A test of this sort may be made substantially as follows: A square or circular column of concrete at least one foot long is placed in a testing machine. A very delicate micrometer mechanism is fastened to the concrete by pointed screws of hardened steel. These points are originally at a known distance apart—say 8 inches. When the concrete is compressed, the distance between these points will be slightly less. A very delicate mechanism will permit this distance to be measured as closely as the ten-thousandth part of an inch, or to about 1of the length. Suppose that the various pressures per 100,000 square inch, and the proportionate compressions, are as given in the following tabular form : We may plot these pressures and compressions as in Fig. 93, using any convenient scale for each. For example, for a pressure of S00 pounds per square inch, we select the vertical line which is at the horizontal distance from the origin 0 of S00, according to the scale adopted. Scaling off on this vertical line the dinate .00045, cording to the scale adopted for pressions, we have the position of one point of the curve. The other pointsare obtained similarly. Although the points thus obtained from the testing of a single block of concrete would not be considered cient to establish the law of the elasticity of concrete in compression, a study of the curves which may be drawn through the series of points obtained for each of a large number of blocks, shows that these curves will average very closely to parabolas that are tangent to the initial modulus of elasticity, which is here represented in the diagram by a straight line running diagonally across the figure.
It is generally considered that the axis of the parabola will be a horizontal line when the curve is plotted according to this method. The position of the vertex of the parabola cannot be considered as definitely settled. Professor Talbot has computed the curve as if the vertex were at the point of the ultimate compression of the con crete, although he conceded that the vertex might be in an imaginary position corresponding to a compression in the concrete higher than that which the concrete could really endure. Mr. A. I.. Johnson, another noted authority, bases his computation of formulte on the assumption that the ultimate compressive strength of the concrete is two-thirds of the value which would be required to produce that amount of compression in case the initial modulus of elasticity were the true value for all compressions. In other words, looking at Fig. 93, if o c is a line representing the initial modulus of elasticity, then, if the elasticity were uniform throughout, it would require a force of about 2,340 pounds (or d I) to produce a proportionate compression of .00132 of the length (represented by o d). Actually that compression will be produced when the pressure equals d e, which is 3- of d f. It
should not be forgotten that the above numerical values are given merely for illustrative purposes. They would, if true, represent a rather weak concrete. The following theory is therefore based on the assumption that the stress-strain curve is represented by the para bolic curve o e (see Fig 93); and that the ultimate stress per square inch in the concrete c' is represented by d e, which is of the com pressive stress that would be required to produce that proportionate compression if the modulus of elasticity of the concrete were uniformly maintained at the value it has for very low pressures.
261. Theoretical Assumptions. 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.
(b) 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 inde terminate; and is not allowed for, except in the appreciation that it adds indef initely 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 abed in the unstrained con dition, is changed to the plane a' b' c' d' when the load is applied. The com pression at the top = a a' = b b'. The neutral axis is unchanged. The concrete at the bottom is stretched an amount = c c' = d d', 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. 95a is given a side view of the beam, with spe cial reference to the deform ation of the fibres. Since the fibres between the neu tral axis and the compressive face are compressed propor tionally, then, if a a' repre sents the linear compression of the outer fibre, the shaded lines represent, at the same scale, the compression of the intermediate fibres.
In Fig. 95b, rn 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 proportion ately 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 pressitre on the intermediate fibres by the ordinates to the curve v N.