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FOOTINGS INTRODUCTION
INTRODUCTION Footings are structural members used to support columns and walls and transmit their loads to the underlying soils. Reinforced concrete is a material admirably suited for footings and is used as such for both reinforced concrete and structural steel buildings, bridges, towers, and other structures. The permissible pressure on a soil beneath a footing is normally a few tons per square foot. The compressive stresses in the walls and columns of an ordinary structure may run as high as a few hundred tons per square foot. It is therefore necessary to spread these loads over sufficient soil areas to permit the soil to support the loads safely. Not only is it desired to transfer the superstructure loads to the soil beneath in a manner that will prevent excessive or uneven settlements and rotations, but it is also necessary to provide sufficient resistance to sliding and overturning.
INTRODUCTION To accomplish these objectives, it is necessary to transmit the supported loads to a soil of sufficient strength and then to spread them out over an area such that the unit pressure is within a reasonable range. If it is not possible to dig a short distance and find a satisfactory soil, it will be necessary to use piles or caissons to do the job. These latter subjects are not considered within the scope of this text. The closer a foundation is to the ground surface, the more economical it will be to construct. There are two reasons, however, that may keep the designer from using very shallow foundations. First, it is necessary to locate the bottom of a footing below the ground freezing level to avoid vertical movement or heaving of the footing as the soil freezes and expands in volume. This depth varies from about 3 to 6 feet. Second, it is necessary to excavate a sufficient distance so that a satisfactory bearing material is reached, and this distance may on occasion be quite a few feet.
TYPES OF FOOTINGS Among the several types of reinforced concrete footings in common use are the wall, isolated, combined, raft, and pile cap types. These are briefly introduced in this section; the remainder of the chapter is used to provide more detailed information about the simpler types of this group. 1. A wall footingis simply an enlargement of the bottom of a wall that will sufficiently distribute the load to the foundation soil. Wall footings are normally used around the perimeter of a building and perhaps for some of the interior walls. 2. Anisolated or single-column footingis used to support the load of a single column. These are the most commonly used footings, particularly where the loads are relatively light and the columns are not closely spaced.
TYPES OF FOOTINGS 3. Combined footingsare used to support two or more common loads. A combined footing might be economical where two or more heavily loaded columns are so spaced that normally designed single-column footings would run into each other. Single-column footings are usually square or rectangular and, when used for columns located right at property lines, would extend across those lines. A footing for such a column combined with one for an interior column can be designed to fit within the property lines. 4. A mat or raft or floating foundationis a continuous reinforced concrete slab over a large area used to support many columns and walls. This kind of foundation is used where soil strength is low or where column loads are large but where piles or caissons are not used. For such cases, isolated footings would be so large that it is more economical to use a continuous raft or mat under the entire area.
TYPES OF FOOTINGS 5. The cost of the formwork for a mat footing is far less than is the cost of the forms for a large number of isolated footings. If individual footings are designed for each column and if their combined area is greater than half of the area contained within the perimeter of the building, it is usually more economical to use one large footing or mat. The raft or mat foundation is particularly useful in reducing differential settlements between columns—the reduction being 50% or more. For these types of footings the excavations are often rather deep. The goal is to remove an amount of earth approximately equal to the building weight. If this is done, the net soil pressure after the building is constructed will theoretically equal what it was before the excavation was made. Thus the building will float on the raft foundation. 6. Pile capsare slabs of reinforced concrete used to distribute column loads to groups of piles.
ACTUAL SOIL PRESSURES The soil pressure at the surface of contact between a footing and the soil is assumed to be uniformly distributed as long as the load above is applied at the center of gravity of the footing. This assumption is made even though many tests have shown that soil pressures are unevenly distributed due to variations in soil properties, footing rigidity, and other factors. A uniform-pressure assumption, however, usually provides a conservative design since the calculated shears and moments are usually larger than those that actually occur.
ACTUAL SOIL PRESSURES As an example of the variation of soil pressures, footings on sand and clay soils are considered. When footings are supported by sandy soils, the pressures are larger under the center of the footing and smaller near the edge. The sand at the edges of the footing does not have a great deal of lateral support and tends to move from underneath the footing edges, with the result that more of the load is carried near the center of the footing. Should the bottom of a footing be located at some distance from the ground surface, a sandy soil will provide fairly uniform support because it is restrained from lateral movement. Just the opposite situation is true for footings supported by clayey soils. The clay under the edges of the footing sticks to or has cohesion with the surrounding clay soil. As a result, more of the load is carried at the edge of the footing than near the middle.
ACTUAL SOIL PRESSURES The designer should clearly understand that the assumption of uniform soil pressure underneath footings is made for reasons of simplifying calculations and may very well have to be revised for some soil conditions. Should the load be eccentrically applied to a footing with respect to the center of gravity of the footing, the soil pressure is assumed to vary uniformly in proportion to the moment, will be discussed later.
ALLOWABLE SOIL PRESSURES The allowable soil pressures to be used for designing the footings for a particular structure are desirably obtained by using the services of a geotechnical engineer. He or she will determine safe values from the principles of soil mechanics on the basis of test borings, load tests, and other experimental investigations. Because such investigations often may not be feasible, most building codes provide certain approximate allowable bearing pressures that can be used for the types of soils and soil conditions occurring in that locality. Next table shows a set of allowable values that are typical of such building codes. It is thought that these values usually provide factors of safety of approximately 3 against severe settlements.
ALLOWABLE SOIL PRESSURES Section 15.2.2 of the ACI Code states that the required area of a footing is to be determined by dividing the anticipated total load, including the footing weight, by a permissible soil pressure or permissible pile capacity determined using the principles of soil mechanics. It will be noted that this total load is theun-factored load, and yet the design of footings described in this chapter is based on strength design, where the loads are multiplied by the appropriate load factors. It is obvious that an ultimate load cannot be divided by an allowable soil pressure to determine the bearing area required. The designer can handle this problem in two ways. He or she can determine the bearing area required by summing up the actual or un-factored dead and live loads and dividing them by the allowable soil pressure. Once this area is determined and the dimensions are selected, an ultimate soil pressure can be computed by dividing the factored or ultimate load by the area provided. The remainder of the footing can then be designed by the strength method using this ultimate soil pressure. This simple procedure is used for the footing examples here.
ALLOWABLE SOIL PRESSURES An alternate method for determining the footing area required that will give exactly the same answers as the procedure just described. By this latter method the allowable soil pressure is increased to an ultimate value by multiplying it by a ratio equal to that used for increasing the magnitude of the service loads. For instance, the ratio for D and L loads would be The resulting ultimate soil pressure can be divided into the ultimate column load to determine the area required.
DESIGN OF WALL FOOTINGS The theory used for designing beams is applicable to the design of footings with only a few modifications. The upward soil pressure under the wall footing tends to bend the footing into the deformed shape shown. The footings will be designed as shallow beams for the moments and shears involved. In beams where loads are usually only a few hundred pounds per foot and spans are fairly large, sizes are almost always proportioned for moment. In footings, loads from the supporting soils may run several thousand pounds per foot and spans are relatively short. As a result, shears will almost always control depths.
DESIGN OF WALL FOOTINGS It appears that the maximum moment in this footing occurs under the middle of the wall, but tests have shown that this is not correct because of the rigidity of such walls. If the walls are of reinforced concrete with their considerable rigidity, it is considered satisfactory to compute the moments at the faces of the walls (ACI Code 15.4.2). Should a footing be supporting a masonry wall with its greater flexibility, the Code states that the moment should be taken at a section halfway from the face of the wall to its center. (For a column with a steel base plate, the critical section for moment is to be located halfway from the face of the column to the edge of the plate.)
DESIGN OF WALL FOOTINGS To compute the bending moments and shears in a footing, it is necessary to compute only the net upward pressure qucaused by the factored wall loads above. In other words, the weight of the footing and soil on top of the footing can be neglected. These items cause an upward pressure equal to their downward weights, and they cancel each other for purposes of computing shears and moments. In a similar manner, it is obvious that there are no moments or shears existing in a book lying flat on a table. Should a wall footing be loaded until it fails in shear, the failure will not occur on a vertical plane at the wall face but rather at an angle of approximately 45 with the wall, as shown in next figure. Apparently the diagonal tension, which one would expect to cause cracks in between the two diagonal lines, is opposed by the squeezing or compression caused by the downward wall load and the upward soil pressure.
DESIGN OF WALL FOOTINGS Outside this zone the compression effect is negligible in its effect on diagonal tension. Therefore, for non pre-stressed sections shear may be calculated at a distance d from the face of the wall (ACI Code 11.1.3.1) due to the loads located outside the section.
DESIGN OF WALL FOOTINGS The use of stirrups in footings is usually considered impractical and uneconomical. For this reason, the effective depth of wall footings is selected so that Vuis limited to the design shear strength φVcthat the concrete can carry without web reinforcing, that is, (from ACI Section 11.3.1.1 and ACI Equation 11-3). The following expression is used to select the depths of wall footings:
DESIGN OF WALL FOOTINGS The design of wall footings is conveniently handled by using 12-in. widths of the wall, as shown in figure below. Such a practice is followed for the design of a wall footing in next example. It should be noted in Section 15.7 of the Code that the depth of a footing above the bottom reinforcing bars may be no less than 6 in. for footings on soils and 12 in. for those on piles. Thus total minimum practical depths are at least 10 in. for the regular spread footings and 16 in. for pile caps.
DESIGN OF WALL FOOTINGS The design of a wall footing is illustrated in next example. Although the example problems use various values, 3000 and 4000 psi concretes are commonly used for footings and are generally quite economical. Occasionally, when it is very important to minimize footing depths and weights, stronger concretes may be used. For most cases, however, the extra cost of higher strength concrete will appreciably exceed the money saved with the smaller concrete volume.
DESIGN OF WALL FOOTINGS Example Solution
DESIGN OF WALL FOOTINGS The determination of a footing depth is a trial-and-errorprocedure. The designer assumes an effective depth d, computes the d required for shear, tries another d, computes the d required for shear, and so on, until the assumed value and the calculated value are within about 1 in. of each other. You probably get upset when a footing size is assumed here. You say, “Where in the world did you get that value?” We think of what seems like a reasonable footing size and start there. We compute the d required for shear and probably find we’ve missed the assumed value quite a bit. We then try another value roughly two-thirds to three-fourths of the way from the trial value to the computed value (for wall footings) and compute d. (For column footings we probably go about half of the way from the trial value to the computed value.) Two trials are usually sufficient.
DESIGN OF SQUARE ISOLATED FOOTINGS Single-column footings usually provide the most economical column foundations. Such footings are generally square in plan, but they can just as well be rectangular or even circular or octagonal. Rectangular footings are used where such shapes are dictated by the available space or where the cross sections of the columns are very pronounced rectangles. Most footings consist of slabs of constant thickness, such as the one shown in next figure, but if calculated thicknesses are greater than 3 or 4 ft, it may be economical to use stepped footings. The shears and moments in a footing are obviously larger near the column, with the result that greater depths are required in that area as compared to the outer parts of the footing. For very large footings, such as for bridge piers, stepped footings can give appreciable savings in concrete quantities.
DESIGN OF SQUARE ISOLATED FOOTINGS Occasionally, sloped footings are used instead of the stepped ones, but labor costs can be a problem. Whether stepped or sloped, it is considered necessary to place the concrete for the entire footing in a single pour to ensure the construction of a monolithic structure, thus avoiding horizontal shearing weakness. If this procedure is not followed, it is desirable to use keys or shear friction reinforcing between the parts to ensure monolithic action. In addition, when sloped or stepped footings are used, it is necessary to check stresses at more than one section in the footing. For example, steel area and development length requirements should be checked at steps as well as at the faces of walls or columns. Before a column footing can be designed, it is necessary to make a few comments regarding shears and moments.
DESIGN OF SQUARE ISOLATED FOOTINGS Shears Two shear conditions must be considered in column footings, regardless of their shapes. The first of these is one-way or beam shear, which is the same as that considered in wall footings in the preceding section. For this discussion, reference is made to the footing of next figure. The total shear (Vu1) to be taken along section 1–1 equals the net soil pressure qutimes the hatched area outside the section. In the expression to follow, bwis the whole width of the footing. The maximum value of Vu1if stirrups are not used equals φVc, which is and the maximum depth required is as follows:
DESIGN OF SQUARE ISOLATED FOOTINGS Shears (Cont’d)
DESIGN OF SQUARE ISOLATED FOOTINGS Shears (Cont’d) The second shear condition is two-way or punching shear, with reference being made to the next figure. The compression load from the column tends to spread out into the footing, opposing diagonal tension in that area, with the result that a square column tends to punch out a piece of the slab, which has the shape of a truncated pyramid. The ACI Code (11.12.1.2) states that the critical section for two-way shear is located at a distance d/2 from the face of the column. The shear force Vu2 consists of all the net upward pressure quon the hatched area shown, that is, on the area outside the part tending to punch out. In the expressions to follow, bois the perimeter around the punching area, equal to 4(a +d). The nominal two-way shear strength of the concrete Vcis specified as the smallest value obtained by substituting into the applicable equations that follow.
DESIGN OF SQUARE ISOLATED FOOTINGS Shears (Cont’d)
DESIGN OF SQUARE ISOLATED FOOTINGS Shears (Cont’d) The first expression is the usual punching shear strength: Tests have shown that when rectangular footing slabs are subjected to bending in two directions and when the long side of the loaded area is more than two times the lengthof the short side, the shear strength Vc = may be much too high. In the expression to follow, βc is the ratio of the long side of the column to the short side of the column, concentrated load, or reaction area.
DESIGN OF SQUARE ISOLATED FOOTINGS Shears (Cont’d) The shear stress in a footing increases as the ratio bo/d decreases. To account for this fact ACI Equation 11-34 was developed. The equation includes a term αswhich is used to account for variations in the ratio. In applying the equation αsis to be used as 40 for interior columns (where the perimeter is four-sided), 30 for edge columns (where the perimeter is three-sided), and 20 for corner columns (where the perimeter is two-sided). The d required for two-way shear is the largest value obtained from the following expressions:
DESIGN OF SQUARE ISOLATED FOOTINGS Moments The bending moment in a square reinforced concrete footing is the same about both axes due to symmetry. It should be noted, however, that the effective depth of the footing cannot be the same in the two directions because the bars in one direction rest on top of the bars in the other direction. The effective depth used for calculations might be the average for the two directions or, more conservatively, the value for the bars on top. This lesser value is used for the examples in this text. Although the result is some excess of steel in one direction, it is felt that the steel in either direction must be sufficient to resist the moment in that direction. It should be clearly understood that having an excess of steel in one direction will not make up for a shortage in the other direction at a 90˚ angle. The critical section for bending is taken at the face of a reinforced concrete column or halfway between the middle and edge of a masonry wall or at a distance halfway from the edge of the base plate and the face of the column if structural steel columns are used (Code 15.4.2).
DESIGN OF SQUARE ISOLATED FOOTINGS Moments The determination of footing depths by the procedure described here will often require several cycles of a trial-and-error procedure. There are, however, many tables and handbooks available with which footing depths can be accurately estimated. One of these is the previously mentioned CRSI Design Handbook. In addition, there are many rules of thumb used by designers for making initial thickness estimates, such as 20% of the footing width or the column diameter plus 3 in. The reinforcing steel area calculated for footings will often be appreciably less than the minimum values and specified for flexural members in ACI Section 10.5.1. In Section 10.5.4, however, the Code states that in slabs of uniform thickness the minimum area and maximum spacing of reinforcing bars in the direction of bending need only be equal to those required for shrinkage and temperaturereinforcement.
DESIGN OF SQUARE ISOLATED FOOTINGS Moments The maximum spacing of this reinforcement may not exceed the lesser of three times the footing thickness, or 18 in. Many designers feel that the combination of high shears and low ρvalues that often occurs in footings is not a good situation. Because of this, they specify steel areas at least as large as the flexural minimums of ACI Section 10.5.1. This is the practice we also follow herein. Example 12.2 illustrates the design of an isolated column footing.
DESIGN OF SQUARE ISOLATED FOOTINGS Example 12.2 Solution Assume depth = 24 ins Solution
DESIGN OF SQUARE ISOLATED FOOTINGS Example 12.2
DESIGN OF SQUARE ISOLATED FOOTINGS Example 12.2
FOOTINGS SUPPORTING ROUND OR REGULAR POLYGON-SHAPED COLUMNS
FOOTINGS SUPPORTING ROUND OR REGULAR POLYGON-SHAPED COLUMNS Sometimes footings are designed to support round columns or regular polygon-shaped columns. If such is the case, Section 15.3 of the Code states that the column may be replaced with a square member having the same area as the round or polygonal one. Then the equivalent square is used for locating the critical sections for moment, shear, and development length.
LOAD TRANSFER FROM COLUMNS TO FOOTINGS All forces acting at the base of a column must be satisfactorily transferred into the footing. Compressive forces can be transmitted directly by bearing, whereas uplift or tensile forces must be transferred to the supporting footing or pedestal by means of developed reinforcing bars or by mechanical connectors (which are often used in precast concrete). A column transfers its load directly to the supporting footing over an area equal to the cross-sectional area of the column. The footing surrounding this contact area, however, supplies appreciable lateral support to the directly loaded part, with the result that the loaded concrete in the footing can support more load. Thus for the same grade of concrete, the footing can carry a larger bearing load than can the base of the column. In checking the strength of the lower part of the column, only the concrete is counted. The column reinforcing bars at that point cannot be counted because they are not developed unless dowels are provided or unless the bars themselves are extended into the footing.
