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Finite Element Benchmarks For 2D Beams And Axisymmetric Shells Involving Geometric Non-Linearity


This report is one of a series commissioned by NAFEMS. 

The purpose of these reports is to provide a set of benchmark tests against which the results of different finite element programs can be compared. The report is concerned with solutions to problems involving thin two-dimensional beams and axisymmetric shells undergoing large displacements and rotations


Problems involving deep beams, large strains or non-conservative loading are specifically excluded. 

Furthermore the beams and axisymmetric shells are assumed to have elastic material properties and a prismatic cross section.The emphasis on the results in this report lies in the area of displaced shapes and thus most of the target results quoted relate to displacements rather than internal stresses. However, stress resultant plots obtained from the Finite Element solutions have been included for all the beam tests along with values for stress resultants obtained by simple statics for those tests which are statically determinate. 

Contents

Terminology

Notation

Quick Reference Table of Tests

List of Figures

List of Tables

Introduction

Layout of Report

Overview of the Elements and Geometric Nonlinearity used

  • Continuum Solutions
  • Thin Beam/Axisymmetric Shell Benchmark Solutions
  • Thick Beam Solutions
  • Other Beam Solutions
  • Solution Techniques Employed

Two Dimensional Beam Benchmark Tests carried out

  • Rigid Body Rotation
  • Straight  Cantilever with End Moment
  • Curved Cantilever with End Moment
  • Straight Cantilever with Transverse End Point Load
  • Straight Cantilever  with Axial End Point Load
  • Symmetrical Shallow Pinned Arch Buckling
  • Unsymmetrical Deep Arch Buckling
  • Lee's Frame Buckling Problem

Axisymmetric Shell Benchmarks tests carried out

  • Built in Circular Plate with Uniformly Distributed Load
  • Annular Plate With Inside Edge Load
  • Truncated Cone with Outside Edge Load
  • Built in Spherical Cap with Central Point Load

Conclusions

References

Appendix A Virtual Work Derivation of Geometrically Nonlinear Analysis

  • Virtual Work Equation
  • Green Strain Displacement Relationship
  • Almansi Strain Displacement Relationship


Appendix B Beam Formulations and Implementation of Geometric Nonlinearity

  • Thin Beam and Axisymmetric Shell Formulations
  • Thick Beam Formulations



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About

P. Lyons & S. Holsgrove

First Published - January 1989

Softback, 274 Pages