2. Benchmark solution#
2.1. Calculation method#
It is possible to calculate the natural frequencies of the first and second modes of flexural vibration of the plate because it functions like a console beam.
The frequency \({f}_{1}\) of the first natural mode is written as:
\({f}_{1}=\frac{\mathrm{3,5156}}{2\pi {L}^{2}}\sqrt{\frac{\mathrm{EI}}{\rho }}\)
with \(L\) the length of the console (1 m here), \(\mathrm{EI}\) l e produces the flexural inertia by Young’s modulus for the complete structure and \(\rho\) the mass of the structure per unit length.
In the same way, frequency \({f}_{2}\) of the second natural mode is written:
\({f}_{2}=\frac{\mathrm{22,0336}}{2\pi {L}^{2}}\sqrt{\frac{\mathrm{EI}}{\rho }}\)
To calculate \({f}_{1}\) and \({f}_{2}\), we break down the parts related to concrete and reinforcements:
\((\mathrm{EI})={(\mathrm{EI})}_{\mathrm{beton}}+{(\mathrm{EI})}_{\mathrm{acier}}\)
where
\({(\mathrm{EI})}_{\mathrm{acier}}=2{E}_{\mathrm{acier}}({\mathrm{sL}}_{2}){e}_{\mathrm{exc}}^{2}\)
with \({E}_{\mathrm{acier}}\) the Young’s modulus of steel, \(s\) the section of the reinforcements per linear meter and \({e}_{\mathrm{exc}}\) the eccentricity of the reinforcing sheets with respect to the mean sheet, and:
\({(\mathrm{EI})}_{\mathrm{beton}}={E}_{\mathrm{beton}}{L}_{2}\frac{{e}^{3}}{12}\)
where \({E}_{\mathrm{beton}}\) is the Young’s modulus of concrete.
For mass per unit length, we break down the mass of concrete and the mass of steel.
By using the preceding equations, it then becomes possible to calculate the frequency of the natural modes considered. The results are:
Frequency |
Reference |
First flexure mode |
\(\mathrm{54,67}\mathrm{Hz}\) |
Second mode of flexure |
\(\mathrm{342,64}\mathrm{Hz}\) |
The center of gravity is located in the center of the console. So his coordinates are: \((0.5,0.05,0)\). The inertia along the y axis of the complete structure is:
\({I}_{\mathrm{yy}}(G)={\int }_{V}({(x-{x}_{G})}^{2}+{(z-{z}_{G})}^{2})\rho \mathrm{dv}\)
with \(({x}_{G},{y}_{G},{z}_{G})\) the coordinates of the center of gravity, \(V\) the volume of the structure, and \(\rho\) its density. By breaking down the elements related to concrete and those related to steel, it is possible to calculate inertia analytically:
\({I}_{\mathrm{yy}}(G)=\mathrm{8,611}{m}^{4}\)
2.2. Reference quantities and results#
The reference results are summarised in the following table.
Sizes |
Reference |
First flexure mode |
\(\mathrm{54,67}\mathrm{Hz}\) |
Second mode of flexure |
\(\mathrm{342,64}\mathrm{Hz}\) |
Inertia along the \(y\) axis |
|
Next moment \(z\) to \(t=\mathrm{0,1}s\) |
|
Next move \(z\) to \(t=\mathrm{0,09}s\) |
|
Next move \(z\) to \(t=\mathrm{0,1}s\) |
|
Total kinetic energy at \(t=\mathrm{0,1}s\) |
|
2.3. Uncertainties about the solution#
Analytical solutions for clean modes.
Comparisons with EUROPLEXUS for temporal responses while moving, reactions, and kinetic energy, for sine wave loading
2.4. Bibliographical references#
HUGHES T.J.R., COHEN M., HAROUN, M.: « Reduced and selective integration techniques in the finite element analysis of plates », Nuclear Engineering and Design, vol. 46, p. 203-222 (1978).
[R3.07.03] — plate elements DKT, DST,, DKQ,, DSQ, and Q4g.