The above designs for footings have been confined solely to the simplest case of the footing required for a continuous wall. A column or pier must be supported by a footing which is offset from the column in all four directions. It is usually made square. The area is very readily obtained by dividing the total load by the allowable pressure per square foot on the soil. The quotient is the required number of square feet in the area of the footing. If a square footing is permissible (and it is usually preferable), the square root of that number gives the length of one side of the footing. Special circumstances frequently require a rectangular footing or even one of special shape. The problem of designing a footing so that the center of pressure of the load on a column shall be vertical over the center of pressure of the subsoil, is usually even more complicated than the problem referred to in section 189. A column placed at the corner of a building which is located at the extreme corner of the property and which cannot extend over the property line, must usually be supported by a cantilever (or by two of them at right angles), balanced at the other end by the load on another pier or column. While the general principle involved in such methods is very simple, a correct solution often requires the exercise of considerable ingenuity.

The determination of the thickness of such a footing depends somewhat upon the method used. When the grillage is constructed of I-beams as illustrated in Fig. 43, the required strength of each series of beams is readily computed from the offset of each layer. If the footing consists of a single block of stone or a plate of concrete, either plain or reinforced, the thickness must be computed on the basis of the mechanics of a plate loaded on one side with a uniformly distributed load and on the other side with a load which is practically concentrated in the center. The theory of the stresses in such a plate is very complicated. It is usually considered safe to design the footing in each direction on the basis of one-half the actual loading.

Fig. 43. Grillage of I Beams.

Fig. 43. Grillage of I-Beams.

189. Example

A column 3 feet 4 inches square, carrying a total load of 630,000 pounds, is to be supported on a soil on which the permissible loading is estimated as three tons per square foot; an I-beam footing is to be used. Required, the design of the I-beams.

Solution. The required area of the footing is evidently 630,000 6,000 = 105 square feet. Using a footing similar to that illustrated in Fig. 43, we shall make the lower layer of the footing, say 10 feet 6 inches by 10 feet wide. The length of the beams being 126 inches, and the column being 40 inches square, the offset from the column is 43 inches (= 3.58 feet) on each side. Looking at a table of standard I-beams, we find that an 8-inch beam weighing 17.75 pounds per linear foot will carry 37,920 pounds on a span of four feet.

For a span of 3.58 feet, the allowable load is 4.00X 37,920, or 42,368

3.58 Pounds

Taking one-fourth of this, as in the example in section 187, we have 10,592 pounds which may be carried by each beam acting as a cantilever. The upward pressure on an offset 3.58 feet long and 1 foot wide, at the rate of 6,000 pounds per square foot, would be 21,500 pounds. Therefore, if two 8-inch beams were placed in each foot of width, they would carry the pressure. Therefore 20 such beams, each 10 feet 6 inches long, would be required in the lower layer. The upper layer must consist of beams 10 feet long on which the offset from the pier is 40 inches on each side. The group of beams under each of these upper offsets carries an upward pressure of 6,000 pounds per square foot on an area 10 feet 6 inches by 3 feet 4 inches; total pressure, 210,000 pounds. The total load on each foot of width of the upper layer is 63,000 pounds. Looking at the tables, a 12-inch I-beam weighing 40 pounds per foot can carry a load, on a 10-foot span, of 43,720 pounds. The permissible load on a cantilever of this length would be one-fourth of this, or 10,930 pounds. The permissible load on a cantilever 3 feet 4 inches long will be in the ratio of 10 feet to 3 feet 4 inches, or, in this case, exactly three times as much, which would be 32,790 pounds. If, therefore, such beams are placed 6 inches apart, their strength would be slightly in excess of that required. Or, as a numerical check, the total of 210,000 pounds, divided by 32,790, will show that although seven such beams will have a somewhat excessive strength, six would be hardly sufficient; therefore seven beams should be used. It should not be forgotten that surrounding all these beams in both layers with concrete adds very largely to the strength of the whole footing, but that no allowance is made for this additional strength in computing dimensions. It merely adds an indefinite amount to the factor of safety.