in which b, the width of section in inches, is equal to the length of the side m, plus kx, where x is, as previously, the distance of the section from the end m; and k is the rate of change in the width of the footing, equal to (n − m)/L. (7) The AREA OF LONGITUDINAL STEEL BETWEEN COLUMNS is computed by Formula (12), Chapter II,

(8) The MAXIMUM BENDING MOMENTS IN THE DIRECTION OF THE LENGTH, IN EACH CANTILEVER, are computed by the formula

M = w(3n- kq) × (q2/6)

(31)

This is the general equation for the bending moments in the cantilevers projecting beyond each column, derived by computing the bending moment of a trapezoidal area about one face of a column; and it can be applied to any section where n represents the length of the adjacent end of the footing and q the distance from the end to the section considered, expressed in feet.

The complete derivation of formula (31) is as follows. The soil-pressure at any section distant q from the wide end of the footing is

By a theorem in mechanics it can be shown that the distance of the center of gravity of this trapezoidal area to the left of this section is

(9) The DEPTH Of footing in eaCH CANTILEVER is computed in two ways:

(a) As governed by the bending moment, computed as in paragraph (8), by Formula (2)

in which b is determined as explained in paragraph (6), (c), above.

(b) As governed by diagonal tension, computed by Formula (1), Chapter IV,

d = V/bjv

in which V is the total reaction on each cantilever at a distance d from the face of column.

(10) The SECTION-AREA OF THE LONGITUDINAL STEEL IN THE CANTILEVERS, IN THE DIRECTION OF THEIR LENGTH is computed by Formula (12), Chapter II,

A. = M/f.jd

For bond, Formula (14), Chapter IV, is used

น = V/Zojd

in which V is the reaction on each cantilever at the face of the column. For instance, if q' represents the projection of the footing beyond the column-face, as in Fig. 14, in relation to column P2, then

in which the expression (2n — kq')/2 is the average width of the area the length of which is q'.

For the anchorage, Formula (16), Chapter IV, is used,

1 = (1⁄44)(fs/u)i

For

(11) The SECTION-AREA OF CROSSWISE STEEL UNDER EACH COLUMN. purposes of computation, and for determining the spacing of the steel, the width of the crosswise distributing beam, in the direction of the length of the footing, should be such that the assumed projections on each side of the column are equal, or the entire distance may be so considered from the end of the footing to the section of maximum moment between columns. The BENDING MOMENTS ACROSS THE FOOTING: Adjacent to end m,

(33)

In both cases the column-width is taken as the dimension across the footing.

The SECTION-AREA OF THE STEEL is computed by Formula (12), Chapter II,

A. = M/f.jd

For the bond, Formula (14), Chapter IV, is used

u = V/Zojd

For the anchorage, Formula (16), Chapter IV, is used

1 = (4)(f./u)i

The depth of the concrete as determined by the bending moment at the columnface, and by the diagonal tension at a distance d from the column-face, is usually less than that required by the shears and bending moments previously determined. They may be computed, however, by Formula (2), this chapter, and Formula (1), Chapter IV, respectively, using the width in inches of the distributing beam, as defined above, for the value of b.

(12) The WEB-REINFORCEMENT is computed by Formula (13), Chapter IV,

14. Typical Design of a Combined Concrete Footing, Rectangular in Plan. The details of a footing of this type are shown in Fig. 15.

It is required to design a footing to support two columns as follows:

Column No. 1, 20 × 20 in in cross-section.

Column No. 2, 24 × 24 in in cross-section.

Load P1 = 200 000 lb

As these values meet the conditions at the site, L is made 22 ft, Q' 4 ft 4 in, and the footing designed rectangular in plan.

(1) The AREA OF THE FOOTING-SLAB. Assuming that no load is transmitted to the footing from the basement-floor, and estimating the weight of the footing to be 40 000 lb, the area of the footing-slab is

Fig. 15. Details of Combined Reinforced-Concrete Rectangular Footing for Two

Columns with Unequal Loads

(2) The CENTER OF GRAVITY OF THE LOADS.

The position of this from the

center of the wall column is computed by Formula (24),

(3) The LENGTHS OF THE SIDES OF THE FOOTING, m and n, are computed by Formula (27),

in which the value of L is equal to Z X 2, since the footing is rectangular in plan.

A width of 4 ft is assumed.

Then, the net unit soil-pressure is

w = (200 000 + 280 000)/(22 × 4)

=

5 450 lb per sq ft

(4) The SECTION OF MAXIMUM BENDING MOMENT BETWEEN COLUMNS. is computed by Formula (29),

This

This is the distance from the end m of the footing to the section of maximum bending moment.

(5) The MAXIMUM BENDING MOMENT BETWEEN COLUMNS is computed by Formula (30),

Since the rate of change k in width for rectangular footings is zero, the first term of the equation reduces to zero. Substituting the numerical values given in the example

The same result is obtained by taking moments, as described on page 239, about the section distant 9.17 ft from the end of the footing:

[200 000 X (9.17 1.66)] [9.17 X 4 X 5 450 X (9.17/2)]

7 032 000 in-lb

(6) The DEPTH OF THE FOOTING BETWEEN COLUMNS is computed in three