Two Dimensional Polar Coordinate System in Airy Stress Functions

Authors:

• S. Senthil (Assistant Professor, Vel Tech Multi Tech Dr. Rangarajan Dr. Sagunthala Engineering College, Avadi, Chennai, Tamil Nadu, India)
• P. Sekar (Associate Professor, C. Kandaswamy Naidu for Men College, Anna Nagar, Chennai, Tamil Nadu, India)

Abstract:

• The paper deals with solving 2-dimensional Airy stress function problems using polar coordinates.
• Equilibrium equations, Airy stress functions, and stress compatibility are expressed in polar coordinates.

Keywords:

• Transformation between Cartesian and Polar Coordinates
• Airy Stress Function
• Symmetric Stress Field
• Circular Hole in Shear Stress

Introduction:

• Introduces a method for solving 2D boundary value problems in polar coordinates.
• Applications include rings, disks, and curved bars.
• Polar coordinates offer advantages in solving Airy stress functions.

Transforming Cartesian to Polar Coordinates:

• Transformation equations:
  • x = r cos θ
  • y = r sin θ
  • r² = x² + y²

Airy Stress Function for Polar Coordinates:

• Applied to plane elasticity problems.
• Stress transformation equations include:
  • σ_rr, σ_θθ, and σ_rθ in terms of Airy stress functions.

Equations in Polar Coordinates:

• Airy stress function expressed in polar coordinates (r, θ).
• Includes the biharmonic equation for stress analysis.

Stress Field Symmetry:

• Discussion of stress fields symmetric about an axis.
• Solution approach involves equi-dimensional equations.

Circular Hole in a Sheet Under Remote Shear:

• Analysis of stress around a circular hole subjected to shear stress.
• Polar coordinate transformation used for stress analysis.

Examples:

1. Tensile Stress in Thin Plate:
   • Uniform tensile stress 0 at the ends.
   • Stress function derived for a plate with and without a circular hole.
2. Plate with Circular Hole in Simple Tension:
   • Boundary conditions applied.
   • Comparisons with solid plates discussed.

Conclusion:

• The paper provides a comprehensive solution approach using polar coordinates for various stress field problems.
• Emphasizes the importance of boundary conditions in determining stress distributions.

References:

• Various scholarly references to previous work on elasticity and stress functions.