Fig. 4-8 Assumed stress–strain relationship for concrete.
Beyond the strain, $\epsilon_o$, the stress is assumed to decrease linearly as the strain increases. An equation for this portion of the relationship can be expressed as
where $Z$ is a constant to control the slope of the line. For this discussion, $Z$ will be set equal to a commonly used value of 150. Lower values for $Z$ (i.e., a shallower unloading slope) can be used if longitudinal and transverse reinforcement are added to confine the concrete in the compression zone.
In tension the concrete is assumed to have a linear stress–strain relationship (Fig. 4-8) up to the concrete modulus of rupture, $f_r$, defined in Chapter 3.
Consider a singly reinforced rectangular section subjected to positive bending, as shown in Fig. 4-9a. In this figure, $A_s$ represents the total area of tension reinforcement, and $d$ represents the effective flexural depth of the section, i.e., the distance from the extreme compression fiber to the centroid of the tension reinforcement. A complete moment–curvature relationship, as shown in Fig. 4-10, can be generated for this section by continuously increasing the section curvature (slope of the strain diagram) and using the assumed material stress–strain relationships to determine the resulting section stresses and forces, as will be discussed in the following paragraphs.
(a) Basic section. (b) Strain distribution. (c) Stress distribution. (d) Internal forces.
Fig. 4-9 Steps in analysis of moment and curvature for a singly reinforced section.