This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
A great deal of T-beam computation is done on the basis that the center of pressure of the concrete is at the middle of the slab, and therefore that the lever-arm of the steel = d - 1/2 t. From these assumptions we may write the approximate formula:
Ms = As (d-1/2t) ..(37)
If the values of Ms and s are known or assumed, we may assume a reasonable value for either A or (d - 1/2 t) and calculate the corresponding value of the other. On the assumption that the slab takes all the compression, the distance between the steel and the center of compression of the concrete varies between (d - 1/2 t) and (d - .14t), which is the approximate value when the beam becomes so small that it merges into the slab. The smaller value (d - 1/2t) is the absolute limit which is never reached. Therefore the lever-arm is always larger than (d - 1/2t). Therefore, if we use Equation 37 to compute the area of steel A for a definite moment Ms and unit steel tension s, the resulting value of A for an assumed depth d, or the resulting value of d for an assumed area A, will be larger than necessary. In either case the result is safe, but uneconomically so.
As an illustration, using the values in Example 2, Article 291, of Ms = 1,350,000; s = 16,000; (d - 1/2 t) = (26.5 - 2) - 24.5, the resulting value of A = 3 44 square inches, which is larger than the more precise value previously computed.
Equation 37 is particularly applicable when the neutral axis is in the rib. Under this condition, the average pressure on the concrete of the slab is always greater than 1/2 c, or at least it is never less than 1/2c. As before explained, the average pressure just equals 1/2c when the neutral axis is at the bottom of the slab. We may therefore say that the total pressure on the slab is always greater than 1/2 c b t. We therefore write the approximate equation:
Mc=1/2 cb't (d- 1/2t)..(38)
As before, the values obtained from this equation are safe, but are unnecessarily so. Applying them to Example 2, Article 291, by substituting Mc = 1,350,000, b' = 60, t = 4, and (d - 1/2t) = 24.5, we compute c = 459. But we know that this approximate value of c is greater than the true value; and if this value is safe, then the true value is certainly safe. The more accurate value of c, computed in Article 291, is 352. If the value of c in Equation 38 is assumed, and the value of d is omputed, the result is a depth of beam unnecessarily great.
If the beam is so shallow that we may know, even without the test of Equation 36, that the neutral axis is certainly within the slab, then we may know that the center of pressure is certainly less than 1/3 t from the top of the slab, and that the lever-arm is certainly less than (d - 1/3t); and we may therefore modify Equation 37 to read:
Ms = As (d - 1/3t) ..(39)
Applying this to Example 1 of Article 291, and substituting Ms = 900,000, s = 16,000, (d - - 1/3 t) = (13.75 - 1.67) = 12.08, we find that A = 4.65, instead of the 4.59 previously computed. This again illustrates that the formula gives an excessively safe value, although in this case the difference is small.
Equations 37 and 38 should be considered as a pair which are applied according as the steel or the concrete is the determining feature. When the percentage of steel is assumed (as is usual), both equations should be used to test whether the unit-stresses in both the steel and the concrete are safe. It is impracticable to form a simple approximate equation corresponding to Equation 39, which will express the moment as a function of the compression in the concrete. Fortunately it is unnecessary, since, when the neutral axis is within the slab, there is always an abundance of compressive strength.