1

RECTANGULAR ISOLATED FOOTINGS

2

RECTANGULAR ISOLATED FOOTINGSAs previously mentioned, isolated footings may be rectangular in

plan if the column has a very pronounced rectangular shape or if the space available for the footing forces the designer into using a rectangular shape. Should a square footing be feasible, it is normally to be desired over a rectangular one because it will require less material and will be simpler to construct.

The design procedure is almost identical with the one used for square footings. After the required area is calculated and the lateral dimensions are selected, the depths required for one-way and two-way shear are determined by the usual methods. One-way shear will very often control the depths for rectangular footings, whereas two-way shear normally controls the depths of square footings.

3

RECTANGULAR ISOLATED FOOTINGSThe next step is to select the reinforcing in the long direction.

These longitudinal bars are spaced uniformly across the footing, but such is not the case for the short-span reinforcing. With reference to the earlier figure, it can be seen that the support provided by the footing to the column will be concentrated near the middle of the footing, and thus the moment in the short direction will be concentrated somewhat in the same area near the column.

As a result of this concentration effect, it seems only logical to concentrate a large proportion of the short-span reinforcing in this area. The Code (15.4.4.2) states that a certain minimum percentage of the total short-span reinforcing should be placed in a band width equal to the length of the shorter direction of the footing. The amount of reinforcing in this band is to be determined with the following expression, in which β is the ratio of the length of the long side to the width of the short side of the footing:

4

RECTANGULAR ISOLATED FOOTINGSThe remaining reinforcing in the short direction should be

uniformly spaced over the ends of the footing but should at least meet the shrinkage and temperature requirements of the ACI Code (7.12).

Next example presents the partial design of a rectangular footing in which the depths for one- and two-way shears are determined and the reinforcement selected.

5

RECTANGULAR ISOLATED FOOTINGSExample 12.5

Solution

6

RECTANGULAR ISOLATED FOOTINGS

7

RECTANGULAR ISOLATED FOOTINGS

8

RECTANGULAR ISOLATED FOOTINGS

9

RECTANGULAR ISOLATED FOOTINGS

10

COMBINED FOOTINGSCombined footings support more than one column. One situation

in which they may be used is when the columns are so close together that isolated individual footings would run into each other [Figure 12.18(a)]. Another frequent use of combined footings occurs where one column is very close to a property line, causing the usual isolated footing to extend across the line. For this situation the footing for the exterior column may be combined with the one for an interior column, as shown in Figure 12.18(b).

On some occasions where a column is close to a property line and where it is desired to combine its footing with that of an interior column, the interior column will be so far away as to make the idea impractical economically. For such a case, counterweights or “deadmen” may be provided for the outside column to take care of the eccentric loading.

11

COMBINED FOOTINGS

12

COMBINED FOOTINGSBecause it is desirable to make bearing pressures uniform

throughout the footing, the centroid of the footing should be made to coincide with the centroid of the column loads to attempt to prevent uneven settlements. This can be accomplished with combined footings that are rectangular in plan. Should the interior column load be greater than that of the exterior column, the footing may be so proportioned that its centroid will be in the correct position by extending the inward projection of the footing, as shown in the rectangular footing of Figure 12.18(b).

Other combined footing shapes that will enable the designer to make the centroids coincide are the trapezoid and strap or T footings shown in Figure 12.19(a) and 12.19(b). Footings with these shapes are usually economical when there are large differences between the magnitudes of the column loads or where the spaces available do not lend themselves to long rectangular footings. When trapezoidal footings are used, the longitudinal bars are usually arranged in a fan shape with alternate bars cut off some distance from the narrow end.

13

COMBINED FOOTINGS

14

COMBINED FOOTINGSYou probably realize that a problem arises in establishing the

centroids of loads and footings when deciding whether to use service or factored loads. The required centroid of the footing will be slightly different for the two cases. The footing areas and centroids with the service loads (ACI Code 15.2.2) can be determined, but the factored loads could be used with reasonable results, too. The important item is to be consistent throughout the entire problem.

The design of combined footings has not been standardized as have the procedures used for the previous problems.

15

COMBINED FOOTINGSFirst, the required area of the footing is determined for the

service loads, and the footing dimensions are selected so that the centroids coincide. Then the various loads are multiplied by the appropriate load factors, and the shear and moment diagrams are drawn along the long side of the footing for these loads. After the shear and moment diagrams are prepared, the depth for one- and two-way shear is determined and the reinforcing in the long direction is selected.

In the short direction it is assumed that each column load is spread over a width in the long direction equal to the column width plus d/2 on each side if that much footing is available. Then the steel is designed, and a minimum amount of steel for temperature and shrinkage is provided in the remaining part of the footing.

16

COMBINED FOOTINGSThe ACI Code does not specify an exact width for these transverse

strips, and designers may make their own assumptions as to reasonable values. The width selected will probably have very little influence on the transverse bending capacity of the footing, but it can affect appreciably its punching or two-way shear resistance. If the flexural reinforcing is placed within the area considered for two-way shear, this lightly stressed reinforcing will reduce the width of the diagonal shear cracks and will also increase the aggregate interlock along the shear surfaces.

Next example is presented to show those parts of the design that are different from the previous examples. A comment should be made about the moment diagram. If the length of the footing is not selected so that its centroid is located exactly at the centroid of the column loads, the moment diagrams will not close well at all since the numbers are very sensitive.

17

COMBINED FOOTINGSNevertheless, it is considered good practice to round off the

footing lateral dimensions to the nearest 3 in. Another factor that keeps the moment diagram from closing is the fact that the average load factors of the various columns will be different if the column loads are different. We could improve the situation a little by taking the total column factored loads and dividing the result by the total working loads to get an average load factor. This value (which works out to be 1.38 in the example) could then be multiplied by the total working load at each column and used for drawing the shear and moment diagrams.

18

COMBINED FOOTINGSExample 12.6

Solution

19

COMBINED FOOTINGS

20

COMBINED FOOTINGS

21

COMBINED FOOTINGS

22

COMBINED FOOTINGS

23

COMBINED FOOTINGS

24

COMBINED FOOTINGS

25

COMBINED FOOTINGS

A similar procedure is used under the exterior column where the steel is spread over a width equal to 18 in. plus d/2, and not 18 in. plus 2(d/2), because sufficient room is not available on the property-line side of the column.

26

COMBINED FOOTINGSThis footing in the long direction has been treated just like a

long beam. If such a procedure is followed, the reader may quite logically think that shear in the long direction should be handled just as it would be in a long beam. In other words, if the shear Vu exceeds φVc/2, stirrups should be used. The Code does not address this particular point. Many designers will use stirrups in such situations where Vu > φVc /2, particularly where the footings are deep and narrow. The author feels that footings should be designed very conservatively because the failure of a footing can result in severe damage to the structure above. Thus he feels that stirrups should be designed for a combined footing as they would be for a regular beam.

27

FOOTING DESIGN FOR EQUAL SETTLEMENTSIf three men are walking along a road carrying a log on their

shoulders (a statically indeterminate situation) and one of them decides to lower his shoulder by 1 in., the result will be a drastic effect on the load supported by the other men. In the same way, if the footings of a building should settle by different amounts, the shears and moments throughout the structure will be greatly changed. In addition, there will be detrimental effects on the fitting of doors, windows, and partitions. Should all the footings settle by the same amount, however, these adverse effects will not occur. Thus equal settlement is the goal of the designer.

The footings considered in preceding sections have had their areas selected by taking the total dead plus live loads and dividing the sum by the allowable soil pressure. It would seem that if such a procedure were followed for all the footings of an entire structure, the result would be uniform settlements throughout—but geotechnical engineers have clearly shown that this assumption may be very much in error.

28

FOOTING DESIGN FOR EQUAL SETTLEMENTSA better way to handle the problem is to attempt to design the

footings so that the usual loads on each footing will cause approximately the same pressures. The usual loads consist of the dead loads plus the average percentage of live loads normally present. The usual percentage of live loads present varies from building to building. For a church it might be almost zero, perhaps 25 to 30% for an office building, and maybe 75% or more for some warehouses. Furthermore, the percentage in one part of a building may be entirely different from that in some other part (offices, storage, etc.).

One way to handle the problem is to design the footing that has the highest ratio of live to dead load, compute the usual soil pressure under that footing using dead load plus the estimated average percentage of live load, and then determine the areas required for the other footings so their usual soil pressures are all the same. It should be remembered that the dead load plus 100% of the live load must not cause a pressure greater than the allowable soil pressure under any of the footings.

29

FOOTING DESIGN FOR EQUAL SETTLEMENTSA student of soil mechanics will realize that this method of

determining usual pressures, though not a bad design procedure, will not ensure equal settlements. This approach at best will only lessen the amounts of differential settlements. He or she will remember first that large footings tend to settle more than small footings, even though their soil pressures are the same, because the large footings exert compression on a larger and deeper mass of soil. There are other items that can cause differential settlements. Different types of soils may be present at different parts of the building; part of the area may be in fill and part in cut: there may be mutual influence of one footing on another; and so forth.

Next example illustrates the usual load procedure for a group of five isolated footings.

30

FOOTING DESIGN FOR EQUAL SETTLEMENTSExample 12.7

Solution

Determine the footing areas required for the loads given in Table 12.2 so that the usual soil pressures will be equal. Assume that the usual live load percentage is 30% for all the footings, qe = 4 ksf.

31

FOOTING DESIGN FOR EQUAL SETTLEMENTS

Computing the areas required for the other footings and determining their soil pressures under total service loads, we show the results in Table 12.3.

32

FOOTINGS SUBJECTED TO LATERAL MOMENTSWalls or columns often transfer moments as well as vertical loads

to their footings. These moments may be due to wind, earthquake, lateral earth pressures, and so on. Such a situation is represented by the vertical load P and the bending moment M shown in Figure 12.23.

Because of this moment, the resultant force will not coincide with the centroid of the footing. Of course, if the moment is constant in magnitude and direction, it will be possible to place the center of the footing under the resultant load and avoid the eccentricity, but lateral forces such as wind and earthquake can come from any direction and symmetrical footings will be needed.

The effect of the moment is to produce a uniformly varying soil pressure, which can be determined at any point with the expression

33

FOOTINGS SUBJECTED TO LATERAL MOMENTS

In this discussion the term kern is used. If the resultant force

strikes the footing base within the kern, the value of -P/A is larger than

+Mc/I at every point and the entire footing base is in compression, as

shown in Figure 12.23

(a). If the resultant force strikes the footing base outside the kern, the

value of +Mc/I will at some points be larger than -P/A and there

will be uplift or tension. The soil cannot resist tension, and the

pressure variation will be as shown in Figure 12.23

(b). The location of the kern can be determined by replacing Mc/I

with Pec/I, equating it to P/A, and solving for e.

34

FOOTINGS SUBJECTED TO LATERAL MOMENTS

35

FOOTINGS SUBJECTED TO LATERAL MOMENTSShould the eccentricity be larger than this value, the method

described for calculating soil pressures [(-P/A) ± (Mc/I)] is not correct. To compute the pressure for such a situation it is necessary to realize that the centroid of the upward pressure must for equilibrium coincide with the centroid of the vertical component of the downward load. In Figure 12.24 it is assumed that the distance to this point from the right edge of the footing is a. Then the soil pressure will be spread over the distance 3a as shown. For a rectangular footing with dimensions l x b, the total upward soil pressure is equated to the downward load and the resulting expression solved for qmax as follows:

36

FOOTINGS SUBJECTED TO LATERAL MOMENTS

37

FOOTINGS SUBJECTED TO LATERAL MOMENTS

Example 12.8 shows that the required area of a footing subjected

to a vertical load and a lateral moment can be determined by trial and

error. The procedure is to assume a size, calculate the maximum soil

pressure, compare it with the allowable pressure, assume another size,

and so on.

Once the area has been established, the remaining design will be

handled as it was for other footings. Although the shears and moments are

not uniform, the theory of design is unchanged. The factored loads are

computed, the bearing pressures are determined, and the shears and

moments are calculated. For strength design, the footing must be

proportioned for the effects of these loads as required in ACI Section 9.2.

38

FOOTINGS SUBJECTED TO LATERAL MOMENTSExample 12.8

Solution

Determine the width needed for a wall footing to support loads: D = 18 k/ft and L = 12 k/ft. In addition, a lateral load of 6 kt/ft is assumed to be applied 5 ft above the top of the footing. Assume the footing is 18 in. thick, its base is 4 ft below the final grade, and qa = 4 ksf.

39

FOOTINGS SUBJECTED TO LATERAL MOMENTS

40

TRANSFER OF HORIZONTAL FORCESWhen it is necessary to transfer horizontal forces from walls or

columns to footings, the shear friction method previously discussed in

Section 8.12 of this text should be used. Sometimes shear keys (see Figure

13.1) are used between walls or columns and footings. This practice is of

rather questionable value, however, because appreciable slipping has to

occur to develop a shear key. A shear key may be thought of as providing

an additional mechanical safety factor, but none of the lateral design force

should be assigned to it.

The next example illustrates the consideration of lateral force

transfer by the shear friction concept.

41

TRANSFER OF HORIZONTAL FORCESExample 12.9

Solution

The six #6 dowels (2.65 in.²) present may also be used as shear friction reinforcing. If their area had not been sufficient, it could have been increased and/or the value of μ could be increased significantly by intentionally roughening the concrete, as described in Section 11.7.4.3 of the Code.

42

TRANSFER OF HORIZONTAL FORCESCheck Tensile Development Lengths of These DowelsAs described in ACI Sections 12.2.3 and 11.7.8, shear friction reinforcing acts in tension and thus must have tensile anchorage on both sides of the shear plane. It must also engage the main reinforcing in the footing to prevent cracks from occurring between the shear reinforcing and the body of the concrete.

43

PLAIN CONCRETE FOOTINGSOccasionally, plain concrete footings are used to support light

loads if the supporting soil is of good quality. Very often the widths and

thicknesses of such footings are determined by rules of thumb such as

“the depth of a plain footing must be equal to no less than the projection

beyond the edges of the wall.” In this section, however, a plain concrete

footing is designed in accordance with the requirements of the ACI.

Chapter 22 of the ACI Code is devoted to the design of structural

plain concrete. Structural plain concrete is defined as concrete that is

completely unreinforced or that contains less than the minimum required

amounts of reinforcing previously specified here for reinforced concrete

members. The minimum compressive strength permitted for such concrete

is 2500 psi, as given in ACI Sections 22.2.4 and 1.1.1.

44

PLAIN CONCRETE FOOTINGS

Structural plain concrete may only be used for

(1) members continuously supported by soil or by other structural

members that are capable of providing continuous support,

(2) walls and pedestals, and

(3) structural members with arch action where compression occurs for all

loading cases (ACI 22.2.2).The Code (22.7.3 and 22.7.4) states that when plain concrete

footings are supported by soil, they cannot have an edge thickness less

than 8 in. and they cannot be used on piles. The critical sections for shear

and moment for plain concrete footings are the same as for reinforced

concrete footings.

45

PLAIN CONCRETE FOOTINGS

In ACI Code Section 22.5, nominal bending and shear strengths

are specified for structural plain concrete. The proportions of plain

concrete members will nearly always be controlled by tensile strengths

rather than shear strengths.

In the equations that follow φ = 0.55 (ACI 9.3.5) for all cases, S is

the elastic section modulus of members, and βc is the ratio of the long side

to the short side of the column or loaded area. In computing the strengths,

whether flexural or shear, for concrete cast against soil the overall

thickness h is to be taken as 2 in. less than the actual thickness. This

concrete is neglected to account for uneven excavation for the footing and

for some loss of mixing water to the soil and other contamination.

46

PLAIN CONCRETE FOOTINGS

47

PLAIN CONCRETE FOOTINGSA plain concrete footing will obviously require considerably more

concrete than a reinforced one. On the other hand, the cost of purchasing

reinforcing and placing it will be eliminated. Furthermore, the use of plain

concrete footings will enable us to save construction time in that we don’t

have to order the reinforcing and place it before the concrete can be

poured. Therefore, plain concrete footings may be economical on more

occasions than one might realize.

Even though plain footings are designed in accordance with the ACI

requirements, they should at the very least be reinforced in the

longitudinal direction to keep temperature and shrinkage cracks within

reason and to enable the footing to bridge over soft spots in the underlying

soil.

48

PLAIN CONCRETE FOOTINGSExample 12.10

Solution

49

PLAIN CONCRETE FOOTINGS

50

PLAIN CONCRETE FOOTINGS

STRIP, GRID AND MAT FOUNDATIONS

51

STRIP, GRID AND MAT FOUNDATIONS As mentioned earlier, continuous foundations are often used to

support heavily loaded columns, especially when a structure is loaded on

relatively weak or uneven soil. The foundation may consist of a continuous

strip footing supporting all columns in a given row, or of two sets of such strip

footings intersecting at right angles so that they form one continuous grid

foundation. For even larger loads or weaker soils, the strips are made to

merge, resulting in a mat foundation.

For the design of such continuous foundations, it is essential that

reasonably realistic assumption be made regarding the distribution of bearing

pressures that act as upward loads on the foundation. For compressible soils,

it can be assumed, as a first approximation, that the deformation or

settlement of the soil at a given location and the bearing pressure at that

location are proportional to each other.52

STRIP, GRID AND MAT FOUNDATIONS If columns are spaced at moderate distances and if the strip, grid, or

mat foundation is quite rigid, the settlements in all portions of the

foundation will be substantially the same. This means that the bearing

pressure, also known as subgrade reaction, will be the same, provided that

the centroid of the foundation coincides with the resultant of the loads. If

they do not coincide, then for such rigid foundation the subgrade reaction

can be assumed to very linearly. Bearing pressure can be calculated based on

statics, as discussed for single footings. In this case, all loads, the downward

column loads as well as the upward-bearing pressures, are known. Hence,

moments and shear forces in the foundation can be found by statics alone.

Once these are determined, the design of strip and grid foundation is similar

to that of inverted continuous beams, and that of mat foundation to that of

inverted flat slabs or plates.53

STRIP, GRID AND MAT FOUNDATIONS On the other hand, if the foundation is relatively flexible and the

column spacing large, settlement will no longer be uniform or linear. For one

thing, the more heavily loaded columns will cause larger settlement, and

thereby larger subgrade reaction, than the lighter ones. Also, since the

continuous strip or slab midway between columns will deflect upward relative

to the nearby columns, the soil settlement, and thereby the subgrade

reaction, will be smaller midway between columns than directly at the

columns. The is shown schematically for a strip footing in Fig. 16.21; the

subgrade reaction can no longer be assumed to be uniform. Mat foundations

likewise require different approaches, depending on whether they can be

rigid when calculating the soil reaction.

54

STRIP, GRID AND MAT FOUNDATIONS

55

STRIP, GRID AND MAT FOUNDATIONS Criteria have been established as a measure of the relative stiffness

of the structure versus the stiffness of the soil (Refs. 16.10 and 16.13). If the

relative stiffness is low, the foundation should be designed as a flexible

member with a nonlinear upward reaction from the soil. For strip footings, a

reasonably accurate but fairly complex analysis can be done using the theory

of beams on elastic foundations (Ref. 16.14). Kramrisch (Ref. 16.8) has

suggested simplified procedures, based on the assumption that contact

pressures very linearly between load points, as shown in Fig. 16.21.

56

For nonrigid mat foundations, great advances in analysis have been

made using finite element methods, which can account specifically for the

stiffnesses of both the structure and the soil.

PILE CAPS If the bearing capacity of the upper soil layers is in sufficient for a

spread foundation, but firmer strata are available at greater depth, piles are

used to transfer the load to these deeper strata. Piles are generally arranged

in groups or clusters, one under each column. The group is capped by a

spread footing or cap that distributions the column load to all piles in the

group. These pile caps are in most ways very similar to footings on soil,

except for two features. For one, reactions on caps act as concentrated loads

at the individual piles, rather than as distributed pressures. For another, if the

total of all pile reaction in a cluster is divided by the area of the footing to

obtain an equivalent uniform pressure (for purposes of comparison only), it is

found that this equivalent pressure is considerably higher in pile caps than for

spread footings.

57

PILE CAPS This means that moments, and particularly shears, are also

correspondingly larger, which requires greater footings depths than for a

spread footing of similar horizontal dimensions. To spread the load evenly to

all piles, it is in any event advisable to provide ample rigidity, i.e., depth, for

pile caps.

58

Allowable bearing capacities of piles Ra are obtained from soil

exploration, pile-driving energy, and test loadings, and their determination is

not within the scope of the present book (see Refs. 16.1 to 16.4). An in spread

footings, the effective portion of Ra available to resist the unfactored column

loads is the allowable pile reaction less the weight of footing, backfill, and

surcharge per pile. This is,

PILE CAPS Where Wf is the total weight of footing, fill, and surcharge divided by

the number od piles.

59

Once the available or effective pile reaction Re is determined, the

number of piles in a concentrically loaded cluster is the integer next larger

than

As far as the effects of wind, earthquake moments at the foot of the

columns, and safety against overturning are concerned, design consideration

are the same as described in Section 16.4 for spread footings. These effects

generally produce an eccentrically loaded pile cluster in which different piles

carry different loads. The number and location of piles in such a cluster are

determined by successive approximations based on the requirement that the

PILE CAPS

60

load on the most heavily loaded pile not exceed the allowable pile reaction

Ra. With a linear distribution of pile loads due to bending, the maximum pile

reaction is

Where P is the maximum load (including weight of cap, backfill, etc.) and M

the moment to be resisted by the pile group, both referred to the bottom of

the cap; lpg is the moment of inertia of the entire pile group about the

centroidal axis about which bending occurs; and c is the distance from that

axis to the extreme pile. lpg = i.e., it is the moment of inertia of

n piles, each country as one unit and located a distance yi from the described

centroidal axis.

PILE CAPS

61

Piles are generally arranged in tight patterns, which minimizes the

cost of the caps, but they cannot be placed closer than conditions of driving

and of undisturbed carrying capacity will permit. A spacing of about 3 times

the butt (top) diameter of the pile but no less than 2 ft 6 in. is customary.

Commonly, piles with allowable reactions of 30 to 70 tons are spread at 3 ft 0

in. (Ref. 16.8).

The design of footings on piles is similar to that of single-column

footings. One approach is to design the cap for the pile reactions calculated

for the factored column loads. For a concentrically loaded cluster, this word

give Ru = (1.2D + 1.6L)/n. However, since the number of piles was taken as the

next-larger integral according to Eq. (16.13), determining Ru in this manner

can lead to a design where the strength of the cap is less than the capacity

PILE CAPS

62

of the pile group. It is therefore recommended that the pile reaction for

strength design be taken as

Ru = Re x average load factor (16.14)

Where the average load factor = (1.2D + 1.6L)/(D + L). In this manner, the cap

is designed to be capable of developing the full allowable capacity of the pile

group. Details of a typical pile cap are shown in Fig. 16.22.

As in single-column spread footings, the depth of the pile cap is

usually governed by shear. ACI Code 15.5.3 specifies that when the distance

between the axis of a pile and the axis of a column is more than 2 times the

distance from the top of the pile cap and the top of the pile, shear design

must follow the procedures for flat slabs and footings, as described in Section

16.6a.

PILE CAPS

63

For closer spacing between piles and columns, the Code specifies either the

use of the procedures described in Section 16.6a or the use of a three-

dimensional strut-and-tie model (ACI Code Appendix A) based on the

principles described in chapter 10. in the latter case, the struts must be

designed as bottle-shaped without transverse reinforcement because of the

difficulty of providing such reinforcement in a pile cap. The use of strut-tie

models to design pile caps is discussed in Ref. 16.15.

When the procedures for flat slabs and footings are used, both

punching or two-way shear and flexural or one-way shear need to be

considered. The critical sections are the same as given in Section 16.6a. The

different is that shear in caps is caused by concentrated pile reactions rather

than by distributed bearing pressures.

PILE CAPS

64