Test 13T: Simply supported thin square plate: transient forced vibration

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

This page discusses:

Elements tested
Problem description
Reference solution
Response predicted by Abaqus/Standard
Response predicted by Abaqus/Explicit
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

S3R

S3RS

S4R

S4RS

S4RSW

S8R5

Problem description

Material and geometry specifications are as given in Test 13: Simply supported thin square plate: frequency extraction.

Mesh:

A coarse mesh, a fine mesh, and a very fine mesh of a quarter of the plate are tested for elements S3R, S3RS, S4R, S4RS, and S4RSW in Abaqus/Explicit. For the quadrilateral element types the mesh densities of the coarse, fine, and very fine meshes are 2 × 2, 3 × 3, and 4 × 4, respectively; for the triangular element types the mesh densities are 2 × 2 × 4, 3 × 3 × 4, and 4 × 4 × 4, respectively.

Forcing function

Suddenly applied pressure.

$F_{0} = $ 100 N/m² over whole plate.

Damping

$ζ = $ 2% [2% of critical damping in the dominant first mode with analytical frequency value $f = $ 2.377 (Hz) or $ω = 2 π f = $ 14.935 (sec⁻¹)].

The damping factors are chosen as $α_{R} = $ 0.299 (sec⁻¹) and $β_{R} = $ 1.339 × 10⁻³ (sec) so that

ζ = \frac{α_{R}}{2 ω} + \frac{β_{R} ω}{2} = 0.01 + 0.01 = 0.02.

For the sole purpose of validation and comparison, the models with a fine mesh are also used to test band-limited damping in Abaqus/Explicit. The damping parameters are chosen as damping factor $α_{R} = $ 0.299 (sec⁻¹), damping ratio $η_{d} = $ 0.01, high-frequency cutoff $f_{h} = $ 10 (Hz), and low-frequency cutoff $f_{l} = $ 1.5 (Hz).

Response location

$u_{z}$ and $σ_{11}$ at center of plate.

General:

Gauss integration is used for the shell cross-section in input file nft1358x.inp.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 13T from NAFEMS “Selected Benchmarks for Forced Vibration,” R0016, March 1993.

Response predicted by Abaqus/Standard

Response predicted by Abaqus/Explicit

Results and discussion

The results are given in Table 1 through Table 7. The values enclosed in parentheses are percentage differences with respect to the reference solution. The static displacement was obtained by creating a second step with a time period of 23 seconds.

Table 1. Element type: S3R, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
Coarse mesh	3.147	0.210	2.079	1.616
Coarse mesh	(−10.67%)	(0.0%)	(−16.30%)	(−11.06%)
Fine mesh	3.313	0.210	2.207	1.701
Fine mesh	(−5.96%)	(0.0%)	(−11.15%)	(−6.38%)
Very fine mesh	3.370	0.210	2.239	1.732
Very fine mesh	(−4.34%)	(0.0%)	(−9.86%)	(−4.68%)

Table 2. Element type: S3RS, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
Coarse mesh	3.263	0.210	2.028	1.679
Coarse mesh	(−7.38%)	(0.0%)	(−18.36%)	(−7.59%)
Fine mesh	3.391	0.210	2.188	1.742
Fine mesh	(−3.75%)	(0.0%	(−11.92%)	(−4.13%)
Very fine mesh	3.429	0.210	2.244	1.762
Very fine mesh	(−2.67%)	(0.0%)	(−9.66%)	(−3.01%)

Table 3. Element type: S4R, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
Coarse mesh	3.366	0.225	1.889	1.760
Coarse mesh	(−4.46%)	(7.14%)	(−23.95%)	(−3.14%)
Fine mesh	3.414	0.215	2.105	1.756
Fine mesh	(−3.09%)	(2.38%)	(−15.26%)	(−3.36%)
Very fine mesh	3.427	0.215	2.186	1.762
Very fine mesh	(−2.72%)	(2.38%)	(−12.00%)	(−3.03%)

Table 4. Element type: S4RS, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
Coarse mesh	3.495	0.240	1.974	1.797
Coarse mesh	(−0.79%)	(14.3%)	(−20.53%)	(−1.1%)
Fine mesh	3.477	0.220	2.185	1.783
Fine mesh	(−1.31%)	(4.8%)	(−12.04%)	(−1.87%)
Very fine mesh	3.461	0.215	2.236	1.785
Very fine mesh	(−1.76%)	(2.4%)	(−9.98%)	(−1.76%)

Table 5. Element type: S4RSW, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
Coarse mesh	2.486	0.205	1.400	1.759
Coarse mesh	(−29.44%)	(−2.4%)	(−43.64%)	(−3.19%)
Fine mesh	3.254	0.215	2.015	1.759
Fine mesh	(−7.64%)	(2.4%)	(−18.88%)	(−3.19%)
Very fine mesh	3.395	0.215	2.181	1.774
Very fine mesh	(−3.63%)	(2.4%)	(−12.20%)	(−2.24%)

Table 6. Band-limited damping with fine mesh, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
S3R	3.264	0.205	2.202	1.701
S3R	(−7.35%)	(−2.4%)	(−11.35%)	(−6.38%)
S3RS	3.293	0.210	2.121	1.716
S3RS	(−6.53%)	(0.0%)	(−14.61%)	(−5.56%)
S4R	3.411	0.215	2.171	1.781
S4R	(−3.18%)	(2.38%)	(−12.60%)	(−1.98%)
S4RS	3.472	0.225	2.239	1.782
S4RS	(−1.45%)	(7.14%)	(−9.86%)	(−1.93%)
S4RSW	3.251	0.220	2.074	1.765
S4RSW	(−7.72%)	(4.76%)	(−16.50%)	(−2.86%)

Table 7. Element type: S8R5, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static displacement
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	3.523	0.210	2.484	1.817
Direct solution	3.467 (−1.59%)	0.212 (0.95%)	2.476 (−2.50%)	1.780 (−2.04%)
Modal solution	3.456 (−1.93%)	0.214 (1.90%)	2.426 (−2.33%)	1.775 (2.37%)

Input files

fv13t_s3r_c.inp: S3R elements, coarse mesh.
fv13t_s3r_f.inp: S3R elements, fine mesh.
fv13t_s3r_vf.inp: S3R elements, very fine mesh.
fv13t_s3r_f_band.inp: S3R elements, fine mesh, band-limited damping.
fv13t_s3rs_c.inp: S3RS elements, coarse mesh.
fv13t_s3rs_f.inp: S3RS elements, fine mesh.
fv13t_s3rs_vf.inp: S3RS elements, very fine mesh.
fv13t_s3rs_f_band.inp: S3RS elements, fine mesh, band-limited damping.
fv13t_s4r_c.inp: S4R elements, coarse mesh.
fv13t_s4r_f.inp: S4R elements, fine mesh.
fv13t_s4r_vf.inp: S4R elements, very fine mesh.
fv13t_s4r_f_band.inp: S4R elements, fine mesh, band-limited damping.
fv13t_s4rs_c.inp: S4RS elements, coarse mesh.
fv13t_s4rs_f.inp: S4RS elements, fine mesh.
fv13t_s4rs_vf.inp: S4RS elements, very fine mesh.
fv13t_s4rs_f_band.inp: S4RS elements, fine mesh, band-limited damping.
fv13t_s4rsw_c.inp: S4RSW elements, coarse mesh.
fv13t_s4rsw_f.inp: S4RSW elements, fine mesh.
fv13t_s4rsw_vf.inp: S4RSW elements, very fine mesh.
fv13t_s4rsw_f_band.inp: S4RSW elements, fine mesh, band-limited damping.
nft1358x.inp: S8R5 elements.

The modal solution in Abaqus/Standard is obtained from Steps 3 and 4 in nfm1358x.inp.