Formulas (12) and (17), Chapter II, are of general application, and Formula A' m (11), Chapter II, and the above Formula (4), are to be used only when it is desired to exactly BALANCE the steel and concrete (see page 21), or, to express it differently, when the depth of the footing is determined by the compressive strength of the concrete. The maximum bending moment in an INDEPENDENT COLUMN-FOOTING of the sloped or stepped-type, is at the face of the column, and may be taken as equal to the moment of a trapezoidal solid of thickness w. In Fig. 10, B' the bending moment about the section AA' is considered as that of the trapezoidal area Imno about the face ol of the column. The static moment of this trapezoid about the axis AA' is equal to the moment of the rectangle lpqo, plus the moments of the two triangles lpm and oqn. Since the normal distance of the center of gravity of each triangle from the axis AA' is % of its altitude, and as the uniform unit soil-reaction is represented by w lb per sq ft, the bending moment about the section AA' is Fig. 10. A n Rectangular Reinforced Concrete Footing Expressing the moments in inch-pounds, when a, a1, c, and c1 are in feet, the moment about the section AA' is in which a and a1 are the lateral dimensions, in feet, of a rectangular column, or the sides of an equivalent square section for a round or octagonal column. The lengths c and c1 are the projections, in feet, of the footing-slab beyond the faces of the column, or of the equivalent square. These are the general formulas applicable to moment-calculations for rectangular footings. Irregular-shaped footings are divided into trapezoidal solids, and the maximum bending moment of each is computed about its proper column-face. It is to be noted that for circular columns, in these formulas α = (diameter of column, in feet) × 0.886 In a square footing, when the projection of the footing-slab is the same on each side of the column, the moments are equal about each face, and the formulas become, M6w(a+1.33c)c2 (7) In the light of the experiments made at the University of Illinois, many designers follow Professor Talbot's recommendations, and write this equation for square footings, M = 6w(a+1.2c)c2 * (8) Having determined the maximum bending moment, the depth of the concrete, which is usually governed by shear, is checked by Formula (9), Chapter II, in which e is the length of the horizontal top of the footing, in feet. If this approximation, determined by the last formula, indicates an excessive stress in the concrete, a more exact computation is made, by using the actual effective cross-section, and computing the required depth d by Formula (10), Chapter II. In square footings, as stated above, the bending moment is the same about each face of the column, but in those oblong in plan, the maximum bending moment is the one used in the computation for depth. The sectional area of the reinforcement, as governed by bending moment, is determined for each direction as in the case of simple wall footings, by either Formula (11) or Formula (12), Chapter II, or their appropriate derivatives. The steel is distributed, with equal spacing, across the entire width of the footing and to within 6 in of each face; or, preferably, through a distance equal to the width of the column, plus twice the depth of the footing, plus half the remaining distance to the face of the footing on each side. If the latter arrangement is followed, additional bars should be placed outside the effective width, at twice the spacing of the computed reinforcement. Whatever arrangement is employed for the placing of the steel, the effective width of the concrete *This formula conforms with the recommendations of the Joint Committee, 1924. section should not be greater than the distance through which the computed amount of reinforcement is placed. The BENDING MOMENTS for combined, cantilever and continuous footings are computed by the same formulas used for a rectangular beam. By substituting wl for W, making the necessary changes in the equation to permit the use of w in lb per sq ft, and expressing M in inch-pounds, the following formulas are derived from those on page 23, and give the values of the maximum bending moments per foot of width. For footings constructed as simple beams, 6. Diagonal Tension in Footings. As in the case of beam-design, the shearing-stress developed in a section is considered to be a measure of the diagonal tension. Let Then, V = total vertical shear at section considered in pounds; v unit shearing-stress in pounds per square inch; b d = = width, or length of perimeter of section in inches; j = 0.875, approximation used for shear-computations; w = unit soil-pressure in pounds per square foot = design-load in pounds divided by footing-area in square feet. v V/jbd (Formula 1, Chapter IV) Or, transposing d and v and substituting the maximum allowable values for the latter, v = 120 lb per sq in, d = (13) V/105b (See Formula (4), Chapter IV) v = 150 lb per sq in, Since the vertical shear at any section of a beam is equal to the sum of the reactions minus the sum of the loads to the left (or right) of the section, V is determined, in the case of footings, by considering the soil-reactions as positive and the column-loads as negative, their algebraic sum being the vertical shear at any section. The MAXIMUM SHEARS are then found for COMBINED and CON d = V/131b (See Formula (5), Chapter IV) TINUOUS FOOTINGS, constructed as simple or continuous beams, at the faces of the columns, and for WALL-FOOTINGS, or sections of combined footings reinforced as cantilevers, at a distance,out from the face of the wall or column equal to the effective depth of the footing. Considering, in the latter case, a length of 1 ft measured parallel to the wall, b is 12 in, and V is the soil-reaction per sq ft w times the length l — (d/12), or from Formula 1, Chapter IV. V [l — (d/12)] × w v = = jbd (14) For INDEPENDENT COLUMN-FOOTINGS supported upon soil, tests show that the critical section for failure by diagonal tension may be taken at a distance out from the face of the column, or pedestal, equal to the effective depth of the footing, and for pile footings, at the inner edge of the first row of piles entirely outside of a section mid-way between the face of the column, or pedestal, and the section described above; but in no case should the critical section be outside of that designated for soil-footings. As it is not practical to use stirrups in stepped or sloped footings, the shear is kept within the limits allowed for concrete without web-reinforcement. The required depth of the footing at the critical section, distance d from the face of the column, is then given by the formula, in which the area, since it is multiplied by w, is taken in square feet, and the perimeter, representing b, in inches. Referring to Fig. 10, the "area outside of distance d," is the shaded area in the drawing, and the depth d' is the effective depth at the section CC'. For SQUARE FOOTINGS UNDER SQUARE COLUMNS the formula may be written In these formulas L is the side of the footing, and a the width of a square column, or, for round columns, the side of an equivalent square. Both dimensions are in feet. In pile footings the shear should be computed not only at the critical section defined above, but also at the inside edge of the exterior row of piles. 7. Punching Shear in Concrete Footings. When designing under certain building ordinances, the depth of a footing at the face of a column is often determined by the unit punching shear which is computed for the whole or a part of the column-section periphery, considering the portion of the total shear that is tributary to the section chosen. The UNIT PUNCHING SHEAR is sometimes taken as high as 200 lb per sq in for a 2000 lb concrete, and computed for the depth jd, while other building ordinances specify a lower stress and base the resistance on the total effective depth of the footing. Following this practice, let Then V = total shear upon part or whole of column-section periphery in pounds; v2 = unit punching shear in pounds per square inch; V2 b d = = length of column-section periphery, or of the part considered in computing V, expressed in inches; effective depth of section in inches; a = side of square column in feet; D = diameter of round column in feet; L = length or side of footing in feet; w = unit soil-pressure in pounds per square foot For SOIL-FOOTINGS punching shear is considered only in relation to the columns and pedestals, where such occur, but when a footing is supported on PILES, it is necessary to check the depth of the concrete over the exterior row adjacent to the edge of the footing-slab, by Formula (18), in which V is the pile-load, or reaction, in pounds, and b the pile-section perimeter in inches. For a group of piles in which the pile-loads are very heavy, the above check may result in an increase of thickness of the concrete at the edge of the footing-slab. 8. Bond Stresses in Reinforced-Concrete Footings. The bond-stress in the reinforcement, which is an important factor in all footing-design, is computed for WALL-FOOTINGS as a function of the vertical shear at the face of the wall, and for INDEPENDENT, COMBINED, or CONTINUOUS COLUMN-FOOTINGS, at |