2. Benchmark solution#
2.1. Bibliographical reference#
The reference solution comes from the results obtained in « Formulas for Stress and Strain » by ROARK’S (6th edition - McGraw-Hill International Editions), which can also be found in the fact sheet on the strength of materials by Jean Goulet and Jean-Pierre Boutin, page 192, combined with « Formulas for Stress, Strains and Structural Matrixes » by W. D. Pilkey, page 969.
2.2. Contactless reference analytical results#
The results obtained were calculated on a plate in simple support subjected to vertical pressure (no contact).
Calculation of the arrow at the center of the plate:
\(\mathrm{max}z=-\frac{\alpha p{l}^{4}}{E{e}^{3}}\)
Calculation of the stress at the center of the plate according to the width of the lower skin plate:
\({\sigma }_{\mathrm{yy}}=+\frac{\beta p{l}^{2}}{{e}^{2}}\)
\(p\) refers to the pressure applied to the plate,
\(E\) Young’s modulus,
\(L\), the length,
\(l\), the width,
\(e\), the thickness,
\(\alpha ,\beta\) being two coefficients obtained from the \(a/b\) ratio.
\(\{\begin{array}{}\alpha =0.1110\\ \beta =0.6102\end{array}\)
Either:
\(\{\begin{array}{}\mathrm{max}z=-0.69375\mathrm{cm}\\ {\sigma }_{\mathrm{yy}}=1.5255{10}^{10}N/{\mathrm{cm}}^{2}\end{array}\)
2.3. Results obtained without contact with Code_Aster#
2.3.1. Modeling DKT#
Modeling: DKTpour test the contact between two plates.
256 finite elements QUAD4 are arranged on the initial contact surface. The mesh has only one layer of elements in the thickness of the plate.
2.3.2. Characteristics of the mesh#
Number of knots: 289 knots
Number of meshes and type: 256 QUAD4
2.3.3. Aster reference values#
Identification |
Reference |
Error |
|
\(\mathrm{DZ}\) in the center of the plate |
|
|
0.35% |
\({\sigma }_{\mathrm{yy}}\) in the center of the plate |
|
|
0.28% |
2.3.4. Modeling COQUE_3D#
Modeling: COQUE_3Dpour test the contact between two plates.
256 finite elements QUAD9 are arranged on the initial contact surface. The mesh has only one layer of elements in the thickness of the plate.
2.3.5. Characteristics of the mesh#
Number of knots: 578 knots
Number of meshes and type: 256 QUAD9
2.3.6. Aster reference values#
Identification |
Reference |
Aster **** (COQUE_3D) ** |
Error |
\(\mathrm{DZ}\) in the center of the plate |
|
|
4.97% |
\({\sigma }_{\mathrm{yy}}\) in the center of the plate |
|
|
7.36% |
2.4. Comments#
It can be seen that the results obtained without contact are very satisfactory for a modeling of the DKT type. The error obtained for a modeling of the COQUE_3D type can be explained by taking into account the effects of transverse shear which should not be negligible for this relatively thick plate since the \(l/e\) ratio is equal to \(1/10\).