Fig. 4-6 Typical doubly reinforced sections in positive bending.
how adding steel in the compression zone to create a doubly reinforced section, as shown in Fig. 4-6, will affect flexural behavior. All of the sections considered here will be under-reinforced. Although this may sound like a bad design, this is exactly the type of cross section we will want to design to obtain the preferred type of flexural behavior. The meaning of an under-reinforced beam section is that, when the section is loaded in bending beyond its elastic range, the tension zone steel will yield before the concrete in the compression zone reaches its maximum useable strain, $\epsilon_{cu}$.
To analytically create a moment–curvature relationship for any beam section, assumptions must be made for material stress–strain relationships. A simple elastic-perfectly plastic model will be assumed for the reinforcing steel in tension or compression, as shown in Fig. 4-7. The steel elastic modulus, $E_s$, is assumed to be 29,000 ksi.
The stress–strain relationship assumed for concrete in compression is shown in Fig. 4-8. This model consists of a parabola from zero stress to the compressive strength of the concrete, $f_c'$. The strain that corresponds to the peak compressive stress, $\epsilon_o$, is often assumed to be 0.002 for normal strength concrete. The equation for this parabola, which was originally introduced by Hognestad [4-4], is
Fig. 4-7 Assumed stress–strain relationship for reinforcing steel.