diagrams, Fig. 15, the loads are considered as if concentrated at the center lines of the columns. The values of the theoretical shears and moments are, therefore, slightly larger than those used in the example, as the latter are computed at the faces of the columns. The procedure in making the diagram is as follows:

The total upward reaction on the footing-slab per foot of length is equal to

The position of the section of zero shear and maximum bending moment, as previously determined, can be checked by dividing either load by the upward reaction per foot of width. For example,

200 000/21 818 = 9.17 ft

In the general case of the trapezoidal footing, the bending moments between columns are plotted from Formula (30) for the different sections distant x from the right end of the footing. The bending moments in the cantilevers are computed by Formula (31) for the different sections distant q from the adjacent end. For rectangular footings, the moments at any sections, either between the columns or in the cantilevers, are more easily determined by taking the algebraic sum of the moments of the loads and reactions to the right or left of the section, considering those acting downward as negative and those acting upward as positive.

The points of inflection, or of zero bending moment, are determined by Formula (30), by making M2 equal to zero and solving for x. In the example, using the data in paragraph (5),

Extracting the square root of both members of the equation,

These two values of x, plotted from the right end of the footing, give the locations of the points of inflection of the footing-beam between the columns.

15. Design-Procedure for Cantilever Footings. Cantilever footings, sometimes called pump-handle footings, are widely used to overcome the eccentricity of exterior columns when an adjacent property line, or building, prevents the construction of a concentric footing. The design is essentially that of two independent footings jointed by a connecting strap which resists the bending moment caused by the eccentricity, e, of the wall-column, see Fig. 16. This strap is considered to act as a balanced cantilever, and assumed to rest upon the center of gravity of the exterior footing. Owing to the restraints exerted by the columns at the ends of the cantilevers, the stresses in the strap are highly indeterminate, but an approximate solution may be obtained by assuming that the ends of the cantilevers are free to rotate which results in a theoretical zero bending moment beneath each column. If the exterior footing reaction be considered as concentrated at the center of gravity of the footing area, a maximum moment occurs at the point of application of R1. As the pressure under soil-footings is assumed to be uniformly distributed over the footing area, the distance from the exterior edge of the footing to the section of zero shear, and of maximum moment, is found by dividing the column load P1, by the soil reaction per linear foot of footing. For footings on piles, the section of maximum moment is likewise at the point of zero shear, found by considering the column load in relation to the pile reactions. This section is usually along the center line of the last row of piles on the side toward the interior footing. The reactions, moments, and shears are found as follows:

Let

P.

=

design load on exterior column in pounds;

p = reaction upon soil-footings per linear foot in a direction parallel

R1

R2

=

=

to the strap, in pounds;

total reaction on exterior footing in pounds;

reaction on interior footing resulting from the load P1, also in pounds;

Mbending moment in the strap in foot-pounds;

e eccentricity of exterior footing, i.e., the distance between the center of pressure of the wall-column and the center of gravity of the footing area, expressed in feet;

Fig. 16. Details of a Reinforced-Concrete Cantilever-Footing

Neglecting the weight of the strap in determining its dimensions, and taking moments about the centers of gravity of each footing area,

If the reaction of the exterior footing is considered as concentrated at its center, the maximum moment occurs at this section and its value is

M =

Pie

(36)

In the case of soil-footings, the reaction is uniformly distributed over the footing area and the section of maximum moment is at a distance P1/p from the exterior edge of the footing, and its value is

The value of the maximum moment for footings supported upon wood, or concrete piles, found by taking moments about the center line of the last row of piles is,

The section of the strap adjacent to the interior face of the exterior column is ordinarily designed for the maximum bending moment, which determines the concrete and steel areas. The shearing stress, as a measure of the diagonal tension, usually determines the area of the strap adjacent to the outside face of the interior column, which is ordinarily designed for a shear equal to

V

=

Pie/l

(39)

This latter section, however, should be checked to resist a positive bending moment due to the restraint of the interior column, which may approach a maximum value of one-half the moment previously computed.

Besides the steel required for the computed moments, it is desirable to guard against the ill effects due to a reversal of stress, owing to unequal settlement, by providing a certain amount of reinforcement at both top and bottom of the strap throughout its entire length. In all designs adequate anchorage should be obtained, particularly at the exterior terminations of the main longitudinal reinforcement in the top of the strap.

The following steps apply to the design of cantilever footings:

(1) Design the interior footing as an independent unit, following the procedure for individual column-footings, section 12 of this chapter, and neglecting any effect of the uplift.

(2) Compute the approximate area of the exterior footing for the load used in the design of the column, plus any additional load in the basement story, and an allowance of 25 per cent to cover the weight of the footing, the weight of the strap, and the increment due to up-lift.

(3) Knowing the distance e, from this preliminary design, calculate the amount of up-lift equal to Pie/l.

=

(4) Compute R1 P1+ Pie/l, and p, the reaction upon the footing per linear foot, in a direction parallel to the strap. Then, a2 Pi/p and a1 is also known.

=

(5) Compute the maximum bending moment in the strap by Formulas,

(6) Assume a width, b, for the strap, usually equal to the column width, and compute the depth and steel area at the interior face of the exterior column by Formula (10), Chapter II,

(7) Compute the vertical shear at the exterior face of the interior footing assuming that

(8) Determine the section of the strap, as governed by shear at the exterior face of the interior footing by Formula (1), Chapter IV,

d = V/vjb

(9) Knowing the value of the up-lift and the weight of the strap, determined by the computations in steps five to eight, check the load assumption in step two, and redesign if necessary.

(10) Check the concrete section and steel area of the strap adjacent to the interior footing, for a positive bending moment in the strap, varying from onequarter to one-half of the maximum negative moment. Add reinforcement, if necessary, along the bottom of the strap, to obtain an amount varying from one-fifth to one-third of that required at the top.

(11) Provide proper anchorage for all the main longitudinal reinforcement at the top of the strap, particularly beneath the exterior column.