Fig. 4-10 Moment–curvature relationship for the section in Fig. 4.9(a) using $f_c' = 4000$ psi and $f_y = 60$ ksi.
The calculation of specific points on the moment–curvature curve follows the process represented in Fig. 4-9b through 4-9d. Each point is usually determined by selecting a specific value for the maximum compression strain at the extreme compression fiber of the section, $\epsilon_c(\text{max})$. From the assumption that plane sections before bending remain plane, the strain distribution through the depth of the section is linear. From the strain diagram and the assumed material stress–strain relationships, the distribution of stresses is determined. Finally, by integration, the volume under the stress distributions (i.e., the section forces) and their points of action can be determined.
After the section forces are determined, the following steps are required to complete the calculation. First, the distance from the extreme compression fiber to the section neutral axis (shown as $x$ in Fig. 4-9b) must be adjusted up or down until section equilibrium is established, as given by Eq. (4-2). When Eq. (4-2) is satisfied, the curvature, $\Phi$, for this point is calculated as the slope of the strain diagram,
The corresponding moment is determined by summing the moments of the internal forces about a convenient point—often selected to be the centroid of the tension reinforcement. This process can be repeated for several values of maximum compression strain. A few maximum compression strain values are indicated at selected points in Fig. 4-10. Exceptions to this general procedure will be discussed for the cracking and yield points.
Cracking Point
Flexural tension cracking will occur in the section when the stress in the extreme tension fiber equals the modulus of rupture, $f_r$. Up to this point, the moment–curvature relationship is linear and is referred to as the uncracked-elastic range of behavior (from $O$ to $C$ in Fig. 4-10). The moment and curvature at cracking can be calculated directly from elastic