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Comparison between the stiffness method and the hybrid method applied to a circular ring

  • Nagore Insausti
  • Itziar Adarraga
  • Javier Urruzola
  • Faustino Mujika
Technical Paper

Abstract

This study deals with the comparison of stresses and displacements of a circular ring obtained by two numerical formulations, namely the stiffness method and the hybrid method, applied to isoparametric quadrilateral elements. After explaining the formulation of the hybrid method in finite elements, an orthogonalization method proposed previously for hybrid elements is applied. Then, the computational cost per element of the stiffness method and of the hybrid method with and without orthogonalization has been evaluated for the first time. A circular ring loaded by two opposing forces is analyzed in order to compare the solution obtained by the hybrid method and the stiffness method with experimental and analytical results. The agreement between analytical and experimental results with those numerical is better in the case of the hybrid method than in the case of the stiffness method. It is observed that the element ratio needed to obtain a given relative error is one magnitude order greater in the stiffness method than in the case of the hybrid method.

Keywords

Force method Stiffness method Hybrid formulation Circular ring 

List of symbols

\(\left\{ {a^{i} } \right\},\left\{ {a^{l} } \right\}\)

Nodal displacements vector of the element and of the structure

a

Thickness of the circular ring

A

Cross sectional area of the circular ring

\(\left[ B \right]\)

Displacement–strain matrix

e

Difference between the average radius and the radius of the neutral surface

\(\left[ E \right],\left[ {\bar{E}^{*} } \right]\)

Equilibrium matrix

E

Young’s modulus of material

\(\left\{ {F^{j} } \right\}\)

Stress parameters vector

\(\left[ G \right]\)

Flexibility matrix

G

Shear modulus of material

\(I^{*}\)

Equivalent moment of inertia

\(\left[ {K^{\text{e}} } \right],\left[ K \right]\)

SFM stiffness matrix of the element and of the structure

k

Stiffness constant of the dynamometer

\(\left[ L \right]\)

Matrix of differential operators

\(M\)

Bending moment

\(\left[ N \right]\)

Matrix of displacement interpolation functions

\(N\)

Axial force

\(N_{i}\)

Displacement interpolation functions

\(\left\{ {P^{i} } \right\},\left\{ {P^{l} } \right\}\)

External forces vector of the element and of the structure

P

Concentrated force acting on the circular ring

ri, ro

Inner and outer radius of the circular ring

(r, θ)

Polar coordinates

\(R_{\text{a}}\)

Average radius of the circular ring

\(R_{\text{e}}\)

Radius of the neutral surface of the circular ring

\(\left[ S \right]\)

Compliance matrix of the material

t

Element thickness

\(\left\{ u \right\}\)

Vector of displacements at any point in the element

U*

Complementary strain energy

X

The redundant unknown

\(\left[ Y \right]\)

Matrix of stress interpolation functions

\(\delta\)

Displacement

\(\left\{ \varepsilon \right\}\)

Strain field vector in element domain

\(\nu\)

Poisson’s ratio

\(\xi ,\eta\)

Natural coordinates

\(\left\{ {\sigma_{i} } \right\}\)

Stress field vector in element domain

Notes

Acknowledgements

The financial support of the University of the Basque Country (UPV/EHU) in the Research Group GIU 16/51 “Mechanics of Materials” is gratefully acknowledged.

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Materials + Technologies Group/Mechanics of Materials, Department of Mechanical Engineering, Faculty of Engineering of GipuzkoaUniversity of the Basque Country UPV/EHUSan SebastiánSpain

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