# Prediction of Post-Yield Performance of Hybrid Precast Beams Based on Strength Degradation
## 3 Analytical Prediction of Post-Yield Deflection
### Prediction of Post-Yield Behavior  
The loads (\(F_{y}\), \(M_{y}\)) at the first yield of the rebar were calculated by iterating neutral axes and the corresponding concrete strain until equilibrium was found [23]. Table 4[23] presents the post-yield behavior of pre-stressed Composite beam #8 with fixed-end, and Figure 5 compares the analytically calculated and experimental post-yield deflections. The post-yield load-displacement relationships for the composite beams were predicted using the following six-step procedure described below and in Figure 2, which was introduced in the authors' previous studies [5; 28]. 1. Calculate the load (\(F_{y}\)) on the composite beams at the yield limit, as shown in Figure 2[23]. 2. Calculate the displacement (\(\Delta_{y}\)) from Equations (4) and (5). 3. Calculate the stiffness \(\left(k_{y}\ =\ \frac{F_{y}}{\Delta_{y}}\right)\) of the composite beams at the yield limit state from Figure 2a. 4. Select the desired ductility \(\left(\mu\ =\ \frac{\Delta}{\Delta_{y}}\right)\) from the load-displacement curve (i.e., \(\mu\ =\ 0.1\)-\(5\)) from Figure 2b. 5. Find \(\left(\frac{k}{k_{y}}\right)\) corresponding to the selected ductility \(\left(\mu\ =\ \frac{\Delta}{\Delta_{y}}\right)\) from Figure 2b. 6. Calculate \(k\)(\(=\left(\frac{k}{k_{y}}\right)\cdot k_{y}\)) and \(\Delta\)(\(=\left(\frac{\Delta}{\Delta_{y}}\right)\cdot\Delta_{y}\)) from Figure 2b in order to determine \(F(=k\cdot\Delta)\). \begin{table}
\begin{tabular}{c} \(\frac{k}{k_{y}}\ =\ -0.45 ln\left(\frac{\Delta}{\Delta_{y}}\right)+1.02\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of stiffness degradation [7]Figure 5: _Cont_. Figure 5: _Cont_. Forces at the yield load limit state were calculated using the strain compatibility-based approach. Deflections of 57.8 mm and 61.5 mm at the yield load limit state were calculated elastically for Specimens #4, 5 and #6, 7, respectively. Table 4[23] describes the overall procedure for calculating the post-yield load-displacement relationship of precast composite beams. The first three columns in Table 4[23] show the calculated values for load (\(F_{y}\)), deflection at the yield limit state (\(\Delta_{y}\)), and stiffness (\(k_{y}\)). The stiffness degradation tendency, obtained by normalizing the effective stiffness of the specimens with respect to the stiffness at first yield of the rebar, is presented in Figure 4. The stiffness degradation regression for non-pre-stressed cantilever beams (Specimens #1 to #3), shown as \(\frac{k}{k_{y}}~{}=~{}-0.603 ln\left(\frac{\Delta}{\Delta_{y}}\right)+1.0071\) in Table 3, was used to obtain the post-yield  
Figure 5: Analytically calculated post-yield deflections vs. experimental post-yield deflections. **(a)** Post-yield deflection of pre-stressed Specimens #1 to #3, calculated using the degradation equation from Table 3 and Figure 4a (Specimens #1 to #3 compared with finite element analysis) [25; 29]: **(a-1)** Specimen #1, **(a-2)** Specimen #2, **(a-3)** Specimen #3. **(b-1)** Specimen #4 (with no sleeve openings), Table 3, Figure 4b[25]. **(b-2)** Specimen #4 (with no sleeve openings), Table 3, Figure 4b[25]. **(b)** Post-yield deflection of pre-stressed Specimens #4 to #5, calculated using the degradation equation for simply supported beams from Figure 4b,e-2. **(b-3)** Specimen #4 (with no sleeve openings), Figure 4e-2[24]; **(b-4)** Specimen #5 (with no sleeve openings), Figure 4e-2[24]. **(c)** Post-yield deflection of pre-stressed Specimens #6 to #7, calculated using the degradation equation for simply supported beams from Figure 4b. **(c-1)** Specimen #6 (with sleeve openings), Table 3, Figure 4b[22]; **(c-2)** Specimens #7 (with sleeve openings), Table 3, Figure 4b[22]. **(d)** Post-yield deflection of Specimens #8 and #9. **(d-1)** Specimen #8, based on the degradation equation of Table 3, Figure 4c-1[7]. **(d-2)** Load-strain relationship for Specimen #8[7]. **(d-3)** Specimen #9, based on the degradation equation of Table 3, Figure 4c-2[7].