Flexural ductility of reinforced and prestressed concrete beams with maximum reinforcement ratios

Abstract

To examine the flexural behavior of reinforced concrete beams with maximum reinforcement ratio, several reinforced concrete sections with different rebars are designed and analyzed using OpenSees. It was found that, although reasonable for rebars with yield point, the determination of the relative depth of the rectangular stress block for balanced failure given in the Chinese Concrete Code (GB50010-2010) is somewhat small for rebars without yield point. Based on the American Concrete Code (ACI318-19) and the work of previous scholars, this paper proposes a more accurate calculation method for the relative depth of the rectangular stress block for balanced failure based on the requirement of a conservative bearing capacity, and further extends it to prestressed medium strength rebars. Two reinforced concrete beams and one prestressed concrete beam were designed and tested with the proposed maximum reinforcement ratios. The outcomes demonstrate the validity of the proposed formula.

Appendix

1.1 Derivation of the relationship between $\xi$ and $\varepsilon _{\rm{s}}^0$

Assume that the measured strain of the steel bar at the time of failure is $\varepsilon _{\rm{s}}^0$. The superscript $^0$ is used below to indicate the measured value of the parameter.

Based on the plane-section assumption, the measured relative depth of the rectangular stress block $\xi$ can be calculated as

$$\begin{aligned} \xi ^0 = \frac{\beta _1}{{1 + \frac{{\varepsilon _{\rm{{s}}}^{{0}}}}{{{\varepsilon _{{{\rm{cu}}}}}}}}} \end{aligned}$$

(A.1)

The equilibrium equations based on the design and measured values can be obtained as

$$\begin{aligned}{} & {} f_{\rm{y}}A_{\rm{s}}=\alpha _1f_{\rm{c}}b\xi {h_0} \end{aligned}$$

(A.2)

$$\begin{aligned}{} & {} f_{\rm{s}}^0A_{\rm{s}}=\alpha _1f_{\rm{c}}^0b\xi ^0{h_0} \end{aligned}$$

(A.3)

Substituting Eq. (A.3) into Eq. (A.2) leads to

$$\begin{aligned} {\xi } = \frac{{{f_{\rm{y}}}}}{{f_{\rm{{s}}}^{{0}}}}\frac{{f_{\rm{c}}^{{0}}}}{{{f_{\rm{c}}}}}\xi ^0 \end{aligned}$$

(A.4)

Equation(A.4) can also be rewritten with the assumption $f_{\rm{c}}^0=f_{ck}$

$$\begin{aligned} {\xi } = \frac{{{f_{\rm{y}}}}}{{f_{\rm{{s}}}^{{0}}}}\frac{{{f_{\rm{{ck}}}}}}{{{f_{\rm{c}}}}}\xi ^0 \end{aligned}$$

(A.5)

Substituting Eqs. (A.1) into (A.5), we can obtain

$$\begin{aligned} \xi = \frac{{\frac{{{f_\mathrm{{y}}}{f_{\mathrm{{ck}}}}}}{{f_\mathrm{{s}}^\mathrm{{0}}{f_\mathrm{{c}}}}}{\beta _1}}}{{1 + \frac{{\varepsilon _\mathrm{{s}}^\mathrm{{0}}}}{{{\varepsilon _{\mathrm{{\rm{cu}}}}}}}}} \end{aligned}$$

(A.6)

In the ultimate state of bearing capacity, the real limit curvature of the section and the ductility increase with the real strain of the rebar. Moreover, if $\xi$ has been determined, then the rebar strain $\varepsilon _{\rm{s}}^0$ at the time of failure can be derived as

$$\begin{aligned} \varepsilon _{\rm{s}}^0=\left( {\frac{f_{\rm{y}}f_{ck}}{f_{\rm{s}}^0f_{\rm{c}}}\beta _1}-\xi \right) \frac{\varepsilon _{\rm{cu}}}{\xi } \end{aligned}$$

(A.7)