STRENGTH OF T=BEA IVIS 2S4. When concrete beams are laid in conjunction with over lying floor-slabs, the concrete for both the beams and the slabs being laid in one operation, the strength of such beams is very much greater than their strength considered merely as plain beams, even though we compute the depth of the beams to be equal to the total depth from the bottom of the beam to the top of the slab. An explanation of this added strength may be made as follows: If we were to construct a very wide beam with a cross-section such as is illustrated in Fig. 104, there is no hesitation about lating such strength as that of a plain beam whose width is b, and whose effective depth to the forcement is d. Our previous study in plain has shown us that the steel in the bottom of the beam takes care of practically all the tension ; that the neutral axis of the beam is somewhat above the center of its height; that the only work of the concrete below the neutral axis is to transfer the stress in the steel to the Fig. 104. T-Beam in Cross-Section. concrete in the top of the beam; and that even in this work it must be assisted somewhat by stirrups or by bending up the steel bars. If, therefore, we cut out from the lower cor ners of the beam two rectangles, as shown by the unshaded areas, are saving a very large part of the concrete, with very little loss in the strength of the beam, provided we can fulfil certain conditiOns. The steel, instead of being distributed uniformly throughout the bottom of the wide beam, is concentrated into the comparatively narrow portion which we shall hereafter call the rib of the beam. The concentrated tension in the bottom of this rib must be transferred to the compres sion area at the top of the beam. We must also design the beam so that the shearing stresses in the plane (nin) immediately below the slab shall not exceed the allowable shearing stress in the concrete. We must also provide that failure shall not occur on account of shear ing in the vertical planes On r and n s) between the sides of the beam and the flanges.
2S5. Resisting Moments of T=Beams. These will be com puted in accordance with straight-line formulw. There are three possible cases, according as the neutral axis is: (1) below the bottom of the slab (which is the most common case, and which is illustrated in Fig. 105); (2) at the bottom of the slab; or (3) above it. All pos sible effect of tension in the concrete is ignored. For Case 1, even the compression furnished by the concrete between the neutral axis and. the under side of the slab is ignored. Such compression is of course zero at the neutral axis; its maximum value at the bottom of the slab is small; the summation of the compression is evidently small; the lever arm is certainly not more than iy; therefore the moment clue to such compression is insignificant compared with the resisting moment due to the slab. The computations are much more com
plicated; the resulting error is a very small percentage of the true figure, and the error is on the side of safety.
2S6 Case 1. If c is the maximum compression at the top of the slab, and the stress-strain diagram is rectilinear, as in Fig. 105, then the compression at the bottom of the slab is e — . The average hi — compression = (e c kd kd) = (1,•c1— 1). The total corn The miter of gravity of the compressive stresses is evidently at the center of gravity of the trapezoid of pressures. The distance x of this center of gravity from the top of the beam is given by the formula: t 3 kd-2 t(33) 3 kd—t If the percentage of steel is chosen at random, the beam will probably be over-reinforced or under-reinforced. In general it will therefore be necessary to compute the moment with reference to the steel and also with reference to the concrete, and, as before with plain beams (Equation 29), we shall have a pair of equations: 2S7. Case 2. If we place kd = t in the equation above Equation 34, and solve for d, we have a relation between d, c, s, r, and t, which holds when the neutral axis is just at the bottom of the slab. The equation becomes: (36) cr A combination of dimensions and stresses which would place the neutral axis exactly in this position, is improbable, although readily possible; but Equation 36 is very useful to determine whether a given numerical problem belongs to Case 1 or Case 3. When the stresses s and c in the steel and concrete, the ratio r of the elasticities, and the thickness t of the slab are all determined, then the solution of Equa tion 36 will give a value of d which would bring the neutral axis at the bottom of the slab. But it should not be forgotten that the coin.. pression in the concrete (c) and the tension in the steel will not simultaneously have certain definite values (say c = 500, and s = 16,000) unless the percentage of steel has been so chosen as to give those simultaneous values. When, as is usual, sonic other percentage of steel is used, the equation is not strictly applicable, and it therefore should not be used to determine a value of d which will place the neutral axis at the bottom of the slab and thus simplify somewhat the numerical calculations. For example, for c = 500, s = 16,000, r = 12, and t = 4 inches, d will equal 14.67 inches. 01 course this particular depth may not satisfy the requirements of the problem. If the proper value for d is less than that indicated by Equation 36, the problem belongs to Case 3; if it is more, the problem belongs to Case 1.