RECTANGULAR ISOLATED
FOOTINGS
1
RECTANGULAR ISOLATED FOOTINGS
As 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.
2
RECTANGULAR ISOLATED FOOTINGS
The 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:
3
RECTANGULAR ISOLATED FOOTINGS
The 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.
4
RECTANGULAR ISOLATED FOOTINGS
Example 12.5
Solution
5
RECTANGULAR ISOLATED FOOTINGS
6
RECTANGULAR ISOLATED FOOTINGS
7
RECTANGULAR ISOLATED FOOTINGS
8
RECTANGULAR ISOLATED FOOTINGS
9
COMBINED FOOTINGS
Combined 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.
10
COMBINED FOOTINGS
11
COMBINED FOOTINGS
Because 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.
12
COMBINED FOOTINGS
13
COMBINED FOOTINGS
You 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.
14
COMBINED FOOTINGS
First, 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.
15
COMBINED FOOTINGS
The 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.
16
COMBINED FOOTINGS
Nevertheless, 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.
17
COMBINED FOOTINGS
Example 12.6
Solution
18
COMBINED FOOTINGS
19
COMBINED FOOTINGS
20
COMBINED FOOTINGS
21
COMBINED FOOTINGS
22
COMBINED FOOTINGS
23
COMBINED FOOTINGS
24
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.
25
COMBINED FOOTINGS
This 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.
26
FOOTING DESIGN FOR EQUAL SETTLEMENTS
If 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.
27
FOOTING DESIGN FOR EQUAL SETTLEMENTS
A 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.
28
FOOTING DESIGN FOR EQUAL SETTLEMENTS
A 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.
29
FOOTING DESIGN FOR EQUAL SETTLEMENTS
Example 12.7
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.
Solution
30
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.
31
FOOTINGS SUBJECTED TO LATERAL MOMENTS
Walls 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
32
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.
33
FOOTINGS SUBJECTED TO LATERAL MOMENTS
34
FOOTINGS SUBJECTED TO LATERAL MOMENTS
Should 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:
35
FOOTINGS SUBJECTED TO LATERAL MOMENTS
36
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.
37
FOOTINGS SUBJECTED TO LATERAL MOMENTS
Example 12.8
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.
Solution
38
FOOTINGS SUBJECTED TO LATERAL MOMENTS
39
TRANSFER OF HORIZONTAL FORCES
When 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.
40
TRANSFER OF HORIZONTAL FORCES
Example 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.
41
TRANSFER OF HORIZONTAL FORCES
Check Tensile Development Lengths of These Dowels
As 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.
42
PLAIN CONCRETE FOOTINGS
Occasionally, 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.
43
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.
44
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.
45
PLAIN CONCRETE FOOTINGS
46
PLAIN CONCRETE FOOTINGS
A 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.
47
PLAIN CONCRETE FOOTINGS
Example 12.10
Solution
48
PLAIN CONCRETE FOOTINGS
49
PLAIN CONCRETE FOOTINGS
50
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.
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.
56
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.
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,
58
PILE CAPS
Where Wf is the total weight of footing, fill, and surcharge divided by
the number od piles.
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
59
PILE CAPS
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.
60
PILE CAPS
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
61
PILE CAPS
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.
62
PILE CAPS
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 threedimensional 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.
63
PILE CAPS
64