# Real Cyclic Load-Bearing Test of a Ceramic-Reinforced Slab
## 5 Vibrational Response
### Estimation of the Elemental Frequency of the Slab Cross Section  
In order to estimate the fundamental frequency of the structure, assuming full continuity of the slab and the beams at both supporting edges in the two cases, the following simplified expression according to [7] may be used:  
\[f=4/\pi\cdot\sqrt{(3\cdot E\cdot I/(0.37\cdot\mu\cdot L^{4})} \tag{13}\]  
where \(E\) is the estimated elastic modulus of the section, \(I\) is the moment of inertia of the cross section, \(\mu\) is the total weight in a width of one meter of slab (including live load), and \(L\) is the span of the slab.  
In the case of one-way slabs supported by steel beams, like in this case, the fundamental frequency of the whole structure (\(f_{\rm{ea}}\)) may be calculated as the combination of both frequencies (the frequency \(f_{1}\) of the slab and the frequency \(f_{2}\) of the I-beam) by means of the Dunkerley [7] expression:  
\[1/f_{\rm{ea}}=1/f_{1}^{2}+1/f_{2}^{2} \tag{14}\]  
We consider a moment of inertia of the equivalent section of 270,843,903 mm\({}^{4}\) (measured from section of Figure 4) and obtain an approximated elasticity modulus \(E\) of the whole slab by using the value of deflection obtained in the experimental test during the first cycle of loading:  
\[\delta=5\cdot W\cdot\frac{L^{4}}{384\cdot E\cdot I}\ \to E\ =5\cdot W\cdot L^{4}/(384 \cdot\delta\cdot I). \tag{15}\]  
Then, by substituting the obtained value of deflection in the first cycle (7 mm) into the latter expression, generated by a uniform load of 7.50 kN/m\({}^{2}\), it is possible to estimate the secant elastic modulus:  
\[E\ \ =5\cdot 350\cdot\frac{4500^{4}}{384\cdot 7\cdot 270843903}\ =97137\ \rm MPa. \tag{16}\]  
Regarding the steel I-beams, it is possible to calculate quite accurately the fundamental frequency by using the same expression [8]. The mentioned beams are IPN400 profiles (typical steel I-beam cross section of 400 mm height, widely used in Europe) with 7 m span, 104 mm\({}^{4}\) moment of inertia, and elastic modulus of 210,000 MPa, characteristic of steel.  
Then, by considering all previous expressions and the real self-weight (slab + pavement = 2.2 + 1.0 = 3.2 kN/m\({}^{2}\)), the fundamental frequencies of the whole structure were estimated for different loading levels.  
It can be observed in Table 2 that the structure is not comfortable without live or service loads, with a fundamental frequency of almost 5 Hz, clearly below the recommended values; this may be directly demonstrated through simple walking on basement level -1, where vibration of the slab becomes notorious for the user. This is mainly due to the low self-weight ratio of the slab compared to its load-bearing capacity.  
Note that the higher the applied load, the higher the fundamental frequency of the structure, although in any case the frequencies are especially high. This is typical of low-weight structures with  
\begin{table}
\begin{tabular}{c c c c} \hline
**Live Load** & \(f_{1}\) **Slab** & \(f_{s}\) **I-Beam** & **f Total** \\ \hline
0 & 8.41 Hz & 0.45 Hz & 4.84 Hz \\
5 kN/m\({}^{2}\) & 5.09 Hz & 0.29 Hz & 11.94 Hz \\
10 kN/m\({}^{2}\) & 4.02 Hz & 0.23 Hz & 19.05 Hz \\ \hline \end{tabular}
\end{table}
Table 2: Calculated fundamental frequencies.  
high load-bearing capacities. These are the reasons why this type of slab was widely used especially for industrial buildings, where heavy machines worked with high frequencies; slabs with high capabilities and low frequencies were needed.