Objective
To determine the stress–strain state of a cylindrical shell subjected to self-weight.
Reference
K.Mallikarjuna Rao, U.Shrinivasa A set of pathological tests to validate new finite elements. Sadhana, Vol. 26, Part 6, Dezember 2001, P. 549-589.
Problem statement
To determine the vertical displacement at the mid-point of the free edge of the shell (point B).
Design model
A cylindrical shell, hinged along its end (curved) edges, subjected to loading due to its self-weight.
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Variant 1 |
Variant 2 |
Geometry
Radius of curvature of the shell R = 25 m.
Shell span length L = 50 m.
Half of the central angle of the shell arc φ = 40°.
Shell thickness h = 0,25 m.
Material properties
Modulus of elasticity Å = 4,32 * 108 Pa.
Poisson's ratio ν = 0,01.
Boundary conditions
Variant 1: A quarter of the shell is analysed.
Fixed displacements and rotations along the arc edge AE – restraints are applied in the directions of DOF X1, Z1, uY1 (u=0, w=0, θY=0).
Symmetry restraints along line ÅÎ – restraints along X1, uY1 (u=0, θY=0).
Symmetry restraints along line ÂÎ – restraints along Y1, uX1 (v=0, θX=0).
Variant 2: The entire shell is analysed.
Fixed displacements and rotations along the arc edge AD and at the opposite end – restraints are applied in the directions of DOF X, Z, uY (u=0, w=0, θY=0).
Loads
Self-weight g = 90 N/m2
Note
The problem is solved in a 3D formulation (model type 5).
The model is generated with FE type 41 – arbitrary rectangular FE of shell.
Variant 1:
To account for the truncated part of the structure, symmetry restraints are applied.
The FE mesh consists of:
– 35 elements along the arc AE;
– 50 elements along the straight line AB.
Nodes: 1836. Elements: 1750.
Variant 2:
The FE mesh consists of:
– 70 elements along the arc AD;
– 100 elements along the straight line AC.
Nodes: 7171. Elements: 7000.
Output data
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Variant 1 |
Variant 2 |
Comparison of calculation results
Without additional side nodes:
| Point | The unknown | Analytical solution | LIRA-FEM | Error, % | |
| Â | Var. 1 | wB, m | -0,3024 | -0,30026 | 0,7077 |
| Var. 2 | wB, m | -0,3024 | -0,30026 | 0,7077 | |
With additional side nodes:
| Point | The unknown | Analytical solution | LIRA-FEM | Error, % | |
| Â | Var. 1 | wB, m | -0,3024 | -0,29415 | 2,7282 |
| Var. 2 | wB, m | -0,3024 | -0,29415 | 2,7282 | |
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