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 con crete in the outer fibre a a' is only one-hall 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' a' 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.
262. Summation of the Compressive Force% The summation of the compressive forces is evidently indicated-by the area of the shaded portion in Fig. 97. The curve v N is a portion of a parabola. The area of the shaded portion between the curve v N and the straight line v N, equals one-third of the area of the triangle ?a N v. The area of the triangle v 9t, N = c kd. Therefore, for the total shaded area, we have : But in this case, ; therefore, Area = 1 kd (i c + s Ec fc ) • (9) In Fig. 9S has been redrawn the parabola of Fig. 93, in which o is the vertex of the parabola. Here c" is the force which would pro duce a compression of provided the concrete could endure such a pressure without rupture. If the initial modulus of elasticity applied to all stresses, the required force would be the line . And e e.
It is one of the well-known properties of the parabola that abscisse are proportional to the squares of the ordinates, or that (in this case): In order to avoid the complication resulting from the attempt to develop formulae which are applicable to all kinds of assumptions, it will be at once assumed, as previously referred to, that the ultimate compressive strength of the concrete is 3 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.
The proof that q will equal 3 under these conditions, is perhaps deter mined most easily by computing the ratio of b h to g h (see Fig. 9S) when o a is assumed to be of o m. In this case, from the properties of the parabola, a b = m n ; e' = nan = c" = E, cc".
But when o = of ono, g h = Therefore c' = g h. But when o a = of o = Therefore, when c' = 3 gh, q = 3 It has already been shown that c" = E. c:, and also that eZ/= . Therefore c, = c"q: It has also been shown that c' c", or that c" = Therefore c'q.
Substituting this value in Equation 11, we have for the summa tion of the compressive forces above the neutral axis, under such conditions: — q)q c'b kd Substituting the further condition that q = 1, we have
.2' X = c'b kd 263. Center of Gravity of Compressive Forces. also called the centroid of compression. The theoretical determination of this center of gravity is virtually the same as the determination of the center of gravity of the shaded area shown in Figs. 96 and 97. The general method of determining this center of gravity requires the use of differential calculus, and is a very long and tedious calculation. But the final result may be reduced to a surprisingly simple form, as expressed in the following equation: Assuming, as explained above, the value of q = 3, this reduces to: x — .357 hid (1 4) When q equals zero, the value of .r equals .333 and, at the other ex treme, when q = 1, x = .375 loci.
There is, therefore, a very small range of inaccuracy in adopting the value of q = for all computations.
264. Position of the Neutral Axis. According to one of the fundamental laws of mechanics, the sum of the horizontal tensile forces must be equal and opposite to the sum of the compressive forces. Ignoring the very small amount of tension furnished by the concrete below the neutral axis, the tension in the steel =As= pbds = the total compression in the concrete. Therefore, applying Equa tion 11, pbds= (1 — q) k b d But s = ; therefore, p = (1 — q) eek Equation 16 is a perfectly general equation which depends for its accuracy only on the assumption that the law of compressive stress to compressive strain is represented by a parabola. The equation shows that k, the ratio determining the position of the neutral axis, depends on three variables—namely, the percentage of the steel (p), the ratio of the moduli of elasticities (r), and the ratio of the deforma tions in the concrete (q). These must all be determined more or less accurately before we can know the position of the neutral axis.
On the other hand, if it were necessary to work out Equation 16, as well as many others, for every computation in reinforced concrete, the calculations would be impracticably tedious. Fortunately the extreme range in k for any one ratio of moduli of elasticities, is only a few per cent, even when q varies from 0 to 1. We shall therefore simplify the calculations by using the constant value q = as ex plained above.