output_mds/page_05.md

(a) Beam.

(b) Bending moment diagram.

(c) Free body diagrams showing internal moment and shear force.

Fig. 4-3 Internal forces in a beam.

(d) Free body diagrams showing internal moment as a compression-tension force couple.

If moments are summed about an axis through the point of application of the compressive force, $C$, the moment equilibrium of the free body gives

$$\begin{array}{cccc}& M=T\times jd& & \text{(4-3a)}\end{array}M\; =\; T\; \backslash times\; jd\; \backslash tag\{4-3a\}$$

Similarly, if moments are summed about the point of application of the tensile force, $T$,

$$\begin{array}{cccc}& M=C\times jd& & \text{(4-3b)}\end{array}M\; =\; C\; \backslash times\; jd\; \backslash tag\{4-3b\}$$

Because $C = T$, these two equations are identical. Eqs. (4-2) and (4-3) come directly from statics and are equally applicable to beams made of steel, wood, or reinforced concrete.

The conventional elastic beam theory results in the equation $\sigma = My/I$, which, for an uncracked, homogeneous rectangular beam without reinforcement, gives the distribution of stresses shown in Fig. 4-4. The stress diagram shown in Fig. 4-4c and d may be visualized as having a "volume"; hence, one frequently refers to the compressive stress