Plain concrete footings are usually stepped off in the shine manner. As the concrete footing is a monolithic structure and capable of carrying small tensile stresses, the projecting step may be considered as a cantilever carrying the upward thrust of the soil upon its lower surface.
Let I =the thickness of the footing at any point; o= the projection of the footing beyond the point where the thickness is t; p= the pressure in pounds per square foot on the bottom of the footing; f =unit stress upon the concrete due to bending.
Then the allowable projection for any given thickness is Thus, if we assume the safe tension on the concrete to he 60 and the pressure upon the foundation soil as 2 tons per square foot, o= .Sot, or the projection should not be greater than .S5 of its thick ness.
The projections for cut stone in which each stone is the full height of the course may be estimated by the above formla, provided the stones may be considered as firmly held in place under the wall. When placed upon compressible soil, however, the pressure will not be uniformly distributed over the base of the stone, and there is like lihood of tipping the block if the projection is too great.
Under brick walls, a bed of concrete is usually employed at the base and the brickwork stepped off on top of this to give the required extensions. The offsets in such work should not be more than three quarters of their heights, which may be composed of two courses of brick.
190. Grillage Foundations.—When a foundation must be spread over an area which is large compared to that of the column or wall resting upon it, a masonry footing becomes uneconomical and a foot ing possessing greater transverse strength and requiring less height becomes desirable. For such foundations, grillages of timber or steel or reinforced concrete slabs are commonly employed.
Steel I-beam grilloges are now very frequently used under heavy buildings. The construction of foundations of this type was begun in Chicago about 1SSO. In founding heavy buildings upon the clay subsoil, it was necessary to spread the footings over considerable areas, and room was not available for masonry footings, as the sub soil was soft at greater depths. A footing consisting of several layers of old steel rails encased in concrete was devised and used for some time. This was soon replaced by I-beams of sufficient depth to carry the loads in a single layer, thus saving space and giving better economy in the use of the metal.
A grillage footing as applied to the foundation of a single column is shown in Fig. 105. Such foundations rest upon a bed of concrete and are enclosed by a filling and surfacing of concrete for the pro tection of the steel. Under heavy loads, the bed of concrete is usually about 12 inches thick and the protective coating from 3 to 6 inches thick. The beams should be held by spacers at least . 3 inches apart in the clear in order to permit filling the spaces with concrete. Under a contin uous wall, a block of plain con crete is usually employed instead of the upper series of I-beams.
In designing a grillage footing, the loads to be carried and the areas of the walls or piers are known and the grillage must be so placed as to bring the center of its area in the line of action of the resultant load. The total load may be considered as distributed uniformly over the base, giving uniform upward pressure upon the beams, while the downward thrust of a pier is taken as uniformly distributed over its section. Usually a grillage is centered under each column or wall, proportioned to the load to be carried, but two or more loads may be carried by a single grillage when it seems desirable.
Fig. 106 shows a footing supporting two piers each 2.5 feet square, one carrying a load of :300,000 pounds and the other 400,000 pounds, spaced 10.5 feet between centers. The soil pressure is limited to -1000 lb. and an area of 175 is required. If this area be assumed as 17.5 feet by 10 feet as the center of gravity of the loads is 4.5 feet from the center of the pier carrying the larger load, the piers will occupy the positions shown, when is uniform upon the soil.
The upper tier of beams under the heavier load carries 400,000 pounds distributed over 2.5 feet at the middle acting downward on its upper surface, and the same load distributed uniformly over the length of 10 feet, acting upward on its lower surface. The maximum moment will be at the mid-section and will be If the allowable unit stress in the steel is 16,000 lb./in.2, The 20-inch beams require less concrete than the 24-ineh, and less steel than the 1S-inch and may be used, although the spacing is rather wide. The flanges are spaced 10 inches apart and 3 inches inside the block of concrete.