## 3 Assessment
### Assessment of Pilotis Columns
#### 3.2.2 Deformation and Strength Indices from Flexure
The secant to yield translational stiffness of the columns is calculated by using \(\text{K}=12\)\(\text{EI}/\text{H}^{3}\). For corner columns \(\text{K}=12\)\(\times\)\(6250/2.7^{3}=3810\)\(\text{kNm}^{-1}\) and for peripheral columns \(\text{K}\approx 4910\)\(\text{kNm}^{-1}\) whereas the elastic and that of the cracked cross section values are 15,790 \(\text{kNm}^{-1}\) and 7895 \(\text{kNm}^{-1}\), respectively. By comparing values of \(\text{K}\), the order of magnitude of the error for the estimated translational stiffness may be large, depending on the adopted approach. This difference has a great impact on the structure period of vibration, and thus on the maximum displacement that is going to be developed during the earthquake. Later, in Section 4, the secant to yield translation stiffness is used in defining the effective translational period \(\text{T}_{\text{eff}}\) of the building (Table 1). The chord rotation at yielding is approximated as \(\theta_{\text{y}}=(\phi_{\text{y}}\text{H})/6=(0.0143\,\times\,2.7)/6=0.64\%\). A more detailed expression of this property is [33]:  
\[\theta_{\text{y}}=1/3\phi_{\text{y}}(\text{a}_{\text{v}}\text{z}+\text{L}_{ \text{V}})+0.0014(1+1.5\text{h}/\text{L}_{\text{V}})+0.125\phi_{\text{y}} \text{D}_{\text{b}}\,\text{f}_{\text{sy}})/\sqrt{\text{f}_{\text{cm}}} \tag{2}\]  
with \(\text{a}_{\text{v}}\text{z}=0\) (it is assumed that shear cracking is not expected to precede flexural yielding), \(\text{L}_{\text{V}}=1350\)\(\text{mm}\) (i.e., half the column clear height), \(\text{h}=350\)\(\text{mm}\), \(\text{D}_{\text{b}}=14\)\(\text{mm}\), \(\text{f}_{\text{sy}}=500\) MPa and \(\text{f}_{\text{cm}}=16\) MPa. Equation (2) results to \(\theta_{\text{y}}=1.15\%\) that is almost double of the simplified estimation (\(\theta_{\text{y}}=0.64\%\)). However, the most conservative value \(\theta_{\text{y}}=0.64\%\) is used in the following calculations because it is testified in relevant tests on substandard columns as a lower bound magnitude [34]. The ultimate chord rotation \(\theta_{\text{u}}\) is estimated by following the three alternative procedures presented in chapter 8 of [4] and are summarized in Table 1. More specifically:  
(a) From basic mechanics:  
\[\theta_{\text{u}}=1/\gamma_{\text{el}}\,\left[\theta_{\text{y}}+\left(\phi_{ \text{u}}-\phi_{\text{y}}\right)\text{l}_{\text{pl}}\,(1-0.5\,\text{l}_{ \text{pl}}/\text{L}_{\text{V}})\right] \tag{3}\]  
Figure 3: Estimation of compression zone depth at ultimate for symmetrically reinforced concrete cross section with variables being the axial load and the amount of reinforcement. where \(\gamma_{\rm el}\) is equal to 1.5 for primary members, \(\theta_{\rm y}\) = 0.64%, \(\phi_{\rm u}\) is the ultimate curvature of the end section evaluated by assigning the concrete ultimate strain \(\varepsilon_{\rm cu}\) = 0.0035 (values are shown in Table 1). The plastic hinge length l\({}_{\rm pl}\) can be estimated from three alternatives, as:  
\[\rm I_{pl}=0.1\ L_{V}+0.17\ h+0.24\ (D_{b}f_{sy})/\surd{f_{cm}} \tag{3a}\] \[\rm I_{pl}=0.2\ h\ [1+1/3\ min\ (9,L_{V}/h)]\] (3b) \[\rm I_{pl}=0.5\ d \tag{3c}\]  
Their implementation results to \(\rm I_{pl(3a)}\) = 615 mm or \(\rm I_{pl(3b)}\) = 160 mm or \(\rm I_{pl(3c)}\) = 161.5 mm, respectively. Note that Equation (3b) considers cyclic loading and Equation (3c) is a simple definition of \(\rm I_{pl}\) for members with old type detailing. The results of Equation (3b,c) coincide. The difference in results for \(\rm I_{pl}\) obtained from Equation (3a-c) is considerably high and affects the estimations of the elements' ductility. For example, for corner columns \(\theta_{\rm u}(\rm I_{pl}=615\ mm)\) = 1.6% or \(\theta_{\rm u}(\rm I_{pl}=160\ mm)\) = 0.8%. The corresponding rotation ductility is \(\rm\mu_{0}=\theta_{u}/\theta_{y}\) = 1.6/0.64 = 2.4 or \(\rm\mu_{0}=0.8/0.64=1.2\). Apparently, the results in Table 1 for \(\rm I_{pl}=160\ mm\) suggest that all columns--except of the middle one--fail close to yielding demonstrating no ductility.