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 m N v. The area of the triangle vn N = 1/2 c kd. Therefore, for the total shaded area, we have:

Fig. 96. Fibre Stresses in Beams.

Fig. 97. Summation of Compressive Forces.

Area = 1/2 c kd + 1/3 (c0 - c) 1/2 kd,

= 1/2 kd (c + 1/3 c0 - 1/3 c),

= 1/2 kd (2/3 c + 1/3 Co).

But in this case, c0= Ecєc; therefore,

Area = 1/2 kd (2/3 c + 1/3 Ec ε c )..(9)

In Fig. 98 has been redrawn the parabola of Fig. 93, in which o is the vertex of the parabola. Here c" is the force which would produce a compression of єc" 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 Ec εc". And c"

=1/2Ecєc".

It is one of the well-known properties of the parabola that abscissę are proportional to the squares of the ordinates, or that (in this case): k'd: mn::

Transforming to the symbols, we have:

(c"-c):c"::(εc" - εc)2 c" -c = c" (εc" - εc)2/ εc"2 c" - c = c" (1 - q)2, since εc/εc" = q. c=c"{l-(l-q) }; = c" (2q-q2) ; = 1/2 Ecεc" (2q - q2), since c"= 1/2 Ec εc"; and also, since εc" = εc/q

= Ecεc (1-1/2q)...(10)

Substituting this value of c in Equation 9, we have:

Area = 1/2 kd {2/3Ecεc(l -1/2q) + 1/3 Ec εc }

= 1/2kd {Ec εc (1 -1/3q)} The summation of the horizontal forces (∑X ) within the shaded area, is evidently expressed by the above "area" mul-tiplied by the breadth of the beam (6). Therefore,

∑X = 1/2 ( 1 - 1/3 q) Ec εc b kd.(11)

In order to avoid the complication resulting from the attempt to develop formulę 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 § 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 2/3 under these conditions, is perhaps determined most easily by computing the ratio of b h to g h (see Fig. 98) when o a is assumed to be 1/3 of o m. In this case, from the properties of the parabola, ab = 1/9 m n; c' = 8/9 m n = 8/9 c" = 4/9 Ec εc".

Fig. 98. Analysis of Compressive Stresses.

But when o a = 1/3 of om, g h = 2/3 Ec εc" 6/9 Ec εc".

Therefore c' = 2/3 g h. But when o a = 1/3 of o m, εc'/εc" Therefore, when c' = 2/3 gh, q = 2/3.

It has already been shown that c" = 1/2 Ec εc", and also that εc"=

εc/q. Therefore 1/2 Ec εc = c"q. It has also been shown that c' =8/9 c", or that c" = 9/8c'. Therefore 1/2 Ec εc=9/8 c'

Substituting this value in Equation 11, we have for the summation of the compressive forces above the neutral axis, under such conditions:

∑x= 9/8 (1 - 1/3q) qc' bkd..(12)

Substituting the further condition that q = 2/3, we have:

∑X = 7/12 c'b kd..(l3)