2. Benchmark solution#
2.1. Calculation method#
The deformation field is obtained from the displacements and the stress field is obtained using Hooke’s law. The elastic energy \({E}_{\mathit{elas}}\) is then calculated by the following formula:
\({E}_{\mathit{elas}}\mathrm{=}{\mathrm{\int }}_{V}\frac{1}{2}\sigma \mathrm{:}\epsilon \mathit{dV}\) equation 2.1-1
2.1.1. Modeling A#
The homogeneous deformation and stress fields on the element are shown in the.
Component |
\(X\) |
|
|
|
|
|
||
Deformation |
0.001 |
-0.002 |
0.002 |
0.003 |
0.003 |
0.005 |
-0.0001 |
-0.0003 |
Constraint (\(\mathit{Pa}\)) |
1.3888889E6 |
-1.111111E6 |
3.0555556E6 |
4.1166667E5 |
-8.3333333E5 |
-8.3333333E4 |
-2.5E5 |
Table 2.1.1-1 : Strain and stress field
2.1.2. Modeling B: plane deformations#
The homogeneous deformation and stress fields on the element are shown in the.
Component |
\(X\) |
|
|
|
|
|
||
Deformation |
0.001 |
-0.002 |
0.002 |
0.0 |
0.0 |
0.0 |
0.0 |
|
Constraint (\(\mathit{Pa}\)) |
5,5555556 E 5 |
-1, 9444444 E6 |
-2,7777778 E 5 |
4,1666667E5 |
-8,3333333E5 |
-8,3333333E4 |
-2,5E5 |
Table 2.1.2-1 : Strain and stress field
2.1.3. C (plane stresses) and D (shell and plate elements) modeling#
The homogeneous deformation and stress fields on the element are shown in the. For shell and plate elements, this is membrane deformation, with the curvature being zero.
Component |
\(X\) |
|
|
|
|
|
||
Deformation |
0.001 |
-0.002 |
-0.002 |
-0.00025 |
0.0005 |
0.0 |
0.0 |
|
Constraint (\(\mathit{Pa}\)) |
6.25 E 5 |
-1, 875 E5 |
-1, 875 E6 |
0.0 |
0.0 |
0.0 |
0.0 |
Table 2.1.3-1 : Strain and stress field
2.1.4. E modeling: axisymmetric.#
The homogeneous deformation and stress fields on the element are shown in the.
Component |
\(X\) |
|
|
|
|
Deformation |
0.0 |
0.0001 |
0.0 |
0.0 |
|
Constraint (\(\mathit{Pa}\)) |
0.0 |
1.0 |
1.0E5 |
0.0 |
0.0 |
Table 2.1.4-1: Deformation and Stress Field
The elastic energy is calculated according to the by expressing the elementary volume \(\mathit{dV}\) in the cylindrical coordinate system for a slice of infinitesimal thickness \(d\theta\):
\({E}_{\mathit{elas}}\mathrm{=}{\mathrm{\int }}_{V}\frac{1}{2}\sigma \mathrm{:}\epsilon \mathit{dV}\mathrm{=}\frac{1}{2}\sigma \mathrm{:}\epsilon {\mathrm{\int }}_{{R}_{1}}^{{R}_{2}}\mathit{rdr}{\mathrm{\int }}_{{H}_{1}}^{{H}_{2}}\mathit{dz}\)
The terminals of the integral are: \({R}_{1}\mathrm{=}\mathrm{1m}\), \({R}_{2}\mathrm{=}\mathrm{2m}\),, \({H}_{1}\mathrm{=}\mathrm{0m}\), and \({H}_{2}\mathrm{=}\mathrm{1m}\).
2.2. Reference quantities and results#
The elastic energy calculated analytically for each of the models is shown in the.
Modeling |
\(A\) |
|
|
|
|
|
Elastic energy |
6680,555556 J |
2430.555556 J |
2395,833333 J |
2395,833333 J |
2395,833333 J |
\(\mathrm{7,5}{\mathit{J.rad}}^{1}\) |
Table 2.2-1: Elastic energy
2.3. Uncertainties about the solution#
None. It is an analytical solution.