2. Benchmark solution#
2.1. Benchmark solution#
The reference solution is:
« ANALYTIQUE « for the resulting efforts and moments,
« NON_REGRESSION « for displacements and constraints (modeling A)
« AUTRE_ASTER « for movements and constraints (B and C models). The results obtained with modeling A serve as a reference solution.
Calculations of the resulting efforts and moments
Line load case \(\mathit{fx}\)
\({R}_{x}\mathrm{=}{\mathrm{\int }}_{s}{f}_{x}\mathit{ds}\mathrm{=}{f}_{x}\mathrm{.}h\mathrm{.}l\mathrm{=}2000N\)
\({M}_{y}\mathrm{=}{\mathrm{\int }}_{s}{f}_{x}z\mathit{ds}\mathrm{=}\frac{({f}_{x}\mathrm{.}{h}^{2})}{2}l\mathrm{=}20000\mathit{N.mm}\)
\({M}_{z}\mathrm{=}{\mathrm{\int }}_{s}{f}_{x}y\mathit{ds}\mathrm{=}\frac{({f}_{x}\mathrm{.}{l}^{2})}{2}h\mathrm{=}200000\mathit{N.mm}\)
Line load case \(\mathit{fy}\)
\({R}_{y}\mathrm{=}{\mathrm{\int }}_{s}{f}_{y}\mathit{ds}\mathrm{=}{f}_{y}\mathrm{.}h\mathrm{.}l\mathrm{=}2000N\)
\({M}_{x}\mathrm{=}{\mathrm{\int }}_{s}{f}_{y}z\mathit{ds}\mathrm{=}\frac{({f}_{y}\mathrm{.}{h}^{2})}{2}l\mathrm{=}20000\mathit{N.mm}\)
\({M}_{z}\mathrm{=}{\mathrm{\int }}_{s}{f}_{y}L\mathit{ds}\mathrm{=}{f}_{y}\mathrm{.}h\mathrm{.}l\mathrm{.}L\mathrm{=}1500000\mathit{N.mm}\)
Line load case \(\mathit{fz}\)
\({R}_{z}\mathrm{=}{\mathrm{\int }}_{s}{f}_{z}\mathit{ds}\mathrm{=}{f}_{z}\mathrm{.}h\mathrm{.}l\mathrm{=}2000N\)
\({M}_{x}\mathrm{=}{\mathrm{\int }}_{s}{f}_{z}y\mathit{ds}\mathrm{=}\frac{({f}_{z}\mathrm{.}{l}^{2})}{2}h\mathrm{=}200000\mathit{N.mm}\)
\({M}_{y}\mathrm{=}{\mathrm{\int }}_{s}{f}_{z}L\mathit{ds}\mathrm{=}{f}_{z}\mathrm{.}h\mathrm{.}l\mathrm{.}L\mathrm{=}1500000\mathit{N.mm}\)
2.2. Reference quantities#
Resulting efforts: \(\mathit{RESULT}\text{\_}X\), \(\mathit{RESULT}\text{\_}Y\), \(\mathit{RESULT}\text{\_}Z\)
Resulting moments: \(\mathit{MOMENT}\text{\_}X\), \(\mathit{MOMENT}\text{\_}Y\), \(\mathit{MOMENT}\text{\_}Z\)
Travel: \(\mathrm{DX}\), \(\mathit{DY}\), \(\mathit{DZ}\)
Constraints: \(\mathit{SIXX}\)
2.3. Benchmark results#
Line load case \(\mathit{fx}\)
Identification |
Reference type |
Reference value |
||
Point |
GROUP_MA |
Components |
||
\(A\) |
GAUCHE |
\(\mathit{RESULT}\text{\_}X\) |
“ANALYTIQUE” |
-2,000. N |
DROITE |
\(\mathit{RESULT}\text{\_}X\) |
“ANALYTIQUE” |
2,000. N |
|
GAUCHE |
\(\mathit{MOMENT}\text{\_}Y\) |
“ANALYTIQUE” |
-20,000. N.mm |
|
GAUCHE |
\(\mathit{MOMENT}\text{\_}Z\) |
“ANALYTIQUE” |
-200,000. N.mm |
|
Identification |
Reference type |
Reference value |
|
Point \(A\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
1.067538 MPa |
Point \(B\) |
\(\mathrm{DX}\) |
“NON_REGRESSION” |
5.676675E-3 mm |
Point \(C\) |
\(\mathit{DX}\) |
“NON_REGRESSION” |
5.837494E-3 mm |
Point \(D\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
1.030237 MPa |
Point \(F\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
1.435778 MPa |
Point \(I\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
1.472049 MPa |
Line load case \(\mathit{fy}\)
Identification |
Reference type |
Reference value |
||
Point |
GROUP_MA |
Components |
||
\(A\) |
GAUCHE |
\(\mathit{RESULT}\text{\_}Y\) |
“ANALYTIQUE” |
-2,000. N |
DROITE |
\(\mathit{RESULT}\text{\_}Y\) |
“ANALYTIQUE” |
2,000. N |
|
GAUCHE |
\(\mathit{MOMENT}\text{\_}X\) |
“ANALYTIQUE” |
20,000. N.mm |
|
GAUCHE |
\(\mathit{MOMENT}\text{\_}Z\) |
“ANALYTIQUE” |
-1.50 E6 N.mm |
|
Identification |
Reference type |
Reference value |
|
Point \(A\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
4.55542676 MPa |
Point \(B\) |
\(\mathit{DY}\) |
“NON_REGRESSION” |
0.15476384 mm |
Point \(C\) |
\(\mathit{DY}\) |
“NON_REGRESSION” |
0.15475096 mm |
Point \(D\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
-4.43905267 MPa |
Point \(F\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
26.8162393 MPa |
Point \(I\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
-26.9483395 MPa |
Line load case \(\mathit{fz}\)
Identification |
Reference type |
Reference value |
||
Point |
GROUP_MA |
Components |
||
\(A\) |
GAUCHE |
\(\mathit{RESULT}\text{\_}Z\) |
“ANALYTIQUE” |
2,000. N |
DROITE |
\(\mathit{RESULT}\text{\_}Z\) |
“ANALYTIQUE” |
-2,000. N |
|
GAUCHE |
\(\mathit{MOMENT}\text{\_}X\) |
“ANALYTIQUE” |
200,000. N.mm |
|
GAUCHE |
\(\mathit{MOMENT}\text{\_}Y\) |
“ANALYTIQUE” |
-1.50 E6 N.mm |
|
Identification |
Reference type |
Reference value |
|
Point \(A\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
-132.297834 MPa |
Point \(B\) |
\(\mathit{DZ}\) |
“NON_REGRESSION” |
-6.2097302 mm |
Point \(C\) |
\(\mathit{DZ}\) |
“NON_REGRESSION” |
-6.2082328 mm |
Point \(D\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
-131.564485 Mpa |
Point \(F\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
37.0401177 MPa |
Point \(I\) |
\(\mathit{SIXX}\) |
“NON_REGRESSION” |
36.1921083 MPa |
2.4. Uncertainty about the solution#
Analytical and numerical solutions.