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Validation Case: Cantilever Plate Subjected to a Follower Pressure

This validation case belongs to solid mechanics. It uses a cantilever plate geometry to validate the follower pressure boundary condition.

SimScale results are compared to simulation results, presented in [SSNV145]\(^1\). The reference uses Code_Aster to perform the analysis.

Geometry

The geometry for this project consists of a cantilever plate, as seen in Figure 1:

cantilever plate follower pressure validation case
Figure 1: Cantilever plate geometry. The results will be assessed at point P.

The dimensions of the geometry are given in Table 1:

CoordinatesABCDEFGHP
x \([m]\)00001010101010
y \([m]\)0.5-0.5-0.50.50.5-0.5-0.50.50
z \([m]\)-0.05-0.050.050.05-0.05-0.050.050.050
Table 1: Coordinates of points within the cantilever plate geometry

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Nonlinear static

Mesh and Element Types: Two meshes are used in this case. The first one is a second-order mesh created in SimScale with the standard algorithm. Case B uses a second-order hexahedral mesh. It was created locally and imported to SimScale.

Table 2 contains details of the resulting meshes:

CaseMesh TypeNodesElement Type
ASecond-order standard9700Standard
BSecond-order hexahedral3502Standard
Table 2: Characteristics of the meshes used for cases A and B

Figure 2 shows the standard mesh, used for case A:

second order standard mesh cantilever beam
Figure 2: Discretization obtained with a second-order standard mesh

Similarly, Figure 3 shows the second-order hexahedral mesh, that was imported to SimScale.

hexahedral mesh imported to simscale med
Figure 3: Hexahedral mesh imported to SimScale. It contains a total of 3502 nodes.

Note

Especially for nonlinear analysis, such as this one, we recommend second-order meshes. They provide more accurate results, due to the higher number of nodes.

The following article provides further information on second-order meshes for finite element analysis.

Simulation Setup

Material:

  • Material behavior: Linear elastic
  • \((E)\) Young’s modulus = 1.2e+7 \(Pa\)
  • \((\nu)\) Poisson’s ratio = 0.3
  • \((\rho)\) Density = 1000 \(kg/m³\)

Boundary Conditions:

The boundary conditions will be defined based on Figure 1:

  • Constraints
    • Fixed support on face ABCD
  • Surface loads
    • Follower Pressure \(P\), applied on face CDHG. The following formulation is used:
      $$P = t \tag{1}$$
      Where \(t\) is the pseudo-time for the nonlinear analysis. This validation case will run until \(t\) = 26.

Result Comparison

SimScale results will be compared against two reference simulations. The first one used Code_Aster, while the remaining one used the SAMCEF software.

The results for both reference simulations are found in [1]. The displacements in the X and Z-directions are evaluated at point P (as seen in Figure 1).

In Figure 4, we compare the results for the displacements in the X-direction against Code_Aster. The reference results were extracted using WebPlotDigitizer.

follower pressure validation case displacement results
Figure 4: Result comparison between SimScale results and the reference simulation for DX

A similar comparison was made for the displacements in the Z-direction:

follower pressure validation case displacement results vertical direction
Figure 5: Result comparison between SimScale results and the reference simulation for DZ

Additionally, still using point P as a reference, the displacements obtained with SimScale are compared to the results from the SAMCEF software\(^1\). Table 1 contains the displacements in the X-direction:

Pseudo-time \([s]\)SAMCEF – DX \([m]\)Case A – DX \([m]\)Error [%]Case B – DX \([m]\)Error [%]
11-7.3664-7.0953-3.68-7.0725-3.99
13-9.03743-8.7143-3.58-8.6851-3.90
22-13.5098-13.2380-2.01-13.2049-2.26
26-14.1513-13.9797-1.21-13.9555-1.38
Table 3: Comparison between SimScale and SAMCEF results, for the displacements in the X-direction

A comparison for the displacements in the Z-direction is also presented:

Pseudo-time \([s]\)SAMCEF – DZ \([m]\)Case A – DZ \([m]\)Error [%]Case B – DZ \([m]\)Error [%]
11-8.44920-8.3929-0.67-8.3872-0.73
13-8.42753-8.43100.04-8.42990.02
22-5.78828-6.07594.97-6.10575.48
26-4.43375-4.76967.57-4.80798.44
Table 4: Comparison between SimScale and SAMCEF results, for the displacements in the Z-direction

The results obtained with SimScale for both directions show a great agreement with the reference simulations.

Figure 6 shows the contours for displacements in the X-direction, for case A. The follower pressure boundary condition is updated after each pseudo time step, based on the current deformed state of the geometry. As a result, the plate gets rolled up:

follower pressure displacement contours
Figure 6: Displacement contours in the X-direction for case A. The initial position of the plate is shaded in blue.

Last updated: August 24th, 2020

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