Objective
To analyse bending in the plane of the ring under distributed load, shear deformations are neglected.
Reference
G.Pisarenko, A.Yakovlev, V.Matveev. Strength of materials handbook. — Kyiv: Naukova Dumka, 1988.
Problem statement
To determine the axial force N in the ring cross-section and the change in the ring diameter δ.
Design model
The ring is subjected to a distributed load q acting in its plane.
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Initial geometry of analytical model
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Initial geometry of FE model
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Geometry
Radius of the ring: R = 1 m
Cross‑sectional area: F = 0,001 m2
Material properties
Modulus of elasticity Е = 2,0 * 1011 Pa
Poisson's ratio μ = 0,3
Loads
Distributed load q = 100 kN/m
Note
Design model – bar 2D model; it consists of 25 bar FEs and 25 nodes.
Local X‑axes at the nodes are directed outward from the circle centre.
Boundary conditions are provided by applying restraints along the degrees of freedom Y, Z, uX, uY, uZ for all nodes.
Output data
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| Diagram of axial force N (kN) | Values of displacements δmax (mm) |
Analytical solution
In the analytical solution, the change in the diameter of the ring in the directions of the x and y axes is determined by the formulas ("Strength of materials" Handbook, p. 384):
Axial force in the ring cross‑section:
Comparison of calculation results
| Parameters | Analytical solution | LIRA-FEM | Error, % |
| Change in the diameter of the ring δ, mm | 0,5 | 0,5 | 0 |
| Axial force in the ring cross‑section N, kN | 100 | 99,3 | 0,7 |
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