68. Standard Notation. The following notation is recommended by the Joint Committee: M=Moment of resistance, or bending moment in general, in inch-pounds.
A=--Steel area, in square inches.
f,=Tensile unit stress in steel.
unit-stress in concrete.
b=Breadth of beam (in a T-beam, breadth of flange), in inches.
b'=Breadth of web or stem in a T-beam, in inches. d=Depth of beam from compressive surface to center of steel, in inches.
t=Thickness of flange, in inches.
jd=Distance between center of compression and center of steel, in inches.
k=Ratio of depth of neutral-axis to effective depth, d. V=Total shear, in pounds.
v=Shearing unit-stress, in pounds per square inch. u=-Bond stress per unit-area of bar. o=Circumference or perimeter of bar, in inches. o=Sum of the perimeters of all bars, in inches. s=Distance apart of the stirrups.
69. Rectangular Beams. In a rectangular beam, neither the allowable working compressive stress in the concrete nor the allowable tension in the steel should be exceeded. These stresses in practice may be determined by the following formulas: M fs— .87 Ad .... (1) 2M f (2) For 0.8 per cent steel, which gives the stresses already suggested for 2,000-pound concrete, j = 0.88, and k = 0.38, so that the value of becomes approximately 6M 70. For a design based on working stresses recom mended in paragraphs 60, 66, and 67 (that is, compres sion in concrete, 650 pounds per square inch; tension in steel, 16,000 pounds per square inch; and ratio of elas ticity of 15), the depth of the beam is -.1 IM 10 b In order to have the specified tension and compression so that the above formula will apply, the percentage of steel must be 0.8 per cent. The area of cross-section of tensile steel reinforcement is therefore: A = 0.008bd (4) In using these formulas, the maximum bending mo ment, M, is computed in inch-pounds, the breadth b is assumed, and the depth to steel, d, is found from the formula. The full depth is obtained by adding to d a thickness of concrete required to imbed and protect the steel. The area of steel is then obtained from formula (4). Shear and bond are considered in succeeding paragraphs.
71. T-Beams. Where adequate bond between slab and web of beam is provided, the slab may be considered as an integral part of the beam, and its effective width may be determined by the following rules : (a) It shall not exceed one-fourth of the span length of the beam; (b) Its overhanging width on either side of the web shall not exceed four times the thickness of the slab.
72. Following the above requirements, and where ample provision has been made for shear, the formulas just given for rectangular beams may be applied without material error so long as the depth d is not greater than 5 times the thickness of the slab.
73. In using the rectangular beam formulas, the width of the slab as above specified is taken as the breadth of beam, b, in computing compression; but in the formu las for shear and bond, the width of the stem or web is used.
74. The area of steel may be obtained (when the depth of beam, d, does not exceed 5 times the thickness of the slab), by the formula : A — M (5) .87f For 16,000-pound stress in steel, this formula becomes: d 75. The width of the stem or web of the T is gov erned by the proper embedding of the rods and the area required to resist diagonal tension, as indicated below. Web reinforcement is considered in paragraphs 86 to 95.
76. Floor-Slabs. Floor-slabs should be designed and reinforced as continuous over the supports. If the length of the slab exceeds 1.5 times its width, the entire load should be carried by transverse reinforcement. Square slabs may well be reinforced in both directions, using one-half the computed reinforcement in each direction.
77. The loads carried to beams by slabs which are reinforced in two directions will not be uniformly dis tributed to the supporting beam, but may be assumed to vary in accordance with the ordinates of a triangle. The moments in the beams should be calculated accordingly.
78. Continuous Beams and Slabs. In computing the positive and negative moments in beams and slabs con tinuous over several supports, due to uniformly dis tributed loads, the following rules are recommended: (a) That for floor-slabs the bending moments at 2 center and at support be taken at w1 for both dead and 12 live loads, where w represents the load per linear foot and 1 the span length.* (b) That for beams the bending moment at center and at support for interior spans be taken at , and for end spans it be taken at center and adjoining support, for both dead and live loads.* (6) * A still more conservative plan is to use 10—at the middle of all ssans.