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SS-V: 2050 Thick Square Plate
Test No. VM06 Determine the first twelve modes and frequencies for a simply supported thick square plate.

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- Static Analysis
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  - SS-V: 2000 Free Vibration of a Wedge
    Test No. VM01 Determine the first fundamental frequency and vibration mode of a wedge constrained at the wide end.
  - SS-V: 2010 In-plane Vibration of a Cantilever Plate
    Test No. VM01 Find the fundamental frequency for in-plane vibration of a cantilevered plate.
  - SS-V: 2020 Cylindrical Shell
    Test No. VM03 Find the fundamental frequency of a cantilever cylindrical shell.
  - SS-V: 2030 Free Thin Square Plate
    Test No. VM04 Determine the first fourteen modal frequencies for a free thin square plate.
  - SS-V: 2040 Thick Skewed Plate
    Test No. VM05 Find the fundamental frequency of a thick clamped skewed plate.
  - SS-V: 2050 Thick Square Plate
    Test No. VM06 Determine the first twelve modes and frequencies for a simply supported thick square plate.
  - SS-V: 2060 In-plane Vibration of Pin-ended Cross
    Test No. VM07 Find the first eight modal frequencies for in-plane vibration of a cross with pin joints at the ends.
  - SS-V: 2070 Cantilever Beam
    Test No. VM08 Perform normal mode dynamic analysis of a cantilevered beam.
  - SS-V: 2080 Circular Ring - In-plane and Out-of-plane Vibration
    Test No. VM09 Perform normal mode dynamic analysis of an unconstrained circular ring.
  - SS-V: 2090 Triangular Wing - Fixed Vibration
    Test No. VM10 Perform a normal mode dynamic analysis of a fixed right triangular plate.
  - SS-V: 2100 Cantilevered Tapered Beam
    Test No. VM11 Shear behavior of a cantilevered tapered beam.
  - SS-V: 2110 Cantilever with Off-Center Remote Mass
    Test No. VM12 Find the frequencies of a cantilever with off center remote masses at the free end.
  - SS-V: 2120 Two-body Symmetric Damped Oscillators System
    Test No. VM13Determine the frequency and damping rate of a 1D damped oscillator.
  - SS-V: 2130 1D Damped Oscillator: Translation or Rotation
    Test No. VM14Determine the frequency and damping rate of a 1D damped oscillator.
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SS-V: 2050 Thick Square Plate

Test No. VM06 Determine the first twelve modes and frequencies for a simply supported thick square plate.

Definition

Figure 1.

The material properties are:

Properties: Value
Modulus of Elasticity: 2e+11 Pa
Poisson's Ratio: 0.3
Mass density: 8.3e+3 kg/m³

Results

The plate has three rigid body modes with zero frequencies. Vertical constraints are applied not at the middle plane of the plate, but to the lower edges. This does not allow for clear separation of out-of-plane and in-plane vibration modes. Also, in full 3D formulation constrained edges introduce stress singularity into the solution.

Additonal comments:

The reference solution has a very rough FEA mesh with just a few elements. Mesh lines orientation and incompleteness of basis functions of iso-parametric elements results in artificial “anisotropy” of the approximation.
Vertical constraints are applied not at the middle plane of the plate, but to the lower edges. This does not allow for clear separation of out-of-plane and in-plane vibration modes. This separation in NAFEMS results is because a very crude approximation is used. Also, in full 3D formulation the constrained edges are the source of a strong singularity in the solution.
The High Order Element (HOE) solution has only one layer of elements across thickness with quadratic basic functions. Quadratic functions do not capture bending (you need 3rd order functions). The Low Order Element (LOE) solution has three layers of bricks across the thickness, but the basis functions are linear incomplete polynomials which also do not capture bending.
An additional OptiStruct model was run with a high density (800,000 element) mesh. Twenty linear HEX elements were used through the thickness (Figure 2). This is felt to be a better comparison.
Figure 2. OptiStruct mesh

The following table summarizes the modes.


	OptiStruct	SimSolid	% Difference	NAFEMS	% Difference
Mode 4	42.47	42.81	0.80%	45.90	-6.73%
Mode 5, 6	100.74	102.14	1.39%	109.44	-6.67%
Mode 7	147.89	151.04	2.13%	167.89	-10.04%
Mode 8	183.70	188.25	2.48%	174.00	8.19%
Mode 9	185.37	189.46	2.21%	Not given	NA
Mode 10	193.59	193.47	-0.06%	193.59	-0.06%

¹ The Standard NAFEMS Benchmarks TNSB, Rev. 3, 5 October, 1990, Test No. FV 52