Figure 7.15 shows a steel stepped cantilever beam. The steps have square cross sections of size \(100 \mathrm{~mm} \times 100 \mathrm{~mm}\) and \(50 \mathrm{~mm} \times 50 \mathrm{~mm}\) each with a length of \(1.2 \mathrm{~m}\). Assuming the Young's modulus as E \(=200 \mathrm{GPa}\) and the density as \(8000 \mathrm{~kg} / \mathrm{m}^{3}\) for the material of the

beam, determine the fundamental natural frequency of bending vibration of the beam using Rayleigh's method. Assume the deflection of the beam as \[y(x)=C\left(1-\cos \frac{\pi x}{2 l}\right)\]
where \(C\) is a constant.

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A h h1 X2 m2 x1 m1 K FIGURE 7.14 Two-story shear building.