Test 21T: Simply supported thick 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

S4R5

S8R

SC8R

Problem description

Material and geometry specifications are as given in Test 21: Simply supported thick square plate: frequency extraction. The plate thickness for the Abaqus/Explicit analyses is 1.0 in.

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 meshes are 2 × 2 × 4, 3 × 3 × 4, and 4 × 4 × 4, respectively.

Forcing function

Suddenly applied pressure.

$F_{0} = $ 1 MN/m² over whole plate.

Damping

$ζ = $ 2% [2% of critical damping in the dominant first mode with analytical frequency value $f = $ 45.897 (Hz) or $ω = 2 π f = $ 288.379 (sec⁻¹)].

The damping factors are chosen as $α_{R} = $ 5.772 (sec⁻¹) and $β_{R} = $ 6.929 × 10⁻⁵ (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} = $ 5.772 (sec⁻¹), damping ratio $η_{d} = $ 0.01, high-frequency cutoff $f_{h} = $ 50 (Hz), and low-frequency cutoff $f_{l} = $ 1.5 (Hz).

Response location

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

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 21T 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 10. 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 about 2 seconds.

Table 1. Element type: S4, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Direct solution	4.569 (0.99%)	0.0108 (0.00%)	58.61 (−5.63%)	2.338 (0.21%)
Modal solution	4.564 (0.88%)	0.0107 (−0.93%)	58.57 (−5.69%)	2.334 (0.04%)

Table 2. Element type: S4R, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Direct solution	4.603 (1.75%)	0.0108 (0.00%)	56.91 (−8.37%)	2.338 (0.21%)
Modal solution	4.534 (0.22%)	0.0107 (−0.93%)	53.82 (−13.35%)	2.334 (0.04%)

Table 3. Element type: S4R5, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Direct solution	4.599 (1.66%)	0.0108	56.90 (−8.68%)	2.339 (0.26%)
Modal solution	4.536 (0.27%)	0.0109 (0.93%)	54.04 (−12.99%)	2.335 (0.09%)

Table 4. Element type: S8R, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Direct solution	4.571 (1.04%)	0.0104 (−3.70%)	64.03 (3.09%)	2.331 (−0.09%)
Modal solution	4.578 (1.19%)	0.0106 (−1.85%)	63.88 (2.85%)	2.331 (−0.09%)

Table 5. Element type: SC8R, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Direct solution	4.627 (2.27%)	0.0110 (1.85%)	57.9 (−6.77%)	2.337 (−0.17%)
Modal solution	4.544 (0.33%)	0.0107 (−0.93%)	54.0 (13.1%)	2.339 (0.26%)

Table 6. Element type: S3R, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Coarse mesh	4.223	0.0110	52.78	2.170
Coarse mesh	(−6.65%)	(1.85%)	(−15.02%)	(−7.00%)
Fine mesh	4.441	0.0107	57.02	2.265
Fine mesh	(−1.83%)	(−0.92%)	(−8.20%)	(−2.91%)
Very fine mesh	4.517	0.0107	58.67	2.296
Very fine mesh	(−0.15%)	(−0.92%)	(−5.54%)	(−1.58%)

Table 7. Element type: S3RS, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Coarse mesh	4.208	0.0109	53.00	2.148
Coarse mesh	(−6.98%)	(0.93%)	(−19.50%)	(−7.93%)
Fine mesh	4.425	0.0106	56.99	2.252
Fine mesh	(−2.19%)	(−1.85%)	(−8.24%)	(−3.47%)
Very fine mesh	4.507	0.0107	58.67	2.288
Very fine mesh	(−0.38%)	(−0.93%)	(−5.54%)	(−1.93%)

Table 8. Element type: S4R, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Coarse mesh	4.466	0.0122	48.21	2.368
Coarse mesh	(−1.28%)	(12.96%)	(−22.38%)	(1.50%)
Fine mesh	4.556	0.0114	55.02	2.348
Fine mesh	(0.71%)	(5.56%)	(−11.42%)	(0.64%)
Very fine mesh	4.577	0.0110	57.33	2.351
Very fine mesh	(1.17%)	(1.85%)	(−7.70%)	(0.77%)

Table 9. Element type: S4RS, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Coarse mesh	4.624	0.0125	49.84	2.381
Coarse mesh	(2.21%)	(15.74%)	(−19.75%)	(2.06%)
Fine mesh	4.625	0.0114	55.75	2.353
Fine mesh	(2.23%)	(5.56%)	(−10.24%)	(0.86%)
Very fine mesh	4.612	0.0110	57.67	2.344
Very fine mesh	(1.95%)	(1.85%)	(−7.15%)	(0.47%)

Table 10. Element type: S4RSW, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
Coarse mesh	4.618	0.0121	49.77	2.381
Coarse mesh	(2.08%)	(12.04%)	(−19.87%)	(2.06%)
Fine mesh	4.623	0.0114	55.74	2.353
Fine mesh	(2.19%)	(5.56%)	(−10.26%)	(0.86%)
Very fine mesh	4.612	0.0109	57.67	2.344
Very fine mesh	(1.95%)	(0.93%)	(−7.15%)	(0.47%)

Table 11. Band-limited damping with fine mesh, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	4.524	0.0108	62.11	2.333
S3R	4.258	0.0105	53.87	2.265
S3R	(−5.88%)	(−2.78%)	(−13.27%)	(−2.91%)
S3RS	4.246	0.0106	54.21	2.252
S3RS	(−6.15%)	(-1.89%)	(−12.72%)	(−3.47%)
S4R	4.381	0.0110	53.04	2.348
S4R	(−3.16%)	(1.85%)	(−14.60%)	(0.64%)
S4RS	4.449	0.0115	54.23	2.353
S4RS	(−1.66%)	(6.48%)	(−12.69%)	(0.86%)
S4RSW	4.448	0.0115	54.22	2.353
S4RSW	(−1.68%)	(6.48%)	(−12.70%)	(0.86%)

Input files

Abaqus/Standard input files

nft21e4x.inp: S4 elements.
nft21f4x.inp: S4R elements.
nft2154x.inp: S4R5 elements.
nft2168x.inp: S8R elements.
nft21_std_sc8r.inp: SC8R elements.

The modal solution in Abaqus/Standard is obtained from Steps 3 and 4 in files whose names begin with nfm21.

Abaqus/Explicit input files

fv21t_s3r_c.inp: S3R elements, coarse mesh.
fv21t_s3r_f.inp: S3R elements, fine mesh.
fv21t_s3r_vf.inp: S3R elements, very fine mesh.
fv21t_s3r_f_band.inp: S3R elements, fine mesh, band-limited damping.
fv21t_s3rs_c.inp: S3RS elements, coarse mesh.
fv21t_s3rs_f.inp: S3RS elements, fine mesh.
fv21t_s3rs_vf.inp: S3RS elements, very fine mesh.
fv21t_s3rs_f_band.inp: S3RS elements, fine mesh, band-limited damping.
fv21t_s4r_c.inp: S4R elements, coarse mesh.
fv21t_s4r_f.inp: S4R elements, fine mesh.
fv21t_s4r_vf.inp: S4R elements, very fine mesh.
fv21t_s4r_f_band.inp: S4R elements, fine mesh, band-limited damping.
fv21t_s4rs_c.inp: S4RS elements, coarse mesh.
fv21t_s4rs_f.inp: S4RS elements, fine mesh.
fv21t_s4rs_vf.inp: S4RS elements, very fine mesh.
fv21t_s4rs_f_band.inp: S4RS elements, fine mesh, band-limited damping.
fv21t_s4rsw_c.inp: S4RSW elements, coarse mesh.
fv21t_s4rsw_f.inp: S4RSW elements, fine mesh.
fv21t_s4rsw_vf.inp: S4RSW elements, very fine mesh.
fv21t_s4rsw_f_band.inp: S4RSW elements, fine mesh, band-limited damping.