2. Reference Solution#
2.1. Calculation method#
We want to check two types of quantities:
the first critical buckling load,
the first natural frequency of the vibrating system.
The reference value of the critical load sought is obtained by a quasistatic calculation (keyword CRIT_STAB of STAT_NON_LINE). We take this value obtained at the last quasistatic calculation step, which corresponds to instant \(t=1s\).
Since the number stored under CHAR_CRIT in the result data structure (this is the minimum multiplying coefficient of the load imposed to obtain the buckling load) being proportional to the imposed load, which is monotonic increasing linearly with time, it is corrected to have the true value at the first moment of the transient dynamic calculation, i.e. \(\mathrm{1,001}s\).
By definition, we have the multiplying coefficient CHAR_CRIT:
\({F}_{\mathrm{critique}}=\text{CHAR\_CRIT}({t}_{i})\mathrm{.}{F}_{\mathrm{ext}}({t}_{i})\)
External force is proportional to time: \({F}_{\mathrm{ext}}({t}_{i})={F}_{\mathrm{ext}}\mathrm{.}{t}_{i}\), so \({F}_{\mathrm{critique}}=\text{CHAR\_CRIT}({t}_{i})\mathrm{.}{F}_{\mathrm{ext}}\mathrm{.}{t}_{i}\).
We hypothesize that over a step, the loading evolves very slowly and therefore that we can assimilate the result of the dynamic calculation to a quasistatic evolution during this step. We can then write, for the first dynamic step, which follows the quasistatic calculation:
\(\begin{array}{c}{F}_{\mathit{critique}}\mathrm{=}{\text{CHAR\_CRIT}}_{\text{STAT\_NON\_LINE}}({t}_{i})\mathrm{.}{F}_{\mathit{ext}}\mathrm{.}{t}_{i}\mathrm{\approx }{F}_{\mathit{critique}}\mathrm{=}{\text{CHAR\_CRIT}}_{\text{DYNA\_NON\_LINE}}({t}_{i})\mathrm{.}{F}_{\mathit{ext}}\mathrm{.}{t}_{i}\\ \mathrm{\Rightarrow }\\ {\text{CHAR\_CRIT}}_{\text{DYNA\_NON\_LINE}}({t}_{i+1})\mathrm{\approx }{\text{CHAR\_CRIT}}_{\text{STAT\_NON\_LINE}}({t}_{i})\mathrm{.}\frac{{t}_{i}}{{t}_{i+1}}\mathrm{=}{\text{CHAR\_CRIT}}_{\text{STAT\_NON\_LINE}}({t}_{i})\mathrm{.}\frac{1}{\mathrm{1,001}}\end{array}\)
For vibration analysis, we will do two tests:
using the elastic stiffness matrix,
using the plastic tangent stiffness matrix.
The two reference values are obtained by two linear modal calculations carried out with the operator CALC_MODES.
To obtain the first natural frequency corresponding to the elastic case, a linear elastic calculation is made with CALC_MODES and the initial material defined above (with Young’s modulus equal to \({2.10}^{4}\mathrm{Mpa}\)).
To obtain the first natural frequency corresponding to the tangent plastic case, a linear elastic calculation is made with CALC_MODES and a fictional elastic material whose Young’s modulus is equal to the plastic tangent modulus defined above: \(200\mathrm{Mpa}\), i.e. 100 times less than the real elastic modulus. We will therefore have a natural frequency 10 times lower than the previous one.
We also know the analytical solution of our problem (cube of length 1 consisting of a single linear finite element) which comes down to a 1D case of tensile compression:
\(\omega =\sqrt{2E/\rho }\approx \{\begin{array}{}\mathrm{0,358128}\mathrm{rad}/s:\text{matériau élastique}\\ \mathrm{0,0358128}\mathrm{rad}/s:\text{matériau plastique}\end{array}\)
2.2. Reference quantities and results#
Sizes |
Values |
Unit |
Multiplying coefficient of the first critical buckling load |
-2.85714E+01/1.001 |
|
First elastic natural frequency |
3.58128E-01 |
|
First plastic natural frequency |
3.58128E-02 |
|