(3) The MAXIMUM BENDING MOMENT is computed by Formula (7),

(4) The DEPTH AS GOVERNED BY BENDING MOMENT is computed by Formula

The depth, 27 in, determined by the punching shear, is adopted.

(5) The DEPTH AS GOVERNED BY DIAGONAL TENSION is computed by Formula (16),

or, substituting numerical values

[10.52 - {2.5+ (27/6)}2] × 3 636 d

1 680[2.5+ (27/6)]

A depth of 19 in is accordingly adopted at the section CC', at the distance d from the face of the column or pedestal.

(6) The SECTION-AREA OF STEEL, in each direction, is computed by Formula (12), Chapter II,

As a trial, nineteen 5-in square bars are selected.

(7) The BOND-STRESS. Since the vertical shear at each face of the column

is

in which 10.5 is the area of the plan of the footing, in square feet, 2.52 the area of the column-section in square feet, and 3 636 lb per sq ft the net unit soilpressure used in the design, by Formula (14), Chapter IV,

u = V/Zojd

=

94 500/(2.5 × 19 × 0.875 X 27)

= 85 lb per sq in

In this equation, 2.5 in is the perimeter of the cross-section of each of the nineteen 5%-in square bars. To illustrate the computation for sections other than those of maximum shear, the bond-stress at a distance from the face of the column equal to the depth of the footing is

in which the numerator is the vertical shear at each of the four faces of the column at a distance d from the face, the value of d being determined by equation (19). The depth, 19 in, is at section CC', Fig. 10.

As these bond-stresses are permissible, the reinforcement is accepted as com

puted, namely, nineteen 5%-in square bars, and as the maximum stress exceeds 80 lb per sq in, a deformed bar is used.

The effective width of the footing is 30+ (2 × 27) + 42/2 = 105 in, or 8 ft 9 in (see page 223). Ordinarily the reinforcing bars should be spaced at equal intervals across this width, with extra bars at double the spacing for the remaining distance. In this case, however, the remaining distance is so small that the computed number of bars are uniformly spaced about 64 in on centers across the full width of footing and to within 6 in of the edge of the footing-slab, as shown in Fig. 12.

By Formula (16), Chapter IV, the embedment required for deformed bars, in order to develop a tensile strength of 16 000 lb per sq in, is 40 diameters in length. If 5%-in bars are used, this distance is 25 in. Consequently we know that the full strength of the bars can be developed at a considerable distance before they reach the face of the pedestal. Fig. 12 illustrates the complete design, in which the thickness at the edge of the footing-slab has been increased from a minimum of 9 in to 15 in, in order to have the required depth of 19 in at section CC'.

30"

C'

6

16"

9'-0"-12'-0"

Critical Section

28"

= 45

15 in, for jg = (d − 3) 24 in, and for fi = (126 – 36)/2

in, and solving for ej,

=

ments as in the previous example, but that the conditions are such that the footing cannot be square.

(1) The AREA OF THE FOOTING-SLAB. The estimated weight of the footing is 40 000 lb. Then the area of the footing is

440 000

=

110 sq ft

A footing-slab, 9 ft X 12 ft in plan is assumed to satisfy the conditions. The

DESIGN-LOAD is the net unit soil-pressure, exclusive of the weight of the footing,

(2) The DEPTH AS GOVERNED BY PUNCHING SHEAR is computed by Formula

In which

(9 +2.5)
2

is the average length, and 4.75 the width, of the trapezoidal portion of the footing tributary to the side of the column, the length of the side of the column being 30 in.

In oblong footings the depths, as computed above, are usually different for the different column-faces, and they should be computed separately and the maximum value used. In this case it is obvious that the side chosen gives the maximum value of d.

or

and

As a trial, d is made 28 in.

(3) The MAXIMUM BENDING MOMENT is computed by Formulas (5) and (6), M6w(a+1.33c1)c2 = 6 x 3 700[2.5+ (1.33 X 3.25)] X 4.752

M6w(a+1.33c)c1 = 6 x 3 700[2.5+ (1.33 X 4.75)] X 3.252

or

M

=

22 200 X 8.8 X 10.56

= 2 063 000 in-lb

(4) The DEPTH AS GOVERNED BY BENDING MOMENT is computed by Formula

in which the numerator is the greater of the two moments. The depth of 28 in, determined by the punching shear, is therefore justified.

(5) The DEPTH AS GOVERNED BY DIAGONAL TENSION is computed by a derivative of Formula (1), Chapter IV,

in which 12.0 X 9.0 ft) is the area of the footing, 7.16 ft the side of a square extending d in, or 28 in outside of the faces of the column, and the first term of the denominator the perimeter of this square in inches.

The depth of the footing at the section CC', at the distance d from the face of the column or pedestal, is therefore made 18 in.

(6) The SECTION-AREA OF STEEL in each direction is computed by Formula (12), Chapter II,

As a trial, for the LONG WAY, twenty-three %-in square bars are assumed to be used, and for the SHORT WAY fourteen %-in square bars.

(7) The BOND-STRESS. Since the vertical shear on the column-face parallel to the shorter dimension is, from paragraph 2,

The shear on the column face parallel to the longer dimension is

As these bond-stresses are permissible, the reinforcement is accepted as computed, that is 23 %-in square bars the long way, and 14 %-in square bars the short way. Deformed bars are used. The steel is located as illustrated in Fig. 13, and the depth of the edge of the footing-slab, 16 in, is determined as in the previous example. Bars in short direction are hooked.

13. Design-Procedure for Combined Concrete Footings. Trapezoidal and rectangular footings of this type are shown in plan in Fig. 14.

Let

P1 = load on wall-column in pounds;

P2

=

load on interior column in pounds;

1 = distance between center lines of columns in feet;

L = length of footing in feet;

S

Q

Q'

=

=

distance from center of wall-column to center of gravity of footingarea in feet;

distance from center of wall-column to exterior edge of footing in feet;

= distance from center of interior column to interior edge of footing

in feet;

m = width of footing-slab at narrow end in feet;

n = width of footing-slab at wide end in feet;

y

=

=

width of any vertical section of footing in feet, at x distance from

end m;

distance in feet from end m to section of maximum bending moment; d effective depth of footing in inches;

b

=

width of any section of footing considered in inches.

Unless the wall-column load is greater than that of the interior column, a

combined footing can usually be designed with a rectangular plan, except when the allowable projection Q' is very small (Fig. 14). Knowing the distance Q', the question whether or not it is possible to use a rectangular type can be quickly decided by applying one of the conditions of equilibrium of vertical forces acting in a plane, that is that the resultant of the two downward forces, or loads, must pass through the center of gravity of the footing-area, through which the resultant of the upward soil-resistances passes. Referring to Fig. 14, it is clear that, if the footing is to be rectangular in plan, L/2 = S + Q. Taking the center of moments at P1, and remembering that the line of action of the resultant of the upward soil-reaction (P1 + P2) acts at G,

Pal (P1 P2)S,

which determines S in terms of the loads and the distance between columns. Substituting this value of S in L/2

S + Q

If this value of L, with the resulting projection Q', is satisfactory, the footing is designed as a rectangle, otherwise a trapezoidal form is used to obtain the required footing-area.

The following procedure is based on the general treatment of trapezoidal footings, and the subsequent example on that of the rectangular form, which is the one commonly used in practice.