neutral plane passes through the centres of gravity of all the normal sections of the beam and, hence, that the neutral axis passes through the passes through the centre of gravity of the section to which it belongs. =78. Fundamental Formulæ of Theory of Beams.=—The fundamental formulæ of the theory of loaded beams may be quite simply written. Fig. 10 exhibits in a much exaggerated manner a bent beam supporting any system of loads _W₁_, _W₂_, _W₃_, etc., while Fig. 11 shows a normal cross-section of the same beam. In Fig. 10 _AB_ is the neutral line, and in Fig. 11 _CD_ is the neutral axis passing through the centre of gravity, _c.g._, of the section. [Illustration: FIG. 10.] [Illustration: FIG. 11.] If _a_ is the amount of force or stress on a square inch (or other square unit), i.e., the intensity of stress, at the distance of unity from the neutral axis _CD_ of the section, then, by the fundamental law already stated, the amount acting on another square inch at any other distance _z_ from the neutral axis will be _az_. This quantity is called the “intensity of stress” (tension or compression) at the distance _z_ from the neutral axis. Evidently it has its greatest values in the extreme fibres of the section, i.e., _ad_ and _ad_₁. At the neutral axis _az_ becomes equal to zero. _FG_ in Fig. 11 represents the same line as _FG_ in Fig. 10. If the line _FH_ in Fig. 11 be laid down equal to _ad_ and at right angles to _FG_, and if _O_ represent the centre of gravity, _c.g._, of the section, then let the straight line _LH_ be drawn. Any line drawn parallel to _FH_ from _FG_ to _LH_ will represent the intensity of stress in the corresponding part of the beam’s cross-section. Obviously, as these lines are drawn in opposite directions from _FG_, those above _O_ will indicate stress of one kind, and those below that point stress of another kind, i.e., if that above be tension, that below will be compression. It can be demonstrated by a simple process that the total tension on one side of the neutral axis is just equal to the total compression on the other side, and from that condition it follows that the neutral axis must pass through the centre of gravity or centroid of the section. Returning to the left-hand portion of Fig. 11, let _dA_ represent a very small portion of the cross-section; then will _az. dA_ be the amount of stress acting on it. The moment of this stress or force about the neutral axis will be _azdA·z = az²·dA_. If this expression be applied to every small portion of the entire section, the aggregate or total sum of the small moments so found will be the moment of all the stresses in the section about the neutral axis. That moment will have the value _M_ = ⌠_az²·dA_ = _a_⌠_z²·dA_ = _aI_. (1) ⌡ ⌡