Fig. 4-14 Mathematical description of compression stress block.
stress block in a beam at the ultimate moment can be expressed mathematically in terms of three constants:
$k_3 =$ ratio of the maximum stress, $f_c''$, in the compression zone of a beam to the cylinder strength, $f_c'$ $k_1 =$ ratio of the average compressive stress to the maximum stress (this is equal to the ratio of the shaded area in Fig. 4-15 to the area of the rectangle, $c \times k_3 f_c'$) $k_2 =$ ratio of the distance between the extreme compression fiber and the resultant of the compressive force to the depth of the neutral axis, $c$, as shown in Figs. 4-14 and 4-15.
For a rectangular compression zone of width $b$ and depth to the neutral axis $c$, the resultant compressive force is
Values of $k_1$ and $k_2$ are given in Fig. 4-15 for various assumed compressive stress–strain diagrams or stress blocks. The use of the constant $k_3$ essentially has disappeared from the flexural theory of the ACI Code. As shown in Fig. 4-12, a large change in the concrete compressive strength did not cause a significant change in the beam section moment capacity. Thus, the use of either $f_c'$ or $f_c'' = k_3 f_c'$, with $k_3$ typically taken equal to 0.85, is not significant for the flexural analysis of beams. The use of $f_c''$ is more significant for column sections subjected to high axial load and bending. Early papers by Hognestad [4-10] and
Fig. 4-15 Values of $k_1$ and $k_2$ for various stress distributions.
(a) Concrete. (b) Triangle. (c) Parabola.