The tapered cantilever beam shown in Fig. 12.15 is used as a spring to carry a load \(P\). Derive the stiffness matrix of the beam using a one-element idealization. Assume \(B=25 \mathrm{~cm}\), \(b=10 \mathrm{~cm}, t=2.5 \mathrm{~cm}, l=2 \mathrm{~m}, E=2.07 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\), and \(P=1000 \mathrm{~N}\).

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B/2 B/2 P FIGURE 12.15 Tapered cantilever beam. b/2 b/2 x