# Prediction of Post-Yield Performance of Hybrid Precast Beams Based on Strength Degradation
## 3 Analytical Prediction of Post-Yield Deflection  
### Stiffness at Yield Limit State  
In Figure 2 proposed by [28], the post-yield hysteretic structural behavior was estimated using the stiffness degradation in terms of normalized effective stiffness, which was calculated by dividing the effective stiffness of the hysteretic cycles by the stiffness at the yield limit state. The drift was also normalized by dividing the deflection of the hysteretic cycles by the deflection at the yield limit [28]. A quadratic equation was derived for the location of the neutral axis (c) relative to the top of Specimen MHS #10 after iteration, given by Equation (1) [23]. Force at the first yield state shown in Figure 3 was then estimated using Equation (2). Equation (4) was used to calculate deflection. The location of the neutral axis relative to the top of the beam was found to be 128.3 mm. Using the location of the neutral axis (c) and Equation (2), the moment of the Specimen MHS #10 at yield limit state was calculated to be 760.8 kN-m [23]. \[ \begin{split}(\alpha f_{c}^{\prime}b+\frac{1}{2}t_{w}\epsilon_{c} E_{s}&+\frac{1}{2}t_{w}F_{y}\frac{\epsilon_{y}}{\epsilon_{c}}+t_{w}F_{y})c^{2}\\ &+\Big{\{}A_{g}^{\prime}\epsilon_{c}E_{s}+A_{f}^{\prime}\epsilon _{c}E_{s}-t_{w}\epsilon_{c}E_{s}\Big{(}d^{\prime\prime\prime}+t_{f}^{\prime} \Big{)}-A_{s}f_{y}-A_{f}F_{y}-t_{w}F_{y}\Big{(}d-d^{\prime}-t_{f}\Big{)} \Big{\}}c-A_{s}^{\prime}d^{\prime\prime}\epsilon_{c}E_{s}\\ &-A_{f}^{\prime}\epsilon_{c}E_{s}\Big{(}d^{\prime\prime\prime}+ \frac{t_{f}^{\prime}}{2}\Big{)}+\frac{1}{2}t_{w}\epsilon_{c}E_{s}\Big{(}d^{ \prime\prime\prime}+t_{f}^{\prime}\Big{)}^{2}\ =\ 0\end{split}  \tag{1}\]  
\[ \begin{split} M_{n}\ =\ \alpha f_{c}^{\prime}bc(c-\gamma c)+A_{g}^{ \prime}E_{s}\frac{\epsilon_{c}}{c}(c-d^{\prime\prime})^{2}+A_{f}^{\prime}E_{s }\frac{\epsilon_{c}}{c}\Big{\{}\Big{(}c-d^{\prime\prime\prime}-t_{f}^{\prime} \Big{)}\Big{(}c-d^{\prime\prime\prime}-\frac{t_{f}^{\prime}}{2}\Big{)}+\frac{t _{f}^{\prime}}{2}\Big{(}c-d^{\prime\prime\prime}-\frac{t_{f}^{\prime}}{3} \Big{)}\Big{\}}\\ +\frac{1}{3}A_{w}^{\prime}E_{s}\frac{\epsilon_{c}}{c}\Big{(}c-d^ {\prime\prime\prime}-t_{f}^{\prime}\Big{)}^{2}+A_{s}f_{y}(d-c)+A_{f}F_{y} \Big{(}d-c-d^{\prime}-\frac{t_{f}}{2}\Big{)}\\ +\frac{1}{2}A_{wp}F_{y}\Big{(}d-c-d^{\prime}-t_{f}+\frac{\epsilon_ {y}}{\epsilon_{c}}c\Big{)}+\frac{1}{3}A_{wny}E_{s}\frac{\epsilon_{(yy)}^{2}}{ \epsilon_{c}}c\ =\ 760.8\ \text{kN.m}\\ \text{where,}\\ A_{w}^{\prime}\ =\ t_{w}\Big{(}c-d^{\prime\prime}-t_{f}^{\prime}\Big{)} \text{, }A_{wny}\ =\ t_{w}\frac{\epsilon_{sy}}{\epsilon_{c}}\text{, }A_{wp}\ =\ t_{w}\Big{(}d-c-d^{\prime}-t_{f}-\frac{ \epsilon_{sy}}{\epsilon_{c}}c\Big{)}\end{split}  \tag{3}\]Symbols used in Equations (1) and (2) are explained as follow: \(\alpha\): Stress factor, \(f_{c}^{\prime}\): concrete compresssive strength, b: beam width, c: depth of concrete compressive zone, d: beam effective depth, \(A_{s}^{\prime}\): area of rebar in compression, \(A_{s}\): area of rebar in tension, \(f_{y}\): yield stress of rebar, \(F_{y}\): yield stress of steel, \(A_{w}\): area of steel web, \(A_{f}\): area of steel flange, \(\varepsilon_{sy}\): yield strain of steel, \(\varepsilon_{c}\): concrete strain of 0.003, \(\gamma\): centroid factor, \(d^{\prime}\): distance from tensile rebar to tensile flange, \(d^{\prime\prime}\): distance from extreme compressive layer of concrete to compressive rebar, \(d^{\prime\prime\prime}\): distance from extreme compressive layer of concrete to compressive steel flange, \(t_{f}^{\prime}\): thickness of steel flange in compression, \(t_{f}\): thickness of steel flange in tension, \(t_{w}\): thickness of steel web in tension, \(b_{f}\): width of steel flange, \(y\): yield, \(ny\): not yield, \(p\): plastic, and \(E_{s}\): [PERSON]'s modulus of steel. The deflection was calculated based on the curvature-strain relationship of the beam section at the first yield of the rebar, as given by Equations (4) and (5) for fixed-end beams and cantilevers, respectively.