- $b_e$ is the effective width of a compression zone for a flanged section with compression in the flange, in.
- $b_w$ is the width of the web of the beam (and may or may not be the same as $b$), in.
- $d$ is the distance from the extreme fiber in compression to the centroid of the longitudinal reinforcement on the tension side of the member, in. In the positive-moment region (Fig. 4-2a), the tension steel is near the bottom of the beam, while in the negative-moment region (Fig. 4-2b) it is near the top.
- $d'$ is the distance from the extreme compression fiber to the centroid of the longitudinal compression steel, in.
- $d_t$ is the distance from the extreme compression fiber to the farthest layer of tension steel, in. For a single layer of tension reinforcement, $d_t = d$, as shown in Fig. 4-2b.
- $f_c'$ is the specified compressive strength of the concrete, psi.
- $f_c$ is the stress in the concrete, psi.
- $f_s$ is the stress in the tension reinforcement, psi.
- $f_y$ is the specified yield strength of the reinforcement, psi.
- $h$ is the overall height of a beam cross section.
- $jd$ is the lever arm, the distance between the resultant compressive force and the resultant tensile force, in.
- $j$ is a dimensionless ratio used to define the lever arm, $jd$. It varies depending on the moment acting on the beam section.
- $\epsilon_{cu}$ is the assumed maximum useable compression strain in the concrete.
- $\epsilon_s$ is the strain in the tension reinforcement.
- $\epsilon_t$ is the strain in the extreme layer of tension reinforcement.
- $\rho$ is the longitudinal tension reinforcement ratio, $\rho = A_s/bd$.
4-2 FLEXURE THEORY
Statics of Beam Action
A beam is a structural member that supports applied loads and its own weight primarily by internal moments and shears. Figure 4-3a shows a simple beam that supports its own dead weight, $w$ per unit length, plus a concentrated load, $P$. If the axial applied load, $N$, is equal to zero, as shown, the member is referred to as a beam. If $N$ is a compressive force, the member is called a beam-column. This chapter will be restricted to the very common case where $N = 0$.
The loads $w$ and $P$ cause bending moments, distributed as shown in Fig. 4-3b. The bending moment is a load effect calculated from the loads by using the laws of statics. For a simply supported beam of a given span and for a given set of loads $w$ and $P$, the moments are independent of the composition and size of the beam.
At any section within the beam, the internal resisting moment, $M$, shown in Fig. 4-3c is necessary to equilibrate the bending moment. An internal resisting shear, $V$, also is required, as shown.
The internal resisting moment, $M$, results from an internal compressive force, $C$, and an internal tensile force, $T$, separated by a lever arm, $jd$, as shown in Fig. 4-3d. Because there are no external axial loads, summation of the horizontal forces gives