9. Pinched Cylinder with Free Ends

This test case is described by Sansour and Bernarczyk[17] and Rebel[18]. A thin cylinder whose ands are free is loaded by two concentrated forces at the centre. Due to symmetry, only one eight of the cylinder has to be modelled.

Pinched cylinder: Test case definition

Figure 38. Pinched cylinder: Test case definition


A nonlinear load-controlled analysis is performed. The case definition is as follows:

cases
  case 1
    analysis               nonlinear
    increment_control_type load
    ebc                    1
    nbc                    1
    max_newton_iterations  50
    max_step               500
    step_size_min          1.0e-10
    step_size_max          0.02
    step_size_init         0.01
  end
end

The deformed configuration at P = 50, using a mesh of 16x8 Q4.S.MITC.E4 elements, is shown in the following figure:

Pinched cylinder: Deformed configuration (Q9).

Figure 39. Pinched cylinder: Deformed configuration (Q9).


The radial displacements at points A and B obtained with various types of MITC shell elements are plotted against the applied load, showing good agreement with the reference values and a the values for very fine mesh of Q4.S.MITC.E4 elements:

Pinched cylinder: Results for Q8.S.MITC shell elements.

Figure 40. Pinched cylinder: Results for Q8.S.MITC shell elements.


Pinched cylinder: Results for Q9.S.MITC shell elements.

Figure 41. Pinched cylinder: Results for Q9.S.MITC shell elements.


[1] Carlo Sansour, Herbert Bednarczyk. The Cosserat surface as a shell model, theory and finite-element formulation, Comput. Methods Appl. Mech. Engrg. 120 (1995), pp 26-27.

[2] Gert Rebel. Finite rotation shell theory including drill rotations and its finite element implementation, PhD thesis, Delft University Press, 1998.



[17] Carlo Sansour, Herbert Bednarczyk. The Cosserat surface as a shell model, theory and finite-element formulation, Comput. Methods Appl. Mech. Engrg. 120 (1995), pp 26-27.

[18] Gert Rebel. Finite rotation shell theory including drill rotations and its finite element implementation, PhD thesis, Delft University Press, 1998.