The working unit-compressions for even the best grade of concrete are seldom allowed to exceed 600 pounds per square inch. An inspection of Fig. 93 will show that the curve from the point o to the point indicating a pressure of 600 pounds, although really a parabola, is so nearly a straight line that there is but little error in considering it to be straight. On this account, many formulae for the strength of reinforced concrete have been developed on the basis of a uniform modulus of elasticity for the concrete. This is virtually the same as assuming that q equals zero in Equation 16. The other equations which are derived from equations involving q, must also be correspondingly modified.

Adopting the same notation as in Article 250, we may say that the triangle mnN in Fig. 97 represents the compressive forces; that the area of the triangle measures the summation of those forces; and, assuming that in this case c = mn, the summation is:

∑X= 1/2 cbkd..(25)

The center of gravity of the triangle, which is the centroid of compression of the concrete, is at 1/3 of the height of the triangle (kd) from the compression face of the concrete. The same value is obtained by making q = 0 in the equation above Equation 14, which gives us: x = 1/3 kd..(26)

Making q = 0 in Equation 16, we have:

270 Straight Line Formulae 0400131

.(27)

From this equation we may deduce Table XV, which corresponds to Table XIII.

Table XV. Value Of K For Various Values Of R And P (Straight-Line FormulŠ

p

r

.020

.018

.016

.014

.012

.010

.008

.006

.004

.003

10

.464

.446

.427

.407

.385

.358

.328

.292

.246

.216

12

.493

.476

.457

.436

.412

.385

.353

.314

.266

.235

15

.531

.513

.493

.471

.446

.418

.384

.343

.291

.258

20

.580

.562

.542

.519

.493

.463

.428

.384

.328

.292

40

.698

.679

.659

.637

.611

.579

.542 '

.493

.428

.384

From an equation in Article 266, by calling q = 0, we may write: k =cr/ cr + s By making q = 0 in Equation 15, we may write pr =

1/2 k / 1 - k-------- By eliminating k from these two equations, we may write: p = 1c/ 2s cr / (cr+s) ..(28)

The similarity of this equation to Equation 18 is readily apparent, the difference being due only to the elimination of the effect of q.

The moment of resistance of a beam equals the total tension in the steel, or the total compression in the concrete (which are equal), times (d - x). Therefore we have the choice of two values (as before):

Mc

= 1/2 cbkd (d - x)

.(29)

Ms

= As (d - x) = pbds (d - x)

If the economical percentage p has already been determined from Equation 28, then either equation may be used, as most convenient, since they will give identical results. If the percentage has been arbitrarily chosen, then the least value must be determined, as was described in Article 267.