## 4 Global Strengthening Requirements
After the increase in the pilots stiffness under the requirement of a shear-type deflection shape using the charts of Figure 4b, one could validate this simple solution by performing the Rayleigh iterative method and considering the stiffness of the exterior infill walls of the upper floors. More specifically, in each of the upper two floors, for two exterior solid walls in y-y direction of a total cross section area \(\rm A_{w}=2\)\(\times\) (10 m \(-\) 3 \(\times\) 0.35 m) \(\times\) 0.12 m \(\approx\) 2.15 m\({}^{2}\) or \(\rm\rho_{mw}=A_{w}/(A_{floor}=120\) m\({}^{2})\approx 0.018\) and for \(\rm h_{i}=2.7\) m, \(\rm f_{bc}=5\) MPa, \(\rm f_{mc}=5\) MPa (each is divided by \(\rm\gamma_{m}=1.5\)), and \(\rm\mu_{y}^{mw}=2.5\), \(\rm\theta_{y}^{mw}=0.15\%\), then the masonry wall stiffness in each floor is: \(\rm K_{mw}\approx A_{floor}/h_{i}\times(0.1\rm f_{bc}^{0.7}\rm f_{mc}^{0.3})/( \rm\mu_{y}^{mw}\rm\phi_{y}^{mw})\times\)\(\rm\rho_{mw,i}\approx 71,100\) kN/m. The available effective stiffness of the 2 nd and 3 rd floor are then: \(\rm K_{2}=K_{3}=K_{eff}+K_{mw}=38,690+71,100\approx 109,800\) kN/m (Figure 5a). Because the available magnitudes \(\rm K_{2}\) and \(\rm K_{3}\) are higher than or close to the required for the chosen shear-type response of the building (i.e., \(\rm K_{2}=100,950\) kN/m and \(\rm K_{3}=146,830\) kN/m), there is no need for global measures in these floors. For the Rayleigh iterative method, the floor mass (i.e., 96.6 tonnes) is increased by adding a share of the total masonry weight \(\rm W_{masony}=33\) tonnes; thus \(\rm W_{masony}\) was distributed as \(\frac{1}{4}\) in each of the ground and the 3 rd floor and \(\frac{1}{2}\) in the 2 nd floor. After such analysis, the deflection shape presented in Figure 5b was obtained (the light green coloured curve titled Rayleigh-final) and is very close to the shear-type on which the addition of stiffness was based. Moreover, from this analysis the calculated period is found 0.46 s, very close to the target value \(\rm T_{trg}=0.45\) s. In addition, the initially assumed lateral deflection shape and the initial approximated shape by using the Rayleigh method are plotted in Figure 5b for comparison reasons (the masonry mass and stiffness of the upper floors is considered). For the selected period \(\rm T_{trg}=0.45\) s (\(\rm<\)\(\rm T_{C}=0.5\) s), the elastic spectral displacement demand is estimated from Equation (9) (from [35]):  
\[\rm S_{d}\left(T\right)=a_{g}\leavevmode\
obreak\ \rm S_{\rm H}\beta_{o}T^{2}/40 \tag{9}\]  
where \(\rm a_{g}=0.24\) g, \(\rm S=1.2\) (soil class B), \(\rm\eta=1\) (\(\rm\xi=5\%\)) and \(\rm\beta_{o}=2.5\), thus \(\rm S_{d}=0.036\) m. Considering that this displacement will be increased by about 20% when transferring from the spectrum (Equivalent Single Degree of Freedom) to the actual structure (Multi-Degree of Freedom), then, the relative drift demand at the ground storey (\(\Delta\Phi=0.5\), Figure 5b) is \(\rm\theta_{dem}=1.2\times 0.5\times S_{d}/H=0.8\%\), which corresponds to a ductility demand of \(\rm\mu_{o}=\mu_{\Delta}=0.80\%/0.64\%=1.25\). This demand is very close to the supply when a short plastic hinge length is considered [procedure (a) in Table 1]. To improve ductility and to alleviate premature shear failure, FRP jacketing is required.