2. Reference solution#
2.1. Method used to calculate the reference solution#
The first reference results for this problem are the results obtained by Bathe and Bolourchi [1] in 1979 with a beam modeling. Bathe obtained essentially the same results in 1990 with 3D modeling, which is therefore completely different from the beam model, which is why its results are widely accepted by the scientific community as a reference value.
Since then, several other researchers have addressed this problem. We therefore choose to compare our results to an average of 8 codes.
2.2. Benchmark results#
We are interested in the coordinates in the global coordinate system of point \(B\) for \(t=0\text{.}5\) (that is, \(F\mathrm{=}\mathrm{300N}\)) and \(t=1\text{.}0\) (\(F\mathrm{=}\mathrm{600N}\)).
Below is a summary table of the results obtained by various researchers since the introduction of the problem as well as the average of these results and the maximum deviation from this mean. The average, shown in bold, is chosen as the reference for each test.
Charge |
\(F\mathrm{=}\mathrm{300N}\) |
\(F\mathrm{=}\mathrm{600N}\) |
||||
\(X\) |
\(Y\) |
\(Z\) |
\(X\) |
\(Y\) |
\(Z\) |
|
ADINA -1 (1979) |
22.5 |
59.2 |
39.5 |
15.9 |
47.2 |
53.4 |
ADINA -2 (1990) |
22.2 |
58.5 |
40.4 |
15.7 |
46.8 |
53.6 |
NACS -1 (1993) |
22.6 |
59.2 |
39.5 |
15.9 |
47.2 |
53.4 |
NACS -2 (1993) |
22.3 |
58.6 |
40.3 |
15.7 |
46.7 |
53.6 |
Cardona and Geradin (1988) |
22.14 |
58.64 |
40.35 |
15.55 |
47.04 |
53.5 |
Crisfield (1990) |
22.16 |
58.53 |
40.35 |
15.61 |
46.84 |
53,71 |
Crivelli and Felippa (1993) |
22.31 |
58.85 |
40.08 |
15.75 |
47.25 |
53,37 |
Shakourzadeh (1994) |
21.99 |
58.4 |
40.49 |
15,24 |
46.91 |
53,55 |
Mean |
22,28 |
58.74 |
40.12 |
15,67 |
46.99 |
53,52 |
Maximum deviation |
1.44 |
0.78 |
1.55 |
2.74 |
0.62 |
0.36 |
ADINA are the results obtained by Bathe in [1], we can find the other results in [2].
2.3. Uncertainty about the solution#
Between 1 and 3% (maximum relative difference to the mean of the results).
2.4. Bibliographical references#
Bathe KJ, Bolourchi S. Large displacement analysis of three-dimensional beam structures. Int J Numer Meth Eng 1979; 14:1961 —86.
Shakourzadeh. Modeling of three-dimensional thin-walled beam structures and simulation of nonlinear geometric and elasto-plastic behavior. Doctoral thesis, University of Compiègne, Compiègne (1994).