Objective
To determine the deformed state of a cylindrical body, free from restraints, under the uniform longitudinal load distributed over the ends.
Reference
P. Germain, Introduction a la mecanique des milieux continus, Paris, Masson, 1986.
Problem statement
A cylindrical body, free from restraints, is subjected to a longitudinal load F/A uniformly distributed over its ends.
To determine the meridional displacements Z(G) and radial displacements X(G) of points E, D, A (C) on the lateral surface of the cylinder; these points are located away from the transverse symmetry plane along the generatrix at distances L/3, 2*L/3, and L, as well as point B at the centre of the end surface.
Design model
The design model is an axisymmetric problem. The FE mesh is divided with a step of 0.25 m in the meridional direction and 0.10 m in the radial direction.
Geometry
Cylinder radius R = 1 m
Cylinder length L = 3 m
Material properties
Modulus of elasticity Е = 2*105 Pa
Poisson's ratio ν = 0,2
Boundary conditions
The geometric stability of the design model is ensured by applying restraints in accordance with its symmetry conditions.
Loads
Uniform longitudinal load distributed over the ends F/A = 100 Pa.
Note
The problem is solved in 2D formulation (model type 1).
The model is generated with FE type 30 – arbitrary quadrilateral FE of 2D problem (wall-beam).
Nodes: 143. Elements: 120.
Output data
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Design model
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Deformed shape
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Mosaic plot of displacements along X(G), mm
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Mosaic plot of displacements along Z(G), mm
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Analytical solution
The meridional displacements Z(G) and radial displacements X(G) can be calculated using the following formulas:
Z(G) = P*X/E; X(G) = v*P*R/E.
Comparioson of calculation results
| Parameter | Theory | LIRA-FEM | Error, % |
| Displacement Z(G) (point E), mm | -0,500 | -0,500 | 0,000 |
| Displacement X(G) (point E), mm | 0,150 | 0,150 | 0,000 |
| Displacement Z(G) (point D), mm | -1,000 | -1,000 | 0,000 |
| Displacement X(G) (point D), mm | 0,150 | 0,150 | 0,000 |
| Displacement Z(G) (point A and C), mm | -1,500 | -1,500 | 0,000 |
| Displacement X(G) (point A and C), mm | 0,150 | 0,150 | 0,000 |
| Displacement Z(G) (point B), mm | -1,500 | -1,500 | 0,000 |
| Displacement X(G) (point B), mm | 0,000 | 0,000 | 0,000 |
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