Smooth Connection on a Free Thin Square Plate

This test lets you study the smooth connection in the case of a modal analysis. You will use 2D meshes.

Reference:

Guide de validation des prologiciels de calcul des structures, AFNOR Technique, pp216-217, SFM.
 

Specifications

Geometry Specifications

Length:
L = 1000 mm  

Thickness:
th = 10 mm


Analysis Specifications

Young Modulus (material):
E = 210 GPa


Mesh formed of parabolic triangle elements (TR6),
with a smooth connection.

Poisson's Ratio (material):
ν = 0.3

Density:
ρ = 7800 kg/m3

Mesh Specifications:
10 x 10 elements

Restraints:

  • smooth connection case:
    a clamp on the point where the smooth connection is applied.

  • no smooth connection case:
    no restraint
 

Results

This test presents the modal frequencies of a plate with a smooth connection in its center, and compares these results with the semi-analytical solution of M.V. Barton (results A). The modal frequencies of the smoothly-connected plate are also compared with the frequencies of a plate without connection (results B).

Reference Solutions

  • Results A: Semi-analytical solution for frequencies of M.V. Barton

    with :
  • Results B: Computed values for a plate without connection:

     Mode*

    Semi-analytical
    solution
    of M.V. Barton

    Computed values

    Linear triangle
    (TR3)

    Parabolic triangle
    (TR6)

    Linear quadrangle
    (QD4)

    Parabolic quadrangle
    (QD8)

    7

    33.7119

    33.5005

    33.3576

    33.6410

    33.5163

    8

    49.4558

    49.2273

    48.9560

    49.3634

    48.9541

    9

    61.0513

    61.1598

    60.6258

    61.4091

    60.6241

    10

    87.5160

    87.3410

    86.2420

    87.5727

    86.5851

    11

    87.5160

    87.3510

    86.2586

    87.5727

    86.5851

    * On a plate without connection, the six first modes of a free modal analysis are rigid. With a smooth connection, these modes do not exist, because of the clamp of the connection. We compare, for the three solutions, the results of the five first free modes.

Computed Results of the Plate with a Smooth Connection

  •  Comparison with Results A

     Mode* 

    Computed values

    Linear triangle
    (TR3)

    Parabolic triangle
    (TR6)

    Linear quadrangle
    (QD4)

    Parabolic quadrangle
    (QD8)

    Computed solution

    Comparison with results A
     [% of error]

    Computed solution

    Comparison with results A
     [% of error]

    Computed solution

    Comparison with results A
     [% of error]

    Computed solution

    Comparison with results A
     [% of error] 

    1

    33.5005

    0.63

    33.3575

    1.05

    33.641

    0.21

    33.5163

    0.58

    2

    49.2271

    0.46

    48.9557

    1.01

    49.3634

    0.19

    48.9541

    1.01

    3

    59.8521

    1.96

    60.3319

    1.18

    59.5249

    2.50

    59.6750

    2.25

    4

    86.6717

    0.96

    86.1295

    1.58

    86.6419

    1.00

    86.1747

    1.53

    5

    86.7038

    0.93

    86.1464

    1.56

    86.6419

    1.00

    86.1747

    1.53

     

  • Comparison with Results B

     Mode* 

    Computed values

    Linear triangle
    (TR3)

    Parabolic triangle
    (TR6)

    Linear quadrangle
    (QD4)

    Parabolic quadrangle
    (QD8)

    Computed solution

    Comparison with results B
     [% of error]

    Computed solution

    Comparison with results B
     [% of error]

    Computed solution

    Comparison with results B
     [% of error]

    Computed solution

    Comparison with results B
     [% of error]

    1

    33.5005

    0.00

    33.3575

    0.00

    33.641

    0.00

    33.5163

    0.00

    2

    49.2271

    0.00

    48.9557

    0.00

    49.3634

    0.00

    48.9541

    0.00

    3

    59.8521

    2.14

    60.3319

    0.48

    59.5249

    3.07

    59.6750

    1.57

    4

    86.6717

    0.77

    86.1295

    0.13

    86.6419

    1.06

    86.1747

    0.47

    5

    86.7038

    0.74

    86.1464

    0.13

    86.6419

    1.06

    86.1747

    0.47

    * On a plate without connection, the six first modes of a free modal analysis are rigid. With a smooth connection, these modes do not exist, because of the clamp of the connection. We compare, for the three solutions, the results of the five first free modes.

To Perform the Test:

The assembly_connection_tr6_smooth.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic triangle elements (TR6), with a smooth connection.

The assembly_connection_tr6_no_smooth.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of parabolic triangle elements (TR6) without a smooth connection.

To compute these two cases with other meshes, proceed as follow:

  1. Open one of the CATAnalysis documents.

  2. Enter the Advanced Meshing Tools workbench.

  3. In the specification tree, double-click on the mesh:
    The Global Parameters dialog box appears.

  4. Change the mesh specifications as indicated below.

  5. Compute the case in the Generative Structural Analysis workbench.