Mat Foundation analysis and design Coduto10.pdf

I. 8 M.ioo.- .I
10
Mats
Chapter 10 Mats
1
Columns
/
T
353
The mere formulation of a problem isfar 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
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 struc-
tural 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 differ-
ential settlements. Again, the structural continuity and flexural strength of the mat
will absorb these irregularities.
352
Figure 10.1 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
a mat will resist such movements. r
• The uplift loads are larger than spread footings can accommodate. The greater
weight and continuity of a mat may provide sufficient resistance .
• The bottom of the structure is located below the groundwater table, so waterproof-
ing 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 mat-
supported 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.

354 Chapter 10 Mats
Building
t t t
Mat
10.1 Rigid Methods
p
~
M
/""P"..
~
355
Figure 10.2 A pile- or shaft-supported mat foundation. The deep foundations are not
necessarily distributed evenly across the mat.
Various methods have been used to design mat foundations. We will divide them
into two categories: Rigid methods and nonrigid methods.
10.1 RIGID METHODS
Figure 10.3 Bearing pressure distribution for rigid method.
Although this type of analysis is appropriate for spread footings, it does not accu-
rately 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 distribu-
tions 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.
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.
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 pres-
sure. 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 lin-
early 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 deflec-
tions (differential settlements) in the mat. For analysis purposes, the mat becomes an in-
verted and simply loaded two-way slab, which means the shears, moments, and
deflections may be easily computed using the principles of structural mechanics. The en-
gineer can then select the appropriate mat thickness and reinforcement.
,
I
I
--1
rm
Rigid Mat
Soil
Bearing
Pressure tJI1JJI1IJID
Nonrigid Mat
(Exaggerated)

356 Chapter 10 Mats 10.2 NonrigidMethods 357
Rock
Where:
k,= coefficient of subgrade reaction
q = bearing pressure
l) = settlement
(a)
Stiff
Soil
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 com-
press 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:
(b)
LP + Wr - Uf) = J qdA = J l) k, dA (10.2)
Figure 10.5 Distribution of bearing pres-
sure under a mat foundation; (a) on bedrock
or very hard soil; (b) on stiff soil; (c) on soft
soil (Adapted from Teng, 1962).
~///////}///////////W-
~iUl.llii111l-J
~: .
(c)
Where:
LP = sum of structural loads acting on the mat
Wr = weight of the mat
Coefficient of Subgrade Reaction
Because nonrigid methods consider the effects of local mat deformations on the distribu-
tion of bearing pressure, we need to define the relationship between settlement and bear-
ing pressure. This is usually done using the coefficient of subgrade reaction, ks (also
known as the modulus of subgrade reaction, or the subgrade modulus):
We overcome the inaccuracies of the rigid method by using analyses that consider defor-
mations in the mat and their influence on the bearing pressure distribution. These are
called nonrigid 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.
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.
(10.1)
Ik, = ~I
10.2 NONRIGID METHODS

The next step up from a Winkler analysis is to use a coupled method, which uses addi-
tional springs as shown in Figure 10.9. This way the vertical springs no longer act inde-
pendently, 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.
~~~~~~~
~
358 Chapter 10 Mats
Un = pore water pressure along base of the mat
q = bearing pressure between mat and soil
A = mat-soil contact area
o = settlement at a point on the mat
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 de-
formations depend on the bearing pressure.
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 at-
tributed to Winkler (1867), so the analytical model is sometimes called a Winkler founda-
tion 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 in-
dependently from the others, and that all of the springs have the same k,. This representa-
tion 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; Hor-
vath, 1983), and suffers from many problems, including the following:
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 real-
ity, 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.
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 repre-
sents the interaction between soil and a mat.
10.2 NonrigidMethods
~
ri
~
iJ
0::
.§
"
"
::c
.~
p
(]
Fj~ure 10.7 The '1-8 relalionship is nonlin-
car. so ks must represent some "equivalent""
linear function.
Coupled Method
t t t t t t t
-,_---....1
Idealized
Linear~ ,/
Function -----... /
,
,
jk,
1/ I
,
,
,
Seulement. (5
True Bchavior
359
Figure to.8 Settlement of a uniformly-loaded mat on a uniform soil: (a) per Winkler
analysis. (b) actual.
Items 2 and 3 are the primary sources of error, and this error is potentially unconser-
vative (i.e., the shears, moments, and deflections in the mat may be greater than those pre-
dicted 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 dish-
shaped settlement, not the uniform settlement pre!iicted by Winkler.
J
PerWinkler Actual

360
Chapter 10 Mats
Mal
Coupling Spring
FiKure 10.9 ModeJing of soil-struclUre inleraction using coupled springs.
10.2 NonrigidMethods 361
Pseudo-Coupled Method
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 concen-
trated 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:
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 ser-
viceability requirements.
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
I
J
Zone A
k, = 50 Ib/in'
Zone B
k, = 75 Iblin'
Zone C
k, = 100 Ib/in'
FiKure to.IO A lypical mal divided inlo zones for a pseudo-coupled analysis. Tbe coef-
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 ele-
ments (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 incorpo-
rated into the model, and thus should be more accurate. However, it has not yet been im-
plemented into readily-available software packages. Therefore, this method is not yet
ready to be used on routine projects.
Finite Element Method
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 Determining the Coefficient of Subgrade Reaction 363
mathematical model of both the mat and the soil, or perhaps the mat, soil, and superstruc-
ture. 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 be-
cause:
1. A three-dimensional finite element model requires tens of thousands or perhaps
hundreds of thousands of elements, and thus place corresponding demands on com-
puter 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, espe-
cially 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 increas-
ingly powerful computers become more widely available.
This method should not be confused with structural analysis methods that use two-
dimensional 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 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:
• 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 Fig-
ure 8.2. Therefore, each has a different k,.
• 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.
• 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.
1
• 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.
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 be-
tween 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 com-
pensate 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 du-
bious 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 elasti-
city, 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 tech-
niques 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.
Example 10.1
A structure is to be supportedon a 30-m wide, 50-m long mat foundation.The averagebear-
ing pressure is 120kPa. According to a settlement analysis conductedusing the techniques
described in Chapter 7, the average settlement, ll, will be 30 mm. Determinethe design val-
ues of k, to be used in a pseduo-coupledanalysis.
Solution
Computeaveragek, using Equation 10.1:
q 120kPa = 4000 kN/m-'
(k.)",.• = 5 = 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

364 Chapter 10 Mats
50m
10.4 StructuralDesign
10.4 STRUCTURAL DESIGN
365
Figure 10.11 Mat foundation for Example 10.1.
Compute the area of each zone:
A, = (25 m)(15 m) = 375 m2
A{j = (37.5 m)(22.5 m) - (25 m)(15 m) = 469 m2
Ac = (50 m)(30 m) - (37.5 m)(22.5 m) = 656 m2
37.5 m
25 m
ZoncA
15 m
22.5 In
Zone B
ZoneC
30 m
General Methodology
The structural design of mat foundations must satisfy both strength and serviceability re-
quirements. This requires two separate analyses, as follows:
Step I: Evaluate the strength requirements using the factored loads (Equations 2.7-
2.15) 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 re-
inforcement is needed.
Step 2: Evaluate mat deformations (which is the primary serviceability requirement)
using the unfactored loads (Equations 2.1-2.4). These deformations are the re-
sult 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
Compute the design k, values:
A, (k,)A + An (k,){j + Ac (k,)c = (A. + AB + Ac) (k,),,,,
375 (k,)A + 469 (1.5)(k,), + 656 (2)(k,)A = 1500 (k,L"
2390 (k,)A = 1500 (k,),,,,
(k,)A = 0.627 (k,),,,g
Because it is so difficult to develop accurate k,. values, it may be appropriate to con-
duct 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.
(k,)A = (0.627)(4000 kN/m-') = 2510 kN/m3
(k,)n = (1.,5)(0.627)(4000 kN/m-') = 3765 kN/m3
(k,)c = (2)(0.627)(4000 kN/m') = 5020 kN/m3
0(= Answer
0(= Answer
0(= Answer
---1
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.
Finite Element Method
Today, most mat foundations are designed with the aid of a computer using the finite ele-
ment 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 spec-
ified stiffness and strength (which may be defined in terms of concrete and steel proper-
ties) 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.
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,

366 Chapter 10 Mats 10.6 Bearing Capacity 367
T .
-tYPlcal Element
10.6 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, espe-
cially 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 Chap-
ter 6. It is good practice to design the mat so the bearing pressure at all points is less than
the allowable bearing capacity.
Plan
111ii111i1i1i"1'11
Prolile
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."
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 superstruc-
ture. In other words, it assumes the superstructure is perfectly flexible and offers no resis-
tance 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 com-
plex 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 TOTAL SETTLEMENT
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 comput-
ing shears, moments, and deformations (differential settlements) in the mat. Total settle-
ment should be computed using the methods described in Chapter 7.
----.-.-tIii...
SUMMARY
Major Points
1. Mat foundations are essentially large spread footings that usually encompass the en-
tire footprint of a structure. They are often an appropriate choice for structures that
are too heavy for spread footings.
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.
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, mo-
ment, 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 inde-
pendent 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, primar-
ily because it does not consider coupling effects.
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.
9. The multiple parameter and finite element methods are more advanced ways of de-
scribing soil-structure interaction.
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.

368 Chapter 10 Mats 10.6 Bearing Capacity 369
11. If the Winkler method is used to describe soil-structure interaction. and the mat
geometry is not too complex, the structural analysis may be performed using
closed-form solutions. However, these methods are generally considered obsolete.
12. Most structural analyses are performed using numerical methods, especially the fi-
nite element method. This method uses finite elements to model the mat, and de-
fines soil-structure interaction using the Winker or pseudo-coupled models. In
principle, it also could use the multiple parameter model.
13. A design could be based entirely on a three-dimensional finite element analysis that
includes the soil, mat, and superstructure. However, such analyses are beyond cur-
rent practices, mostly because they are difficult to set up and require especially
powerful computers.
14. The total settlement is best determined using the methods described in Chapter 7.
Do not use the coefficient of subgrade reaction to determine total settlement.
15. Bearing capacity is not a problem with sands and gravels, but can be important in
silts and clays. It should be checked using the methods described in Chapter 6.
Vocabulary
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.
10.6 An office building is to be supported on 150-ft x 300-ft mat foundation. The sum of the col-
umn loads plus the weight of the mat will be 90.000 k. According to a settlement analysis con-
ducted 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 pseudo-
coupled method. divide the mat into zones and composite each zone. Then indicate the high-
end and low-end valves of k, that should be used in the analysis.
Beam on elastic foundation
Bed of springs
Coefficient of subgrade
reaction
Coupled method
Finite element method
Mat foundation
Multiple parameter method
Nonrigid method
Pile-supported mat
Pseudo-coupled method
Raft foundation
Rigid method
Shaft-supported mat
Soil-structure interaction
Winkler 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 con-
tinues?
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.
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Mat Foundation analysis and design Coduto10.pdf

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