Table 10 Simplified bending capacity formulas development for ECC-concrete composite and ECC specimens

I-Strain and stress distribution

II-Force equilibrium

\( \alpha_{\text{c,n}} f_{\text{c,n}} bx = f_{\text{sy}} A_{\text{s}} + f_{\text{etc,n}} bh_{\text{e}} \)

\( \alpha_{\text{e,n}} f_{\text{ecp,n}} bx = f_{\text{sy}} A_{\text{s}} + f_{\text{etc,n}} bh_{\text{t}} \)

(27)

III-Strain compatibility

\( x_{\text{c}} ( = x/\beta_{\text{c,n}} ) + h_{\text{t}} = h \)

\( x_{\text{e}} ( = x/\beta_{\text{e,n}} ) + h_{\text{t}} = h \)

(28)

IV-Equivalent resultant force

\( \alpha_{\text{c,n}} f_{\text{c,n}} b\beta_{\text{c,n}} x_{\text{c}} = \int_{0}^{{x_{\text{c}} }} {\sigma_{\text{c}} (y)bdy} \)

\( \alpha_{\text{e,n}} f_{\text{ecp,n}} b\beta_{\text{e,n}} x_{\text{e}} = \int_{0}^{{x_{\text{e}} }} {\sigma_{\text{ec}} (y)bdy} \)

(29)

V-Equivalent resultant moment

\( \alpha_{\text{c,n}} f_{\text{c,n}} b\beta_{\text{c,n}} x_{\text{c}} (x_{\text{c}} - \beta_{\text{c,n}} x_{\text{c}} /2) = \int_{0}^{{x_{\text{c}} }} {\sigma_{\text{c}} (y)bydy} \)

\( \alpha_{\text{e,n}} f_{\text{ecp,n}} b\beta_{\text{e,n}} x_{\text{e}} (x_{\text{e}} - \beta_{\text{e,n}} x_{\text{e}} /2) = \int_{0}^{{x_{\text{e}} }} {\sigma_{\text{ec}} (y)bydy} \)

(30)

VI-Neutral axis formula

\( x = \frac{{f_{\text{sy}} A_{\text{s}} }}{{\alpha_{\text{c,n}} f_{\text{c,n}} b}} + \frac{{f_{\text{etc,n}} h_{\text{e}} }}{{\alpha_{\text{c,n}} f_{\text{c,n}} }} \)

\( x = \frac{{f_{\text{sy}} A_{\text{s}} /b + f_{\text{etc,n}} h}}{{\alpha_{\text{e,n}} f_{\text{ecp,n}} + f_{\text{etc,n}} /\beta_{\text{e,n}} }} \)

(31)

VII-Moment capacity formula

\( \begin{aligned} M_{\text{u}} = f_{\text{sy}} A_{\text{s}} (h_{0} - x/2) + \hfill \\ \, f_{\text{etc,n}} bh_{\text{e}} (h - h_{\text{e}} /2 - x/2) \hfill \\ \end{aligned} \)

\( \begin{aligned} M_{\text{u}} = f_{\text{sy}} A_{\text{s}} (h_{0} - x/2) + \hfill \\ \, f_{\text{etc,n}} bh_{\text{t}} (h - h_{\text{t}} /2 - x/2) \hfill \\ \end{aligned} \)

(32)