Test 5H: Deep simply supported beam: harmonic 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
Results and discussion
Input files

ProductsAbaqus/Standard

Elements tested

C3D4
C3D4H
C3D6
C3D6H
C3D8
C3D8H
C3D8I
C3D8R
C3D8RH

C3D10
C3D10H
C3D10HS
C3D10M
C3D10MH
C3D15
C3D15H
C3D15V
C3D15VH

C3D20
C3D20H
C3D20R
C3D20RH
C3D27
C3D27H
C3D27R
C3D27RH

Problem description

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

Forcing function

Steady-state harmonic.

$F = F_{0}$ $\sin (ω t)$

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

$ω = 2 π f$

$f = $ 40 to 45 Hz

Damping

$ζ = $ 2%

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 5H from NAFEMS “Selected Benchmarks for Forced Vibration,” R0016, March 1993.

Response predicted by Abaqus

Results and discussion

The results are shown in Table 1 and Table 2. The values enclosed in parentheses are percentage differences with respect to the reference solution.

Table 2. Direct solution.
	Peak displacement (mm)	Peak stress (N/mm²)	Frequency (Hz)
Reference solution	13.45	241.9	42.65
B21	13.87 (3.12%)	247.83 (2.45%)	42.63 (−0.05%)
B22	13.97 (3.87%)	249.25 (3.04%)	42.63 (−0.05%)
B23	12.19 (−9.37%)	243.46 (0.64%)	45.33 (6.28%)
B31	13.99 (4.01%)	248.08 (2.55%)	42.68 (0.07%)
B32	13.98 (3.94%)	249.25 (3.04%)	42.63 (−0.05%)
B33	12.19 (−9.37%)	243.46 (0.64%)	45.33 (6.28%)
C3D4	12.89 (−4.16%)	235.83 (−2.51%)	43.06 (0.96%)
C3D4H	12.89 (−4.16%)	235.23 (−2.76%)	43.06 (0.96%)
C3D6	11.43 (−13.93%)	204.56 (−15.44%)	45.76 (7.29%)
C3D6H	11.43 (−13.93%)	204.08 (−15.63%)	45.76 (7.29%)
C3D8	13.54 (0.68%)	214.94 (−11.15%)	42.55 (−0.23%)
C3D8H	13.54 (0.68%)	214.93 (−11.15%)	42.55 (−0.23%)
C3D8I	13.00 (−3.34%)	235.82 (−2.51%)	42.65 (0%)
C3D8IH	13.00 (−3.34%)	235.80 (−2.52%)	42.65 (0%)
C3D8R	14.62 (8.70%)	209.11 (−13.56%)	41.02 (−3.82%)
C3D8RH	14.62 (8.70%)	209.01 (−13.60%)	41.02 (−3.82%)
C3D10	13.13 (−2.38%)	243.66 (0.73%)	42.75 (0.23%)
C3D10H	14.12 (4.98%)	238.68 (−1.33%)	41.43 (2.86%)
C3D10HS	13.13 (−2.38%)	242.29 (0.16%)	42.76 (0.26%)
C3D10M	13.66 (1.56%)	259.82 (7.41%)	41.94 (−1.66%)
C3D10MH	13.66 (1.56%)	259.86 (7.42%)	41.94 (−1.66%)
C3D15	13.31 (−1.04%)	242.08 (0.07%)	42.76 (0.26%)
C3D15V	13.31 (−1.04%)	242.08 (0.07%)	42.76 (0.26%)
C3D15VH	13.30 (−1.12%)	244.53 (1.09%)	42.76 (0.26%)
C3D15H	13.31 (−1.04%)	241.35 (–0.23%)	42.76 (0.26%)
C3D20	13.38 (−0.52%)	242.62 (0.30%)	42.65 (0%)
C3D20H	13.43 (−0.15%)	238.13 (–1.56%)	42.55 (−0.23%)
C3D20R	13.51 (0.45%)	237.53 (−1.81%)	42.45 (−0.46%)
C3D20RH	13.59 (1.04%)	236.71 (−2.15%)	42.35 (−0.70%)
C3D27	13.37 (−0.59%)	242.37 (0.19%)	42.65 (0%)
C3D27H	13.53 (0.59%)	241.44 (−0.19%)	42.24 (−0.96%)
C3D27R	13.48 (0.22%)	237.58 (−1.79%)	42.55 (−0.23%)
C3D27RH	13.77 (2.37%)	241.11 (−0.33%)	42.04 (−1.43%)

Input files

nfh5x21x.inp: B21 elements.
nfh5x22x.inp: B22 elements.
nfh5x23x.inp: B23 elements.
nfh5x31x.inp: B31 elements.
nfh5x32x.inp: B32 elements.
nfh5x33x.inp: B33 elements.
nfh5xf4x.inp: C3D4 elements.
nfh5xh4x.inp: C3D4H elements.
nfh5xf6x.inp: C3D6 elements.
nfh5xh6x.inp: C3D6H elements.
nfh5xf8x.inp: C3D8 elements.
nfh5xi8x.inp: C3D8I elements.
nfh5xj8x.inp: C3D8IH elements.
nfh5xr8x.inp: C3D8R elements.
nfh5xy8x.inp: C3D8RH elements.
nfh5xfax.inp: C3D10 elements.
nfh5xhax.inp: C3D10H elements.
nfh5xiax.inp: C3D10HS elements.
nfh5xkax.inp: C3D10M elements.
nfh5xlax.inp: C3D10MH elements.
nfh5xffx.inp: C3D15 elements.
nfh5xffv.inp: C3D15V elements.
nfh5xhfv.inp: C3D15VH elements.
nfh5xhfx.inp: C3D15H elements.
nfh5xfkx.inp: C3D20 elements.
nfh5xhkx.inp: C3D20H elements.
nfh5xrkx.inp: C3D20R elements.
nfh5xykx.inp: C3D20RH elements.
nfh5xfrv.inp: C3D27 elements.
nfh5xhrv.inp: C3D27H elements.
nfh5xrrv.inp: C3D27R elements.
nfh5xyrv.inp: C3D27RH elements.