13. Spherical Cap

A simply supported spherical cap loaded with an apex load is analyzed under static load. The structure is axisymmetric and can thus be modelled with axisymmetric 2D elements. A reference solution can be found in the publications by Stricklin et.al.[22][23].

The simply supported spherical cap is modeled by axi-symmetric 2D solid elements as shown in the figure below: Higher order (quadratic) 2D elements are sufficient to model the shell behavior of the structure.

Spherical cap test: Mesh (Q8 axi-symmetric elements).

Figure 47. Spherical cap test: Mesh (Q8 axi-symmetric elements).


The non-linear static response is calculated with the analysis control parameters defined in the case command (see input file cap.mdl):

cases
   case 1
      analysis                nonlinear  # Nonlinear analysis
      ebc                     1          # 'Displacement constraints'
      nbc                     1          # 'Forces'
      step_size_init          0.1        # Try with step size 0.1
      step_size_min           0.01       # Smallest step size
      step_size_max           1.0        # Largest step size
      max_newton_iterations 100          # Max. n. of Newton iterations
   end
end

The solver will, by default, increment the load (or time) parameter from 0.0 to 1.0. It will try to start with a step size of 0.1 (option step_size). If the steps are too large, steps will be cut down, but not less that 0.01 (option step_size_min). If convergence is 'good' the step size can be increased, but not more that 1.0 (option step_size_max). Note that the steps refer to the path length. Thus, the step size can be larger than 1.0. Observe that the step size refers to the path parameter and thus has a different meaning, depending on the increment control option selected for the current case.

The limitation given by max_newton_iterations will influence the number of increments required to solve the problem: If max_newton_iterations is large, the number of increments decreases but to the cost of a larger number of iterations. With max_newton_iterations set to 100 the solution displayed below is obtained:

Spherical cap test: Load-displacement curve for max_newton_iterations set to 100.

Figure 48. Spherical cap test: Load-displacement curve for max_newton_iterations set to 100.


With max_newton_iterations set to 10 the number of steps increases slightly:

Spherical cap test: Load-displacement curve for max_newton_iterations set to 10.

Figure 49. Spherical cap test: Load-displacement curve for max_newton_iterations set to 10.




[22] J.A. Stricklin; Geometrically Non-Linear Static and Dynamic Analysis of Shells of Revolution; High Speed Computing od Elastic Structures, Proceedings IUTAM, University of Liège, 1970

[23] W.E. Haisler, J.A. Stricklin, J. E: Key; Displacement Incrementation in Non-Linear Structural Analysis by the Self-Correcting Method; Int. J. Num. Meth. Eng. 11, pp 3-10, 1977