The various types of REINFORCED CONCRETE FOOTINGS designed to meet the different conditions encountered in building-construction, may be roughly

divided into five groups: (1) Wall Footings (Fig. 5); (2) Independent Columnfootings of either sloped or stepped design (Fig. 6); (3) Combined Footings carrying usually two columns, and constructed as inverted beams (Fig. 7); (4)

Cantilever Footings, in which the eccentricity of an exterior footing is resisted by a strap connected with an adjacent interior footing (Fig. 8); (5) Continuous Footings, constructed as inverted beams and carrying a number of columns (Fig. 9). All of these footings may be used with or without piles, and the design procedure in both cases is substantially the same.

A reinforced-concrete WALL-FOOTING is ordinarily built as a slab, projecting on each side of the wall, and the projection is designed as an inverted cantilever beam, uniformly loaded by the soil-reaction. The depth is usually governed by the shear, as a measure of the diagonal tension, which in turn is based upon the resisting stresses developed in plain concrete without web-reinforcement, as it is not practicable to use stirrups in concrete work of this kind. The section for which the maximum shear is computed is taken at a distance from the wall equal to the effective depth of the footing. The maximum bending moment is at the face of the wall, and its value is used to compute the sectional area of the reinforcement, and to check the depth as dependent upon the crushing strength of the concrete. The steel, in the form of small bars, is placed at rightangles to the wall, and extends to within 3 or 4 in of the edge of the footing. The bond-stress is tested as a function of the maximum shear at the face of the wall, and the values of the bars as governed by embedment, or anchorage, are tested for intermediate sections between the face of wall and edge of footing. Here, as in most footing-design, bond is a governing factor and almost invariably requires the use of smaller bars than would otherwise be chosen. There should be 3 or 4 in* of concrete below the reinforcement, for insulation, and the edge of the footing-slab should not be less than 6 in in thickness above it. Over piles the minimum thickness above the reinforcement should not be less than 12 in.

The INDEPENDENT COLUMN-FOOTING, square or oblong in plan, and of the sloped or stepped type, offers the most satisfactory means of supporting a column-load where there is ample space to permit of a concentric-load design. The choice between a stepped or sloped footing is largely one of individual preference, as the increased cost of form-work for the latter is just about offset by its more economical section. The depth in both cases is usually governed either by (1), the so-called PUNCHING SHEAR, which is computed for a sectionarea equal to the product of the perimeter of the cross-section of the column or pedestal by the maximum thickness of the footing, and expresses the resistance of the footing to the tendency of the column to punch its way through it; or by (2), the vertical shear, as a measure of the diagonal tension, computed for a section at a distance from the face of the column, or pedestal, equal to the effective depth of the footing. As in the wall-footing, stirrups are not used, and the unit shearing stress is based upon the value of concrete without web-reinforcement, ordinarily 40 lb per sq in, except when the reinforcement is anchored by hooking the ends, under which condition most designers allow 60 lb per sq in for 2 000 lb concrete. The maximum bending moment in each direction is at the face of the column or pedestal. The resulting compressive stress in the *New York City requires 4 in, the Joint Committee, 1924, 3 in, and this insulation in both cases applies to all types of footings.

concrete is checked, and the required reinforcement, in the form of small or medium-sized bars, is placed, preferably as a two-way reinforcement parallel to the sides of the footing, and with the same insulation as that noted for wallfootings. The bond is tested at the face of the column, and the bars investigated for security against slipping as previously described.

The use of a pedestal between the footing-slab and the base of the superimposed column is of value, not only in reducing the unit compression on the top surface of the footing, but also in furnishing an easy means of leveling up the footings, when they are of different heights, and thus obtaining greater uniformity in column-construction. The horizontal cross-sectional area of the pedestal is determined by dividing the column-load by the allowable working unit stress in axial compression, and it is often advisable to make this pedestal area as much as twice that of the cross-section of the column. Similarly, where no pedestal is used, the flat top of a sloped, or stepped, footing must usually be made of considerably greater area than that of the supported column, as the average compressive stress, fe, in the column-shaft, especially for spirally reinforced columns, is higher than the compression permitted in the concrete of the footing. In any case, the projections of the flat top of a sloped, or stepped, footing should be at least 3 in on all sides of the column or pedestal.

The Joint Committee, 1924, limits the compressive stress on top of the pedestal, or footing, directly under the column, to the value given by the following formula:

permissible working stress over the loaded area, in pounds per square inch;

total area at the top of the pedestal, or footing, in square inches; loaded area at the column-base, in square inches;

ultimate compressive strength of the concrete, in pounds per square inch.

In sloped or stepped footings A may be taken as the area of the top horizontal surface of the footing, or as the area of the lower base of the largest frustum of a pyramid or cone contained wholly within the footing, and having for its upper base the loaded area A', and having side slopes of 1 vertical to 2 horizontal.

The allowable compressive unit stress on the gross area of the concentricallyloaded pedestal, or on the minimum area of a pedestal-footing, shall not exceed 0.25 f'e, unless reinforcement is provided and the member designed as a reinforced-concrete column. In practice it is usually more convenient to design column-pedestals of sufficient sectional area to carry the imposed loads without other reinforcement than the necessary number of dowels unless their height exceeds three times their least lateral dimension.

In any case pedestals should be thoroughly bonded to the footing below and the column above. For this purpose it is customary to use dowel-rods of the same size and number as the longitudinal reinforcement in the column. The Joint Committee, 1924, requires that such dowels extend, on either side

of the joint, 50 bar-diameters for plain bars and 40 bar-diameters for deformed bars.

The COMBINED FOOTING, rectangular or trapezoidal in plan, and constructed as an inverted beam, may be used where space is limited, as in the case of a wall column when the adjoining property-line makes a footing designed for a concentric load impossible. In such cases an eccentrically loaded isolated footing is avoided by joining the exterior wall-column footing to that of an interior column, as one footing, and so proportioning it in plan that the resultant of the two loads passes through its center of gravity. Ordinarily such a footing is rectangular or trapezoidal in plan, and supports an exterior column and its adjacent interior column, but under some conditions it may be designed to support four columns, as at the corner of a building.

THE CANTILEVER FOOTING is used under the same conditions as the ordinary combined footing. When the allowable soil-pressure does not require the greater bearing area of the former, a cantilever footing is usually more economical. In this type of footing the connecting strap is designed to resist the bending moment due to the eccentricity of the exterior footing and the shear which resists the tendency to uplift. The areas of the two footings are found in the usual manner except that the amount of uplift should be added to the columnload when determining the size of the exterior footing.

THE CONTINUOUS FOOTING can often be used to advantage for exterior columns when the distance beyond the building-line is insufficient for an individual footing, concentrically loaded, and when the loads on the wall columns are light enough to be carried without eccentrically loading the available area. The footing-slab is designed as a continuous beam supporting a series of wall columns, or two continuous footings may be constructed at right angles to each other, forming a mat over the entire area of the basement. The design is developed by the same formulas, or their derivatives, and follows a procedure. similar to that employed for rectangular beams, except that it is good practice to use slightly larger moment factors, and a more general distribution of steel when the supporting material is a compressible soil or pile-group.

5. Bending Moments in Footings. The following discussion treats of the bending moments in concrete footings:

Let

M bending moment in inch-pounds;

=

=

W = unit soil-pressure design-load divided by footing-area, in pounds

d

ι

=

=

per square foot;

length of the projection of the footing-slab, or distance between center lines of columns in combined footings, in feet;

effective depth of section in inches;

L = length of footing in feet;

b

=

width of footing or of the section considered parallel to the wall or face of column in inches;

k ratio of distance of neutral axis of cross-section from extreme fibers in compression to effective depth of section;

kd

j

=

= distance of neutral axis from extreme fibers in compression in inches; ratio of distance between center of compression of concrete and center of tension of steel to effective depth of section; ratio of arm of resisting couple to d;

=

jd distance between center of compression in concrete and center of tension in steel; arm of resisting couple in inches;

compressive unit stress in extreme fibers of concrete in pounds per square inch;

tensile unit stress in the steel in pounds per square inch;

area of cross-section of main tensile reinforcement in square inches; modulus of elasticity of steel divided by modulus of elasticity of concrete, E./Ec;

=

The general equation for the maximum bending moment occurring at the support in a UNIFORMLY LOADED CANTILEVER is M WL/2, in which W represents the total distributed load and L the length of the projection. Making the appropriate changes in this equation to permit the use of w in pounds per square foot, with l equal to L in feet, W equal to wl, and M in inch-pounds,

Having found the width of footing,

This formula will be used for all footings treated as simple cantilevers, the maximum moment being at the face of the wall or column. maximum bending moment by Formula (1) for a 1-ft the depth of the concrete, determined by this maximum bending moment, is given by Formula (10), Chapter II:

In Table I, Chapter II, other values of j, k, and p, corresponding to other stresses in the steel and concrete are given. The sectional area of the steel in 1 ft of width is given by Formula (12), Chapter II.

or by Formula (11), Chapter II,

A. = M/f.jd

A, = pbd