf042h.dat (parabolic axisymmetric, lumped mass)

• nf042j.dat (parabolic axisymmetric, coupled mass)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




• 10 axisymmetric solid linear quadrilateral (CQUADX) elements -α = 5°

Boundary Conditions

• Z′ displacement = 0 at all grid points

• Grid points at the same R′ are constrained to have the same r′ displacement

• One constraint set

Solution Type







Results





1 369.9 Linear

Parabolic

370.6

370.0

369.3

369.7

369.6

369.7

2 838.0 Linear

Parabolic

841.2

838.1

828.1

836.2

837.7

837.7

3 1451. Linear

Parabolic

1473.

1453.

1416.

1445.

1468.

1451.

4 2117. Linear

Parabolic

2192.

2132.

2023.

2100.

2186.

2117.

5 2796. Linear

Parabolic

2976.

2853.

2595.

2764.

2967.

2799.


References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C. Selected Benchmarks for Natural Frequency Analysis Test No. 42. Glasgow: NAFEMS,Nov., 1987.



12.3 Simply Supported Annular Plate — Axisymmetric VibrationThis test is a normal mode dynamic analysis of a simply supported annular plate meshed withaxisymmetric elements. This document provides the input data and results for NAFEMSSelected Benchmarks for Natural Frequency Analysis, Test 43.




Input Files

• nf043a.dat (lumped mass)

• nf043b.dat (coupled mass)

• nf043c.dat (lumped mass)

• nf043d.dat (coupled mass)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




Two tests:

• 60 axisymmetric solid linear quadrilateral (CQUADX) elements

• 5 axisymmetric solid parabolic quadrilateral (CQUADX) elements

Boundary Conditions

• Z = 0 at A

Solution Type







Results





1 18.54 Linear

Parabolic

18.71

18.58

18.23

18.48

18.27

18.55

2 150.2 Linear

Parabolic

145.5

145.6

140.9

135.9

142.6

138.6

3 224.2 Linear

Parabolic

224.2

224.2

224.2

224.1

224.2

224.2

4 358.3 Linear

Parabolic

385.6

374.1

366.3

345.3

376.5

360.3

5 629.2 Linear

Parabolic

689.3

686.0

647.7

592.7

677.8

640.2


References




12.4 Deep Simply Supported "Solid" BeamThis test is a normal mode dynamic analysis of a deep, simply supported beam meshed withbrick elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis , Test 51.


• Skewed coordinate system

• Skewed restraints


Input Files

• nf051l.dat (linear brick)

• nf051b.dat (parabolic brick)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




Two tests:

• 30 solid linear brick (CHEXA) elements

• 5 solid parabolic brick (CHEXA) elements

Boundary Conditions

• X′ = Z′ = 0 along AA′

• Z′ = 0 along BB′

• Y′ = 0 at all grid points on the plane Y′ = 2.0 m



Solution Type





Results





1 38.20 linear

parabolic

42.88

38.82

37.96

37.85

38.28

38.24

2 85.21 linear

parabolic

93.82

88.45

83.38

87.12

83.95

87.52

3 152.2 linear

parabolic

170.7

159.4

152.7

151.8

157.6

157.0

4 245.5 linear

parabolic

286.1

259.2

251.6

248.5

264.9

258.2

5 297.1 linear

parabolic

318.9

307.9

288.0

289.6

298.3

305.6


References

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles,N. C., Selected Benchmarks for Natural Frequency Analysis Test No. 51. Glasgow: NAFEMS,Nov., 1987.



12.5 Simply Supported "Solid" Square PlateThis test is a normal mode dynamic analysis of a simply supported square plate meshed withbrick elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 52.




• Kinematically incomplete suppressions


Input Files


• nf052b.dat (parabolic brick)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




Two tests:

• 64 solid linear brick (CHEXA) elements

• 16 solid parabolic brick (CHEXA) elements

Boundary Conditions

Z = 0 along the four edges on the plane Z = –0.5 m

Solution Type

SOL 103 normal modes






Results





4 45.90 Linear

Parabolic

51.65

44.76

44.04

43.81

45.24

44.16

5, 6 109.4 Linear

Parabolic

132.7

110.5

106.5

105.2

113.7

107.9

7 167.9 Linear

Parabolic

194.4

169.1

155.5

156.3

172.3

163.9

8 193.6 Linear

Parabolic

197.2

193.9

193.6

194.0

196.8

193.9

9, 10 206.2 Linear

Parabolic

210.6

206.6

200.1

193.5

209.6

206.6


References




12.6 Simply Supported "Solid" Annular PlateThis test is a normal mode dynamic analysis of a simply supported annular plate meshed withbrick elements. This document provides the input data and results for NAFEMS SelectedBenchmarks for Natural Frequency Analysis, Test 53.



• Constraint equations


Input Files


• nf053h.dat (parabolic brick)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 0.3




• 60 solid linear brick (CHEXA) elements — α = 5°

• 5 solid parabolic brick (CHEXA) elements — α = 10°

Boundary Conditions• θ displacement = 0 at all grid points

• Z displacement = 0 at all grid points along AA

• Grid points at same R and Z are constrained to have same z displacement

• One constraint set







Results




NX Nastran Result(coupled mass)(Hz)

1 18.58 Linear

Parabolic

19.66

18.58

18.57

18.45

18.61

18.58

2 140.2 Linear

Parabolic

146.4

140.4

138.8

135.9

140.5

140.3

3 224.2 Linear

Parabolic

224.3

224.2

224.2

223.7

224.4

224.2

4 358.3 Linear

Parabolic

386.7

374.0

361.8

351.2

372.1

371.9

5 629.2 Linear

Parabolic

689.5

686.0

643.8

624.7

674.7

679.6

References




12.7 Cantilevered Solid BeamThis test is a normal mode dynamic analysis of a cantilevered solid beam meshed using brickelements. This document provides the input data and results for NAFEMS Selected Benchmarksfor Natural Frequency Analysis, Test 72.


• Highly populated stiffness matrix


Input Files

• nf072a.dat (conventional)

• nf072b.dat (unconventional)

Units

SI

Material Properties

• E = 200E09 N/m2

• ρ = 8000 kg/m3

• ν = 3




Two tests — both use solid parabolic brick (CHEXA) elements

• Test 1: conventional grid point numbering

• Test 2: unconventional grid point numbering

Boundary Conditions

• X = Y = Z = 0 at all grid points on X = 0 plane

• Y = 0 at grid points on Y = 1 m plane



Solution Type





Results

Mode # Mesh NAFEMSTarget Value(Hz)



1 Test 1

Test 2

16.01

16.01

15.82

15.82

15.99

15.99

2 Test 1

Test 2

87.23

87.23

83.18

83.18

87.09

87.09

3 Test 1

Test 2

126.0

126.0

125.5

125.5

126.0

126.0

4 Test 1

Test 2

209.6

209.6

193.5

193.5

209.1

209.1

5 Test 1

Test 2

351.1

351.1

310.1

310.1

349.9

349.9

6 Test 1

Test 2

375.8

375.8

364.2

364.2

375.8

375.8

References



Part

VI Verification Test Cases from theSociete Francaise des Mecaniciens

Overview of Verification Test Cases Provided by the Societe Francaise des Mecaniciens . . . 13-1

Mechanical Structures — Linear Statics Analysis with Beam or Rod Elements . . . . . . . . . 14-1

Mechanical Structures — Linear Statics Analysis with Shell Elements . . . . . . . . . . . . . . 15-1

Mechanical Structures — Linear Statics Analysis with Solid Elements . . . . . . . . . . . . . . 16-1

Mechanical Structures — Normal Mode Dynamics Analysis . . . . . . . . . . . . . . . . . . . . . . 17-1

Mechanical Structures — Normal Mode Dynamics Analysis and Model Response . . . . . . . 18-1

Stationary Thermal Tests — Heat Transfer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-1

Thermo-mechanical Tests — Linear Statics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1


Chapter

13 Overview of VerificationTest Cases Provided by the SocieteFrancaise des Mecaniciens

The purpose of these linear statics test cases is to verify the function of NX Nastran usingstandard benchmarks published by SFM (Societe Francaise des Mecaniciens. Paris, France) inGuide de validation des progiciels de calcul de structures.

Included here are:

• Tests cases on mechanical structures using linear statics analysis, normal mode dynamicsanalysis, and model response.

• Stationary thermal test cases using heat transfer analysis.

• Thermo-mechanical test cases using linear statics analysis.

Results published in Guide de validation des progiciels de calcul de structures are compared withthose computed using NX Nastran.



– Input files

– Units

– Material properties


– Boundary conditions (loads and restraints)

– Solution type

• Results

• Reference


Chapter 13 Overview of Verification Test Cases Provided by the Societe Francaisedes Mecaniciens


Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990.


Chapter

14 Mechanical Structures —Linear Statics Analysis with Beamor Rod Elements

14.1 Short Beam on Two Articulated SupportsThis test is a linear statics analysis of a short, straight beam with plane bending and shearloading. It provides the input data and results for benchmark test SSLL02/89 from Guide devalidation des progiciels de calcul de structures.

• Area = 31E–04 m2

• Inertia = 2810E–08 m4

• Shear area ratio = 2.42


Input Files

ssll02.dat

Units

SI

Material Properties

• E = 2E11 Pa

• ν = 0.3


Chapter 14 Mechanical Structures — Linear Statics Analysis with Beam or RodElements



• 11 grid points

The mesh is shown in the following figure:

Boundary Conditions

• Restrain both free ends of the beam in translation DOF.

– Edge load = 1E05 N/m in –Y direction


Solution Type


Results


Displacement at point B v (m) (Grid point 7) –1.259E–3 –1.249E–3

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SSLL02/89.


Mechanical Structures — Linear Statics Analysis with Beam or Rod Elements

14.2 Clamped Beams Linked by a Rigid ElementThis test is a linear statics analysis of a straight, cantilever beam with plane bending and a rigidelement. It provides the input data and results for benchmark test SSLL05/89 from Guide devalidation des progiciels de calcul de structures.


Input File

ssll05.dat

Units

SI

Material Properties

• E = 2E11 Pa

• I = (4/3)E–08 m4



• 1 rigid element

• 26 grid points




Boundary Conditions

• Points A and C: Clamped

• Point D: Set nodal force = 1000 N in –Y direction


Solution Type


Results

Type Grid point Point BenchValue

NXNastran

v (m) Disp. Y Grid point 6 B –0.1250 –0.1250

v (m) Disp. Y Grid point 3 D –0.1250 –0.1250

V force (N) Y Grid point 1 A 500.0 500.0

M moment (Nm) Rz Grid point 1 A 500.0 500.0

V force (N) Y Grid point 4 C 500.0 500.0

M moment (Nm) Rz Grid point 4 C 500.0 500.0

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990 Test No. SSLL05/89.



14.3 Transverse Bending of a Curved PipeThis test is a linear statics analysis (three-dimensional problem) of a curved pipe with transversebending and bending-torque loading. It provides the input data and results for benchmark testSSLL07/89 from Guide de validation des progiciels de calcul de structures.

• R = 1 m

• de = 0.02 m

• di = 0.016 m

• A = 1.131E-0–04 m2

• Ix = 4.637E–09 m4


Input Files

• ssll07a.dat linear beam

• ssll07b.dat curved beam

Units

SI

Material Properties

• E = 2E11 Pa

• ν = 0.3



Finite Element ModelingTest 1


• 91 grid points

Test 2

• 90 curved beam (CBEND) elements

• 91 grid points

To obtain the point where θ = 15° with accuracy, use surface mapped meshing on 1/4 of a cylinder.Then mesh a curved edge with the Surface Coating command and undo the mesh on the surface.

The mesh for Test 1 is shown in the following figure:

Boundary Conditions• Clamp point A.

• Grid point force F = 100 N in Z direction.


Solution TypeSOL 101 — Linear Statics



Results

Type Grid point Point BenchValue

TestNumber

NX Nastran

u (m) Disp. Z Grid point 1 B 0.1346 1 0.1346

2 0.1346

Mt (Nm)* Grid point 1 θ = 15° 74.12 1 76.67

2 77.51

Mf (Nm) –96.59 1 –96.37

2 –95.70

Mf = bending moment

Mt = torsional moment

*See "Post Processing" below.

Post Processing

Linear Beam (CBAR) Elements

List beam forces on element 167, second end:

• Mf = torque

• Mt = bending moment

Curved Beam (CBEND) Elements

List beam forces on element 166, second end:

• Mf = torque

• Mt = bending moment

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990.Test No. SSLL07/89..



14.4 Plane Bending Load on a Thin ArchThis test is a linear statics analysis (plane problem) of a thin arc with plane bending. It providesthe input data and results for benchmark test SSLL08/89 from Guide de validation des progicielsde calcul de structures.

• R = 1 m

• de = 0.02 m

• di = 0.016 m

• A = 1.131E–04 m2

• Ix 4.637E–09 m4


Input File

ssll08.dat

Units

SI

Material Properties

• E = 2E11 Pa

• ν = 0.3



Finite Element Modeling• 10 linear beam (CBAR) elements

• 11 grid points

Boundary Conditions• Point A: Articulated Z

• Point B: Sets Y and Z displacement to 0

• Force = 100N in –Y direction


Solution TypeSOL 101 — Linear Statics

Results

Type Grid Point Point Bench Value NX Nastran

Rz (rad) 2 A –3.077E–2 –3.110E–2

Rz (rad) 1 B 3.077E–2 3.110E–2

Y (m) 7 C -1.921E–2 –1.934E–2

X (m) 1 B 5.391E-2 5.374E–2



References




14.5 Grid Point Load on an Articulated CONROD TrussThis test is a linear statics analysis of a plane truss with an articulated rod. It provides the inputdata and results for benchmark test SSLL11/89 from Guide de validation des progiciels decalcul de structures.


Input File

ssll11.dat

Units

SI

Material Properties

• E = 1.962E11 Pa




• 4 rod (CONROD) elements

• 4 grid points


Element Length (m) Area (m2)

AC 2.000E–4

CB 2.000E–4

CD 1.000E–4

BD 1.000E–4

Boundary Conditions

• Point A and B: Articulated

• Point D: Set Nodal force = 9.81 E3 N in –Y direction


Solution Type




Results


X (m) 18.00 C 0.2652E–3 0.2652E–3

Y (m) 18.00 C 0.08839E–3 0.08839E–3

X (m) 2.000 D 3.479E–3 3.479E–3

Y (m) 2.000 D –5.601E–3 –5.600E–3

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.(Paris, Afnor Technique, 1990..) Test No. SSLL11/89.



14.6 Articulated Plane TrussThis test is a linear statics analysis of a straight cantilever beam with plane bending andtension-compression. It provides the input data and results for benchmark test SSLL14/89 fromGuide de validation des progiciels de calcul de structures.

• I1 = 5E–04 m4

• I2 = 2.5E–04 m4


Input Files

• ssll14a.dat (4 elements)

• ssll14b.dat (10 elements)

Units

SI

Material Properties

• E = 2.1E11 Pa




Test 1


• 5 grid points

Test 2


• 11 grid points

The mesh for Test 2 is shown in the following figure:



Boundary Conditions

• Point A and B: Articulate

• Set forces and moments to the following numeric values:

– p = –3,000 N/m

– F1 = –20,000 N

– F2 = –10,000 N

– M = –100,000 Nm


Solution Type


Results

Type GridPoint

Point Bench Value TestNumber

NX Nastran

Vertical reaction(N)

1.000 A 3.150E4 1

2

3.150E4

3.320E4

Hortizontalreaction (N)

1.000 A 2.024E4 1

2

1.920E4

2.061E4

VerticalDisplacement (m)

8.000 C 0.03072 1

2

–0.02100

–0.03161

NX Nastran takes shear effect into account.



References




14.7 Beam on an Elastic FoundationThis test is a linear statics analysis (plane problem) of a straight beam with plane bending andan elastic support. It provides the input data and results for benchmark test SSLL16/89 fromGuide de validation des progiciels de calcul de structures.


Input Filessll16.dat

UnitsSI

Material Properties• E = 2.1E11 Pa

• K = 8.4E05 N/m2

• Each spring stiffness is set to: K * L/(number of spring elements).


• 49 spring (CBUSH) elements

• 51 grid points




Boundary Conditions

• Point A and B: Articulated

• Set forces and moments to the following numeric values:

–F = –10000 N

–p = –5000 N/m

–M = 15000 Nm.


Solution Type




Results

Type Point Bench Value NX Nastran

Rotation(rad) Rz A —0.003050 –0.003034

Vertical Reaction force (N) 1.167E4 1.158E4

Vertical Disp. (m) D –0.4233E–2 –0.4216E–2

M moment (Nm)* 3.384E4 3.369E4

*List beam forces on element 26, first end, z bending moment.

References



Chapter

15 Mechanical Structures —Linear Statics Analysis withShell Elements

15.1 Plane Shear and Bending Load on a PlateThis test is a linear statics analysis (plane problem) of a plate with plane bending. It providesthe input data and results for benchmark test SSLP01/89 from Guide de validation des progicielsde calcul de structures.

• Thickness = 1 mm


Input File

sslp01.dat

Units

SI

Material Properties

• E = 3E10 Pa

• ν = 0.25


Chapter 15 Mechanical Structures — Linear Statics Analysis with Shell Elements


• 100 linear quadrilateral thin shell (CQUAD4) elements

• 126 grid points


Boundary Conditions

• Clamped Plate

• Set a shear force with parabolic distribution on width and constant distribution on thickness.

• Resultant force: p = 40 N.


Solution Type



Mechanical Structures — Linear Statics Analysis with Shell Elements

Results

Type Grid point # Location Bench Value NX Nastran

Y (mm) Grid point 3 (L,y) 0.3413 0.3408

Displacement is shown in the following figure:

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990 Test No. SSLP01/89.



15.2 Infinite Plate with a Circular HoleThis test is a linear statics analysis (plane problem) of a plate with tension-compression and amembrane effect. It provides the input data and results for benchmark test SSLP02/89 fromGuide de validation des progiciels de calcul de structures.


Input File

sslp02.dat

Units

SI

Material Properties

• E = 3E10 Pa

• ν = 0.25





• 121 grid points

The plate is meshed using the biasing option.


Boundary Conditions

• u (0,y) = 0, Ry (y) = 0, Rz (y) = 0, (z = 0, all grid points)

• ν (x,0) = 0, Rx (x) = 0, Rz (x) = 0

• Tension force P = 2.5 N/mm**2 (in plane force of 2500 N/m)




Solution Type


Results

Type Point Bench Value NX Nastranσθθ (a, 0) 7.500E7 7.528E7σθθ (a, π/4) 2.500E7 2.511E7σθθ (a, π/2) –2.500E7 –2.452E7

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SSLP02/89.



15.3 Uniformly Distributed Load on a Circular PlateThis test is a linear statics analysis (three-dimensional problem) of a circular plate fixed at theedge with transverse bending and a uniform load. It provides the input data and results forbenchmark test SSLS03/89 from Guide de validation des progiciels de calcul de structures.


Input Files

• ssls03a.dat linear quadrilateral

• ssls03b.dat linear triangle

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3




Test 1


• 50 grid points

Test 2

• 53 linear triangular thin shell (CTRIA3) elements

• 38 grid points

Meshing is only done on 1/4 of the plate.

The meshes are shown in the following figure:



Boundary Conditions

• Clamp free edges.

• Uniform pressure p = –1000 Pa.

• Symmetric conditions are applied to the sides.


Solution Type


Results

Result Grid Point Point Bench Value Test Number NX Nastran

Z 1.000 Center O –0.006500 1 –0.006600

w (m) 1.000 2 –0.006500

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SSLS03/89.



15.4 Torque Loading on a Square TubeThis test is a linear statics analysis (three-dimensional problem) of a thin-walled tube loaded intorsion by pure shear at the free end. It provides the input data and results for benchmark testSSLS05/89 from Guide de validation des progiciels de calcul de structures.


Input File

ssls05.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• = 0.3




• 160 CQUAD4 elements

• 219 grid points


Boundary Conditions

• Plane X = 0

• Clamped beam

• Apply a torque equal to 10 Nm on the free end.


Solution Type




Results

Result Grid Point Bench Value NX Nastran

Disp. Y (m) 193.0 –6.170E–7 .6.170E–7

Disp. Rx (rad) 1.230E–5 1.230E–5

Stress XY Shear (Pa) –11.00E4 –11.00E4

Disp. Y (m) 208.0 –9.870E–7 –9.870E–7

Disp. Rx (rad) 1.970E–5 1.970E–5

Stress XY Shear (Pa) –11.00E4 –11.00E4

Results are post-processed using the Shell surface: Bottom option.

References




15.5 Cylindrical Shell with Internal PressureThis test is a linear statics analysis of a thin cylinder loaded by internal pressure. It provides theinput data and results for benchmark test SSLS06/89 from Guide de validation des progiciels decalcul de structures.


Input Files

• ssls06a.dat

• ssls06b.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3





Test 1


• 121 grid points

Test 2

• 400 liinear quadrilateral thin shell (CQUAD4) elements

• 441 grid points



Boundary Conditions

• Free conditions:

To set free boundary conditions, use symmetry about XZ, XY, and YZ planes.

• Internal pressure = 10000 Pa.


Solution Type


Results

Type Point Bench Value TestNumber

NX Nastran

σ11(Pa) All 0 1 1.720

2 4.960

σ22(Pa) 5.000E5 1 4.950E5

2 4.990E5

ΔR(m) 2.380E–6 1 2.370E–6

2 2.380E–6

ΔL(m) –1.430E–6 1 –1.420E–6

2 –1.430E–6

All results are averages.



Post Processing

• σ11 is the stress of z at grid point 11 (test 1) and grid point 21 (test 2)

• σ22 is the stress of x at grid point 111 (test 1) and grid point 421 (test 2)

• ΔR(m) is the displacement of x at grid point 121 (test 1) and grid point 441 (test 2)

• ΔL(m) is the displacement of z at grid point 121 (test 1) and grid point 441 (test 2)

References




15.6 Uniform Axial Load on a Thin Wall CylinderThis test is a linear static analysis of a thin cylinder loaded axially. It provides the input dataand results for benchmark test SSLS07/89 from Guide de validation des progiciels de calcul destructures.


Input Files

• ssls07a.dat – parabolic quadrilateral, thin shell

• ssls07b.dat – parabolic triangle, thin shell

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3




Test 1

• 200 parabolic quadrilateral thin shell (CQUAD8) elements

• 661 grid points

Test 2

• 400 parabolic triangular thin shell (CTRIA6) elements




Boundary Conditions

• Axial displacement = 0 in. X = 0 section

• Uniform axial load q = 10000 N/m


Solution Type

SOL 101 — Linear Sstatics

Results

Type Point Bench Value Test Number NX Nastran

σ11(Pa) Any 5.000E5 1 5.000E5

2 5.790E5

σ22(Pa) Any 0 1 0

2 3.080E4

ΔL(m) Any 9.520E–6 1 9.520E–6

2 9.560E–6

ΔR (m) Any –7.140E–7 1 –7.140E-7

2 –7.330E–7

All results are averages.



Post Processing

• σ11 is the stress of z at grid point 641 in coordinate system 2.

• σ22 is the stress of y at grid point 641 in coordinate system 2.

• ΔR is the displacement of x at grid point 641 in coordinate system 2.

• ΔL is the displacement of z at grid point 641 in coordinate system 2.

References




15.7 Hydrostatic Pressure on a Thin Wall CylinderThis test is a linear statics analysis of a thin cylinder loaded by hydrostatic pressure. It providesthe input data and results for benchmark test SSLS08/89 from Guide de validation des progicielsde calcul de structures.


Input File

ssls08.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3





• 661 grid points




Boundary Conditions

• Restrain the grid points on side A (from grid point 21 to grid point 661) in the X translationand the Y and Z rotations.

• Restrain the grid points on side B (from grid point 1 to grid point 641) in the Y translationand X and Z rotations.

• Internal pressure p = p0 * Z/L with p0 = 20000 Pa.


Solution Type


Results


σ11(Pa) 321.0 Any 0 8.800E3

L/2σ22 (Pa) 321.0 x = L/2 5.000E5 4.970E5

ΔR (m) 321.0 x = L/2 2.380E–6 2.380E–6

ΔL (m) 1.000 x = L –2.860E–6 2.860E–6

(rad) 321.0 1.190E–6 1.190E–6

represents the rotation of a generator.



Post Processing

• σ11is the stress of z at grid point 321 in coordinate system 2

• σ22 is the stress of y at grid point 321 in coordinate system 2

• ΔR is the displacement of x at grid point 321 in coordinate system 2

• ΔL is the displacement of z at grid point 1 in coordinate system 2

• is the rotation of y at grid point 321 in coordinate system 2

References




15.8 Gravity Loading on a Thin Wall CylinderThis test is a linear statics analysis of a thin cylinder loaded by its own weight. It provides theinput data and results for benchmark test SSLS09/89 from Guide de validation des progiciels decalcul de structures.


Input Files

• ssls09a.dat linear quadrilateral, thin shell

• ssls09b.dat axisymmetric

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3

• γ = 7.85 x 104 N/m3

• Mass = 8002 kg/m3




Test 1


• 84 grid points

Test 2

• 20 linear axisymmetric (CCONEAX) elements

• 21 grid points




Boundary Conditions

• Axial displacement = 0 in. Z = 0 section.

• Gravity loading; gravity acts in the Z direction.


Solution Type


Results

Type Gridpoint

Point Bench Value TestNumber

NX Nastran

σ22(Pa) 2.000 Any 0 1 –34.66

2 0

σ11 (Pa) 2.000 x = 0 3.140E5 1 3.020E5

2 3.060E5

Δz (m) 1.000 x = L 2.990E–6 1 2.990E–6

2 2.990E–6

ΔR (m) 2.000 x = 0 –4.490E–7 1 –4.390E–76

2 –4.480E–7

(rad) 10.00 x – L 1.120E–7 1 –1.120E–7

2 –1.120E–7



Post Processing

Test 1

• σ11 is the stress of z at grid point 2 in coordinate system 2

• σ22is the stress of x at grid point 2 in coordinate system 2

• Δz is the displacement of z at grid point 1 in coordinate system 2

• ΔR is the displacement of x at grid point 2 in coordinate system 2

• is the rotation of y at grid point 10 in coordinate system 2

References




15.9 Pinched Cylindrical ShellThis test is a linear statics analysis of a cylindrical shell with grid point forces, F, pinching asshown. It provides the input data and results for benchmark test SSLS20/89 from Guide devalidation des progiciels de calcul de structures.


Input Files

• ssls20a.dat linear triangle thin shells

• ssls20b.dat linear quadrilateral thin shells

Units

SI

Material Properties

• E = 10.5 x 106 Pa

• ν = 0.3125




Test 1


• 173 grid points

Test 2


• 165 grid points




Boundary Conditions

• Free conditions

To set free boundary conditions, use symmetry about XY, XZ, and YZ planes.

• Grid point forces Fy = –25 N at point D


Solution Type


Results

Type Point Bench Value Test Number NX Nastran

Disp. Y (Grid point 3) ν(m) D –113.9E-3 1 –114.4E–3

Disp. Y (Grid point 3) 2 –113.3E–3

Post Processing

• ν(m) is the displacement of y at grid point 3 (quadrilateral).

References




15.10 Spherical Shell with a HoleThis test is a linear statics analysis of a spherical shell with a hole with grid point forces. Itprovides the input data and results for benchmark test SSLS21/89 from Guide de validation desprogiciels de calcul de structures.


Input Files

• ssls21a.dat – linear quadrilateral thing shells

• ssls21b.dat – linear triangle thin shells

• ssls21c – parabolic quadrilateral thin shells

Units

SI

Material Properties

• E = 6.285 x 107 Pa

• ν = 0.3




Test 1


• 121 grid points

Test 2


• 121 grid points

Test 3


• 441 grid points




Boundary Conditions

• Free conditions

To set free boundary conditions, use symmetry about XY and YZ planes.

• Grid point forces F = 2 Newtons

Due to the symmetric boundary conditions, only half of the load is applied.


Solution Type


Results

Result Point BenchValue

TestNumber

NX Nastran

u (m) grid point 111 A(R,0,0) 9.400E–2 1 102.0E–3

Grid point 111 2 102.1E–3

Grid point 421 3 100.9E–3

References




15.11 Bending Load on a Cylindrical ShellThis test is a linear statics analysis of a cylindrical shell with bending and membrane effect. Itprovides the input data and results for benchmark test SSLS23/89 from Guide de validation desprogiciels de calcul de structures.


Input Files

• ssls23a.dat (Test 1, linear)

• ssls23b.dat (Test 2, parabolic)

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3




Test 1


• 78 grid points

Test 2


• 215 grid points


Boundary Conditions

• AB side: Clamped in local system coordinates.

• AD and BC sides: Restrain Z translation, θx and θy.

• DC side: Set bending moment CZ to 1000 Nm/m. Set in plane force to 0.6E6 N/m.

• ABCD surface: Set internal pressure to 0.6E06 N/m**2.

• AD and DC sides are restrained in the global coordinate system.


Solution Type




Results

Use coordinate system 3 (the cylindrical coordinate system) to display the results.

Results are post-processed using the Shell surface middle option.

Result Point Bench Value Test Number NX Nastran

Grid point 35 E 60.00 MPa 1 60.70

Grid point 93 2 59.60

References




15.12 Uniformly Distributed Load on a Simply-SupportedRectangular PlateThis test is a linear statics analysis of a plate with pressure loading and simple supports. Itprovides the input data and results for benchmark test SSLS24/89 from Guide de validation desprogiciels de calcul de structures.


Input Files

• ssls24a.dat (Test 1, coarse mesh)

• ssls24b.dat (Test 2, fine mesh)

• ssls24c.dat (Test 3, very fine mesh)

Units

SI

Material Properties

• E = 1.0 x 107 Pa

• ν = 0.3




Test 1 — a/b = 1


• 121 grid points

Test 2 — a/b = 2


• 231 grid points

Test 3 — a/b = 5


• 561 grid points




Boundary Conditions

Restraints

• All edges: w = 0

• One corner fixed

Loads

• Set pressure = 1 N/m**2 in the –Z direction


Solution Type




Results

Result a/b Parameters BenchValue

TestNumber

NX Nastran

61z direction 1.000 1.000α 0.004440 1 0.004500

116z direction 2.000 2.000α 0.01110 2 0.01110

281z direction 5.000 5.000α 0.01417 3 0.01406

61x component topsurface

1.000 1.000β 2874. 1 2867.

116x componenttop surface

2.000 2.000β 6102. 2 6034.

281x componenttop surface

5.000 5.000β 7476. 3 7331.

Where:

q = distributed load

b = dimension

t = thickness

E = elastic modules

β values of reference from the Guide de Validation are incorrect. The correct values are extractedfrom Formulas for Stress and Strain (Roark/Young).

Note that the shell top surface corresponds to the side of the plate with negative globalZ coordinates.

References




15.13 Uniformly Distributed Load on a Simply-Supported RhomboidPlateThis test is a linear statics analysis (three-dimensional problem) of a plate with pressure andtransverse bending. It provides the input data and results for benchmark test SSLS25/89 fromGuide de validation des progiciels de calcul de structures.


• b = 1.0 m

• a = 2.0 m


Input Files

• ssls25a.dat (Test 1)

• ssls25b.dat (Test 2)

Units

SI

Material Properties

• E = 36.0 x 106 Pa

• ν = 0.3




• a/b = 2

• Linear quadratic thin shell (CQUAD4) elements

Test 1

• θ = 30°

Test 2

• θ = 45°


Boundary Conditions

• All edges: w = 0, one corner fixed

• Pressure = 1 N/m2 in the –Z direction


Solution Type




Results

Parameter Test Case Bench Value NX Nastranα Test 1

ssls25a

z displacement

–3.277E–3 m

116z displacement

–2.963E–3 mβ Y stress

–5.700E3 N/m2

116y stress

–5.831E3 N/m2α Test 2

ssls25b

z displacement

–3.000E-3 m

116z displacement

–2.720E–3 mβ Y stress

–5.390E3 N/m2

116Y stress

–5.441E3 N/m2

Where:

q = distributed load

b = dimension

t = thickness

E = elastic modulus

Values of reference from the Guide de validation are incorrect. The correct values are extractedfrom Formulas for Stress and Strain (Roark/Young), table 26, case number 14a.

References




15.14 Shear Loading on a PlateThis test is a linear statics analysis of a thin plate with torque and shear loading. It provides theinput data and results for benchmark test SSLS27/89 from Guide de validation des progiciels decalcul de structures.


Input Files

• ssls27a.dat (Test 1, Mindlin)

• ssls27b.dat (Test 2, Kirchoff)

• ssls27c.dat (Test 3, Mindlin)

Units

SI

Material Properties

• E = 1.0 x 107 Pa

• ν = 0.25




Test 1 — Mindlin


• 14 grid points

Test 2 — Kirchhoff


• 14 grid points

Test 3 — Mindlin


• 75 grid points


All tests are executed with mapped meshing

Boundary Conditions

• Clamp AD side

• Point B: grid point force Fz = –1N

• Point C: grid point force –Fz = 1N


Solution Type




Results at Location C

Displacement atGrid point

Bench Value Test Number NX Nastran

14.00 3.537E–2 1 3.585E–2

14.00 3.537E–2 2 3.573E–2

75.00 3.537E–2 3 3.603E–2

References



Chapter

16 Mechanical Structures —Linear Statics Analysis withSolid Elements

16.1 Solid Cylinder in Pure TensionThis test is a linear statics analysis of a solid cylinder with tension-compression. It provides theinput data and results for benchmark test SSLV01/89 from Guide de validation des progiciels decalcul de structures.


Chapter 16 Mechanical Structures — Linear Statics Analysis with Solid Elements


Input Files

• sslv01a.dat (Test 1)

• sslv01b.dat (Test 2)

• sslv01c.dat (Test 3)

• sslv01d.dat (Test 4)

• sslv01e.dat (Test 5)

Units

SI

Material Properties

• E = 2.0 x 1011 Pa

• ν = 0.30


Mechanical Structures — Linear Statics Analysis with Solid Elements

Finite Element ModelingTest 1

• 155 parabolic tetrahedron (CTETRA) elements

• 342 grid points

Test 2

• 144 linear brick (CHEXA) elements & 48 linear solid wedge (CPENTA) elements

• 307 grid points

Test 3

• 48 linear quadrilateral axisymmetric solid (CQUADX) elements

• 65 grid points

Test 4

• 96 linear triangular axisymmetric solid (CTRIA6) elements

• 65 grid points

Test 5 — Mapped meshing

• 18 parabolic quadrilateral axisymmetric solid (CQUADX) elements

• 95 grid points




Boundary Conditions

• Uniaxial deformation of the cylinder section

• Set uniformly distributed force –F/A on the free end in the Z direction

• F/A = 100 MPa


Solution Type


Results

linear statics

Point GridPoint

Displacement Bench Value TestNumber

NX Nastrans

A & C 6.000 u (m) 1.500E–3 1 1.500E–3

A & C 279.0 2 1.500E–3

A & C 1.000 3 1.500E–3

A & C 4.000 4 1.500E–3

A & C 1.000 5 1.600E–3

B 4.000 u (m) 1.500E–3 1 1.500E–3



Point GridPoint

Displacement Bench Value TestNumber

NX Nastrans

B 307.0 2 1.500E–3

B 53.00 3 1.450E–3

B 3.000 4 1.500E–3

B 39.00 5 1.600E–3

D 37.00 u (m) 1.000E-3 1 1.000E–3

D 189.0 2 1.000E–3

D 5.000 3 1.000E–3

D 25.00 4 1.000E–3

D 7.000 5 1.000E–3

E 41.00 u (m) 0.5000E-3 1 0.5000E–3

E 99.00 2 0.5000E–3

E 9.000 3 0.5000E–3

E 29.00 4 0.5000E–3

E 13.00 5 0.5000E–3

A & C 6.000 w (m) –0.1500E–3 1 0.1500E–3

A & C 279.0 2 0.1500E–3

A & C 1.000 3 0.1500E–3

A & C 4.000 4 0.1500E–3

A & C 1.000 5 0.1500E–3

D 37.00 w (m) –0.1500E-3 1 0.1500E–3

D 189.0 2 0.1500E–3

D 5.000 3 0.1500E–3

D 25.00 4 0.1500E–3

D 7.000 5 0.1500E–3

E 41.00 w (m) –0.1500E–3 1 0.1500E–3

E 99.00 2 0.1500E–3

E 9.000 3 0.1500E–3

E 29.00 4 0.1500E–3

E 13.00 5 0.1500E–3

Post Processing

To view the results for Test 1 and Test 2, use coordinate system 2 (cylindrical). All results areaveraged.



References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SSLV01/89.



16.2 Internal Pressure on a Thick-Walled Spherical ContainerThis test is a linear statics analysis of a thick sphere with internal pressure. It provides theinput data and results for benchmark test SSLV03/89 from Guide de validation des progiciels decalcul de structures.


Input Files





Units

SI

Material Properties

• E = 2 x 105 Pa

• ν = 0.3




Test 1

• 1600 linear brick (CHEXA) elements & linear solid wedge (CPENTA) elements

• 1898 grid points

Test 2

• 200 parabolic brick (CHEXA) elements & 50 solid wedge (CPENTA) elements


Test 3


• 451 grid points

Test 4

• 400 parabolic quadrilateral axisymmetric solid (CQUADX) elements


The meshes from these tests are shown in the following figure:



Boundary Conditions

• The equivalent of the center of the sphere being fixed is modeled via symmetric boundaryconditions.

• Uniform radial pressure = 100 MPa.


Solution Type




Results

Point GridPoint

DisplacementStress

Bench Value TestNumber

NX Nastran

r=1 m 1.000 σrr (MPa) –100.0 1 –90.15

1.000 2 –97.29

451.0 3 –97.07

451.0 4 –99.27

1.000 σθ (MPa) 71.43 1 72.09

1.000 2 77.23

451.0 3 68.62

451.0 4 71.52

1.000 u (m) 0.4000E–3 1 0.4000E–3

1.000 2 0.4000E–3

451.0 3 0.4000E–3

451.0 4 0.4000E-3

r=2 m 1826. σrr (MPa) 0 1 –0.02800

2221. 2 0.2241

411.0 3 –0.2860

411.0 4 –0.04100

1826. σθ (MPa) 21.43 1 21.18

2221. 2 21.18

411.0 3 22.18

411.0 4 21.38

1826 u (m) 1.500E–4 1 1.500E–4

2221. 2 1.500E–4

411.0 3 1.500E–4

411.0 4 1.500E–4

All results are averaged. Use the spherical coordinate system for the stress results.

References




16.3 Internal Pressure on a Thick-Walled Infinite CylinderThis test is a linear statics analysis of a thick cylinder with internal pressure. It provides theinput data and results for benchmark test SSLV04/89 from Guide de validation des progiciels decalcul de structures.


Input Files





Units

SI

Material Properties

• E = 2 x 105 mPa

• ν = 0.3


Test 1

• 400 linear brick (CHEXA) elements

• 902 grid points

Test 2

• 240 parabolic brick (CHEXA) elements




Test 3


• 656 grid points

Test 4

• 600 parabolic quadrilateral axisymmetric solide (CQUADX) elements





Boundary Conditions

• Unlimited cylinder

• Internal pressure p = 60 MPa


Solution Type


Results

All results are averaged.

Test Case Grid Point Displacement /Stress

Bench Value NX Nastran

sslv04a 411.0 σr –60.00 (MPa) –57.00

sslv04b 977.0 σr –60.00

sslv04c 616.0 σr ( –59.00



Test Case Grid Point Displacement /Stress


sslv04d 1831. –127.0

sslv04a 411.0 σθ 100.0 (MPa) 99.70

sslv04b 977.0 102.0

sslv04c 616.0 99.40

sslv04d 1831. 145.0

sslv04a 411.0 τmax 80.00 (MPa) 79.34

sslv04b 977.0 81.00

sslv04c 616.0 79.40

sslv04d 1831. 79.98

sslv04a 411.0 ur 59.00E–6 (m) 59.00E–6

sslv04b 977.0 59.00E–6

sslv04c 616.0 59.00E–6

sslv04d 1831. 59.00E–6

sslv04a 451.0 σr 0 (MPa) –0.006500

sslv04b –0.04480

sslv04c –0.1684

sslv04d .003417

sslv04a σθ 40.00 (MPa) 39.66

sslv04b 40.39

sslv04c 40.09

sslv04d 40.19

sslv04a τmax 20.00 (MPa) 20.08

sslv04b 20.17

sslv04c 20.07

sslv04d 19.99

sslv04a ur 40.00E–6 (m) 40.00E–6

sslv04b 40.00E–6

sslv04c 40.00E–6

sslv04d 40.20E–6

References




16.4 Prismatic Rod in Pure BendingThis test is a linear statics analysis of a solid rod with bending. It provides the input data andresults for benchmark test SSLV08/89 from Guide de validation des progiciels de calcul destructures.


Input Files





Units

SI

Material Properties

• E = 2 x 105 MPa

• ν = 0.3




Test 1

• 198 linear solid pyramid (CTETRA) elements

• 76 grid points

Test 2

• 198 parabolic solid pyramid (CTETRA) elements

• 409 grid points

Test 3

• 48 linear parabolic brick (CHEXA) elements

• 117 grid points

Test 4 — Mapped meshing

• 48 linear parabolic brick (CHEXA) elements

• 381 grid points




Boundary Conditions

• Clamp Point B.

• Other points of B section: Set Z-displacement to 0. NOTE: In these tests some grid points ofsection B are also restrained in the x direction about the x-axis at the free end of the rod.

• Set moment Mx equal to (4/3)E+7 N.m


Solution Type




Results

Test # Point GridPoint

DisplacementStress


1 F or G 5.000 σzz –10.00E6 (Pa) –4.268E6

2 5.000 –10.03E6

3 75.00 –10.00E6

4 245.0 –9.995E6

1 A 26.00 uA 4.000E–4 (m) 2.964E–4

2 90.00 4.000E–4

3 77.00 4.000E–4

4 251.0 4.000E–4

1 H 19.00 wB 2.000E–4 (m) 2.000E–4

2 40.00 2.000E–4

3 76.00 2.000E–4

4 249.0 2.000E–4

1 F or G 5.000 vF = -vG 0.1500E-4 (m) 0.07450E–4

2 5.000 0.1508E–4

3 75.00 0.1500E–4

4 245.0 0.1503E–4

1 D or E 8.000 vD = -vE -0.1500E-4 (m) –6.262E–4

2 8.000 –0.1505E–4

3 73.00 –0.1500E–4

4 241.0 –0.1503E–4

References




16.5 Thick Plate Clamped at EdgesThis test is a linear statics analysis of a thick plate with pressure and transverse bending. Itprovides the input data and results for benchmark test SSLV09/89 from Guide de validation desprogiciels de calcul de structures.


Input Files



Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3




Test 1

• 25 parabolic linear brick (CHEXA) elements

• 228 grid points

• λ =10, 20, 50, 75, 100

Test 2


• 36 grid points

• λ =10, 20, 50, 75, 100

Test 2 is done using CQUAD4 elements with the thicknesses specified in the physical propertytable.




Boundary Conditions

• AB and AD sides: clamped

• BC and DC sides: symmetry

• Load case 1:

Pressure p = 1E06 Pascals in –Z direction

• Load case 2: Point C

Grid Point force F = 1E06 N in –Z direction


Solution Type

SOL 101 Linear statics



Results

Test Case 1 (z displacement at location C)

PartName

Load Case Grid Point Analytical Reference FEM NX Nastran

10 Pressure 242.0 –0.6552E-4 –0.7620E–4 –0.7379E–4

Force 242.0 –0.2915E-3 –0.4300E–3 –0.3684E–3

20 Pressure 242.0 –0.5242E-3 –0.5383E–3 –0.5266E–3

Force 242.0 –0.2332E–2 –0.2535E–2 –0.2456E–2

50 Pressure 242.0 –0.8190E–2 –0.8029E–2 –0.7935E–2

Force 242.0 –0.3643E–1 –0.3574E–1 –0.3602E–1

75 Pressure 242.0 –0.2764E–1 –0.2690E–1 –0.2666E-1

Force 242.0 –0.1230 –0.1184 –0.1206

100 Pressure 242.0 –0.6552E–1 –0.6339E–1 –0.6305E–1

Force 242.0 –0.2915 –0.2779 –0.2849

Test Case 2 (z displacement at location C)

PartName

Load Case Grid Point Analytical Reference FEM NX Nastran

10 Pressure 1.000 –0.6552E–4 –0.7866E–4 –0.8131E–4

Force 1.000 –0.2915E–3 –0.4109E–3 –0.4050E–3

20 Pressure 36.00 –0.5242E–3 –0.5557E–3 –0.5775E–3

Force 36.00 –0.2332E–2 –0.2595E–2 –0.2668E–2

50 Pressure 36.00 –0.8190E–2 –0.8348E–2 –0.8669E–2

Force 36.00 –0.3643E–1 –0.3745E–1 –0.3878E–1

75 Pressure 36.00 –0.2764E–1 –0.2805E–1 –0.2906E–1

Force 36.00 –0.1230 –0.1253 –0.1292

100 Pressure 36.00 –0.6552E–1 –0.6639E–1 –0.6864E–1

Force 36.00 –0.2915 -0.2958 –0.3042

References



Chapter

17 Mechanical Structures —Normal Mode Dynamics Analysis

17.1 Lumped Mass-Spring SystemThis test is a normal mode dynamics analysis of an elastic link with lumped mass. It provides theinput data and results for benchmark test SDLD02/89 from Guide de validation des progiciels decalcul de structures.


Input File

sdld02.dat

Units

SI

Material Properties

Spring constant


Chapter 17 Mechanical Structures — Normal Mode Dynamics Analysis


• 8 lumped mass (CONM2) elements

• 9 spring (CBUSH) elements

• 8 grid points


Boundary Conditions

• Clamp points A and B

• Other points:

ν = 0 ; θ = 0


Solution Type

SOL 103 — Normal Modes


Mechanical Structures — Normal Mode Dynamics Analysis

Results

Frequency Results (Hz)

Normal Mode Bench Value NX Nastran

1 5.527 5.527

2 10.89 10.89

3 15.92 15.92

4 20.46 20.46

5 24.38 24.38

6 27.57 27.57

7 29.91 29.91

8 31.35 31.35

Mode Shapes Results

The mode shapes results are exact. The multiplication coefficient is 3.162.

Normal Mode Point Bench Value NX Nastran

1 P1 0.1612 0.05100

P2 0.3030 0.09580

P3 0.4082 0.1291

P4 0.4642 0.1468

P5 0.4642 0.1468

P6 0.4082 0.1291

P7 0.3030 0.09580

P8 0.1612 0.05100

8 P1 0.1612 0.05100

P2 –0.3030 –0.09580

P3 0.4082 0.1291

P4 –0.4642 –0.1468

P5 0.4642 0.1468

P6 –0.4082 –0.1291

P7 0.3030 0.09580

P8 –0.1612 –0.05100

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SDLD02/89, p. 178.



17.2 Short Beam on Simple SupportsThis test is a modal analysis of a straight short beam with simple supports both inline and offset.It provides the input data and results for benchmark test SDLL01/89 from Guide de validationdes progiciels de calcul de structures.


Input Files

• sdll01a.dat (Test 1)

• sdll01b.dat (Test 2)

Units

SI

Material Properties

• E = 2 x 101111 Pa

• ν = 0.3

• ρ = 7800 kg/m3





• 11 grid points




Boundary Conditions

Problem 1

• Point A (grid point 1): Constrain in all directions, except the Z rotation.

• Point B (grid point 2): Constrain in the Y and Z translations and X and Y rotations.

• All other grid points (3-11): Constrain in the Z translation and X and Y rotations.

• No load case.

Problem 2

• Point C (grid point 1): Constrain in all directions, except the Z rotation.

• Point D (grid point 2): Constrain in the Y and Z translations and X and Y rotations.

• All other grid points (3-11): Constrain in the Z translation and X and Y rotations.

• No load case.


Solution Type




Results

Problem 1: Frequency results (Hz)


Bending 1 431.6 437.2

Tension 1 1266. 1265.

Bending 2 1498. 1539.

Bending 3 2871. 2925.

Tension 2 3798. 3763.

Bending 4 4378. 4328.


Mode Number Bench Value NX Nastran

1 392.8 398.5

2 902.2 927.3

3 1592. 1666.

4 2629. 2815.

5 3126. 3266.

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SDLL01/89.



17.3 Axial Loading on a RodThis test is a modal analysis of a simply-supported beam with stress stiffening. It provides theinput data and results for benchmark test SDLL05/89 from Guide de validation des progiciels decalcul de structures.


Input Files

• sdll05a.dat

• sdll05b.dat

Units

SI

Material Properties

• E = 2 x 1011 Pa

• ρ = 7800 kg/m3



• 11 grid points




Boundary Conditions

• Points A: u = v = 0

• Points B: v = 0

• Load case 2 (grid point 2): Fx = 1E05N in –X direction

• Stress stiffening on


Solution Type


Results

Frequency results (Hz)

Load Case Normal Mode Bench Value NX Nastran

1 Bending 1 28.70 28.68

Bending 2 114.8 114.4

2 Bending 1 22.43 22.40

Bending 2 109.1 108.7

References




17.4 Cantilever Beam with a Variable Rectangular SectionThis test is a modal analysis of a straight cantilever beam with a variable section. It provides theinput data and results for benchmark test SDLL09/89 from Guide de validation des progiciels decalcul de structures.


Input Files

• sdll09a.dat

• sdll09b.dat

Units

SI

Material Properties

• E = 2 x 1011 Pa

• ρ = 7800 kg/m3




• 10 tapered beam (CBEAM) elements

• 11 grid points


Boundary Conditions

• Clamp point A

• No load case


Solution Type




Results


β Normal Mode Bench Value NX Nastran

4 1 54.18 54.24

2 171.9 172.4

3 384.4 384.9

4 697.2 695.4

5 1112. 1104.

5 1 56.55 56.59

2 175.8 176.3

3 389.0 389.5

4 702.4 700.6

5 1118. 1109.

References




17.5 Thin Circular RingThis test is a modal analysis of a thin curved beam. It provides the input data and results forbenchmark test SDLL11/89 from Guide de validation des progiciels de calcul de structures.


Input File

sdll11.dat

Units

SI

Material Properties

• E = 7.2 x 1010 Pa

• ν = 0.3

• ρ = 2700 kg/m3





• 36 grid points


Boundary Conditions

• Free conditions

• Create one constraint set (kinematic DOF) to fully constrain the three grid points shownbelow (grid points 7, 21, 30).

• No load case




Solution Type


Results


Normal Mode Bench Value NX Nastran ADS #

Plane mode 1,2,3 0 0 1.000, 2.000, 3.000

Plane mode 4,5 318.4 319.0 7.000, 8.000

Plane mode 6,7 900.5 900.9 11.00, 12.00

Plane mode 8,9 1727. 1724. 15.00, 16.00

Plane mode 10,11 2792. 2781. 17.00, 18.00

Transverse Mode1,2,3

0 0 4.000, 5.000, 6.000

Transverse Mode 4,5 511.0 511.0 9.000, 10.00

Transverse Mode 6,7 1590. 1585. 13.00, 14.00

Transverse Mode 8,9 3184. 3159. 19.00, 20.00

References




17.6 Thin Circular Ring Clamped at Two PointsThis test is a modal analysis of a thin curved beam. It provides the input data and results forbenchmark test SDLL12/89 from Guide de validation des progiciels de calcul de structures.


Input File

sdll12.dat

Units

SI

Material Properties

• E = 7.2 x 1010 Pa

• ν = 0.3

• ρ = 2700 kg/m3





• 29 grid points




Boundary Conditions

• Points A and B: Clamped in local coordinate system

• No load case


Solution Type


Results



1 235.3 235.9

2 575.3 575.2

3 1106. 1103.

4 1406. 1399.

5 1751. 1743.

6 2557. 2536.

7 2802. 2723.

References




17.7 Vibration Modes of a Thin Pipe ElbowThis test is a modal analysis of a straight cantilever beam, and a thin curved beam. It providesthe input data and results for benchmark test SDLL14/89 from Guide de validation des progicielsde calcul de structures.


Input Files

• sdll14a.dat (test 1, L=0)

• sdll14b.dat (test 2, L=0.6)

• sdll14c.dat (test 3, L=2.0)

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3

• ρ = 7800 kg/m3




• L = 0 or L = 0.6:

18 linear beam (CBAR) elements

19 grid points

• L = 2:

28 linear beam (CBAR) elements

29 grid points

Two of the meshes are shown in the following figure:



Boundary Conditions

• Clamp points C and D

• Point A: v = 0; w = 0

• Point B: u = 0; w = 0


Solution Type




Results


L Normal Mode Bench Value NX Nastran ADS#0 Transverse 1 44.23 44.07 1.000

Plane 1 119.0 119.2 2.000

Transverse 2 125.0 125.4 3.000

Plane 2 227.0 225.0 4.000

0.6000 Transverse 1 33.40 33.15 1.000

Plane 1 94.00 94.42 2.000

Transverse 2 100.0 98.50 3.000

Plane 2 180.0 183.7 4.000

2.000 Transverse 1 17.90 17.65 1.000

Plane 1 24.80 24.40 3.000

Transverse 2 25.30 24.94 2.000

Plane 2 27.00 26.67 4.000

References




17.8 Cantilever Beam with Eccentric Lumped MassThis test is a modal analysis of a straight cantilever beam and a lumped mass. It provides theinput data and results for benchmark test SDLL15/89 from Guide de validation des progiciels decalcul de structures.


Input Files

• sdll15a.dat

• sdll15b.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ρ = 7800 kg/m3




Test 1:

• 10 linear beam (CBEAM) elements

• 1 rigid (RBAR) element from point B to point C

• 1 lumped mass (CONM2) element at point C

• 11 grid points

Test 2:


• 1 rigid (RBAR) element from point B to point C

• 1 lumped mass (CONM2) element at point C

• 11 grid points

The mesh both tests is is shown in the following figure:

Boundary Conditions

• Clamp point A


Solution Type

SOL 103 normal mode dynamics — SVI



Results


yc Normal Mode Bench Value NX Nastran

0 Transverse 1,2 1.650 1.650

Transverse 3,4 16.07 15.88

Transverse 5,6 50.02 48.64

Tension 1 76.47 76.42

Torsion 1 80.47 80.68

Transverse 7,8 103.2 97.89

1 1 1.636 1.633

2 1.642 1.638

3 13.46 13.36

4 13.59 13.59

5 28.90 29.20

6 31.96 31.57

7 61.61 59.85

8 63.93 61.72

Mode shapes results

yc NormalMode

ModalDisplacement


1 1 wc/wb 1.030 1.030

2 uc/vb 0.1480 –0.1480

3 uc/vb 2.882 –2.904

4 wc/wb –0.9220 –0.9800

• wc = z displacement at point C

• wb = z displacement at point B

• uc = x displacement at point C

• vb = y displacement at point B

References




17.9 Thin Square Plate (Clamped or Free)This test is a normal mode dynamics analysis (three-dimensional problem) of a thin plate. Itprovides the input data and results for benchmark test SDLS01/89 from Guide de validation desprogiciels de calcul de structures.


Input Files

• sdls01a.dat

• sdls01b.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3

• ρ = 7800 kg/m3





• 121 grid points


Boundary Conditions

• Problem 1: AB side clamped

• Problem 2: Free plate; 1 kinematic DOF set (grid points 1, 11, 111)


Solution Type




Results



1 8.727 8.638

2 21.30 20.89

3 53.55 52.42

4 68.30 65.77

5 77.74 75.14

6 136.0 127.8



7 33.71 32.91

8 49.46 47.42

9 61.05 59.19

10,11 87.52 83.08

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SDLS01/89..



17.10 Simply-Supported Rectangular PlateThis test is a normal mode dynamics analysis (three-dimensional problem) of a thin plate. Itprovides the input data and results for benchmark test SDLS03/89 from Guide de validation desprogiciels de calcul de structures.


Input Files

sdls03.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3

• ρ = 7800 kg/m3





• 176 grid points


Boundary Conditions

• Z-displacement = 0 on all sides of the plate

• One DOF set

• No load case


Solution Type




Results



1 35.63 35.27

2 68.51 67.29

3 109.6 108.5

4 123.3 120.8

5 142.5 138.2

6 197.3 188.2

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SDLS03/89.



17.11 Thin Ring Plate Clamped on a HubThis test is a normal mode dynamics analysis (three-dimensional problem) of a thin plate. Itprovides the input data and results for benchmark test SDLS04/89 from Guide de validation desprogiciels de calcul de structures.

• Re = 0.1 m

• Ri = 0.2 m

• Thickness = .001 m


Input Files

sdls04.dat

Units

SI

Material Properties

• E = 2 x 1011 Pa

• ν = 0.3

• ρ = 7800 kg/m3





• 440 grid points


Boundary Conditions

• If r = Ri: Clamp in local coordinate system.

• No load case.




Solution Type


Results



1 79.26 79.22

2,3 81.09 80.72

4,5 89.63 88.83

6,7 112.8 111.3

8,9 Not available 152.7





18 518.9 510.9

19,20 528.6 519.7

21,22 559.1 546.2

23 609.7 590.3

References




17.12 Vane of a Compressor - Clamped-free Thin ShellThis test is a normal mode dynamics analysis (three-dimensional problem) of a cylindrical thinshell. It provides the input data and results for benchmark test SDLS05/89 from Guide devalidation des progiciels de calcul de structures.

• α = 0.5 rad

• AD = L = 0.3048m

• r = 2L = 0.6096m

• thickness = 3.048 x 10–3 m


Input Files

• sdls05a.dat (coarse mesh)

• sdls05b.dat (fine mesh)

Units

SI

Material Properties

• E = 2.0685 x 1011 Pa

• ν = 0.3

• ρ = 7857.2 kg/m3



Finite Element Modeling — Coarse Mesh


• 121 grid points

The coarse mesh is shown in the following figure:

Finite Element Modeling — Fine Mesh


• 256 grid points

The fine mesh is shown in the following figure:



Boundary Conditions

• AD side: Clamped in local coordinate system.


Solution Type


Results


Normal Mode Bench Value NX Nastran coarsemesh

NX Nastran fine mesh

1 85.60 84.60 85.30

2 134.5 137.1 137.8

3 259.0 240.7 243.9

4 351.0 333.3 338.1

5 395.0 370.0 378.3

6 531.0 503.7 515.4

References




17.13 Bending of a Symmetric TrussThis test is a normal mode dynamics analysis (plane problem) of a straight cantilever beamstructure. It provides the input data and results for benchmark test SDLX01/89 from Guide devalidation des progiciels de calcul de structures.

• h = 0.0048 m

• b = 0.029 m

• A = 1.392 x 10–4 m2

• Iz = 2.673 x 10–10 m4


Input File

sdlx01.dat

Units

SI



Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3

• ρ = 7800 kg/m3



• 24 grid points


Boundary Conditions





Solution Type


Results



1 8.800 8.769

2 29.40 29.34

3 43.80 43.82

4 56.30 56.25

5 96.20 95.43

6 102.6 102.5

7 147.1 146.2

8 174.8 173.1

9 178.8 177.4

10 206.0 202.9

11 266.4 262.4

12 320.0 309.7

13 335.0 321.9

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SDLX01/89.



17.14 Hovgaard’s Problem — Pipes with Flexible ElbowsThis test is a normal mode dynamics analysis (three-dimensional problem) of a straightcantilever beam structure. It provides the input data and results for benchmark test SDLX02/89from Guide de validation des progiciels de calcul de structures.

• A = 0.3439 x 10E–2 m2

• R = 0.922 m

• e = 0.00612 m

• Re = 0.0925 m

• Ri = 0.08638 m

• Iy = Iz = 0.1377x10–4 m4 (straight elements)

• Iy = Iz = 0.5887x10–5 m4 (curved elements)


Input Files

sdlx02.dat

Units

SI



Material Properties

• E = 1.658 x 1011 Pa

• ν = 0.3

• ρ = 13404.106 kg/m3



• 26 grid points




Boundary Conditions



Solution Type


Results



1 10.18 10.39

2 19.54 19.85

3 25.47 25.32

4 48.09 47.74

5 52.86 51.78

6 75.94 83.00

7 80.11 85.12

8 122.3 125.8

9 123.2 127.7

References




17.15 Rectangular PlatesThis test is a normal mode dynamics analysis (three dimensional problem) of a thin plate withrigid body modes. It provides the input data and results for benchmark test SDLX03/89 fromGuide de validation des progiciels de calcul de structures.


Input Files

sdlx03.dat

Units

SI

Material Properties

• E = 2.1 x 1011 Pa

• ν = 0.3

• ρ = 7800 kg / m3



Finite Element Modeling• 300 linear quadrilateral thin shell (CQUAD4) elements

• 320 grid points


Boundary Conditions• Free plate

• One DOF set





Results



1 584.0 577.0

2 826.0 813.0

3 855.0 844.0

4 911.0 895.0

5 1113. 1062.

6 1136. 1118.

References



Chapter

18 Mechanical Structures —Normal Mode Dynamics Analysisand Model Response

18.1 Transient Response of a Spring-Mass System with AccelerationLoadingThis test is an undamped transient response by modal superposition. It provides the inputdata and results for benchmark test SDLD04/89 from Guide de validation des progiciels decalcul de structures.

Where:

• m = 1 kg

• k = 1000 N/m


Input Files

sdld04.dat

Units

SI

Material Properties

Spring constant.


Chapter 18 Mechanical Structures — Normal Mode Dynamics Analysis and ModelResponse


• 3 lumped mass (CONM) elements

• 3 translational spring (CELAS) elements


Boundary Conditions

• Points A: Clamped (u = v = 0 : θ - 0)

• Points B, C and D: v = 0 ; = 0

• Point A: Set acceleration: u(t) = 2E5 * (t2) ; (0 < t < 0.1 s)

• Initial condition: u(0) = 0 ; u(0) = 0 at every point

The mesh and the boundary conditions are shown in the following figure:

Solution Type

SOL 112 — Modal Transient Response

Results

The mode shapes results are exact.



1 2.239 2.239

2 6.275 6.275

3 9.069 9.069


Mechanical Structures — Normal Mode Dynamics Analysis and Model Response

Mode shapes results

Normal Mode Point Bench Value NX Nastran

1 B 0.4450 0.4450

C 0.8019 0.8019

D 1.000 1.000

2 B 1.000 1.000

C 0.4450 0.4450

D –0.8019 –0.8019

B –0.8019 –0.8019

C 1.000 1.000

D –0.4450 –0.4450

Transient response (Point D: X-displacement in meters)

Time (sec) Bench Value NX Nastran

0.02000 –0.002700 –0.002670

0.04000 –0.04260 –0.04270

0.05000 –0.1041 –0.1041

0.06000 –0.2158 –0.2160

0.08000 –0.6813 –0.6818

0.1000 –1.658 –1.659



References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. SDLD04/89.



18.2 Transient Response of a Clamped-free PostThis test is a transient response of a straight cantilever beam with acceleration and forceloadings, and modal damping. It provides the input data and results for benchmark testSDLL06/89 from "Guide de validation des progiciels de calcul de structures."


Input Files

sdll06.dat

Units

SI

Material Properties

• E = 4 x 1010 Pa

• Iz = 3.285 x 10–1 m4

• ρ = 0



• 9 grid points




Boundary Conditions

To apply an acceleration üA(t) at point A, we can do the following:

• Points A: Clamped (u = v = 0 : θ - 0)

• Point B: Set nodal force Fx(t) equal to mB * üA(t) in the -X direction

Fx(t) = –m * üA(t)

• Initial conditions: u(0) = 0 ; u (0) = 0 at every point

The mesh and the boundary conditions are shown in the following figure:



Solution Type

SOL 109 — Direct Transient Response

Results

uB displacement (mm)

Time (s) Bench Value NX Nastran

0.01000 –0.06500 –0.06570

0.02000 –0.5130 –0.5152

0.03000 –1.679 –1.682

0.04000 –3.457 –3.464

0.05000 –5.316 –5.333

0.06000 –6.764 –6.804

0.07000 –7.609 –7.682

0.08000 –7.774 –7.891

0.09000 –7.244 –7.413

0.1000 –6.068 –6.289

0.1200 –2.242 –2.542

0.1400 2.367 2.070

0.1600 6.149 5.977

0.1800 7.783 7.847

0.2000 6.698 7.042

The problem with damping is not computed.



References



Chapter

19 Stationary Thermal Tests —Heat Transfer Analysis


Chapter 19 Stationary Thermal Tests — Heat Transfer Analysis

19.1 Hollow Cylinder - Fixed TemperaturesThis test is a steady-state heat transfer analysis of a 2D axisymmetric cylinder with fixedtemperatures. It provides the input data and results for benchmark test TPLA01/89 from "Guidede validation des progiciels de calcul de structures."

• Re = 0.30 m

• Ri = 0.35 m


Input Files

htpla01.dat

Units

SI

Material Properties

• λ = 1 W/m °C


• 10 linear axisymmetric solid (CTRIAX6) elements



Stationary Thermal Tests — Heat Transfer Analysis

Boundary Conditions

• One temperature set:

– Internal temperature Ti = 100 °C

– External temperature Te = 20 °C

Solution Type

SOL 153 — Steady State Heat Transfer

Results

Temperature results

Radius (m) Bench Value (°C) NX Nastran (°C)0.3000 100.0 100.0

0.3100 82.98 82.98

0.3200 66.51 66.51

0.3300 50.54 50.54

0.3400 35.04 35.04

0.3500 20.00 20.00

Flux results

Radius (m) Bench Value (W/m2) NX Nastran (W/m2)0.3000 1730. 1702.

0.3100 1674. 1666.

0.3200 1622. 1614.

0.3300 1573. 1565.

0.3400 1526. 1519.

0.3500 1483. 1505.



References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. TPLA01/89.



19.2 Hollow Cylinder - ConvectionThis test is a steady-state heat transfer analysis of a 2D axisymmetric cylinder with convection.It provides the input data and results for benchmark test TPLA03/89 from "Guide de validationdes progiciels de calcul de structures."

• Re = 0.300 m

• Ri = 0.391 m


Input Files

htpla03.dat

Units

SI

Material Properties

• λ = 40.0 W/m °C






Boundary Conditions• Convection on internal surface:

hi = 150.0 W/m2 / °C

Ti = 500 °C

• Convection on external surface:

he = 142.0 W/m2 / °C

Ti = 20 °C

Solution TypeSOL 153 — Steady State Heat Transfer

Results

Temperature / Flux Bench Value NX Nastran

Ti (°C) 272.3 272.5

Te (°C) 205.1 204.6

i (W/m2) 3.416E4 3.378E4

e (W/m2) 2.628E4 2.642E4

φ / l = φ * 2 * π * R

So: φ / l= 34173.82 * 2 * π * 0.300 = 64416.13 W/m

ReferencesSociete Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. TPLA03/89.



19.3 Cylindrical Rod - Flux DensityThis test is a steady-state heat transfer analysis of a 2D axisymmetric rod with fixedtemperatures and flux density. It provides the input data and results for benchmark testTPLA05/89 from "Guide de validation des progiciels de calcul de structures."


Input Files

htpla05.dat

Units

SI

Material Properties

• λ = 33.33 W/m °C



• 42 grid points




Boundary Conditions• z = 0

Set temperature to 0 °C

• z = 1


• Cylindrical surface

Set flux to –200 W/m2




Solution Type

SOL 153 — Steady State Heat transfer

Results

Temperature results (°C)

Grid point # z (m) Bench Value NX Nastran

Grid point 3 0 0 0

Grid point 41 0.1000 –4.000 –4.020

Grid point 39 0.2000 4.000 3.980

Grid point 37 0.3000 24.00 23.97

Grid point 35 0.4000 56.00 55.97

Grid point 33 0.5000 100.0 99.97

Grid point 31 0.6000 156.0 156.0

Grid point 29 0.7000 224.0 224.0

Grid point 27 0.8000 304.0 304.0

Grid point 25 0.9000 396.0 396.0

Grid point 4 1.000 500.0 500.0

Results are post-processed on the internal surface. NX Nastran does not make theapproximation, T = cte when r is fixed.

References




19.4 Hollow Cylinder with Two Materials - ConvectionThis test is a steady-state heat transfer analysis of a 2D axisymmetric cylinder with twomaterials and convection. It provides the input data and results for benchmark test TPLA08/89from "Guide de validation des progiciels de calcul de structures."

• Ri = 0.30 m

• Rm = 0.35 m

• Re = 0.37 m


Input Files

htpla08.dat

Units

SI

Material Properties

• Material 1: λ1 = 40.0 W/m °C




• 16 grid points




Boundary Conditions• Convection on internal surface

hi = 150.0 W/m2 °C

Ti = 70 °C

• Convection on external surface

he = 200.0 W/m2 °C

Te = –15 °C

Solution TypeSOL 153 — Steady State Heat Transfer

Results

Grid point # Temperature Flux Bench Value NX Nastran

Grid point 9 Ti (°C) 25.42 25.45

Grid point 14 Tm (°C) 17.69 17.68

Grid point 16 Te (°C) 12.11 12.09

Grid point 9 Grid point 9 i(W/m2)

6687. 6609.

Grid point 14 Grid point 14 m(W/m**2)

5732. 5768.

Grid point 16 Grid point 16 e(W/m2)

5422. 5497.

φ/l = φ * 2 * π * R



So: φ/l= 5733.33 * 2 * π * 0.35 = 12608.25 W/m

References




19.5 Wall-ConvectionThis test is a steady-state heat transfer analysis of a 1D wall with fixed convection. It providesthe input data and results for benchmark test TPLL03/89 from "Guide de validation desprogiciels de calcul de structures."


Input Files

htpl03.dat

Units

SI

Material Properties

• λ = 1.0 W/m °C


• 1 linear quadrilateral thin shell (CQUAD4) element

• 4 grid points

The thin shell element thickness is set to 1 m.




Boundary Conditions

• Convection on internal surface

hA = 20.0 W/m2 °C

TA = –20.0 °C

• Convection on external surface

hB = 10.0 W/m2 °C

TB = 500.0 °C

• Convection coefficient is defined as

energy / (length * time * temperature) in the current system of units.




Solution Type


Results

Temperature Results

Grid point # Temperature /Flux


Grid point 2 TA (°C) 21.71 21.71

Grid point 4 TB (°C) 416.6 416.6

Grid point 1 (W/m2) 834.2 834.3

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. TPLL03/89.



19.6 Wall-Fixed TemperaturesThis test is a steady-state heat transfer analysis of a 1D wall with fixed temperatures. Itprovides the input data and results for benchmark test TPLL01/89 from "Guide de validation desprogiciels de calcul de structures."



Input Files

htpl01.dat

Units

SI

Material Properties

• λ = 0.75 W/m °C



• 6 grid points



Boundary Conditions

• Internal temperature Ti = 100 °C

• External temperature Te = 20 °C

Solution Type


Results


Grid point # Length: x (m) Bench Value NX Nastran

Grid point 1 0 100.0 100.0

Grid point 3 0.01000 84.00 84.00

Grid point 4 0.02000 68.00 68.00

Grid point 5 0.03000 52.00 52.00

Grid point 6 0.04000 36.00 36.00

Grid point 2 0.05000 20.00 20.00

The flux calculated with NX Nastran is exact:

= 1200 Ω / μ2

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No. TPLL01/89.



19.7 L-PlateThis test is a steady-state heat transfer analysis of a 2D L-plate with fixed temperatures. Itprovides the input data and results for benchmark test TPLP01/89 from "Guide de validation desprogiciels de calcul de structures."


Input Files

htpp01a.dat - linear quadrilateral thin shell elements

htpp01b.dat - parabolic quadrilateral thin shell elements

Units

SI

Material Properties

• λ = 1.0 W/m °C


• Test 1 – 12 linear quadrilateral thin shell (CQUAD4) elements

• Test 2 – 12 parabolic quadrilateral thin shell (CQUAD8) elements




Boundary Conditions• AF side


• DE side


Solution TypeSOL153 — Steady State Heat Transfer

Results


Node Bench Value NX Nastran CQUAD4 NX Nastran CQUAD88 7.869 7.924 7.882



Node Bench Value NX Nastran CQUAD4 NX Nastran CQUAD89 5.495 5.613 5.519

10 2.816 2.885 2.834

19 8.018 8.043 8.015

18 5.680 5.821 5.665

20 2.881 2.963 2.877

17 8.514 8.425 8.518

6 6.667 6.667 6.667

16 2.972 3.148 2.962

21 9.001 8.992 9.107

15 8.640 8.356 8.668

14 9.316 9.189 9.282

5 9.009 8.773 8.961

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No.TPLP01/89.



19.8 Orthotropic SquareThis test is a steady-state heat transfer analysis of a square plate with orthotropic conductionand convection. It provides the input data and results for benchmark test TPLP02/89 from"Guide de validation des progiciels de calcul de structures."


Input Files

htpp02.dat

Units

SI

Material Properties

• λx = 1.00 W/m °C

• λx =.75 W/m °C



• 121 grid points

The thin shell element thickness is set to 1 m.




Boundary Conditions

• Flux density y = 60 W/m2 for face y = –0.1. (Entry)

• Flux density y = –60 W/m2 for face y = 0.1. (Exit)

• Convection on the faces X = –0.1 and x = 0.1:

– h = 15.0 W/m2 °C

• Linear variation of the external temperatures:

– Te = 30 – 80y on the face x = –0.1

– Te = 15 – 80y on the face x = 0.1

• Convection coefficient is defined as:

– Energy / (length * time * temperature)

• Flux density is defined as:

– Energy / (length * time) in the current system of units




Solution Type


Results

Temperature Results

Point Bench Value (°C) NX Nastran (°C)0 22.50 22.50

A 35.00 34.80

B 26.00 25.80

C 10.00 10.20

D 19.00 19.20

E 30.50 30.50

F 18.00 18.00

G 14.50 14.50

H 27.00 27.00

Flux Results (W/m2)

Grid Point Bench Value NX Nastran

61 X 45.00 45.00

61 Y 60.00 59.55



References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No.TPLP02/89.



19.9 Hollow Sphere - Fixed Temperatures, ConvectionThis test is a steady-state heat transfer analysis of a 3D sphere with fixed temperatures andconvection. It provides the input data and results for benchmark test TPLV02/89 from "Guide devalidation des progiciels de calcul de structures."

• Ri = 0.30 m

• Re = 0.35 m


Input Files

htpv02.dat

Units

SI

Material Properties

• λ = 1 W/m °C


• 500 linear brick (CHEXA) and linear wedge (CPENTA) elements

• 766 grid points

The test is executed on 1/8 mapped meshed sphere.




Boundary Conditions

• Convection on internal surface

hi = 30 W/m2 °C

Ti = 100 °C

• Set external surface temperature Te to 20 °C

The load sets are shown in the following figure:

Solution Type




Results


Radius r (m) Bench Value NX Nastran

0.3000 65.00 64.88

0.3100 54.84 54.75

0.3200 45.31 45.25

0.3300 36.36 36.33

0.3400 27.94 27.92

0.3500 20.00 20.00

Flux results (W/m2): (X-direction)

Radius r (m) Bench Value NX Nastran

0.3000 1050. 1013.

0.3100 983.4 981.4

0.3200 922.9 921.2

0.3300 867.5 866.3

0.3400 817.5 816.3

0.3500 771.4 792.4

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No.TPLV02/89.



19.10 Hollow Sphere with Two Materials - ConvectionThis test is a steady-state heat transfer analysis of a 3D sphere with two materials andconvection. It provides the input data and results for benchmark test TPLV04/89 from "Guide devalidation des progiciels de calcul de structures."

• Ri = 0.30 m

• Rm = 0.35 m

• Re = 0.37 m


Input Files

htpv04a.dat (CHEXA & CPENTA) elements

htpv04b.dat (CTETRA) elements

htpv04c.dat (CTRIAX6) elements

Units

SI

Material Properties




• Test 1 - 700 solid linear brick (CHEXA) & solid linear wedge (CPENTA) elements

• Test 2 - 2192 solid parabolic tetrahedron (CTETRA) elements

• Test 3 - 8 axisymmetric parabolic (CTRIAX6) elements



The test is executed on a 1/8 meshed sphere


Boundary Conditions

• Convection on internal surface:

hi = 150.0 W/m2 °C

Ti = 70 °C

• Convection on external surface:

he = 200.0 W/m2 °C

Te = –9 °C




Solution Type


Results

Temperature Rresults

TemperatureFlux (°C)

Bench Value NX NastranLinear brick

NX NastranParabolicTetrahedron

NX Nastranaxisymmetric

Ti 25.06 N1 25.02 N19 25.06 N2 24.98

Tm 17.84 N556 17.84 N9 17.84 N6 17.72

Te 13.16 N778 13.18 N5 13.15 N5 13.15

i (W/m2) 6741. N1 6487. N19 5865. N2 6390.

m (W/m2) 4952. N556 4931. N9 4765. N6 5148.

e (W/m2) 4431. N778 4531. N5 4551. N5 4547.

φ = φ * 4 * π * R2

So: φ = 4931.20 * 4 * π * 0.352 = 7590.00 W

Flux is in the x direction

References



Chapter

20 Thermo-mechanical Tests —Linear Statics Analysis

20.1 Orthotropic CubeThis test is a steady-state heat transfer analysis of a 3D cube with convection and flux density. Itprovides the input data and results for benchmark test TPLV07/89 from "Guide de validation desprogiciels de calcul de structures."


Input Fileshtpv07.dat

UnitsSI

Material Properties• λx = 1.00 W/m °C

• λy = 0.75 W/m °C

• λz = 0.50 W/m °C


Chapter 20 Thermo-mechanical Tests — Linear Statics Analysis


• 512 linear brick (CHEXA) elements

• 729 grid points


Boundary Conditions

• Flux density y = 60 W/m2 for face y = –0.1 (Entry)

• Flux density y = –60 W/m2 for face y = 0.1 (Exit)

• Flux density z = 30 W/m2 for face z = –0.1 (Entry)

• Flux density z = –30 W/m2 for face z = 0.1 (Exit)

• Convection on the faces X = –0.1 and x = 0.1:

–h = 15.0 W/m2 °C

• Linear variation of the external temperatures:

–Te = 30 – 80y – 60z on the face x = –0.1

–Te = 15 – 80y – 60z on the face x = 0.1


Thermo-mechanical Tests — Linear Statics Analysis


Solution Type


Results


Point Bench Value NX Nastran

(A) 35.00 34.70

(B) 26.00 25.70

(C) 10.00 10.30

(D) 19.00 19.30

S 30.50 30.40

F 18.00 18.00

M 14.50 14.60

H 27.00 27.00

N 29.00 29.00

P 20.00 20.00

J 4.000 4.600

I 13.00 13.60

E 16.50 16.60

R 41.00 40.40

Q 32.00 31.40

K 16.00 16.00



Point Bench Value NX Nastran

L 25.00 25.00

G 28.50 28.40

References




20.2 Thermal Gradient on a Thin PipeThis test is a thermo-mechanical linear statics analysis of a thin pipe with thermal gradient andplane strain. It provides the input data and results for benchmark test HSLA01/89 from "Guidede validation des progiciels de calcul de structures."

• Ri = 0.020 m

• Re = 0.025 m


Input Files

hsla01.dat

Units

SI

Material Properties

• E = 1.0 x 1011 Pa

• = 0.3

• Coefficient of expansion: α = 1.0 x 10–5/C°



• 561 grid points




Boundary Conditions

• Articulate AB side

• Radial temperature with Ti = 100 °C, and Te = 0 °C.




Solution Type

SOL 101 Linear statics

Results

Point Stress Bench Value NX Nastran

r=Ri σr (Pa) –265x 0 –14.03 E6

σθ (Pa) –265y –74.07 E6 –79.84 E6

r = (Re + Ri) / 2 σr (Pa) –270x –3.950 E6 –3.908 E6

σθ (Pa) –270y 1.306 E6 1.469 E6

r = Re σr (Pa) –275x 0 –11.31 E6

σθ (Pa) –275y 68.78 E6 73.69 E6

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No.HSLA01/89.



20.3 Simply-Supported ArchThis test is a thermo-mechanical linear statics analysis of a thin curved beam with thermalgradient and articulation. It provides the input data and results for benchmark test HSLL01/89from "Guide de validation des progiciels de calcul de structures."

• R = 10 m

• A = 144 x 10–4 m2

• I = 1.728 x 10–5 m4

Beam cross section:


Input Fileshsll01.dat

UnitsSI

Material Properties• E = 0.2 x 1011 Pa

• = 0.3

• Coefficient of expansion: α = 11.0 x 10–6/C°




• 51 grid points


Boundary Conditions

• Articulate point A and B

• Top temperature Ts = 160 °C

• Middle temperature Tm = 100 °C

• Bottom temperature Ti = 40 °C


Solution Type




Results

Point Force Moment Bench Value NX Nastran

θ = π/2 M 0 4.040 e–5

N 0 15.10

T –479.2 –527.6

θ = π/4 M 3388. 3729.

N –338.8 –373.2

T –338.8 –373.2

θ = 0 M 4792. 5277.

N –479.2 527.5

T 0 15.00

Post Processing

List the beam forces

• M - Z bending moment

• N - axial force

• T - Y shear force

References

Societe Francaise des Mecaniciens. Guide de validation des progiciels de calcul de structures.Paris, Afnor Technique, 1990. Test No.HSLL01/89.


Part

VII Material Nonlinear (Plasticity)Verification Using StandardNAFEMS Benchmarks

Overview of the Material Nonlinear (Plasticity) Verification Using NAFEMS Test Cases . . 21-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1


Chapter

21 Overview of the MaterialNonlinear (Plasticity) VerificationUsing NAFEMS Test Cases

The purpose of this section is to verify the accuracy and robustness of NX Nastran. The plasticityverification uses test cases published by the National Agency for Finite Element Methods andStandards (NAFEMS) in Fundamental Tests for Two and Three Dimensional, Small Strain,Elastoplastic Finite Element Analysis. (See Reference.)

To perform the tests, input data is applied to single elements including plane strain elements,plane stress elements, axisymmetric solid elements, and solid elements. Results are tabulatedand compared to results published by NAFEMS.

The plasticity verification includes perfect plasticity and isotropic hardening tests. Within thesecategories, results for uniaxial, biaxial, and triaxial displacement tests are provided.

21.1 Understanding the Verification FormatThe format for the nonlinear section of the Solver Verification document looks somewhatdifferent from the linear section. Each test case in this section provides a brief description of thetest including input data. The results are then displayed in the form of a graph comparing NXNastran Nonlinear results published by NAFEMS for the same test case.

21.2 ReferenceThe following reference has been used in the NX Nastran Plasticity verification problems:

Hinton, E., and Ezatt, M.H. Fundamental Tests for Two and Three Dimensional, Small Strain,Elastoplastic Finite Element Analysis. East Kilbride, Glasgow, UK: National Agency for FiniteElement Methods and Standards, April, 1987.


Chapter

22 Test Cases

22.1 Plane Strain Elements - Perfect Plasticity TestsThis article provides input data and results for perfect plasticity tests including prescribeduniaxial and prescribed biaxial displacement tests. The tests were run on these plane stresselements:

• Linear triangle (CTRIA3) elements

• Linear quadrilateral (CQUAD4) elements

The material description and initial boundary conditions are the same for the uniaxial andbiaxial displacement tests.


Input Filesnlspls89.dat (uniaxial)

nlspls90.dat (biaxial)

UnitsInch

AttributesLoad Control

Material Properties• E = 250000.0



• = 0.25

• σy = 5.0

• H = 0.0

• o = 0.000025 (strain at first yield)

Boundary Conditions

The following figure shows the plane strain elements and the boundary conditions applied toeach. The strain state is completely defined as a function of time since all degrees of freedom aresuppressed or prescribed.

These boundary conditions represent initial conditions only and do not illustrate the time historyof the applied conditions.


Test Cases

Results

Uniaxial Displacement Test — Applied Strain History

The following graph shows results of the uniaxial displacement test for the plane strain elements.

Results are exactly the same for both elements. The graph shows the NX Nastran Nonlinear testresults compared with NAFEMS test results for plane strain with perfect plasticity.

History Strain XX Strain YY Strain ZZ1 0.2500D–4 0D+00 0D+00

2 0.5000D–4 0D+00 0D+00

3 0.2500D–4 0D+00 0D+00

4 0D+00 0D+00 0D+00

10 increments per strain history step



Biaxial Displacement Test — Applied Strain History

The following graph shows results of the biaxial displacement test for the plane strain elements.

Results are exactly the same for both elements. The graph shows the NX Nastran Nonlinear testresults (points) compared to NAFEMS test results for plane strain with perfect plasticity.

HistoryStage

Strain XX Strain YY Strain ZZ

1 0.2500D–4 0D+00 0D–00

2 0.5000D–4 0D+00 0D–00

3 0.5000D–4 0.2500D–4 0D–00

4 0.5000D–4 0.5000D–4 0D–00

5 0.2500D–4 0.5000D–4 0D–00

6 0D+00 0.5000D–4 0D+00

7 0D+00 0.2500D–4 0D+00

8 0D+00 0D+00 0D+00



Test Cases

References

Hinton, E., and Ezatt, M.H. Fundamental Tests for Two and Three Dimensional, Small Strain,Elastoplastic Finite Element Analysis. East Kilbride, Glasgow, UK: National Agency for FiniteElement Methods and Standards, April, 1987 pp. 2.3-2.25.



22.2 Plane Strain Elements - Isotropic Hardening TestsThis article provides input data and results for isotropic hardening tests including prescribeduniaxial and prescribed biaxial displacement tests. The tests were run on these plane strainelements:





Input Files

nslpls91.dat (uniaxial)

nlspls92.dat (biaxial)

Units

Inch

Attributes

Load Control

Material Properties

• E = 250000.0

• = 0.25

• σy = 5.0

• H = 62500.0



Test Cases

Boundary Conditions





Results



Results are exactly the same for both elements. The graph shows the NX Nastran Nonlinear testresults (points) compared to NAFEMS test results for plane strain with isotropic hardening.

History Strain XX Strain YY Strain ZZ1 0.2500D–4 0D+00 0D+00

2 0.5000D–4 0D+00 0D+00

3 0.2500D–4 0D+00 0D+00

4 0D–00 0D+00 0D+00



Test Cases



Results are exactly the same for both elements. The graph shows the NX Nastran Nonlinear testresults (points) compared to NAFEMS test result for plane strain with isotropic hardening.

HistoryStage


1 0.2500D–4 0D+00 0D+00

2 0.5000D–4 0D+00 0D+00

3 0.5000D–4 0.2500D–4 0D+00

4 0.5000D–4 0.5000D–4 0D+00

5 0.2500D–4 0.5000D–4 0D+00

6 0D–00 0.5000D–4 0D+00

7 0D–00 0.2500D–4 0D+00

8 0D–00 0D–00 0D+00




References

Hinton, E., and Ezatt, M.H. Fundamental Tests for Two and Three Dimensional, Small Strain,Elastoplastic Finite Element Analysis. East Kilbride, Glasgow, UK: National Agency for FiniteElement Methods and Standards, April, 1987 pp. 2.26 - 2.35.


Test Cases

22.3 Plane Stress Elements - Perfect Plasticity TestsThis article provides input data and results for perfect plasticity tests including prescribeduniaxial and prescribed biaxial displacement tests. The tests were run on these plane strainelements:




The following figure shows the geometry:


Input Files

nlspls61.dat (uniaxial test), linear quadrilateral (CQUAD4) elements

nlspls62.dat (uniaxial test), linear triangle (CTRIA3) elements

nlspls65.dat (biaxial test), linear quadrilateral (CQUAD4) elements

nlspls66.dat (biaxial test), linear triangle (CTRIA3) elements

Units

Inch

Material Properties

• E = 250000.0

• = 0.25

• σy = 5.0

• H = 0.0

• o = 0.2080126 x 10–4 (strain at first yield)



Boundary Conditions




Test Cases

Results



Results are exactly the same for both elements. The graph shows the NX Nastran Nonlinear testresults (points) compared to NAFEMS test results for plane stress with perfect plasticity.

History Strain XX Strain YY Strain ZZ1 0.2080D–4 0D+00 -0.6934D–5

2 0.4160D–4 0D+00 -0.2538D–4

3 0.2080D–4 0D+00 -0.1835D–4

4 0.4235D–21 0D+00 -0.1128D–4






Results are exactly the same for both elements. The graph shows the NX Nastran Nonlinear testresults (points) compared to NAFEMS test results.

HistoryStage


1 0.2080D–4 0D+00 -0.6934D–5

2 0.4160D–4 0D+00 -0.2528D–4

3 0.4160D–4 0.2080D–4 -0.4284D–4

4 0.4160D–4 0.4160D–4 -0.6513D–4

5 0D–00 0.4160D–4 -0.5035D–4

6 0D–00 0.1872D–4 -0.3871D–4

7 0D–00 0D–00 -0.1867D–4



Test Cases

References




22.4 Plane Stress Elements - Isotropic Hardening TestsThis article provides input data and results for isotropic hardening tests including prescribeduniaxial and prescribed biaxial displacement tests. The tests were run on the these plane stresselements:




The following figure shows the geometry:


Input Files

nlspls71.dat (uniaxial test), linear quadrilateral (CQUAD4) elements

nlspls72.dat (uniaxial test), linear triangle (CTRIA3) elements

nlspls75.dat (biaxial test), linear quadrilateral (CQUAD4) elements

nlspls76.dat (biaxial test), linear triangle (CTRIA3) elements

Units

Inch

Material Properties

• E = 250000.0

• = 0.25

• σy = 5.0

• H = 62500.0

• o = 0.2080126 x 10–4 (strain at first yield)


Test Cases

Boundary Conditions


These boundary conditions represent initial conditions only and do not show the time historyof the applied conditions.



Results




History Strain XX Strain YY Strain ZZ1 0.2080D–4 0D+00 -0.6934D–5

2 0.4160D–4 0D+00 -0.2249D–4

3 0.2080D–4 0D+00 -0.1555D–4

4 0.4235D–21 0D+00 -0.8619D–5



Test Cases




HistoryStage


1 0.2080D–4 0D+00 -0.6934D–5

2 0.4160D–4 0D+00 -0.2249D–4

3 0.4160D–4 0.2080D–4 -0.3569D–4

4 0.4160D–4 0.4160D–4 -0.5406D–4

5 0.2080D–4 0.4160D–4 -0.4712D–4

6 0D–00 0.4160D–4 -0.4102D–4

7 0D–00 0.2080D–4 -0.3408D–4

8 0D–00 0D–00 -0.2715D–4




References



Test Cases

22.5 Solid Element - Perfect Plasticity TestsThis article provides input data and results for perfect plasticity tests including prescribeduniaxial, biaxial, and triaxial displacement tests. The tests were run on the solid parabolicbrick element.


Input Files

nlspls08.dat

Units

Inch

Material Properties

• E = 250000.0

• = 0.25

• y = 5.0

• H = 0.0


Boundary Conditions

The following figure shows the parabolic brick (CHEXA) element and the boundary conditionsapplied to it. The strain state is completely defined as a function of time since all degrees offreedom are suppressed or prescribed.




Results


History Strain XX Strain YY Strain ZZ1 2.500E–5 0E+00 0E+00

2 5.000E–5 0E+00 0E+00

3 2.500E–5 0E+00 0E+00

4 0E+00 0E+00 0E+00



Test Cases

The following graph shows results of the uniaxial displacement test for the solid brick element.It shows the NX Nastran Nonlinear test results (points) compared to NAFEMS test results.




The following graph shows results of the biaxial displacement test for the solid brick element. Thegraph shows the NX Nastran Nonlinear test results (points) compared to NAFEMS test results.


2 5.000E–5 0E+00 0E+00

3 5.000E–5 2.500E–5 0E+00

4 5.000E–5 5.000E–5 0E+00

5 2.500E–5 5.000E–5 0E+00

6 0E+00 5.000E–5 0E+00

7 0E+00 2.500E–5 0E+00

8 0E+00 0E+00 0E+00

- 10 increments per strain history step


Test Cases

Triaxial Displacement Test — Applied Strain History

The following graph shows results of the triaxial displacement test for the solid brick element.The graph shows the NX Nastran Nonlinear test results (points) compared to NAFEMS testresults.


2 5.000E–5 0E+00 0E+00

3 5.000E–5 2.500E–5 0E+00

4 5.000E–5 5.000E–5 0E+00

5 5.000E–5 5.000E–5 2.500E–5

6 5.000E–5 5.000E–5 5.000E–5

7 2.500E–5 5.000E–5 5.000E–5

8 0E+00 5.000E–5 5.000E–5

9 0E+00 2.500E–5 5.000E–5

10 0E+00 0E+00 5.000E–5

11 0E+00 0E+00 2.500E– 5



History Strain XX Strain YY Strain ZZ12 0E+00 0E+00 0E+00


References



Test Cases

22.6 Solid Element - Isotropic Hardening TestsThis article provides input data and results for isotropic hardening tests including prescribeduniaxial, biaxial, and triaxial displacement tests. The tests were run on the solid parabolic brickelement (CHEXA), which has 20 grid points.


Input Filesnlspls09.dat

UnitsInch

Material Properties• E = 250000.0

• = 0.25

• y = 5.0

• H = 62500.0


Boundary ConditionsThe following figure shows the parabolic brick element and the boundary conditions applied toit. The strain state is completely defined as a function of time since all degrees of freedom aresuppressed or prescribed.




Results


The following graph shows results of the uniaxial displacement test for the solid brick element.It shows the NX Nastran Nonlinear test results (points) compared to NAFEMS test results.


2 5.000E–5 0E+00 0E+00

3 2.500E–5 0E+00 0E+00

4 0E+00 0E+00 0E+00



Test Cases


The following graph shows results of the biaxial displacement test for the solid brick element. Thegraph shows the NX Nastran Nonlinear test results (points) compared to NAFEMS test results.


2 5.000E–5 0E+00 0E+00

3 5.000E–5 2.500E–5 0E+00

4 5.000E–5 5.000E–5 0E+00

5 2.500E–5 5.000E–5 0E+00

6 0E+00 5.000E–5 0E+00

7 0E+00 2.500E–5 0E+00

8 0E+00 0E+00 0E+00




Triaxial Displacement Test — Applied Strain History

The following graph shows results of the triaxial displacement test for the solid brick element.The graph shows the NX Nastran Nonlinear test results (points) compared to NAFEMS testresults.


2 5.000E–5 0E+00 0E+00

3 5.000E–5 2.500E–5 0E+00

4 5.000E–5 5.000E–5 0E+00

5 5.000E–5 5.000E–5 2.500E–5

6 5.000E–5 5.000E–5 5.000E–5

7 2.500E–5 5.000E–5 5.000E–5

8 0E+00 5.000E–5 5.000E–5

9 0E+00 2.500E–5 5.000E–5

10 0E+00 0E+00 5.000E–5

11 0E+00 0E+00 2.500E–5


Test Cases

History Strain XX Strain YY Strain ZZ12 0E+00 0E+00 0E+00


References



Part

VIII Geometric NonlinearVerification Using StandardNAFEMS Benchmarks

Overview of the Geometric Nonlinear Verification Using NAFEMS Test Cases . . . . . . . . . 23-1

Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-1


Chapter

23 Overview of the GeometricNonlinear Verification UsingNAFEMS Test Cases

This section verifies the accuracy and robustness of the arc-length method of NX Nastran. Thegeometric nonlinear verification uses test cases published by the National Agency for FiniteElement Methods and Standards (NAFEMS) in NAFEMS Non-Linear Benchmarks and AReview of Benchmark Problems for Geometric Non-linear Behaviour of 3-D Beams and Shells.(See References.)

23.1 Understanding the Verification FormatEach test case includes the following information.


- Units

- Material properties

- Finite element modeling information

- Boundary conditions (loads and restraints)

- Solution type

• Results — time history versus Load Factor plots are presented. (Note that in NX Nastran,the load factor is displayed as "eigenvalue".)

• Reference

23.2 ReferenceThe following references have been used for these verification problems:

• NAFEMS Non-Linear Benchmarks. Glasgow: NAFEMS, Oct., 1989., Rev. 1. Test No. NL6.

• NAFEMS, A Review of Benchmark Problems for Geometric Non-Linear Behaviour of 3-DBeams and Shells (Summary) (Glasgow: NAFEMS, Ref. R0024.)


Chapter

24 Test Cases

24.1 Straight Cantilever with End MomentThis test is a nonlinear analysis of a single row of equal-sized elements. This document providesthe input data and results for NAFEMS Non-linear Benchmarks NL5.

Test attributes:

• Bending action only

• Initially straight elements

• Load control


Input Filesnfnl05a.dat (load control)

nfnl05b.dat (arc-length control)

UnitsSI

Material Properties• E = 210 x 109 N / m2



• = 0.0

SI


32 linear beam (CBEAM) elements

Boundary Conditions

• U = V = = 0 at point B

• Concentrated load at Point A applied in equal increments up to a maximum value of M L/ 2 E I = 1.0

Solution Type

SOL 106 — Geometric Nonlinear

• Loading method — arc-length control.

• Adaptive search control:

– Initial increment factor = 0.05

– Target number of iterations = 6


Test Cases

– Maximum number of splits = 3

– Max increment factor = 1

– Number of reporting steps = 18

Geometric nonlinear 2

• Loading method — load control.

• 6 equal steps.

Results

Normalizing Constants

• 2 E I / L = 3436.12 x 103 Nm

• L = 3.2 m

• 2 = 6.28319

Graphs of Results

• Free end axial displacement vs. Load Factor



• Vertical displacement at grid point 33 vs. Load Factor


Test Cases

• Rotational displacement at grid point 33 vs. Load Factor

References

National Agency for Finite Element Methods and Standards, NAFEMS Non-Linear Benchmarks(Glasgow: NAFEMS, Oct., 1989., Rev. 1). Test No. NL5.



24.2 Straight Cantilever with Axial End Point Load - Brick ElementsThis test is a nonlinear analysis of a straight cantilever with an axial end point load, madeup of a single row of straight elements. This document provides the input data and resultsfor NAFEMS Non-linear Benchmarks NL6.


• Combined bending and membrane action.

• Presence of bifurcation.

• Initially straight elements.

• Load control.


Input Files

nlsarg07.dat

Units

SI

Material Properties

• E = 210 x 109 N/m2

• = 0.0


Test Cases


256 solid parabolic brick (CHEXA) elements.

Boundary Conditions

• U = V = θ = 0 at point B.

• Concentrated load at Point A applied in increments up to a value of PL2 / EI = 22.493.

Solution Type


Results

Normalizing Constants:

• EI / L2 = 170898 N

• L + 3.2 m

• = 3.14159



Graphs of results:

• X-displacement at cantilever end point vs. Load Factor.


Test Cases

• Y-displacement at cantilever end point vs. Load Factor

Reference

National Agency for Finite Element Methods and Standards. NAFEMS Non-Linear Benchmarks.Glasgow: NAFEMS, Oct., 1989., Rev. 1. Test No. NL6.



24.3 Straight Cantilever with Axial End Point Load - BEAM ElementsThis test is a nonlinear analysis of a single row of straight elements. This document provides theinput data and results for NAFEMS Non-linear Benchmarks NF6.


Input Filesnlsarp01.dat

UnitsSI

Material Properties• E = 210 x 109 N/m2

• = 0.0

Finite Element Modeling32 linear (CBEAM) elements


Test Cases

Boundary Conditions

• U = V = θ = 0 at point B.

• Concentrated load at Point A applied in increments up to a value of PL2 / EI = 22.493or P = –3.85 x 106 N

Solution Type


Results

Normalizing Constants:

• EI / L 2 = 170898 N

• π = 3.14159



Graphs of results:

• X-displacement at grid point 33 vs. Load Factor


Test Cases

• Y-displacement at grid point 33 vs. Load Factor



• Rz-displacement at grid point 33 vs. Load Factor

Reference

National Agency for Finite Element Methods and Standards, NAFEMS Non-Linear Benchmarks.Glasgow: NAFEMS, Oct., 1989., Rev. 1. Test No. NL6.


Test Cases

24.4 Lee’s Frame Buckling ProblemThis test is a nonlinear analysis of a single row of straight elements. This document provides theinput data and results for NAFEMS Non-linear Benchmarks NF7. Attributes of this test are:


Input Files

nlsarg01.dat

Units

SI

Material Properties

• E = 71.74 x 109 N/m2

• = 0.0




20 linear beam (CBEAM) elements

Boundary Conditions

• U = V = 0; θ ≠ 0 at points B and C

• Concentrated load at Point A applied incrementally using arc-length constraint withautomatic adjustment of arc length (P = –20000 N)

Solution Type



Test Cases

Results

Normalizing Constants EI / L2 = 996.389 N, L = 1.2 m

Graphs of results: Y-displacement at grid point 13 vs. Load Factor

Reference

National Agency for Finite Element Methods and Standards. NAFEMS Non-Linear Benchmarks.Glasgow: NAFEMS, Oct., 1989., Rev. 1. Test No. NL7.



24.5 Large Displacement Elastic Response of a Hinged SphericalShell Under Uniform Pressure LoadingThis test is a nonlinear analysis of a hinged spherical shell element under uniform pressureloading. This document provides the input data and results for A Review of Benchmark Problemsfor Geometric Non-Linear Behaviour of 3-D Beams and Shells (Summary) 3DNLG-7.


Input Files

nlsarg05.dat

Units

SI

Material Properties

• E = 69

• = 0.3


• The shell midsurface is defined in terms of the global Cartesian coordinate system where Z =2.0285 x 10 –4 [X (1570 – X) + Y (1570 – Y)].


Test Cases

Boundary Conditions

• Evenly distributed follower pressure load normal to shell surface. Maximum pressure = 0.1.Pressure follows the deformation of the shell surface.

Solution Type


• Loading method:

– Arc-length control

• Adaptive search control:

– Initial increment factor = 0.3

– Target number of iterations = 6

– Maximum number of splits = 3

– Maximum increment factor = 1

– Number of reporting steps = 18



24.6 ResultsMagnitude displacement at grid point 145 vs. Load Factor

24.7 Reference• National Agency for Finite Element Methods and Standards. A Review of Benchmark

Problems for Geometric Non-Linear Behaviour of 3-D Beams and Shells (Summary) Glasgow:NAFEMS, Ref. R0024. Test No. 3DNLG-7


Documents

NX Nastran Verification Manual