The theory of reinforced-con-crete beams is based on the usual assumptions that:

(a) The loads are applied at right angles to the axis of the beam. The usual vertical gravity loads supported by a horizontal beam fulfil this condition.

(6) There is no resistance to free horizontal motion. This condition is seldom, if ever, exactly fulfilled in practice. The more rigidly the beam is held at the ends, the greater will be its strength above that computed by the simple theory. Under ordinary conditions the added strength is quite indeterminate; and is not allowed for, except in the appreciation that it adds indefinitely to the safety.

(c) The concrete and steel stretch together without breaking the bond between them. This is absolutely essential.

(d) Any section of the beam which is plane before bending is plane after bending.

In Fig. 94 is shown, in a very exaggerated form, the essential meaning of assumption d. The section a b c d in the unstrained condition, is changed to the plane a' b' c' d' when the load is applied. The compression at the top = a a' = b b'. The neutral axis is unchanged. The concrete at the bottom is stretched an amount = c c' = dd', while the stretch in the steel equals g g'. The compression in the concrete between the neutral axis and the top is proportional to the distance from the neutral axis.

In Fig. 95 a is given a side view of the beam, with special reference to the deformation of the fibres. Since the fibres between the neutral axis and the compressive face are compressed proportionally, then, if a a' represents the linear compression of the outer fibre, the shaded lines represent, at the same scale, the compression of the intermediate fibres.

In Fig. 956, m n indicates the stress there would be in the outer fibre if the initial modulus of elasticity applied to all stresses. But since the force required to produce the compression a a' is proportionately so much less than that required for the lesser compressions, the actual pressure in pounds on the outer fibre may be represented by a line v n, and the pressure on the intermediate fibres by the ordinates to the curve v N.

Fig. 94. Plane Section of Beam before and after Bending.

Fig. 94. Plane Section of Beam before and after Bending.

Fig. 95. Fibre Stresses in Beams.

Fig. 95. Fibre Stresses in Beams.

In Fig. 96, a and b, are shown a pair of figures corresponding with those of Fig. 95, except that the compressive deformation of the concrete in the outer fibre a a' is only one-half of the value in Fig. 95. But it will require about three-fourths as much pressure to produce one-half as much compression. In Fig. 96, v' n' is therefore three-fourths of v n in Fig. 95. The student should note that k' here differs slightly from k, which means that the position of the neutral axis varies with the conditions.