Table of Contents
LN2 BEAM-TIMO element cantilever validation
This validation aims at assessing the behavior of the SesamX LN2 BEAM-TIMO element (Timoshenko beam element) against the equivalent Abaqus B31 element.
Validation model
The model studied for this comparison is made of a cantilever beam assembly.
The beam is clamped at one end and subjected to an arbitrary load at the other end, where:
- the beam is $1 m$ long and made of 101 nodes and 100 elements.
- the beam properties come from a square section of $10 cm$ side. The area of the section is $10000 mm^2$, the torsion constant is $1.406e7 mm^4$, the area moments of inertia are $8.33e6mm^4$ and the shear correction factors are $0.44$.
- a linear elastic material is applied with $E = 200 GPa$ and $\nu = 0.33$.
This case is solved as a linear static resolution.
Validation results
The following figure gives an overview of the cantilever beam displacements, as well as the node numbers.
The following table gives the comparison of the nodal translations between Abaqus and SesamX.
| Abaqus | SesamX | Comparison | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Node id | $u_x (m)$ | $u_y (m)$ | $u_z (m)$ | Magnitude $(m)$ | $u_x (m)$ | $u_y (m)$ | $u_z (m)$ | Magnitude $(m)$ | Magnitude error |
| 25 | 1.20E-06 | 5.92E-04 | 3.26E-04 | 6.76E-04 | 1.20E-06 | 5.92E-04 | 3.26E-04 | 6.76E-04 | -0.01% |
| 50 | 2.45E-06 | 2.32E-03 | 1.13E-03 | 2.58E-03 | 2.45E-06 | 2.32E-03 | 1.13E-03 | 2.58E-03 | 0.00% |
| 75 | 3.70E-06 | 4.99E-03 | 2.14E-03 | 5.43E-03 | 3.70E-06 | 4.99E-03 | 2.14E-03 | 5.42E-03 | 0.00% |
| 101 | 5.00E-06 | 8.56E-03 | 3.09E-03 | 9.10E-03 | 5.00E-06 | 8.56E-03 | 3.09E-03 | 9.10E-03 | 0.00% |
The following table gives the comparison of the nodal rotations between Abaqus and SesamX.
| Abaqus | SesamX | Comparison | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Node id | $r_x (rad)$ | $r_y (rad)$ | $r_z (rad)$ | Magnitude (rad) | $r_x (rad)$ | $r_y (rad)$ | $r_z (rad)$ | Magnitude $(rad)$ | Magnitude error |
| 25 | 1.14E-03 | -2.36E-03 | 4.70E-03 | 5.38E-03 | 1.14E-03 | -2.36E-03 | 4.70E-03 | 5.38E-03 | 0.00% |
| 50 | 2.32E-03 | -3.72E-03 | 8.85E-03 | 9.88E-03 | 2.32E-03 | -3.72E-03 | 8.85E-03 | 9.88E-03 | 0.00% |
| 75 | 3.50E-03 | -3.95E-03 | 1.23E-02 | 1.33E-02 | 3.50E-03 | -3.95E-03 | 1.23E-02 | 1.33E-02 | 0.00% |
| 101 | 4.73E-03 | -3.00E-03 | 1.50E-02 | 1.60E-02 | 4.73E-03 | -3.00E-03 | 1.50E-02 | 1.60E-02 | 0.00% |
The following table gives the comparison of the strain on the neutral axis between Abaqus and SesamX.
| Abaqus | SesamX | Comparison | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Element id | $\varepsilon_{11} (\%)$ | $\varepsilon_{13} (\%)$ | $\varepsilon_{12} (\%)$ | Magnitude $(\%)$ | $\varepsilon_{11} (\%)$ | $\varepsilon_{13} (\%)$ | $\varepsilon_{12} (\%)$ | Magnitude $(\%)$ | Magnitude error |
| 1 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 0.00% |
| 25 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 0.00% |
| 50 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 0.00% |
| 75 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 5.00E-04 | 9.07E-03 | 6.05E-03 | 1.09E-02 | 0.00% |
The following table gives the comparison of the curvatures about the 2nd local axis and the 3rd local axis, as well as the twist of the beam, between Abaqus and SesamX.
| Abaqus | SesamX | Comparison | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Element id | $\kappa_1 (\%/m)$ | $\kappa_2 (\%/m)$ | $\psi (\%/m)$ | Magnitude $(\%/m)$ | $\kappa_1 (\%/m)$ | $\kappa_2 (\%/m)$ | $\psi (\%/m)$ | Magnitude $(\%/m)$ | Magnitude error |
| 1 | -1.19E+00 | 2.09E+00 | 4.73E-01 | 2.46E+00 | -1.19E+00 | 2.09E+00 | 4.73E-01 | 2.46E+00 | 0.00% |
| 25 | -7.59E-01 | 1.81E+00 | 4.73E-01 | 2.02E+00 | -7.59E-01 | 1.81E+00 | 4.73E-01 | 2.02E+00 | 0.00% |
| 50 | -3.09E-01 | 1.51E+00 | 4.73E-01 | 1.61E+00 | -3.09E-01 | 1.51E+00 | 4.73E-01 | 1.61E+00 | 0.00% |
| 75 | 1.41E-01 | 1.21E+00 | 4.73E-01 | 1.30E+00 | 1.41E-01 | 1.21E+00 | 4.73E-01 | 1.30E+00 | 0.00% |
The following table gives the comparison of the section forces: the axial force, the shear force along the 3rd local axis and the shear force along the 2nd local axis, between Abaqus and SesamX.
| Abaqus | SesamX | Comparison | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Element id | $F_1 (N)$ | $F_3 (N)$ | $F_2 (N)$ | Magnitude $(N)$ | $F_1 (N)$ | $F_3 (N)$ | $F_2 (N)$ | Magnitude $(N)$ | Magnitude error |
| 1 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 0.00% |
| 25 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 0.00% |
| 50 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 0.00% |
| 75 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 1.00E+04 | 3.00E+04 | 2.00E+04 | 3.74E+04 | 0.00% |
The following table gives the comparison of the section moments: the bending moment about the 2nd local axis, the bending moment about the 3rd local axis and the twisting moment, between Abaqus and SesamX.
| Abaqus | SesamX | Comparison | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Element id | $M_2 (Nm)$ | $M_3 (Nm)$ | $M_1 (Nm)$ | Magnitude $(Nm)$ | $M_2 (Nm)$ | $M_3 (Nm)$ | $M_1 (Nm)$ | Magnitude $(Nm)$ | Magnitude error |
| 1 | -1.99E+04 | 3.49E+04 | 5.00E+03 | 4.05E+04 | -1.99E+04 | 3.49E+04 | 5.00E+03 | 4.05E+04 | 0.00% |
| 25 | -1.27E+04 | 3.01E+04 | 5.00E+03 | 3.30E+04 | -1.27E+04 | 3.01E+04 | 5.00E+03 | 3.30E+04 | 0.00% |
| 50 | -5.15E+03 | 2.51E+04 | 5.00E+03 | 2.61E+04 | -5.15E+03 | 2.51E+04 | 5.00E+03 | 2.61E+04 | 0.00% |
| 75 | 2.35E+03 | 2.01E+04 | 5.00E+03 | 2.08E+04 | 2.35E+03 | 2.01E+04 | 5.00E+03 | 2.08E+04 | 0.00% |
The results are identical between SesamX and Abaqus. The SesamX LN2 BEAM-TIMO element implementation is the same as the Abaqus B31 implementation.