Simply-supported Square Plate

This test is used to study the convergence and the absence of any locking in the case of very thin shells (th / L = 0.5e-4).
You will use 2D meshes.

Reference:

TIMOSHENKO S.P., WOINOWSKY-KRIGER S., Theory of Plates and Shells, 1959.
 

Specifications

Geometry Specifications

Length:
L = 200 mm

Thickness:
th = 0.01 mm  

For symmetry reasons, only a quarter of the plate is modeled (ABCD).

Analysis Specifications

Young Modulus (material):
E = 78000 MPa

Poisson's Ratio (material):
ν= 0.3

Mesh Specifications:
Imposed numbers of nodes (see the tables of results) 

Restraints (User-defined):

  • Point A (position fixed): Tx=Ty=0

  • Edge AB (simply supported edge): Tz=Ry=0

  • Edge DA (simply supported edge): Tz=Rx=0

  • Edge BC (symmetry plane edge): Tx=Ry=Rz=0

  • Edge CD (symmetry plane edge): Ty=Rx=Rz=0

Loads:

  • Case 1 (Distributed force):
    Fz = 0.001 N at point C
    (equivalent to a concentrated vertical force)

  • Case 2 (Pressure):
    P=0.1 Pa on the plate
 

Results

  • The analytical solutions for the z displacement of the point C are:
    Case 1: Tz=64.96 mm
    Case 2: Tz=90.942 mm

  • The tables below present the analysis results for cases 1 and 2.
    The normalized results (computed results divided by analytical solution) are listed.

    • Case 1: 

    Nodes   

    Analytical solution 
    [mm]

    Values

    Linear triangle
    (TR3)

    Parabolic triangle
    (TR6)

    Linear quadrangle
    (QD4)

    Parabolic quadrangle
    (QD8)

    Computed results
     [mm]

    Normalized results

    Computed results
     [mm]

    Normalized results

    Computed results
     [mm]

    Normalized results

    Computed results
     [mm]

    Normalized results

    3 x 3

    64.96

    62.15

    0.957

    62.09

    0.956

    67.44

    1.038

    63.03

    0.970

    5 x 5

    64.96

    64.66

    0.995

    64.61

    0.995

    66.37

    1.022

    64.58

    0.994

    7 x 7

    64.96

    64.81

    0.998

    64.83

    0.998

    65.77

    1.013

    64.80

    0.997

    9 x 9

    64.96

    64.87

    0.999

    64.89

    0.999

    65.49

    1.008

    64.87

    0.999

    11 x 11

    64.96

    64.91

    0.999

    64.92

    0.999

    65.33

    1.006

    64.91

    0.999

    • Case 2:

    Nodes   

    Analytical solution 
    [mm]

    Values

    Linear triangle
    (TR3)

    Parabolic triangle
    (TR6)

    Linear quadrangle
    (QD4)

    Parabolic quadrangle
    (QD8)

    Computed results
     [mm]

    Normalized results

    Computed results
     [mm]

    Normalized results

    Computed results
     [mm]

    Normalized results

    Computed results
     [mm]

    Normalized results

    3 x 3

    90.942

    83.621

    0.919

    87.039

    0.957

    90.308

    0.993

    86.767

    0.987

    5 x 5

    90.942

    89.196

    0.981

    91.117

    1.002

    91.576

    1.007

    90.977

    1.000

    7 x 7

    90.942

    90.276

    0.993

    91.053

    1.001

    91.366

    1.005

    90.998

    1.001

    9 x 9

    90.942

    90.596

    0.996

    91.030

    1.001

    91.231

    1.003

    90.998

    1.001

    11 x 11

    90.942

    90.746

    0.998

    91.019

    1.001

    91.155

    1.002

    90.998

    1.001

To Perform the Test:

The simplysupportedplate11nodesqd8.CATAnalysis document presents a complete analysis of this case, computed with a mesh formed of 11 nodes and parabolic quadrangle elements (QD8).

To compute the case with other types of elements and other number of nodes, proceed as follow:

  1. Open the CATAnalysis document.

  2. In the Advanced Meshing Tools workbench, change the mesh specifications as indicated above.

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