Chapter 10 Mats

353

Columns

10

Mats

I.
The mere formulation of a problem is far more often essential than its
solution, which may be merely a matter of mathematical or experimental
skill. To raise new questions, new possibilities, to regard old problems
from a new angle requires creative imagination and marks real
advances in science.
Albert Einstein

8 M.ioo.Figure

10.1

.I

A mat foundation supported directly on soil.

Lateral loads are not uniformly distributed through the structure and thus may cause
differential horizontal movements in spread footings or pile caps. The continuity of
r
a mat will resist such movements.
The uplift loads are larger than spread footings can accommodate. The greater
weight and continuity of a mat may provide sufficient resistance .

The second type of shallow foundation is the mat foundation, as shown in Figure IO.!. A
mat is essentially a very large spread footing that usually encompasses the entire footprint
of the structure. They are also known as raft foundations.
Foundation engineers often consider mats when dealing with any of the following
conditions:
The structural loads are so high or the soil conditions so poor that spread footings
would be exceptionally large. As a general rule of thumb, if spread footings would
cover more than about one-third of the building footprint area, a mat or some type
of deep foundation will probably be more economical.
The soil is very erratic and prone to excessive differential settlements. The structural continuity and flexural strength of a mat will bridge over these irregularities.
The same is true of mats on highly expansive soils prone to differential heaves.
The structural loads are erratic, and thus increase the likelihood of excessive differential settlements. Again, the structural continuity and flexural strength of the mat
will absorb these irregularities.
352

The bottom of the structure is located below the groundwater table, so waterproofing is an important concern. Because mats are monolithic, they are much easier to
waterproof. The weight of the mat also helps resist hydrostatic uplift forces from
the groundwater.
Many buildings are supported on mat foundations, as are silos, chimneys, and other
types of tower structures. Mats are also used to support storage tanks and large machines.
Typically, the thickness, T, is 1 to 2 m (3-6 ft), so mats are massive structural elements.
The seventy five story Texas Commerce Tower in Houston is one of the largest matsupported structures in the world. Its mat is 3 m (9ft 9 in) thick and is bottomed 19.2 m
(63ft) below the street level.
Although most mat foundations are directly supported on soil, sometimes engineers
use pile- or shaft-supported mats, as shown in Figure 10.2. These foundations are often
called piled rafts. and they are hybrid foundations that combine features of both mat and
deep foundations. Pile- and shaft-supported mats are discussed in Section 1!.9.

Chapter 10 Mats

354

355

10.1 Rigid Methods

M

Building

/""P"..

Mat

Figure 10.3

Figure 10.2 A pile- or shaft-supported mat foundation.

necessarily distributed evenly across the mat.

The deep foundations

Although this type of analysis is appropriate for spread footings, it does not accurately model mat foundations because the width-to-thickness ratio is much greater in
mats, and the assumption of rigidity is no longer valid. Portions of a mat beneath columns
and bearing walls settle more than the portions with less load, which means the bearing
pressure will be greater beneath the heavily-loaded zones, as shown in Figure 10.4. This
redistribution of bearing pressure is most pronounced when the ground is stiff compared
to the mat, as shown in Figure 10.5, but is present to some degree in all soils.
Because the rigid method does not consider this redistribution of bearing pressure, it
does not produce reliable estimates of the shears, moments, and deformations in the mat.
In addition, even if the mat was perfectly rigid, the simplified bearing pressure distributions in Figure 10.3 are not correct-in reality, the bearing pressure is greater on the edges '-c
and smaller in the center than shown in this figure.

are not

Various methods have been used to design mat foundations. We will divide them
into two categories: Rigid methods and nonrigid methods.

10.1

Bearing pressure distribution for rigid method.

RIGID METHODS
The simplest approach to structural design of mats is the rigid method (also known as the
conventional method or the conventional method of static equilibrium) (Teng, 1962). This
method assumes the mat is much more rigid than the underlying soils, which means any
distortions in the mat are too small to significantly impact the distribution of bearing pressure. Therefore, the magnitude and distribution of bearing pressure depends only on the
applied loads and the weight of the mat, and is either uniform across the bottom of the mat
(if the normal load acts through the centroid and no moment load is present) or varies linearly across the mat (if eccentric or moment loads are present) as shown in Figure 10.3.
This is the same simplifying assumption used in the analysis of spread footings, as shown
in Figure 5.10e.
This simple distribution makes it easy to compute the flexural stresses and deflections (differential settlements) in the mat. For analysis purposes, the mat becomes an inverted and simply loaded two-way slab, which means the shears, moments, and
deflections may be easily computed using the principles of structural mechanics. The engineer can then select the appropriate mat thickness and reinforcement.

rm
Rigid Mat

,
I

--1

Soil
Bearing
Pressure

tJI1JJI1IJID
Nonrigid Mat
(Exaggerated)

Figure 10.4 The rigid method assumes there are no flexural deflections in the mat. so
the distribution of soil bearing pressure is simple to define. However. these detlections are
important because they influence the bearing pressure distribution.

Chapter 10

356

Mats

357

10.2 Nonrigid Methods

Where:

k, = coefficient of subgrade reaction

q = bearing pressure
Rock

l) = settlement
The coefficient k, has units of force per length cubed. Although we use the same units to
express unit weight, k, is not the same as the unit weight and they are not numerically
equal.
The interaction between the mat and the underlying soil may then be represented as
a "bed of springs," each with a stiffness k, per unit area, as shown in Figure 10.6. Portions
of the mat that experience more settlement produce more compression in the "springs,"
which represents the higher bearing pressure, whereas portions that settle less do not compress the springs as far and thus have less bearing pressure. The sum of these spring
forces must equal the applied structural loads plus the weight of the mat:

(a)

Stiff
Soil
(b)

LP

Wr -

Uf)

J qdA

l)

k, dA

Where:

~///////}///////////WFigure 10.5 Distribution of bearing pressure under a mat foundation; (a) on bedrock
or very hard soil; (b) on stiff soil; (c) on soft
soil (Adapted from Teng, 1962).

10.2

LP = sum of structural loads acting on the mat

~iUl.llii111l-J
~:

Wr

= weight of the mat

(c)

NONRIGID METHODS
We overcome the inaccuracies of the rigid method by using analyses that consider deformations in the mat and their influence on the bearing pressure distribution. These are
called non rigid methods, and produce more accurate values of mat deformations and
stresses. Unfortunately, nonrigid analyses also are more difficult to implement because
they require consideration of soil-structure interaction and because the bearing pressure
distribution is not as simple.
Coefficient of Subgrade Reaction
Because nonrigid methods consider the effects of local mat deformations on the distribution of bearing pressure, we need to define the relationship between settlement and bearing pressure. This is usually done using the coefficient of subgrade reaction, ks (also
known as the modulus of subgrade reaction, or the subgrade modulus):

Ik, = ~I

(10.1)

Figure 10.6 The coefficient of subgrade reaction forms the basis for a "bed of springs"
analogy to model the soil-structure interaction in mat foundations.

(10.2)

358

Chapter 10 Mats

10.2 Nonrigid Methods

359

= pore water pressure along base of the mat

q = bearing pressure between mat and soil
A = mat-soil contact area

Un

Idealized
Function
Linear~

-----... / ,/

o = settlement at a point on the mat

ri

This method of describing bearing pressure is called a soil-structure interaction

analysis because the bearing pressure depends on the mat deformations, and the mat deformations depend on the bearing pressure.

iJ
0::

.
"

Winkler Method

(]

The "bed of springs" model is used to compute the shears, moments, and deformations in
the mat, which then become the basis for developing a structural design. The earliest use
of these "springs" to represent the interaction between soil and foundations has been attributed to Winkler (1867), so the analytical model is sometimes called a Winkler foundation or the Winkler method. It also is known as a beam on elastic foundation analysis.
In its classical form the Winkler method assumes each "spring" is linear and acts independently from the others, and that all of the springs have the same k,. This representation has the desired effect of increasing the bearing pressure beneath the columns, and
thus is a significant improvement over the rigid method. However, it is still only a coarse
representation of the true interaction between mats and soil (Hain and Lee, 1974; Horvath, 1983), and suffers from many problems, including the following:

Fj~ure 10.7

,,,

1/ I

The '1-8 relalionship is nonlin-

car. so ks must represent

some "equivalent""

linear function.

Seulement. (5

Coupled Method
The next step up from a Winkler analysis is to use a coupled method, which uses additional springs as shown in Figure 10.9. This way the vertical springs no longer act independently, and the uniformly loaded mat of Figure 10.8 exhibits the desired dish shape. In
principle, this approach is more accurate than the Winkler method, but it is not clear how
to select the k, values for the coupling springs, and it may be necessary to develop custom
software to implement this analysis.

1. The load-settlement behavior of soil is nonlinear, so the k, value must represent

some equivalent linear function, as shown in Figure 10.7.
2. According to this analysis, a uniformly loaded mat underlain by a perfectly uniform
soil, as shown in Figure 10.8, will settle uniformly into the soil (i.e., there will be no
differential settlement) and all of the "springs" will be equally compressed. In reality, the settlement at the center of such a mat will be greater than that along the
edges, as discussed in Chapter 7. This is because the 6.CT: values in the soil are
greater beneath the center.

ttttttt

3. The "springs" should not act independently. In reality, the bearing pressure induced
at one point on the mat influences more than just the nearest spring.
4. Primarily because of items 2 and 3, there is no single value of k, that truly represents the interaction between soil and a mat.
Items 2 and 3 are the primary sources of error, and this error is potentially unconservative (i.e., the shears, moments, and deflections in the mat may be greater than those predicted by Winkler). The heart of these problems is the use of independent springs in the
Winkler model. In reality, a load at one point on the mat induces settlement both at that
point and in the adjacent parts of the mat, which is why a uniformly mat exhibits dishshaped settlement, not the uniform settlement pre!iicted by Winkler.

True Bchavior

jk,

"

::c
.~

-,_---....1
PerWinkler

Figure to.8 Settlement

analysis. (b) actual.

~~~~~~~
~

Actual
of a uniformly-loaded

mat on a uniform soil: (a) per Winkler

361

10.2 Nonrigid Methods

Chapter 10 Mats

360

Mal

Coupling Spring
FiKure 10.9

ModeJing of soil-struclUre inleraction using coupled springs.

Zone A

k,
50
100Iblin'
Ib/in'
Ib/in'
k, = 75

Pseudo-Coupled

Method

Zone C

The pseudo-coupled method (Liao, 1991; Horvath, 1993) is an attempt to overcome the
lack of coupling in the Winkler method while avoiding the difficulties of the coupled
method. It does so by using "springs" that act independently, but have different k, values
depending on their location on the mat. To properly model the real response of a uniform
soil, the "springs" along the perimeter of the mat should be stiffer than those in the center,
thus producing the desired dish-shaped deformation in a uniformly-loaded mat. If concentrated loads, such as those from columns, also are present, the resulting mat deformations
are automatically superimposed on the dish-shape.
Model studies indicale that reasonable results are obtained when k, values along the
perimeter of the mat are about twice those in the center (ACI, 1993). We can implement
this in a variety of ways, including the following:

Zone Binlo zones for a pseudo-coupled analysis. Tbe coefFiKure to.IO

A lypical mal divided
ficient of suhgradc real.'tioll. k,. progressively increases from the innermost zone to the
outermost lone.

pseudo-coupled method. Given the current state of technology and software availability,
this is probably the most practical approach to designing most mat foundations.
Multiple-Parameter

Method

Another way of representing soil-structure interaction is to use the multiple parameter

method (Horvath, 1993). This method replaces the independently-acting linear springs of
the Winkler method (a single-parameter model) with springs and other mechanical elements (a multiple-parameter model). These additional elements define the coupling effects.
The multiple-parameter method bypasses the guesswork involved in distributing the
k, values in the psuedo-coupled me!hod because coupling effects are inherently incorporated into the model, and thus should be more accurate. However, it has not yet been implemented into readily-available software packages. Therefore, this method is not yet
ready to be used on routine projects.

1. Divide the mat into two or more concentric zones, as shown in Figure 10.10. The
innermost zone should be about half as wide and half as long as the mat.
2. Assign a k, value to each zone. These values should progressively increase from the
center such that the outermost zone has a k, about twice as large at the innermost
zone. Example 10.1 illustrates this technique.
3. Evaluate the shears, moments, and deformations in the mat using the Winkler "bed
of springs" analysis, as discussed later in this chapter.
4. Adjust the mat thickness and reinforcement as needed to satisfy strength and serviceability requirements.

Finite Element Method

ACI (1993) found the pseudo-coupled method produced computed moments 18 to
25 percent higher than those determined from the Winkler method, which is an indication
of how unconservative Winker can be.
Most commercial mat design software uses the Winkler method to represent the
soil-structure interaction, and these software packages usually can accommodate the

All of the methods discussed thus far attempt to model a three-dimensional soil by a series
of one-dimensional springs. They do so in order to make the problem simple enough to
perform the structural analysis. An alternative method would be to use a three-dimensional

362

Chapter 10

Mats

10.3

The position on the mat-To model the soil accurately, k, needs to be larger near
the edges of the mat and smaller near the center.
Time-Much
of the settlement of mats on deep compressible soils will be due to
consolidation and thus may occur over a period of several years. Therefore, it may
be necessary to consider both short-term and long-term cases.

mathematical model of both the mat and the soil, or perhaps the mat, soil, and superstructure. This can be accomplished using theftnite element method.
This analysis method divides the soil into a network of small elements, each with
defined engineering properties and each connected to the adjacent elements in a specified
way. The structural and gravitational loads are then applied and the elements are stressed
and deformed accordingly. This. in principle. should be an accurate representation of the
mat, and should facilitate a precise and economical design.
Unfortunately, such analyses are not yet practical for routine design problems because:

Actually, there is no single k, value, even if we could define these factors because
the q-S relationship is nonlinear and because neither method accounts for interaction between the springs.
Engineers have tried various techniques of measuring or computing k,. Some rely
on plate load tests to measure k, in situ. However, the test results must be adjusted to compensate for the differences in width, shape, and depth of the plate and the mat. Terzaghi
(1955) proposed a series of correction factors, but the extrapolation from a small plate to a
mat is so great that these factors are not very reliable. Plate load tests also include the dubious assumption that the soils within the shallow zone of influence below the plate are
comparable to those in the much deeper zone below the mat. Therefore, plate load tests
generally do not provide good estimates of k, for mat foundation design.
Others have used derived relationships between k, and the soil's modulus of elasticity, E (Vesic and Saxena, 1970; Scott, 1981).Although these relationships provide some
insight, they too are limited.
Another method consists of computing the average mat settlement using the techniques described in Chapter 7 and expressing the results in the form of k, using Equation
10.1. If using the pseudo-coupled method, use k, values in the center of the mat that are
less than the computed value, and k, values along the perimeter that are greater. This
should be done in such a way that the perimeter values are twice the central values, and
the integral of all the values over the area of the mat is the same as the produce of the
original k, and the mat area. Example 10.1 describes this methodology.

1. A three-dimensional finite element model requires tens of thousands or perhaps

hundreds of thousands of elements, and thus place corresponding demands on computer resources. few engineers have access to computers that can accommodate
such intensive analyses.
2. It is difficult to determine the required soil properties with enough precision, especially at sites where the soils are highly variable. In other words, the analysis
method far outweighs our ability to input accurate parameters.
Nevertheless, this approach may become more usable in the future, especially as increasingly powerful computers become more widely available.
This method should not be confused with structural analysis methods that use twodimensional finite elements to model the mat and WinkleI' springs to model the soil. Such
methods require far less computational resources, and are widely used. We will discuss
this use of finite element analyses in Section lOA.

10.3

363

Determining the Coefficient of Subgrade Reaction

DETERMINING THE COEFFICIENT OF SUBGRADE REACTION

Most mat foundation designs are currently developed using either the Winkler method or
the pseudo-coupled method, both of which depend on our ability to define the coefficient
of subgrade reaction, k,. Unfortunately, this task is not as simple as it might first appear
because k, is not a fundamental soil property. Its magnitude also depends on many other
factors, including the following:

Example 10.1
A structure is to be supported on a 30-m wide, 50-m long mat foundation.The average bearing pressure is 120 kPa. According to a settlement analysis conducted using the techniques
described in Chapter 7, the average settlement, ll, will be 30 mm. Determine the design values of k, to be used in a pseduo-coupledanalysis.

The width of the loaded area-A

wide mat will settle more than a narrow one
with the same q because it mobilizes the soil to a greater depth as shown in Figure 8.2. Therefore, each has a different k,.

Solution

Compute average k, using Equation 10.1:

The shape of the loaded area-The

stresses below long narrow loaded areas are
different from those below square loaded areas as shown in Figure 7.2. Therefore, ks
will differ.

(k.)", . =

The depth of the loaded area below the ground surface-At

greater depths, the
change in stress in the soil due to q is a smaller percentage of the initial stress, so
the settlement is also smaller and k, is greater.

5q =

120kPa = 4000 kN/m-'

0.030

ID

Divide the mat into three zones, as shown in Figure lO.l!, with (k,lc = 2(k,.)Aand
(k,)B = 1.5(k,lA

Chapter 10

364

Mats

10.4 Structural Design

10.4

50m
37.5 m

15 m

22.5 In

365

STRUCTURAL DESIGN
General Methodology

25 m
ZoneC

The structural design of mat foundations must satisfy both strength and serviceability requirements. This requires two separate analyses, as follows:

ZoncA

Zone B

30 m

Figure

10.11

Mat foundation for Example 10.1.

Compute the area of each zone:

A, =
A{j

(25 m)(15 m) = 375 m2

= (37.5 m)(22.5 m) - (25 m)(15 m) = 469

Ac = (50 m)(30 m) - (37.5 m)(22.5 m)

m2

A, (k,)A + An
+ 469

(1.5)(k,),

(k,){j

+ Ac

+ 656

(k,)c

(2)(k,)A

2390 (k,)A
(k,)A

(k,)A

(k,)n

= (0.627)(4000

= (1.,5)(0.627)(4000

(k,)c = (2)(0.627)(4000

kN/m-')

= 2510

kN/m-')
kN/m')

1500 (k,L"

1500 (k,),,,,

kN/m3

= 3765 kN/m3
= 5020 kN/m3

Step 2:

Evaluate mat deformations (which is the primary serviceability requirement)

using the unfactored loads (Equations 2.1-2.4). These deformations are the result of concentrated loading at the column locations, possible non-uniformities
in the mat, and variations in the soil stiffness. In effect, these deformations are
the equivalent of differential settlement. If they are excessive, then the mat must
be made stiffer by increasing its thickness.

Closed-Form Solutions

= (A. + AB + Ac)

= 0.627

Evaluate the strength requirements using the factored loads (Equations 2.72.15) and LRFD design methods (which ACI calls ultimate strength design).
The mat must have a sufficient thickness, T, and reinforcement to safely resist
these loads. As with spread footings, T should be large enough that no shear reinforcement is needed.

656 m2

When the Winkler method is used (i.e., when all "springs" have the same k,.) and the
geometry of the problem can be represented in two-dimensions, it is possible to develop
closed-form solutions using the principles of structural mechanics (Scott, 1981; Hetenyi,
1974). These solutions produce values of shear, moment, and deflection at all points in
the idealized foundation. When the loading is complex, the principle of superposition may
be used to divide the problem into multiple simpler problems.
These closed-form solutions were once very popular, because they were the only
practical means of solving this problem. However, the advent and widespread availability
of powerful computers and the associated software now allows us to use other methods
that are more precise and more flexible.

Compute the design k, values:

375 (k,)A

Step I:

(k,),,,,

(k,),,,g

0(=

Answer
0(=

0(=

Answer

Finite Element Method

Answer

Today, most mat foundations are designed with the aid of a computer using the finite element method (FEM). This method divides the mat into hundreds or perhaps thousands of
elements, as shown in Figure 10.12. Each element has certain defined dimensions, a specified stiffness and strength (which may be defined in terms of concrete and steel properties) and is connected to the adjacent elements in a specified way.
The mat elements are connected to the ground through a series of "springs," which
are defined using the coefficient of subgrade reaction. Typically, one spring is located at
each corner of each element.

Because it is so difficult to develop accurate k,. values, it may be appropriate to conduct a parametric studies to evaluate its effect on the mat design. ACI (1993) suggests
varying k, from one-half the computed value to five or ten times the computed value, and
basing the structural design on the worst case condition.
This wide range in k,. values will produce proportional changes in the computed
total settlement. However, we ignore these total settlement computations because they are
not reliable anyway, and compute it using the methods described in Chapter 7. These
changes in k,. have much less impact on the shears, moments, and deflections in the mat,
and thus have only a small impact on the structural design.

The loads on the mat include the externally applied column loads, applied line loads,
applied area loads, and the weight of the mat itself. These loads press the mat downward,

---1

366

Chapter 10 Mats

-t

10.6 Bearing Capacity

T .

10.6

YPlcal Element

SUMMARY

ii

Major Points

111 111 1 1i"1'11

1. Mat foundations are essentially large spread footings that usually encompass the entire footprint of a structure. They are often an appropriate choice for structures that
are too heavy for spread footings.

Prolile

2. The analysis and design of mats must include an evaluation of the flexural stresses
and must provide sufficient flexural strength to resist these stresses.

Figure 10.12 Use of the finite elemeIll method to analyze mat foundations. The mat is
divided into a series of elements which arc connected in a specified IJ./ay. The elements
are connected to the ground through a "bed of springs."

3. The oldest and simplest method of analyzing mats is the rigid method. It assumes
that the mat is much more rigid than the underlying soil. which means the magnitude
and distribution of bearing pressure is easy to determine. This means the shears, moment, and deformations in the mat are easily determined. However, this method is
not an accurate representation because the assumption of rigidity is not correct.
4. Nonrigid analyses are superior because they consider the flexural deflections in the
mat and the corresponding redistribution of the soil bearing pressure.
S. Nonrigid methods must include a definition of soil-structure interaction. This is
usually done using a "bed of springs" analogy, with each spring having a linear
force-displacement function as defined by the coefficient of subgrade reaction, k,.
6. The simplest and oldest nonrigid method is the Winkler method, which uses independent springs, all of which have the same k,. This method is an improvement over
rigid analyses, but still does not accurately model soil-structure interaction, primarily because it does not consider coupling effects.

and this downward movement is resisted by the soil "springs." These opposing forces,
along with the stiffness of the mat, can be evaluated simultaneously using matrix algebra,
which allows us to compute the stresses, strains, and distortions in the mat. If the results of
the analysis are not acceptable, the design is modified accordingly and reanalyzed.
This type of finite element analysis does not consider the stiffness of the superstructure. In other words, it assumes the superstructure is perfectly flexible and offers no resistance to deformations in the mat. This is conservative.
The finite element analysis can be extended to include the superstructure, the mat,
and the underlying soil in a single three-dimensional finite element model. This method
would, in principle, be a more accurate model of the soil-structure system, and thus may
produce a more economical design. However, such analyses are substantially more complex and time-consuming, and it is very difficult to develop accurate soil properties for such
models. Therefore, these extended finite element analyses are rarely performed in practice.
10.5

BEARING CAPACITY
Because of their large width, mat foundations on sands and gravels do not have bearing
capacity problems. However, bearing capacity might be important in silts and clays, especially if undrained conditions prevail. The Fargo Grain Silo failure described in Chapter 6
is a notabl6 example of a bearing capacity failure in a saturated clay.
We can evaluate bearing capacity using the analysis techniques described in Chapter 6. It is good practice to design the mat so the bearing pressure at all points is less than
the allowable bearing capacity.

Plan

ii

367

7. The coupled method is an extension of the Winkler method that considers coupling
between the springs.
8. The pseudo-coupled method uses independent springs, but adjusts the k, values to
implicitly account for coupling effects.

TOTAL SETTLEMENT

9. The multiple parameter and finite element methods are more advanced ways of describing soil-structure interaction.

The bed of springs analyses produce a computed total settlement. However, this value is
unreliable and should not be used for design. These analyses are useful only for computing shears, moments, and deformations (differential settlements) in the mat. Total settlement should be computed using the methods described in Chapter 7.

10. The coefficient of subgrade reaction is difficult to determine. Fortunately, the mat
design is often not overly sensitive to global changes in k,. Parametric studies are
often appropriate.

----.-.-tIii...

368

Chapter

11. If the Winkler

method

is used to describe

geometry is not too complex,

closed-form solutions. However,

soil-structure

the structural
these methods

interaction.

fines

analyses are performed using numerical

method. This method uses finite elements

soil-structure

principle,

interaction

using

the Winker

it also could use the multiple

parameter

the soil, mat, and superstructure.

rent practices,
powerful

mostly

because

and the mat

methods, especially the fito model the mat, and de-

or pseudo-coupled

models.

However,

they are difficult

In

finite element analysis that

such analyses are beyond cur-

to set up and require

is best determined

Do not use the coefficient

capacity

of subgrade

is not a problem

using the methods

reaction

to determine

with sands and gravels,

silts and clays. It should be checked

using the methods

described

umn loads plus the weight of the mat will be 90.000 k. According to a settlement analysis conducted using the techniques described in Chapter 7. the total settlement will be 1.8 inches. The
groundwater table is at a depth of 10ft below the bottom of the mal. Using the pseudocoupled method. divide the mat into zones and composite each zone. Then indicate the highend and low-end valves of k, that should be used in the analysis.

7.

total settlement.
but can be important

described

in Chapter

in

6.

Vocabulary
Beam on elastic foundation

Mat foundation

Bed of springs

Multiple

parameter

Coefficient

Nonrigid

method

of subgrade

reaction
Coupled

Pile-supported
method

Finite element

method

Pseudo-coupled
Raft foundation

Rigid method
method

mat

Shaft-supported
mat
Soil-structure
interaction
Winkler

369

10.6 An office building is to be supported on 150-ft x 300-ft mat foundation. The sum of the col-

especially

in Chapter

Bearing Capacity

10.5 A 25-m diameter cylindrical water storage tank is to be supported on a mat foundation. The
weight of the tank and its contents will be 50,000 kN and the weight of the mat will be
12.000 kN. According to a settlement analysis conducted using the techniques described in
Chapter 7. the total settlement will be 40 mm. The groundwater table is at a depth of 5 m
below the bottom of the mat. Using the pseudo-coupled method. divide the mat into zones and
compute k for each ZOlle.Then indicate the high-end and low-end values of k. that should be
used in the analysis.

computers.

14. The total settlement

15. Bearing

10.6

model.

13. A design could be based entirely on a three-dimensional

includes

Mats

analysis may be performed

using
are generally considered obsolete.

12. Most structural

nite element

10

method

method

COMPREHENSIVE QUESTIONS AND PRACTICE PROBLEMS

10.1 Explain the reasoning behind the statement in Section 10.6: "Because of their large width, mat
foundations on sands and gravels do not have bearing capacity problems."
10.2 How has the development of powerful and inexpensive digital computers affected the analysis
and design of mat foundations? What changes do you expect in the future as this trend continues?
10.3 A mat foundation supports forty rwo columns for a building. These columns are spaced on a
uniform grid pattern. How would the moments and differential settlements change if we used
a nonrigid analysis with a constant k, in lieu of a rigid analysis?
10.4 According to a settlement analysis conducted using the techniques described in Chapter 7, a
certain mat will have a total settlement of 2.1 inches if the average bearing pressure is
5500 lb/ft'. Compute the average k, and express your answer in units of Ib/in.1.

---'-