Any discussion of flexural behavior of a reinforced concrete beam section usually involves a discussion of ductility, i.e., the ability of a section to deform beyond its yield point without a significant strength loss. Ductility can be expressed in terms of displacement, rotation, or curvature ratios. For this discussion, section ductility will be expressed in terms of the ratio of the curvature at maximum useable compression strain to the curvature at yield. The maximum useable strain can be expressed as a specific value, as done by most codes, or it could be defined as the strain at which the moment capacity of the section has dropped below some specified percentage of the maximum moment capacity of the section. By either measure, the moment–curvature relationship given in Fig. 4-10 represents good ductile behavior.
Effect of Major Section Variables on Strength and Ductility
In this subsection, a series of systematic changes are made to section parameters for the beam given in Fig. 4-9a to demonstrate the effect of such parametric changes on the moment–curvature response of the beam section. Values of material strengths and section parameters are given for seven different beams in Table 4-1. The first column (Basic Section) represents the original values that correspond to the $M-\Phi$ curve given in Fig. 4-10. Each successive beam section (represented by a column in Table 4-1) represents a modification of either the material properties or section dimensions from those for the basic section. Note that for each new beam section (column in table) only one of the parameters has been changed from those used for the basic section.
$M-\Phi$ plots that correspond to the first three sections given in Table 4-1 are shown in Fig. 4-11. The only change for these sections is an increase in the area of tension reinforcement, $A_s$. It is clear that increasing the tension steel area causes a proportional increase in the strength of the section. However, the higher tension steel areas also causes a less ductile behavior for the section. Because of this loss of ductility as the tension steel area is increased, the ACI Code places an upper limit on the permissible area of tension reinforcement, as will be discussed in detail in Section 4-6.
Figure 4-12 shows $M-\Phi$ plots for the basic section and for the sections defined in the last four columns of Table 4-1. It is interesting from a design perspective to observe how changes in the different section variables affect the flexural strength, stiffness, and ductility of the beam sections. An increase in the steel yield strength has essentially the same effect as increasing the tension steel area—that is the section moment strength increases and the section ductility decreases. Increases in either the steel yield strength or the tension steel area have very little effect on the stiffness of the section before yield (as represented by the elastic slope of the $M-\Phi$ relationship).
Increasing the effective flexural depth of the section, $d$, increases the section moment strength without decreasing the section ductility. Increasing the effective flexural depth
TABLE 4-1 Material and Section Properties for Various Beam Sections
| Primary Variables | Basic Section | Moderate* $A_s$ | High* $A_s$ | High* $f_y$ | Large* $d$ | High* $f_c'$ | Large* $b$ |
|---|---|---|---|---|---|---|---|
| $A_s$ (sq.in.) | 2.5 | 4.5 | 6.5 | 2.5 | 2.5 | 2.5 | 2.5 |
| $f_y$ (ksi) | 60 | 60 | 60 | 80 | 60 | 60 | 60 |
| $d$ (in.) | 21.5 | 21.5 | 21.5 | 21.5 | 32.5 | 21.5 | 21.5 |
| $f_c'$ (psi) | 4000 | 4000 | 4000 | 4000 | 4000 | 6000 | 4000 |
| $b$ (in.) | 12 | 12 | 12 | 12 | 12 | 12 | 18 |
*Relative to values in the basic section.