Benchmark C4

VMC4
Simply-Supported Thin Annular Plate

Overview

Reference:

J. Barlow, G. A. O. Davis, "Selected FE Benchmarks in Structural and Thermal Analysis", NAFEMS Rept. FEBSTA, Rev. 1, October 1986, Test No. 14 (modified).

Analysis Type(s):

Mode-frequency Analysis (ANTYPE = 2)

Element Type(s):

4-Node Structural Shell Elements (SHELL181)

8-Node Structural Shell Elements (SHELL281)

Input Listing:

vmc4.dat

VMC4 requires supplemental .cdb input files which are too long to include full input listings. This file must be downloaded and placed in your working directory for the test case to run properly. Additionally, the geometry and mesh should be regenerated.

Download link: MAPDL Test Case Files for 2024 R2

vmc4-1.cdb

vmc4-2.cdb

vmc4-3.cdb

vmc4-4.cdb

vmc4-5.cdb

vmc4-6.cdb

Test Case

A simply-supported thin annular plate of thickness t is to be analyzed to determine the first nine natural frequencies. Determine the natural frequencies for two mesh densities using both quadrilateral and triangular element shapes with Block Lanczos eigenvalue extraction.

Figure 593: Thin Annular Plate Problem Sketch

Material Properties

Geometric Properties

Loading and Boundary Conditions

E = 200 x 10⁹ N/m²

υ = 0.3

ρ = 8000 kg/m³

a = 1.8 m

b = 6.0 m

t = 0.06 m

Parameter Definitions

N1 = No. elements along radial direction

N2 = No. elements along circumferential direction

All nodes UX = UY = ROTZ = 0 in cylindrical coordinate system;

At x = b UZ = 0

At x = b ROTX = 0

Figure 594: Representative Mesh Options

(a) Quadrilateral Mesh (N1 = 3, N2 = 16)
(b) Quadrilateral Mesh (N1 = 5, N2 = 32)
(c) Triangle Mesh (N1 = 3, N2 = 16)
(d) Triangle Mesh (N1 = 5, N2 = 32)

Target Solution

Mode				1	2	3	4	5	6	7	8	9
Frequency (Hz)				1.880	5.187	5.195	9.784	9.791	15.227	15.861	15.877	18.942

Results Comparison - Block Lanczos Extraction Method Frequency Ratio

ETYP	N1	N2	DOF	1	2	3	4	5	6	7	8	9
Quadrilateral Mesh
181	3	16	480	0.980	1.289	1.289	1.399	1.399	1.080	1.390	1.390	1.219
181	5	32	1296	0.994	1.037	1.037	1.064	1.064	1.033	1.086	1.086	1.042
281	3	16	1224	1.001	0.996	0.996	1.011	1.011	1.006	1.044	1.044	0.998
281	5	32	3528	1.001	0.998	0.998	1.001	1.001	1.001	1.000	1.000	0.996
Triangular Mesh
281	3	16	1992	1.020	1.040	1.041	1.035	1.038	1.054	1.059	1.069	1.093
281	5	32	4968	1.006	1.010	1.011	1.011	1.012	1.025	1.018	1.020	1.030

Assumptions, Modeling Notes, and Solution Comments

The problem is designed to test a solution involving repeated eigenvalues.