Test 5T: Deep simply supported beam: 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

B21

B22

B31

B32

Problem description

Material and geometry specifications are as given in Test 5: Deep simply supported beam: frequency extraction.

Mesh:

A coarse mesh and a fine mesh are tested in the Abaqus/Explicit analyses.

Forcing function

Suddenly applied step load, transverse to the beam.

$F_{0} = $ 1 MN/m over whole length of beam.

Damping

$ζ = $ 2% [2% of critical damping in the dominant first mode with analytical frequency value $f = $ 42.650 (Hz) or $ω = 2 π f = $ 267.98 (sec⁻¹)].

The damping factors are chosen as $α_{R} = $ 5.36 (sec⁻¹) and $β_{R} = $ 7.46 × 10⁻⁵ (sec) so that

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

Response

$u_{y}$ and extreme fiber bending stress.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 5T 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 5. The values enclosed in parentheses are percentage differences with respect to the reference solution. The static displacement was obtained by running the analysis for a second step with a time period of ten seconds.

Table 1. Element type: B21, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	1.043	0.0117	18.76	0.538
Coarse mesh	0.991	0.0120	17.35	0.510
Coarse mesh	(−4.99%)	(2.56%)	(−7.52%)	(−5.20%)
Fine mesh	1.009	0.0120	18.04	0.520
Fine mesh	(−3.24%)	(2.56%)	(−3.84%)	(−3.35%)

Table 2. Element type: B22, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	1.043	0.0117	18.76	0.538
Coarse mesh	1.042	0.0119	17.80	0.532
Coarse mesh	(−0.09%)	(1.70%)	(−5.12%)	(−1.12%)
Fine mesh	1.043	0.0116	18.17	0.537
Fine mesh	(0.00%)	(−0.85%)	(−3.14%)	(−0.19%)

Table 3. Element type: B31, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	1.043	0.0117	18.76	0.538
Coarse mesh	0.991	0.0120	17.37	0.510
Coarse mesh	(−4.99%)	(2.56%)	(−7.41%)	(−5.20%)
Fine mesh	1.010	0.0120	18.06	0.518
Fine mesh	(−3.16%)	(2.56%)	(−3.73%)	(−3.72%)

Table 4. Element type: B32, Abaqus/Explicit analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	1.043	0.0117	18.76	0.538
Coarse mesh	1.042	0.0117	17.80	0.535
Coarse mesh	(−0.09%)	(0.00%)	(−5.11%)	(−0.56%)
Fine mesh	1.043	0.0116	18.17	0.539
Fine mesh	(0.00%)	(−0.85%)	(−3.14%)	(0.19%)

Table 5. Element type: B32, Abaqus/Standard analysis.
	Peak displacement		Peak stress	Static disp.
	$δ_{p}$ (mm)	$t_{p}$ (sec)	$σ$ (N/mm²)	$δ_{s}$ (mm)
Reference solution	1.043	0.0117	18.76	0.538
Direct solution	1.043	0.0118	18.29	0.536
Direct solution		(0.85%)	(−2.50%)	(−0.37%)
Modal solution	1.041	0.0116	18.09	0.507
Modal solution	(−0.19%)	(−0.85%)	(−5.57%)	(−5.76%)

Input files

Abaqus/Standard input files

nft5x32x.inp: B32 elements.

The modal solution in Abaqus/Standard is obtained from Steps 3 and 4 in nfm5x32x.inp.

Abaqus/Explicit input files

fv5t_b21_c.inp: B21 elements, coarse mesh.
fv5t_b21_f.inp: B21 elements, fine mesh.
fv5t_b22_c.inp: B22 elements, coarse mesh.
fv5t_b22_f.inp: B22 elements, fine mesh.
fv5t_b31_c.inp: B31 elements, coarse mesh.
fv5t_b31_f.inp: B31 elements, fine mesh.
fv5t_b32_c.inp: B32 elements, coarse mesh.
fv5t_b32_f.inp: B32 elements, fine mesh.