Raft foundations _design_and_analysis_with_a_practical_approach

Raft
Foundations
Design and Analysis with a
Practical Approach

SHARAT CHANDRA CUPTA
Advisor, Indian Buildings Congress,
Former Chief Engineer
Central Public Works Department

PUBLISHING FOR ONE WORLD

NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS

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New Delhi Bangalore Calcutta Chennai Guwahati Hyderabad
Lukhnow Mumbai Pune

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Copyright O 1997 New Age International (P) Limited, Publishers
NEW A G E INTERNATIONAL (P) LIMITED, PUBLISHERS
NEW DELHI
BANGALORE

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CALCUTTA
CHENNAI

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GUWAHATI
HYDERABAD

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LUCKNOW
MUMBAI

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PUNE

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4835124, Ansari Road, Daryaganj, New Delhi-110 002
35, Annapoorna Complex, South End Road,
Basavangudy, Bangalore-560 004
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Chennai-600 020
Pan Bazar, Rani Bari, Guwahati-781 001
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Hyderabad-500 029
18, Madan Mohan Malviya Marg, Lucknow-226 001
1281A. Noorani Building, Block No. 3, First Floor.
L.J. Road, Mahim, Mumbai-400 016.
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Kothrud, Pune-4 1 1029.

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CONTENTS
Preface
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1.

2.

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NEED OF RAFT FOUNDATION

3.

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INTRODUCTION

TYPES OF RAFT FOUNDATION

4.

SURVEY OF AVAILABLE LITERATURE
Foundation Engineering by Peck, Hansen and Thornburn
1
Foundation Design and Practice by Elwyn. E.S. Seelye
4.2
4.3
Foundation Design by Teng
Foundation of Structures by Dunham
4.4
4.5
Indian Standard Code of Practice for Design and Construction of Raft
Foundation - IS 2950-1965
Raft Foundation - The Soil Line Method of Design by A.L.L. Baker
Indian Standard Code of Practice for Design and Construction of Raft Foundation
1.S : 2950 (Part-I) 1973
Foundation Engineering Handbook Edited by' Hans F. Winterkorn & Hsaiyang Fang
Foundation Analysis and Design by Joseph. E. Bowels
Building Code Requirements for Reinforced Concrete (ACI 318 - 77)
Foundation Design and Construction by M.J. Tomlinson
Design of Combined Footings and Mats ACI Committee 336
Pile Foundation Analysis and Design by H.G.Poulos and E.H. Davis 1980
Reinforced Concrete Designers Handbook by Charles E. Reynolds and
James C. Steedman - 9th Edition 1981
IS 2950 (Part I) 1981 -Code for Design and Construction of Raft Foundation Part I
~esi~n
Eleventh International Conference of Soil Mechanics and Foundation Engineering
San Francisco, August 12 - 16,1985
Foundation Design and Construction by M.J. Tomlinson, 5th Edition, 1986

CONTENTS

Handbook of Concrete Engineering -Mark Fintel - 2nd Edition, 1986
Reinforced Concrete Designer Handbook by Charles E. Reynolds and James
Steedman, 10th Edition, 1988
Building Code Requirements in Reinforced Concrete - ACI - 3 18 - 1989
Foundation Engineering Hand book by Hsai-Yang-Fang 2nd Edition, 1991
Design of Combined Footings and Mats - ACI committee 336 2R - 88
Published in ACI Manual 1993
Foundation Analysis and Design by Bowles, 4th Edition, 1988
Proceedings of Indian Geo-Technical Conference 1992, Calcutta, December, 1992
Designs of Foundation Systems - Principles and Practices by Nainan P. Kwian, 1992
13th International Conference on Soil Mechanics and Foundation Engineering,
New Delhi, January, 1994
Soil Structure Inter-action -The Real Behaviour of Structures, published by the
Institution of Structural Engineers, U.K. The Institution of Civil Engineers,
U.K. International Association for Bridge and Structural Engineering in March, 1989

5.

DESIGN APPROACH AND CONSIDERATIONS
5.1
Rigid Approach
5.2
Flexible Approach
5.3
Parameters for Raft Design
Pressure Distribution Under the Raft
5.4
5.5
Rigidity Criteria
5.5.1
Proposed by IS : 2950 (Part I) 1981
5.5.2 ACI Committee, 336
5.5.3 Hetenyi's Criteria
5.6
Modulus of Sub-Grade Reaction
5.6.1
Recommended by Bowles
5.6.2 IS : 2950 Part I Indian Standard Code of Practice for Design and
Construction of Raft Foundation 2950 - 1981
5.6.3 I.S. 9214-1979 - Method of Determination of Modulus of Subgrade
Reaction (k value) of Soils in Field
5.6.4. IS 8009 - Part I - 1978. Code of Practice for Calculation of Settlements of
Foundations - Part I - Shallow Foundations. Subjected to Sy_mmetrical
Static Vertical Load.
5.6.5
Recommendation by Alpan and Prof. Alarn Singh
5.6.6
Summary

6.

STRUCTURALDESIGNERS DILEMMA

7.

STUDIES CARRIED OUT ON EFFECT OF VARIOUS PARAMETERSON DESIGN OF RAFT 38
7.1

Study 1
7.1.1
Examples Selected
7.1.2
Raft Size

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41
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CONTENTS

7.1.3

5

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Soil Investigation

7.1.4
Load Considered in Study
7.1.5
Analysis
7.1.6
Discussions of Results
7.1.7
Conclusions
Study 2 -Effect of Horizontal Loads
7.2.1
Example Selected
7.2.2
Discussion of Results
7.2.3
Conclusion
Study 3: Comparison with Conventional Rigid Methods
Details of Conventional Method: Combined Footing Approach
7.3.1
7.3.2 Examples Selected
7.3.3
Discussion of Results
7.3.4
Inverted Floor Method
7.3.5
Conclusions
Study 4. Another Office Building
7.4.1
Example Details
7.4.2
Comparison of Results
7.4.3
Discussions of Results
7.4.4
Conclusions
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7.2

7.3

7.4

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STUDIES CARRIED OUT ON ANALYSIS AND DESIGN OF PILED RAFTS
8.1
8.2

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Design Procedures being Used
Example Selected

8.3
8.4

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Soil Data
Methods of Analysis Studied
8.4.1 Conventional Rigid Method with Simplified Models
8.4.1.1 Combined footing approach
8.4.1.2
Continuous beam analogy :inverted floor
8.4.1.3 Comparison of results
8.4.2
Piled RafPAnalysis Based on Finite Element Approach
Study of Parameters Influencing the Raft Behaviour
8.5.1
Effect of Raft Stiffness on the Pile Loads and Raft Moments
8.5.2
Effect of Superstructure and Retaining Walls on Foundation Stiffness
8.5.3
Effect of Earthquake Loads and Moments
8.5.4
Effect of End Bearing and Friction Piles
8.5.5
Summary of Results

8.5

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8.6
8.7

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10.

Discussions
Conclusions

JOINTS IN RAFl'S
SUMMARY OF STUDIES

CONTENTS

11.

FACTORS AFFECTING CHOICE OF MET,HODOF ANALYSIS

12. GUIDELINES
APENDM - ILLUSTRATIVE EXAMPLES
A.l Conventional Rigid Method - Combined footing approach
A.2 Flexible Raft - Beam on elastic foundation
A.3 Piled Raft-Plate on elastic foundation

INTRODUCTION
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In 1957, when the author was a student of Civil Engineering at the Indian Institute of Technology, Kharagpur,
the first institute of national importance, one of his professors of Civil Engineering at his first lecture in the
class said:
"Civil Engineering is 50% common sense but common sense is that sense which is quite uncommon. "
After 34 years of experience in Civil Engineering construction and design, the author only wonders how
true the statement of his Professor was and how much more it is true in case of foundation engineering.

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1.1 Foundation engineering has been practised as an art, without help of science, since time immemorial upto
1920 when it had achieved a considerable amount of refinement. It was in the earlier 1920s that a concerted
effort was made to study and undentand the physical laks governing the behaviour of sub surface materials,
i.e.. soil from which foundations derived their support and on whose behaviour its own behaviour depends.
This is the time when study of soil mechanics was started and it was in 1919 when Karl Terzaghi, popularly
known as 'father of soil mechanics', made successful attempt to explain the phenomenon of settlement oti a
scientific basis. Though study of soil mechanics has provided us with new techniques for selecting appropriate
type of foundation and predicting the behaviour of completed structures, it has not been able to decrease the
importance of the accumulated experience of the ages. Amount of uncertainty and degree of variation in the
properties of soil and number of parameters on which performance of a foundation depends, make exact
solution impractical, if not impossible. With so much of advancement in science and computer application,
structural design is still defined as:I5
a creation of a structuralfonn to satisfy a number of requirements. It is a combination of art and science.
As a rule, there is no direct procedure leading to the solution of a specific problem. An engineer uses all
his resources of knowledge experience and imagination to produce a trial scheme. He then constructs a
mathematical model of such a solution to assess its adequacy and ifnecessary, modifies the original concept
in the light of analytical results. The process is repeated until the designer is satisfied with thefinalproduct,
taking into account not only structural adequacy but also such non-quantifiablefactors as aesthetics, ease
of construction and performance. The design process is characterised by a complex interaction of
parameters and the need to arrive at decisions based on incomplete data Intuitive decisions which have to
be taken, appear to be diametrically opposite to the logical nature of ... '

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RAFT FOUNDATIONS-DESIGN AND ANALYSIS

Foundation design and analysis is, at a stage behind structural analysis and design for superstructure, and
even now continues to be practised more as an art and will probably continue to be done so, for many years
to come.

1.2 Available textbooks, handbooks, various publications and papers give widely different approaches to
design of raft foundations. A designer, when faced with a task of designing a raft foundation, finds himself in
a precarious position where he has to balance the time available for design, the cost of design, the need of
adequate safety and, above all, acceptance of the design by the client and the professional community in general
and decide the method of design to be followed by him. Generally, it is not practical for any designer to go
through the various approaches as available in engineering literature at a particular time, compare their merits
and demerits and select the most suitable for his purpose. He, therefore, perforce selects a particular textbook
and applies the same to his problem, quite often little realising that the theoretical problem dealt with in the
textbook is widely different from his practical problem relating to an actual building. Resulting solution may
not be as satisfactory as he feels.
An effort has been made in the following chapters to explain the various approaches suggested in literature,
give their comparative limitations, examine the implications of the so-called more sophisticated approaches
and finally make recommendation for the method which can be followed by a designer till he accumulates
enough experience so as to select his own method particularly applicable to his problem. Intention of this
publication is not to hinder initiative of an individual in going deeper in any problem, but to give him a
comparative idea of available approaches with sufficient number of references which he can study during the
beginning of his profession and formulate his own opinion in due course but still continuing to design
satisfactory raft foundations.
This publication should, therefore, be studied in this background.

NEED OF RAFT FOUNDATION.
Raft or Mat foundation is a combined footing that covers the entire area beneath a structure and supports all
walls and columns. This raft or mat normally rests directly on soil or rock, but can also be supported on piles
as well.
Raft foundation is generally suggested in the following situations:
(a) Whenever building loads are so heavy or the allowable pressure on soil so small that individual
footings would cover more than floor area.
(b) Whenever soil contains compressible lenses or the soil is sufficiently erratic and it is difficult to
define and assess the extent of each of the weak pockets or cavities and, thus, estimate the overall
and differential settlement.
(c) When structures and equipment to be supported are very sensitive to differential settlement.
(d) Where structures naturally lend themselves for the use of raft foundation such as silos, chimneys,
water towers, elc.
(e) Floating foundation cases wherein soil is having very poor bearing capacity and the weight of the
super-structure is proposed to be balanced by the weight of the soil removed.
(f) Buildings where basements are to be provided or pits located below ground water table.
(g) Buildings where individual foundation, if provided, will be subjected to large widely varying
bending moments which may result in differential rotation and differential settlement of individual
footings causing distress in the building.
Let us now examine each of the above situations in greater detail.

2.1 In case of soil having low bearing pressure, use of raft foundation gives three-fold advantage:
(a) Ultimate bearing capacity increases with increasing width of the foundation bringing deeper soil
layers in the effective zone.
(b) Settlement decreases with increased depth.
(c) Raft foundation equalises the differential settlement and bridges over the cavities. Every structure
has a limiting differential settlement which it can undergo without damage. The amount of
differential settlement between various parts of a structure supported on a mat foundation is much
lower than that if the sarne.structure was supported on individual footings and had undergone the
same amount of maximum settlement. With these considerations, maximum total settlement which


can be allowed for a particular structure on mat foundation is more than what is permitted when the
structure is resting on individual footings. This, therefore, allows a higher bearing capacity for such
situations.
It may, however, be noted that if in a case deeper layers of soil are of very poor quality, increase in width
of the foundation may not always lead to higher bearing capacity. In situation where comparatively s h a l l y
top layers of soil are underlain with deeper layers of much poorer soils, it may be advantageous to provide
individual footings so that the zone of influence of the footings remains within the top stronger layer. In such
a situation, provision of a mat foundation may be disadvantageous.

2.2 Some designers work on the rule that if more than 50%of the area of the structure is occupied by individual
footings, it is necessary to provide an overall raft. This is not true and quite often, the quantity of reinforcing
steel and concrete required to avoid excessive deflection and cracking of a raft carrying unequal column loads,
necessitating carry-over of stresses from one part of the raft to the other part, may be large and may make raft
foundation uneconomical. In such situations, it may be more economical to excavate the entire site to a level
formation, construct individual closed space footings (sometimes touching each other) and then backfill around
them. In these cases, however, one must weigh form work costs against the extra footing material required by
using mat foundation. It should be considered that it is possible to construct alternate footings by using spacer
pads against already laid footings and thus save form work cost.
Quite often, doubt exists about the structural behaviour of individual footings touching each other. This
problem of interaction of footings has been studied by many researchers. It has been reported that the effect
of adjacent footings may vary considerably with the angle of shearing resistance. For low values, they are
negligible though for high values they appear to be significant, particularly if a footing is surrounded by other
footings on both sides. It is also stated that these effects are considerably reduced as length over breadth ratio
of the footings approaches unity. There are practically no such effects in the case of punching shear failure.
For these and other reasons, it has been recommended that interference effects need not be considered in
designs. Adesigner should, however, be aware of the possibility of their existence in some special circumstances 11.
2.3 Situations exist in practice w h p a soil stratum contains compressible lenses or the soils have a formation
where individual layers of soil are neither parallel nor can be reasonably stratified into different layers of known
properties to enable calculations of settlement to a reasonable accuracy. In such situations, individual footings,
if provided, would undergo widely varying settlements resulting in large differential settlement which cannot
be tolerated by the structure.
2.4 Situations, as mentioned in (c) and (d) above, are explicit and do not require further explanation. These
are special cases, and adoption of raft foundation is more or less necessary by the particular nature of the
problem involved.
2.5 In cases where soil is very soft and highly compressible and the buildings cannot be founded on such soils
in normal circumstances, it may be possible to provide the building with a basement in such a manner that
weight of the structure is equal to the weight of the soil removed and, thus, there being no change in the stresses
in the soil beneath the basement and, therefore, little settlement. However, in practice it is rarely possible to
balance the loading so that no additional pressure comes on the soil. However, in such cases still, it is only a
part of the total load which comes on the bottom soil and, thus, it is possible to construct a building inducing
a much larger load than the soil would have otherwise supported. The basement provided, gives additional
space in the building for the owner and can be made use of. However while constructing such foundations,

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NEED OF RAFT FOUNDATION

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reconsolidation of the soil, which has swelled as a result of removal of over burden pressure in excavating for
the sub-structure, should always be considered and necessary steps be taken to prevent detrimental effects.

2.6 Basements located below ground water table should use a mat as their base to provide water tight

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c.onstruction.The alternative of having individual columns footings connected by thin slabs has not proved to
be successful in most of the cases; presents difficulties in water proofing; causes concentration of stresses at
the junction of the thin slabs and footings and also at the junction of basement walls and raft causing cracks
to develop. This arrangement, therefore, should not be resorted to unless the economy is of such a magnitude
as to outweigh all other considerations.
Even in cases where sub-soil water level is low and basement does not extend below ground water table,
long-term built up of surface water accumulating against basement walls and bottom should be allowed for.
This is particularly so in case of impermeable soils (permeability co-efficient below 0.1 mm per second) or of
large surface areas draining towards the building. i.e., areas on sloping ground near hillocks. The basement
walls should also normally be designed as self-supporting cantilever retaining walls even though they may
eventually be strutted by floor construction. It is inconvenient and often impossible to provide temporary raking
struts to support a basement retaining wall until such time as strutting given by ground floor or intermediate
basement floor is completed.

2.7 Situations also arise when isolated footings are subjected to very large eccentric loadings, and one is faced
with the possibility of excessive footing rotation, excessive differential settlement or possibility of exceeding
the allowable bearing capacity of the soil at some location. This can happen when the building consists of shear
walls and columns, shear walls sharing most of the horizontal load subjecting its footings to larger settlements
and rotation, decreasing the effectiveness of the shear walls and also creating difficulties by way of large
differential settlements. Raft, if provided, will even out these deformations.
Mats or rafts are supported on piles'in cases where sub-soil conditions warrant provision of piles, but one
has to have the basement. In such situations, raft also helps in making the basement water tight.
It would, therefore, be seen that it is not possible to lay down hard and fast rules defining situations wherein
a raft foundation is required. The author, therefore, opines that every designer should learn all that he can
within reason about the conditions at site, determine the types of foundations that are practical, compare their
cost, suitability, ease of construction, safety and select a type which in his judgement would serve the purpose
well. There can always be differences of opinion about the solution decided by him, but as already mentioned
in chapter I , it cannot be helped because foundation design still continues to be practised more as an art than
an exact science. Two artists seldom agree.

Raft can be classified into various types on the basis of criteria used for classification.

3.1 Based on the method of their support, raft can be:
(a) Raft supported on soil,
(b) Raft supported on piles, and
(c) Buoyancy raft.
3.2 On the basis of structural system adopted for the structure of the raft, these can be classified as:
(a) Plain slab rafts which are flat concrete slabs having a uniform thickness throughout. This can be
with pedestals or without pedestals.
(b) Beam and slab raft which can be designed with down stand beam or upstand beam systems.
(c) Cellular raft or framed raft with foundation slab, walls, columns and one of the floor slabs acting
together to give a very rigid structure.
Raft of uniform depth is most popular due to its simplicity of design and construction. This type is most
suitable where the column loads are moderate and the column spacing fairly small and uniform. Pedestals are
utilised to distribute the load on a bigger area in case of heavy column loads.
3.3 Slab and beam raft is used as a foundation for heavy buildings where stiffness is the principal requirement
to avoid excessive distortion of the super structure as a-resultof variation in the load distribution over the raft
or the compressibility of the supporting soil. These rafts, however, have many obvious difficulties. If the beams
are deep, ribs placed below the basement floor or raft, the bottom of the excavation becomes badly cut up with
trenches, impairing the bearing value of the soil because of its disturbance.Water proofing in case of basements
becomes more complicated arid involved. If the beams are projecting up, usefulness of the basement is
destroyed unless the entire foundation is lowered and the gap filled up or an upper slab is provided supported
on these inverted beams to form the ground floor of the structure.

3.4 Buoyancy raft are necessarily to be provided with a basement so that the weight of the soil removed
balances to a large extent, the imposed load. Cellular raft consisting of foundation slabs, walls, columns and
ground floor slab can be designed, but it creates considerable amount of uncertainties, difficulty of construction
and quite often even in such cases, raft is designed as a slab of uniform rhickncss.

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Raft, as a slab of uniform thickness, has an additional advantage of providing better water-proofing treatment
ease of reinforcement fabrication and laying of concrete. This type of raft is most commonly used.
Various types of rafts are shown in Fig. 3.1

--------------------- R A------- -------------- F T S U P P O R T E D ON SOIL

RAFT SUPPORTED ON PILE

BUOYANCY RAFT
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FLAT PLATE RAFT

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FLAT PLATE WITH PEDESTALS

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BEAM AND SLAB RAFT
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Fig. 3.1 Various types of rafts

FRAMED RAFT

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SURVEY OF AVAILABLE
LITERATURE
Testbooks and design manuals by various authors suggest varying approaches to analysis and design of raft
foundation. Differences of opinion exist in the method of analysis proposed to be adopted while determining
moments, shear forces for the design of raft. Once the bending moments and shear forces are known, structural
design does not present any difficulty and there exists no difference of opinion in this respect except very minor
difference relating to desired thickness of slab and the effectiveness of the shear reinforcement
Methods suggested by different authors are summarised below. These have been arranged chronologically
with reference to date of publication of the testbooktdesign handbook.
4.1

Foundation Engineering b y Peck, Hansen and

hornb burn^

Raft is usually regarded and designed as an inverted continuqus flat slab floor supported without any upward
deflections at the columns and walls. The soil pressure acting against the slab is commonly assumed to be
uniformly distributed and equal to the total of all column loads multiplied by appropriate load factors and
divided by the area of the raft. The moment and shears in the slabs are determined by the use of appropriate
coefficient listed in the specifications for the design of flat slab floors. On account of erratic variation in
compressibility in almost every soil deposit, there are likely to be correspondingly erratic deviations of the soil
pressure from the average value. Since the moment and the shears are determined on the basis of the average
pressure, it is considered good practice to provide this slab with more than theoretical amount of reinforcement
and to use the same percentage of steel at top and bottom. This method has been widely used, often with
complete success. On the other hand, it has also sometimes led to structural failure not only of the slab but also
of the super structure. Therefore, its limitations must be clearly understood. The analogy follows only if the
differential settlement between columns will be small and if the pattern of the differential settlement will be
erratic rather systematic. The method is valid when the columns are more or less equally loaded and equally
spaced. If the downward loads on some areas are on the average much heavier than on others, differential
settlementsmay lead to substantial re-distribution of moments in the slabs resulting in unconservative design.
Rafts are sometimes designed as if they rested on a bed of closely and equally spaced elastic springs of
equal stiffness. The contact pressure beneath any small area is then proportional to the deflection of the spring


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in that area and thus to the settlement. The constant of proportionality 'K' is called the modulus of sub-grade
reaction. Although, the theory has been well developed but the value of 'K' for real soils is not constant and
depends not only on the stress deformation characteristics of the soil but also in a complex manner on the shape
for
and size of the loaded area and the magnitude and position of nearby loaded areas. Evaluation of 'K' design
is difficult and fraught with uncertainty. Whatever method may be adopted for design, there is no guarantee
that the deflections of the raft will actually be unimportant. In case, the structure covers a fairly large area with
possibilities of differential settlements, it is not enough to provide great strength in the slab. It is also necessary
to provide sufficient stiffness. However, a stiff foundation is likely to be subjected to bending moments far in
excess of those corresponding to the flat slabsubgrade modulus analysis.
There appears to.be no further edition of this book after 1954.
4.2

Foundation Design and Practice b y Elwyn. E.S. seelye9

According to Seelye after determining the soil pressures at various points of raft, shear and moment diagrams
can be constructed for bands assumed from centre of bay to centre of bay. However, 65% of the moment is
assumed to be resjsted by half the width of the band. There has not been any further edition of this book after
1956.
4.3

Foundation Design b y

en^'

In the conventional method, it is assumed that the mat is infinitely rigid and that the bearing pressure against
bottom of the mat follows the planner distribution. The mat is analysed as a whole in each of two perpendicular
directions. Thus the total shear forces acting on any section cutting across the entire mat is equal to the
arithmetic sum of all forces and reactions (bearing pressure) to the left orright of the section. The total bending
moments acting on such section is equal to the sum of all moments to the left or right of this section.
Although the total shear and moments can be determined by the principles of simple statics, the distribution
along this section is a problem of highly indeterminate nature, the average moment not being indicative of the
sign and the magnitude of the bending moments in the individual strip in either direction. In order to obtain
some idea as to the upper limit of these values, each strip bounded on central line of the column bays, may be
analysed as independent continuous or combined footings. If the column loads are used, the soil reaction under
each strip is determined without reference to the planner distribution determination for the mat as a whole.
This method, undoubtedly, gives very high stress because it ignores the two way action of the mat. Therefore,
certain arbitrary reduction in values (15% to 33113%) is made.
The author gives other method like Finite Difference Method also for the design of the raft. There has not
been any further edition of this book after 1962. The book, however, has been reprinted in 1992.

The recommendation in this book can be summarised in the following words:
A great refinement of calculations is not always justified or practicable in case of raft.foundations because
of the uncertainties of the action of soil and of short thick members that are arranged in complicated and
multiple systems. It is reasonable to assume that the mat is so stiff and the load so constant that plastic soil will
compress and adjust itself so that each column load will spread almost uniformly under the mat in the general
vicinity of that particular column. For example, the total unit pressure under the rectangular area D, E, F, G
shown in Fig. 4.1 may be assumed equal to 114th of the total loads on the columns at D, E, F and G divided
by the area of D, E, F, G plus the weight of the mat per sq m. For the purpose of computing average pressure

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under the slabs, near the walls, the outer column loads are treated as though they were concentrated at the
columns. For this method, however, the load on adjacent columns should not differ very much and the bays
in either direction should be reasonably, equal in length, the larger spacing not exceeding 1.2 time, the smaller
one and the columns should be arringed in reasonably straight rows.

Fig. 4.1 Plan of assumed columns strips and distribution of loads
One method of making a preliminary analysis of such a mat is on the basis of an assumed supporting system
of columns strips that constitute a grid of beam along the column rows in each direction. The portion of the
slabs in the central areas is taken up to be supported by this grid. The effective width of these strips or shallow
beams has to be assumed and it is normal to take it slightly more than, what is determined by 45 degrees fiom
the pedestal or column, to the lower reinforcement in themat. Technically the top reinforcement of a central
panel may be less than of the bottom. However, it may be advisable to reinforce both sides equally because
any yielding of end restraint will increase the, tension in the top of the mat above the computed value. Each
column strip may be analysed by moment dishbution if the variation of loading or spans make this desirable,
the entire thing being designed as an inverted floor. The effect of hydrostatic pressure has to be considered
wherever it is present. There has been no further edition of this book after 1962.
4.5

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Indian Standard Code of Practicefor Design and Construction of Raft Foundation IS 2950-1965'

There are two approaches for design-conventional method and the elastic method. In the conventional method,
the foundation is considered infinitely rigid and pressure distribution independent of the deflection of the raft.
Soil pressures are also assumed to be planner so that the centroid of the soil pressure coincides with the line
of action of the resulting forces of all the loads acting on the foundation. The method is normally used in design
because of its simplicity . A generous amount of reinforcement is provided to safeguard uncertainties caused

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i

I!


by differential settlement. The raft is anabjsed as a whole in each of the two perpendicular directions. Thus,
total shear forces and total bending moments acting on any section cutting across the entire raft is equal to the
arithmetic sum of all forces and reactions/moments to the left or right of the section. The actual reinforcement
provided shall be twice that worked out theoretically.
Elastic method has two approaches. In one, the soil is replaced by an infinite number of isolated springs.
In the other, the soil is assumed as a continuous elastic medium obeying Hook's Law. These methods are
applicable in case the foundation is comparatively flexible and the loads tend to concentrate over small areas.
The actual reinforcement can be one-and-a-half times that required theoretically.The famous soil line method
falls in this category.
As limitations to applicability of the methods, code mentions that the coda1 provisions:
(1) do not apply to large and heavy industrial construction where special considerations of the base
pressure distribution will be required.
(2) apply only to fairly uniform soil conditions and for fairly horizontal planes of separation of layer
below.
(3) foundations in seismic area and/or to vibrating load shall be given special considerations.
This code has been revised in 1973. Kindly see para 4.7.
4.6

'

'

11

-

RafL Foundation The Soil Line Method of Design by A.L.L. ~ a k e q

According to Mr. Baker, the design of raft as a reversed floor is dangerous. Engineers being aware of this, who.
therefore, normally adopt the second method in which earth pressure is assumed to be uniform throughout and
moments are obtained at any section by statics. He, however, feels that in the second method also high values
of moments are obtained, which may or may not be present, and it is irrational or wasteful to provide for such
moments without investigating the deflections and variation in soil pressure. Mr. Baker has, therefore,
suggested the soil line method which takes into account the variations in soil pressure and its relation to
deflection but in order to simplify the calculations, it is assumed that the earth pressure varies throughout a
beam according to straight line law.
There is no further edition of this book after 1969.

4.7

Indian Standard Code of Practice for Design and Construction of Raft Foundation
1.S :2950 (Part-I) 1973~

In the revised version of the code, following methods of analysis have been proposed:
(a) Assumption of linearly varying contact pressure
(b) Perfectly rigid structures
(c) Perfectly flexible structures
(d) Structures stiffened along one axis
(e) Structures stiffened along both the axis
(f) General methods:
(i) Based on modulus of subgrade reaction, and
(ii) Based on modulus of compressibility (half space theory).
Method (a) corresponds to the conventional method in the earlier version of the code and has similar
limitations. In method (b), contact pressure distribution is to be calculated based on Boussineq's Equation for
Elastic Isotropic half space and is applicable when deformations of raft under loads are small as compared to
the mean settlement of the structure.

12


Method (c) is applicable for structures which have relatively less stiffening members specially resting on
very stiff foundation soil. In this case, the deflections of the raft are same as the settlements of the foundation
soil under external load.
Method (d) is something in between methods (b) and (c) . Here in the direction of the stiffened axis the
contact pressure distribution is determined by Boussineq's Equation as in method (b). In perpendicular
direction distribution is determined as given in (f).
Method (e) is same as method (b). The two methods under (f) are elastic methods and are used when
simplified methods from (a) to (e) are not applicable. Details given in the codedo not provide enough guidance
to enable the analysis and design 10 be completed by the designer. Apart from the limitations applicable in
earlier version of the code it is stated that:
(i) Allowable settlement both total and differential shall satisfy the requirement of the super-structure
(ii) The approximate values of permissible settlements as given in earlier code have been deleted.
This code has further been revised. Please see para 4.15.
4.8

Foundation Engineering Handbook Edited by Hans F. Witerkorn & Hsaiyang

an^''

Dr. Joseph E. Bowles and Wayne C. Teng are authors of chapters on spread footings, combined and special
footings and mat foundation respectively. Chapter on floating foundation has been written by Dr. H.Q. Golder.
This book classifies the method of design of mat foundation according to assumptions used. The rigid
method which is the conventional method assumes that:
(a) Mat is extremely rigid as compared to the sub-soil and, therefore, the flexural deflection of the mat,
does not alter the contact pressure.
(b) The contact pressure or the pile reaction are distributed in a straight line or a plain surface such that
the centroid of the contact pressure coincides with the line of action of the resultant force of all the
loads acting on the mat. When mat foundation is supported on piles, piles are assumed to be perfectly
elastic. Raft is considered to be rigid when the column spacing is less than 1.751h or when the mat
is supporting a rigid super-structure. his same as defined by Heteny. The mat is analysed as a whole
in each of the two perpehdicular directions. The mat is divided into perpendicular bands of width
between centre lines of adjacent column rows. Each band is assumed to act as an independent beam
subjected to common contact pressure and known column loads.
The simplified elastic method assumes that the soil behaves like an infinite number of individual elastic
springs each of which is not affected by others. This foundation model is also referred to as Winkler foundation.
Analysis procedures have also been developed for the beams on the simplified elastic foundation concept.
The mat is considered as a plate and the effect of each column load is considered in area surrounding the load.
Using the method of super-imposition,effect of all the column loads within the zone of influence is calculated.
Among computer-oriented methods suggested is finite difference method, based on the assumption that the
sub-grade can be substituted by a bed of uniformly distributed elastic springs with a spring constant equal to
coefficient of sub-grade reaction. For this purpose, the mat is divided into square areas. The deflection at the
nodal points of these areas is expressed by a differential equation in terms of deflection at the adjacent points
to the right, left, top and bottom. These simultaneous equations are solved with an electronic computer and
deflection at all the points are determined. Once deflections are known, the bending moment at any point in
each direction is determined from theory of elasticity.
The finite element method transforms the problem of plates on elastic foundation into a computer-oriented
procedure of matrix structural analysis. The mat is idealised as a mesh of finite elements inter-connected only

13


at the comers and the soil may be modeled as a set of isolated springs or as an elastic isotropic half space. The
matrix structural analysis can be extended to include the influence of the super-structure as well, thus the
interaction between the super-structure, the foundation and the soil is accounted for.
It is further suggested that in a mat supported on hard rock, the column loads are transmitted to the rock on
relatively small areas directly under the column. A greater economy may be achieved by designing the mat by
elastic methods. On very soft soils, the contact pressure against the mat foundation approaches planer
distribution and, therefore, it is commonly justified to design a mat on mud, soft clay, peat, organic soils or
even medium clays by the conventional rigid method. A generous amount of reinforcement running in both
directions at top and bottom is suggested regardless of method of design used in view of the likelihood that
the stresses actually introduced would bedifferent from those calculated irrespective of the method used foi
analysis.
Second edition of this book is published in 1991. Please see para 4.21.

4.9

"

Foundation Analysis and Design by Joseph. E. ~ o w e l s '

The mat may be designed as rigid structures thereby soil pressure are computed as Q = V/A in the case where
the resultant of the forces coincide with the centre of the mat area. If resultant has eccentricity with respect to
geometric centre, soil pressure is calculated by the relation

In case, however, if the eccentricity is very large, the resulting internal stresses may be seriously in error.
Once the dimensions of the mat are established, soil pressures at various locations beneath the base may be
computed. With the pressure distribution known, the mat is sub-divided into a series of continuous beams
(strips) centred on the appropriate column lines as shown in Fig. 4.2. For the series of beams, shear and moment
diagram may be established using either combined footinglanalysis or beam moment coefficient. The depth is
selected to satisfy shear stresses and is usually constant but the steel reinforcement vary from strip to strip.
The perpendicular direction is analysed similarly, to complete the design.

Fig. 4.2

14


When the soil bearing pressure is low say 0.5 ~ i ~ s l f(25 K N I ~or less and if the deformation of the mat
t2
~)
surface can be tolerated, the mat may be designed as an inverted flat slab, using heavy beams from column to
column. The portion between beams is designed as a conventional one or two way slabs.
When footings are designed as flexible members, the computation takes some form of the solution of a
beam on an elastic foundation. The experience has indicated that the solution obtained are generally reliable
when the data are satisfactory. Possibly the reasons, as to why the methods have not been widely used in the
past, are ease of making conventional solution, which have been generally satisfactory and usually not much
different from elastic solution. Second reason is that the soil data are generally obtained using the standard
penetration test for which no straight forward conversion to a value of modulus of sub-grade reaction exists.
Various methods for elastic analysis like finite element and finite differences have also been explained in this
book.
New edition of this book is publisheg in 1988. Kindly see para 4.23
4~10 Building Code Requirements for Reinforced Concrete (ACI 318 - 7 ) 8
71

Matters relating to design of footings are included in this code in Chapter 15. paragraph 15.10 relates to
combined footings and mats. This paragraph reads as under:
15.10.1 Footings supporting more than one column, pedestal, or wall (combinedfootings or mats) shall be
proportioned to resist the factored loads and induced reactions, iir accordance with appropriated design
requirements of this code.
15.10.2 The Direct Design Method of Chapter 13 shall not be usedfor design of combinedfootings and mats.
15.10.3 Distribution of soil pressure under combined footings and mats shall be consistent with propemees
of the soil and the structure and with establishedprinciples ofsoil mechanics.
It would be seen that this code does not provide for much guidance in design of raft foundation.
This code has been revised several times. Final being in 1989. Please see Para 4.20.
4.11

Foundation Design and Construction by M .J . ~ o m l i n s o n ' ~

Mr. Tomlinson states that it is wrong in principal to assume that araft acts as an inverted floor slab on unyielding
supports and to design the slab on the assumption that its whole area is loaded to the maximum safe bearing
pressure on the soil as this canlead to wasteful and sometimes dangsrous designs. Allowance must be made
for deflection under the most favourable combination of dead and live load and variation in soil compressibility.
Guidance is required from the soil mechanics engineer on the estimated total and differential settlement for
dead and live load considered separately. Some flexibility is desirable to keep bending moments and shear
stresses to a minimum, but the degree of flexibility must be related to the allowable distortion of the
super-structure.Basement rafts carrying heavy building on weak soils are often founded on piles. The normal
function of the piles is to transfer the loading to stronger and less compressible soil at greater depth or if
economically possible, to transfer the load to bed rock or other relatively incompressible strata. The piles also
have the effect of stiffening the raft and reducing or eliminating re-consolidation of ground heave, thereby
reducing differential settlement or tilting. In such cases, considerable heave takes place with further upward
movement caused by displacement due to pile driving. After completion of piling, the swelled soil should be
trimmed off to the finished level. The basement walls should generally be designed as self-supportingcantilever
retaining walls even though they may eventually be supported by the floor construction and additional stability
against overturning given by super-structure loading on top of the wall. The basement floor slabs must be able


15

to withstand pressure on the underside of the slab together with stresses caused by differential settlement,
non-uniform column loads, reaction from the retaining walls. If the columns are provided with independent
bases with only a light slab between them, there would be likelihood of failure of the slab from the pressure
of the underlying soil.
Fifth edition of this book has been out in 1986. Please see para 4.17.

t
g

4.12

i

a

1

The committee observes that no authentic method has been devised that can evaluate all the factors involved
in the problem and allow carrying out determination of contact pressures under combined footings and mats.
Simplifying assumption must, therefore, be made based on the knowledge of the interaction of the various
elements of the system. The following factors should be considered while examining any problem:
(1) Soil type immediately below the footing
(2) Soil type at the greater depth
(3) Size of footing
(4) Shape of footing
(5) Eccentricity of loading
(6) Rigidity of footing
(7) Rigidity of the super-structure
(8) Modulus of sub-grade reaction
The committee suggests procedure to be followed for design of footings under two columns: grid
foundations and smp footings supporting more than two columns and mat foundation. Linear soil pressure
distribution is suggested for footings which can be considered rigid to the extent that only very small relative
deformations result from the loading. The rigidity may result from the spacing of the columns on the footing
from the rigidity of the footing itself or the rigidity of the super-structure.Limitations which must be fulfilled
to make this assumption valid have been discussed in the report.
Distribution of soil pressure by means of sub-grade reaction has been suggested where sub-soils are of such
character that the deformations are localised in the general vicinity of the loads and when the maximum contact
pressure is smaller than about one and a half times the ultimate bearing capacity. In case of rigid footings, it
is suggested that uniform or linear distribution of soil pressure can be assumed and the design based on statics.
Flexible footing procedure is divided into 2 parts i.e. uniform condition and general condition. Uniform
conditions are considered to be those where the variation in adjacent column loads and spans is not greater
than 20%. For cases where supporting columns are at random location with varying intensities of loads a
detailed design procedure based on plate theories has been recommended.
4.13

:

Design of Combined Footings and Mats ACI Committee 33614

Pile Foundation Analysis and Design by H.G.Poulos and E.H.Davis 1 9 8 0 ~ ~

In this book, Chapter 10 deals with piled raft systems. The author says that, "in design of foundation for a large
building on a deep deposit of clay it may be found that a raft foundation would have an adequate factor of
safety against ultimate bearing capacity failure but the settlement would be excessive; traditional practice
would then be, to pile the foundation and to choose the number of piles to give an adequate factor of safety
assuming the piles take all the load; however it is clearly illogical to design the piles on an ultimate load basis
when they have only been introduced in order to reduce the settlement on other-wise satisfactory raft."
According to the author, once the
have been introduced solely for the purpose of reducing the settlement

16


design question becomes not "how many piles are required to carry the weight of the structure" but "how many
piles are required to reduce the settlement to an acceptance level".
However, in Chapter 5, the settlement behaviour of a free standing pile is obtained from the elastic-based
analysis. The pile is divided into number of elements and the expressions for vertical settlement of the pile and
the soil at each element in terms of unknown stresses on the piles are obtained and solved, imposing the vertical
displacement compatibility condition, to arrive at the settlement behaviour of the pile. As a further extension,
the unit consisting of a single pile with an attached cap resting on the soil surface is considered. It is assumed
that purely elastic condition prevails upto the load at which the pile would fail if no cap were present and
thereafter any additional load is taken entirely by the cap. The book gives charts indicating interaction factor
between the raft and the pile for various values of length of the piles, diameter of the pile, poisson ratio of soil,
height of soil layer over the rigid stratum and the cap diameter.The method is further extended to group of
piles upto about 40 numbers. Curves are drawn which are applicable only for rigid rafts or perfectly flexible
rafts. The entire emphasis is to work out the ratio of the load carried out by the piles and the raft soil system.
No details are given on &e method to determine the bending moment and shear forces in the raft. It is only
mentioned that none of the simple methods are satisfactory and a proper analysis of plate on piles and
continuum is desirable.

4.14

Reinforced Concrete Designers Handbook by Charles E. Reynolds and JamesC. Steedman
9th Edition 1981"

-

This book suggests the analysis of a raft foundation supporting a series of symmetrically arranged equal loads
on the assumption of uniformly distributed pressure on the ground considering the structure as an inverted
reinforced concrete floor acted upon by the load of earth pressure from bottom. It is further suggested that
when the columns on the raft are not equally loaded or are not symmetrically arranged, the raft should be so
designed that the centroid coincides with the centre of gravity of the loads. If this coincidence of centre of
gravity is impracticable owing to the extent of the raft being limited on one or more sides, the plan of the raft
should be made so that the eccentricity of the total loading is a minimum, though this may produce a raft which
is not rectangular in plan.

4.15

-

IS 2950 (Part I) 1981 Code for Design and Construction of Raft Foundation Part I ~ e s i ~ n ~

In the second revision of the code, two methods of analysis have been suggested depending upon the
assumption involved. Conventional method assuming planner distribution of contact pressure is applicable to
foundations which are rigid relative to supporting soil and the compressible soil layer is relatively shallow.
The rigidity of the foundation is determined with a relative stiffness factor K > 0.5 or columns spacing less
than 1.75A. Methods of determining value of K and hare given in the code. Conventionalmethod is applicable
when either of the two conditions are satisfied. The value of K depends upon the flexural rigidity of the
super-structure, modulus of the compressibilityof the foundationsoil, thickness of the raft, length of the section
in the bending axis and length perpendicular to the section under investigation. Value of h depends upon
modulus of sub-grade reaction for the footing of the width of the raft, modulus of elasticity of concrete and
moment of inertia of the raft. In this method, the rf is analysed as a whole in each of the two perpendicular
at
directions on the basis of statics.
In case of flexible footings, simplified methods are applicable when variation in adjacent column load is
not more than 20% of the higher value and the structure (combined action of the super-structure and raft) may
be considered as flexible, ie., relative stiffness factor K is greater than 0.5. In this method, it is assumed that

17

SURVEY OF AVAILABLE I-ITERATURE

&'
i
:

the sub-grade conslsts of an infinite array of individual elastic springs each of which is not affected by others.
This method is more or less same as the famous soil line method.
When conditions, as mentioned above, for flexible foundations are not satisfied ,a method based on closed
form of solution of elastic plate theory has been suggested. The distribution of deflection and contact pressure
on the raft due to a column load is determined by the plate theory. Since the effect of a column load on the
elastic foundation is damped out rapidly. It is possible to determine the total effect at a point of all column
loads with~nthe zone of influence by the method of super-imposition. The computation of the effect at any
point is restricted to columns of two adjoining bays in all directions.
The code also lays down that:
(a) Size and shape of the foundation adopted affects the magnitude of subgrade modulus which should
be taken into consideration.
(b) Consideration must be given to the increased contact pressure developed along the edges of the raft
on cohesive soils and the opposite effect on granular soils.
(c) Expansion joint should be provided when the structure supported by the raft consists of several parts
with varying heights and loads or there is a change in the direction of the raft.
(d) This code does not explicitly provide any guidance as to how factors emphasised in (a) and (b) above
should be allowed for. The second part of the code relating to construction aspect is still not printed.
There has not been any further revision and this code was reaffirmed in 1987.

4.16

Eleventh Intenationul Conference of Soil Mechanics a d Foundation Engineering San Francisco,
August 12 1 6 , 1 9 8 5 ~ ~

-

In the conference while two papers were presented on instrumentation of pile raft foundation and cap pile soil
interaction, there was no recommendation or paper on design of raft foundation.

4.17

Foundation Design and Construction by M.J. Tomiinson, 5th Edition, 1986"

There is no significant change in this edition from what was recommended in 4th edition

4.18

-

-

Handbook of Concrete Engineering Mark Fintei 2nd Edition, 1986%

This book makes no recommendation about raft foundation.

4.19

Reinforced Concrete Designer Handbook by Charles E. Reynolds and James Steedman,
10th Edition, 1988~'

There is no change in recommendations from what was done in the earlier edition published in 1981

4.20

-

- -

Building Code Requirements in Reinforced Concrete ACI 318 1989~'

Building code requirements since their second edition in 1977 have gone in for further revision 1983, 1989
and 1992. In the latest revision there is no change in the code requirements for design of combir.ed footings
and mats, but in commentary a reference has been made to 'design procedure for combined footings and mat
i per report prepared by ACI committee 336'and also to a paper 'simplified design of footings by' Kramrisch,
s
Fritz and Rpgers Paul published in American Society of Civil Engineers Proceeding, V. 87, NOSM 5, October
1961, p. 19.

18

4.21

RAFT FOUNDKTIONS-DESIGN AND ANALYSIS

Foundation Engineering Handbook by Hsai-Yang-Fang2nd Edition, 1 9 9 1 ~ ~

This edition has omitted the chapter on mat foundation which was originally'included in first edition.

4.22

-

-

Design of Combined Footings and Mats ACI committee 336 2R 88 Published in ACI
Manual 1
~
3
~
~

1966 report mentioned in para 4.12 above was reaffirmed in 1980 but has been completely revised and
elaborated in 1988. This report suggests that:
(a) Maximum unfactored design contact pressure should not exceed the available soil pressure determined by geotechnical engineer. Where wind or earthquake forces form a part of the load
combination, the allowable soil pressure may be increased as allowed by the local code and in
consultation with geo-technical engineer.
(b) Combined footings and mats are sensitive to time dependent sub surface response. Many structural
engineers analyse and design mat foundations by computer using the finite element method. Soil
response can be estimated by modelling with coupled or uncoupled "Soil springs". The spring
properties are usually calculated using a modulus of subgrade reaction, adjusted for footing size,
tributary area to the node, effective depth, and change of modulus with depth. The use of uncoupled
springs in the model is a simplified approximation. The time dependent characteristics of the soil
response, consolidation settlement or partial consolidation settlement, often can significantly
influence the subgrade reaction values. Thus, the use of a single constant modulus of subgrade
reaction can lead to misleading results.
(c) Caution should be exercised when using finite element analysis for soils. Without good empirical
results, soil springs derived form values of subgrade reaction may only be a rough approximation
of the actual response of soils. Some designers perform several finite element analyses with soil
springs calculated from a range of subgrade moduli to obtain an adequate design.
(d) The response of a footing is a complex interaction of the footing itself, the superstructure above,
and the soil. That interaction may continue for a long time until final equilibrium is established
between the superimpos&lloads and the supporting soil reactions. Moments, shears, and deflections
can only be computed if these soil reactions can be determined.
(e) No analytical method has been devised that can evaluate all of the various factors involved in the
problem of soil-structure interaction and allow the accurate determination of the contact pressures
and associated subgrade response.
(f) For mat foundationsmodulus of subgrade reaction cannot be reliably estimated on the basis of field
plate load tests because the scale effects are too severe.
(g) Mats may be designed and analysed as either rigid bodies or as flexible plates supported by elastic
foundation. A combination analysis is common in current practice. An exact theoretical design of
mat as plate on an elastic foundation can be made. However a number of factors like, difficulty in
projecting subgrade responses, variation in soil properties both horizontal and vertical, mat shape,
variety of superstructure loads and assumption in their development and effect of superstructure
stiffness on mat rapidly reduce exactness to a combination of approximations.The design is further
affected by excavation heave.
(h) After propottioning the mat size, compute the minimum mat thickness based on punching shear at
critical columns based on column load and shear perimeter. It is common practice not to use shear
reinforcement so that mat depth is maximum.

I

I

1
I

*

;

i
I

19


(i) In case column spacing is less than 1.75 divided by h or the mat is very thick and variation of column
loads and spacing is not over 2096, mat may be designed by treating it as a rigid body and considering
strips both ways. These strips are analysed as combined footings with multiple column loads and
loaded with the soil pressure on the strip and column reactions equal to loads obtained from the
superstructure analysis. Since a mat transfers load honzontally, any given strip may not satisfy
vertical load summation.
Q) In case the criteria is not met with an approximate analysis can be made using the method suggested
by ACI Committee 336 in 1966.
(k) Computer aided finite differences,finite grid or finite element methods can be used where computers
are available. The report gives details of these 3 methods. In any of these 3 methods node pressure
should not exceed the safe bearing pressure value recommended by the geotechnical engineer.
(1) A mat analysis is only as good as the soil parameters. Since it is very difficult for the geotechnlcal
engineer to provide accurate vdues of moGulus of subgrade reaction, the structural designer may
do the parametric study, varying the value of K over range of one half the furnished value to 5 or
10 times the furnished value.
(m) The analysis and design of combined footings and mats is a soil-structure interaction effort in which
there is no unique method to determine mat deflection. The determination of mat deflection extends
far beyond the analysis of a beam or finite element model to the prediction of subgrade response.
The prediction of subgrade response, though part of the structural analysis of the mat, is more elusive
than designers wish to admit. Experience with extensive measurements of both foundation loadings
and subgrade response are needed to develop a high degree of confidence in the method selected.
A very close working relationship must exist between the geotechnical and structural engineers to
properly analyse comb~ned
footings and mats.

4.23

Foundation Analysis and Design by Bowles, 4th Edition, 1 9 8 8 ~ ~

In this edition analysis of mat foundation has further been elaborated considerably. Among the design methods
included are conventional or rigid methods as explained in earlier edition stating that this method is not
recommended at present because of substantial amount of approximations and the wide availability of
computer programmes which are relatively easy to use and mat being generally too expensive and important
not to use most refined analytical method available.
The approximate flexible procedure suggested by ACI Committee 436 (1966) has been retained and
elaborated. Further details have been given for finite difference method, finite element method and finite grid
method applicable with computer.
4.24

Proceedings of Indian Geo-Technical Conference 1992, Calcutta, December, 1 9 9 2 ~ ~

This conference does not have papers relating to design and analysis of raft foundation.

4.25

-

Designs of Foundation Systems Principles and Prrictices by Nainan P. Kurian, 1 9 9 2 ~ ~

The book details conventional approach to raft design as a flat slab and beam and slab raft, following the Indian
Standard Code of Practice, more on the inverted floor approach. The book only mentions that an integrated
analysis of the beam and slab on the computer by the finite element method using package programmes such
as SAP IV which will give exact results based on the actual behaviour of the system can be carried out. This
book also mentions about the design of raft foundation by the Soil line method stating that this method has

20

RAW FOilNDATlONSDESlGNAND ANALYSIS

1I

rather become obsolete in the wake of possibility of using more refined flexible methods with the aid of
computer.
4.26

13th International Conference on Soil Mechanics and Foundation Engineering, New Delhi
January, 1 9 9 4 ~ ~

A paper by M.F. Randolph was presented as a special lecture on design methods for Pile Groups and Piled

Rafts.
The paper recalls that in majority of the cases where piles form part of the foundation for a building or other
structures, the primary reason for inclusion of the piles is to reduce settlements. However, once the decision
has been made that piles are required the traditional design approach has been to ensure that the total structural
load can be carried out by the piles, with adequate factor of safety against bearing failure. However, there is
elastic interaction'between the raft and soil below, between piles and piles as the performance of a pile within
a group is affected by the presence of other piles. The key question that arises in the design of pile rafts concerns
the relative proportion of load carried out by raft and the piles and the effect of additional pile support on
absolute and differentialsettlements.,Thepaper suggests that this distribution of load between the raft and piles
be taken into account. The paper also gives methods by which this proportion of load between the two
components are carried out.
4.27

-

Soil Structure Inter-action The Real Behaviour of Structures, published by the Institution of
Structure Engineers, U.K. The Institution of Civil Engineers, U.K. International Association
for Bridge and Structural Engineering in March, 1 9 ~ 9 ~ ~

The above institutions constituted a joint committee under Dr. Sam Thornborn which prepared this report.
Pointing out that,
(i) Red behaviour of structures in contact with ground involves an inter-active process beginning with
the construction phase and ending with a state of balance after a period of adjustment of stresses
and strains within the structure and within the ground influenced by the structure.
(ii) Actual behaviour of the structure relates to the inherent spatial variations in the ground and it should
be appreciated that these variations are not always readily identifiable by occasional and local
boring, sampling and testing.
The report deals with the question of soil structure interaction in 2 parts. Pari I relates to structures supported
by ground and Part I1 for ground supported by structures.
(a) Under structures supported by ground, the report points out that engineers could estimate the
settlements for a perfectly flexible load or they could estimate the avenge settlement of a rigid load
but in between these limits, the engineers could say nothing.
(b) Analytical methods have been developing so rapidly over the last few years that it is now possible
to obtain solution to many complex problems which a few years ago would have been quite out of
reach. If used sensibly and with discernment, these powerful analytical methods can be of considerable assistance enabling a designer to gain a feel for the behaviour of soil structure system.
However, if used blindly, such methods cause menace and can be extremely misleading. The key
to successful use is to gain a clear understanding of the idealisations that are being made and to be
aware of, how far they may be, from reality.
(c) For a framed building founded on a raft, during excavation some heave of the soil will occur. The
raft will then be constructed and will be influenced by the differential settlement there after. As the

II
i

i

,

I

1

I

i

t

(d)

(e)

(f)

(g)

structural load is applied short term settlements take place, the part of the structure in existence
distorts and the overall stiffness gradually increases. The cladding is then added and may substantially increase the stiffness of the building. Finally, the imposed load is applied. Not all the
components of the buildings are subject to the same relative deflection. The relative deflections
experienced by the raft will be the largest. Those experienced by the structural members will vary
with location and elevation in the building. The likelihood of damage will diminish, the larger the
proportion of medium and long-term settlements,the smaller the ratio of imposedldead loads and
later the stage at which the finishes are applied.
The report has an appendix which has reviewed currently available techniques for the analysis of
the total soil structure system. More readily available computer packages that utilise these techniques, have been listed in the appendix.
The manner in which and the limitations with which super-structure can be modelled have been
singled out. For soil model, it is pointed out that commonly known approach of treating the soil as
a set of liner unconnected springs cannot be recommended for the analysis of rafts and continuous
footings although this model has the advantage of being easily included in standard computer
programmes for structural analysis. It is a poor physical model. The results of analysis based on use
of this model may be excessively sensitive to the pattern of applied load.
The half space continuum using elastic theory for both stresses and strains has severe limitations
because it does not take into account, the soil layering or the variation of soil modulus with depth
within a given layer. In an extension of this method where elastic theory is used for strains only and
then stresses are calculated using the various deformation moduli of the soil is better approximation.
In a further improvement of a layered coniinuum the exact stresses and strains in a layered soil mass
are calculated.
Super structure stiffness has a marked influence on the behaviour of the raft and should not be
ignored although the quantitative assessment of all but the simplest of the wall system connected
to the raft may prove difficult. However, often the raft is itself a major contributor to the overall
stiffness of the building. Since the raft is in intimate contact with the supporting soil, the inter-active
effects are perhaps most marked in consideration of its own behaviour. In the design of raft
foundation, it is totally unrealistic to ignore deformation and rely on moment and shears obtained
from the analysis of the conventional flat slab method. It is equally unrealistic to compute
deformation without consideration of the structural stiffness and then to design on the basis of the
corresponding stress resultants. Rational design approach must be based on the results of an
interactive analysis.

DESIGN APPROACH AND
CONSIDERATIONS
Summary of methods suggested by various authors discussed in Chapter 4 would indicate that basically two
approaches have been suggested for analysing the behaviour of raft foundation:
A. Rigid foundation approach
B. Flexible foundation approach
5.1

Rigid Approach

In rigid foundation approach, it is presumed that raft is rigid enough to bridge over non-uniformities of soil
structure. Pressure distribution is considered to be either uniform or varying linearly. Design of rigid raft
follows convkntional methods where again following two approaches have been suggested:
(a) Inverted floor system
(b) Combined footing approach
In rigid rafts, differential settlements are comparatively low but bending moment and shear forces to which
raft is subjected are considerably high.

5.2

Flexible Approach

In flexible foundation approach, raft is considered to distribute load in the area immediately surrounding the
column depending upon the soil characteristics. In this approach differential settlements are comparatively
larger but bending moments and shear forces to which the raft is subjected are comparatively low. Analysis is
suggested basically on two theories
(a) Flexible plate supported on elastic foundation, i.e., Hetenyi's Theory
(b) Foundation supported on bed of uniformly distributed elastic springs with a spring constant
determined using coefficient of sub-grade reaction. Each spring is presumed to behave independently, i.e., Winklers's foundation
Based on these two basic approaches, methods suggested include simplified methods subject to certain
limitations which can be carried out by manual computation. Also now available are computer based methods

23


like finite element and finite differences methods. Finite differences method is based on the second approach
uf uniformly distributed elastic springs and can consider one value of sub-grade modulus for the entire area.
Finite element method transforms the problem of plates on elastic foundation into a computer oriented method
of matrix structural analysis. In this method, plate is idealised as a mesh of finite elements inter-connected
only at the nodes (corners), and the soil may be modelled as a set of isolated springs or as an elastic isotropic
half space. The matrix structural analysis can be extended to include the influence of the super-structure as
well. Thus, the interaction between the super-structure, the foundation and the soil can be accounted for. It is
possible to consider different values of sub-grade modulus in different areas of the raft foundation.
In case of piled rafts against the usual assumption of entire load being carried by piles alone, emphasis is
now being laid on sharing of load between raft supported on soil, i.e., raft soil system and raft pile system.
Sufficiently accurate methods for practical distribution of these loads are not yet available.
As a simplification of treating the entire raft as a plate, concept of beam on elastic foundation is also being
used. For this purpose raft is considered to consist of beams in both the directions. Each of these beams is
treated as supported on springs having spring constant calculated using modulus of subgrade reaction and
carrying column loads. The beam is then analysed as a bean1 on elastic foundation.

5.3

Parameters for Raft Design

In all these methods, however, three basic parameters, i.e., rigidity of the raft, pressure distribution under the
raft and value of sub-grade modulus become important in addition to whatever other info&ation'is received
from soil investigation report. These three parameters and method of their determination are discussed in
subsequent paragraphs.

A problem which has to be solved while designing a raft foundation is to evaluate the actual contact pressure
of the soil against the raft. This problem has occupied many researchers theoretically and a lesser number
experimentally with no exact values being known. Contact pressure, settlement of foundation, soil characteristics and its behaviour are so much inter-related and their relationship so complex, that soil foundation structure interaction is not clear even now. Considering all these aspects it can be said that the contact pressure
distribution under the raft depends upon:
(1) The nature of the soil below the raft, i.e., a single homogenous mass or a layered formation,
thicknesses of various layers and their relative locations
(2) Properties of the soil
(3) The nature of the foundation, i.e., whether rigid, flexible or soft
(4) Rigidity of the super-structure
(5) The quantum of loads and their relative magnitude
(6) Presence of adjoining foundation
(7) Size of raft
(8) Time at which pressure measurements are taken
The total settlement under the raft foundation can be considered to be made up of three components, i.e.,

S = Sd+Sc+Ss
where Sd is the immediate or distortion settlement, Sc the consolidation settlement and S is the secondary
s
compression settlement. The immediate component is that portion of the settlement which occurs simul-

-,

24


taneously with the load application,primarily as aresultof distortion within the foundation soils. Thesettlement
is generally not elastic although it is calculated using elastic theory. The remaining components result from
the gradual expulsion of water from the void and corresponding compression of the soil skeleton. The
distinction between the consolidation and secondary compression settlement is made on the basis of physical
process which control the time rate of settlement. Consolidation settlements are largely due to primary
consolidation in which the time rate of settlement is controlled by the rate at which water can be expelled horn
the void spaces in the soil. The secondary compression settlement,the speed of settlement is controlled largely
by the rate at which the soil skeleton itself yields and compresses. The time rate and the relative magnitude of
the 3 components differ for different soil types. Water flows so readily through most clean granular soil that
the expulsion of water from the pores for all practical purposes is instantaneous and thus foundation settles
almost simultaneously with the application of load. In cohesive soil, it takes considerable time for water to
escape and thus settlement in cohesive soils continue much longer. In fact, it has been reported that the pressure
under a mat foundation on clay may vary from time to time.
It is usual to assume that the soil below the foundation is an isotropic homogeneous material for its entire
depth. But normally this is not the situation and we get different layers in varying thickness, having different
properties below foundation. If the thickness of the upper most layer is large relative to the dimension of the
loaded area, it would probably be sufficient if the soils were considered as a homogenous layer of indefinite
depth. However, if the upper stratum is relatively thin ignoring theeffect of layering, it may have an appreciable
influence on the contact pressure distribution and consequently settlements. This is likely to be of special
importance when a compressive stratum is underlain by rock or a very hard or dense soil. Such presence
decreases the settlement considerably. It is very significant when this occurs within a depth equal to width of
the footings. Incase, there is a stiff stratum underlain by a soft stratum like layer of sand over soft clay layer,
effect is negligible if depth is greater or equal to 3.5 b2.1n case of raft, dimensions of raft are generally such
that the possibilities of encountering a different soil layer within the significant depth are quite large and as
such it would be necessary to account for the different soil layers within the significant depth. Moreover it is
to be remembered that properties of soil constituting each layer which determine the shear strength characteristics and settlement characteristicsof the soil become more important as rafts are generally adopted in areas
where soils of poorer types are'~ncounteredand which some years ago might have not been taken up for
construction at all.
Effect of groundwater table is appreciable on the load carrying capacity of the soil and consequently
settlements. It is, therefore, necessary to consider the expected ground water table in life time of the structure
including the temporary rises as during floods. Even in areas where sub-soil water table is not present, it is
necessary to consider long term built up water for design of basement and raft foundation. If permeability
coefficient of the soil is below 0.1 mm per second, soil is cohesive and probability of surface water accumulated
against basement walls exist'. In such situations, it may be necessary to design raft foundations of basement
for water uplift also.
The conventional analysis of footings, in general, uses the concept of a rigid fcotings and with rigid footing
are associated the concept of uniform soil pressure. Actually to have a uniform soil pressure distribution, we
require a very flexible footing. If simultaneously we accept the concept of soil being elastic (modulus of
elasticity or coefficient of sub-grade modulus), settlementof rigid footing will be uniform and that for a flexible
footing the settlement would be non-uniform and but if this is the case then how can the contact pressure be
uniform (under a rigid footing). In reality we have a soil snucture interaction problem and there is a non-uniform
soil pressure and differential settlements within the footings. It has been suggested that in case of square footing
resting on clay on average contact pressure of 0.6 PIA with additional 0.1 PIAalong edges would be reasonable

25

. DESIGN APPROACH AND CONSIDERATIONS

1
I
1
:

i-

i

:

1

:

pressure distribution. For a rectangular footing of large length it is suggested that it would be reasonable to
have an average pressure equal to 0.8 P average + 0.1 PIB for the edges. Here P is total load, A, area and B,
length of the footing.
For footings on sands a pressure distribution of uniform soil pressure is reasonable.
Rigidity of foundation gets modified by the rigidity of super-structure. Arigid super-structurewill not allow
differential settlement to take place in foundation. Situation can arise when a particular column of the building
may be hanging from the super-structure and even transmitting the weight of attached soil mass to the super
structure rather than transmitting any load from the super-structure to the foundation soil. In fact, a rigid
foundation with a rigid super structure means less differential settlement, large variation of contact pressure
and high bending and shear stress in foundation members. A flexible foundation with flexible super structure
means large differential settlements, uniform contact pressure and lower values of bending and shear stresses
in foundation members.
Quantum of loads and their relative magnitude affect the contact pressure. When the loads are so high that
bearing pressures are increased to the point of shear failure in the soil, the contact pressure is changed leading
to an increase in pressure over the centre of the loaded area in all cases.
The consolidation pressure involves expulsion of water from the soil being compressed. This takes time
and at any time between the application of the load producing consolidation and the time at which essentially
ultimate or 100 per cent consolidation has occurred, the measured settlements and consequently contact
pressure distribution would be different. Many times it may take several years to achieve final settlement.
There are situations in engineering practice where footings are placed so close to each other that their zones
of influence overlap. Studies have shown that effect of adjacent footings may vary considerably with angle of
shearing resistance. For low values they are negligible. For higher values they appear to be significant
particularly if footing is surrounded by others on all sides. There are practically no effects in case of punching
shear failure. It is generally recommended that interference effect may be neglected.
In view of various factors affecting the pressure distribution under a raft foundation and difficulties in
determining affect of each, it is generally believed that contact pressure distribution under a raft could be of
the following type as shown in Fig. 5.1.
,

( c ) SOFT SOIL

-------

Fig. 5.1 Contact pressure distribution under a raft

26


Fig. 5.1 (a) is applicable when the mat is supported on hard rock and column loads are transmitted to the
rock on areas of relatively small size directly under the columns. If the raft rests on a stiff dense soil, then loads
are distributed to the sub-soil in relatively large areas, as shown in Fig. 5.1 (b). It is only on very soft soils that
the contact pressure against the mat foundation approaches linear distribution as shown in Fig. 5.1 (c).
Therefore, it is commonly justified to design a mat on mud, soft clay, peet or organic soil by the conventional
rigid method using uniform pressure. In fact assumption of rigid footings with uniform soil pressure results in
designing the raft for assumed bending moments which are larger than the actual bending moments. The
resulting design is conservative generally but may not be economical. A greater economy can, perhaps, be
achieved by designing the mat with elastic methods, but at what risk and is it really so ? Actual pressure
distribution under the raft, therefore, remains unanswered.

5.5

Rigidity Criteria

Whether a structure behaves as rigid or flexible, it depends on the relative stiffness of the structure and the
foundation soil.The behaviour of the foundation as rigid or flexible will also depend upon the rigidity of the
super-structure above and properties of soil below. In physical terms, a rigid foundation would mean a
foundation which is capable of bridging over pockets of soil with different properties and thus try to even out
the settlements at various points. A rigid foundation would, therefore, have comparatively lower values of
differential settlement but higher values of stresses. A rigid foundation with a rigid super-structure on a
comparatively compressible soil will result in uniform settlements of structure.
A flexible foundation with a flexible super-structures and a comparatively rigid soil below will behave as
a flexible foundation and would result in large differential settlements and low stresses. Thus:
(i) Arigid member is characterisedby high bending moments and relatively small, uniform deflections.
Over all differential settlements are small.
(ii) An intermediate member, as the term implies, has intermediate bending and deflection values.
(iii) The flexible member has comparatively smaller bending moments and deflection is maximum in
vicinity of the loads and small values else where. Overall differential settlement would be of higher
orders.

-

Rigidity criteria proposed by various authorities are discussed below:

5.5.1 Proposed by IS :
2950 (Part I) 19813
Appendix C of this standard gives the method of deciding rigidity of super-structure and foundation. This is
reproduced below:
Rigidity of Superstructure and Foundation

C-1

Determination of the Rigidity of the Structure

C-1.1 Theflexural rigidity El of the structure of any section may be estimatcdaccotding to the relation given
below (see also Fig. 5.2):


Fig. 5.2 Determination of rigidity of a structure
where

El = modulus of elasticity of the infilling material (wall material) in kg/crn2,
I, = Moment of,inertia of the infilling in cm4,
b = length or breadth of the s ructure in the direction of bending.
H = total~height the infill In cm,
of
E, =modulus of elasticity of frame material in kg/cm2
Ib = moment of inertia of the beam in cm4

J

/
.

where
1 = Spacing of columns in cm,
h, = Length of upper column in cm,
hl = Length of lower column in cm,

I,, = Moment of inertia of upper column in cm4,
Il = Moment of inertia of lower column in cm4
If = hioment of inertia of foundation beam or raft in cm4,

28

RAFT FOUNDATIONS-DESIGN
AND ANALYSIS

Note :The summation is to be done over all the storeys, including the foundation beam of raft. In the case of
the' foundation I;replaces Pb and 1, becomes zero, whereas for the topmost beam 1'" become zero
Relative Sti#hess Factor K:

C-2

C-2.1 Whether a structure behave as rigid orflexible depends on the relative s t i m s s ofthe structure and
thefoundation soil. This relation is expressed by the relative stimess factor K given below:
(a) For the whole structure

(b) For rectangular rafts or beams

(c) For circular rafts

where
El = Flexible rigidity of the structure over the length (a) in kg/cm2
E, = Modulus of compressibility of the foundation soil in kg/cm2
b = Length of the section in the bending axis in cm,
a = Length perpendicular to the section under investigation in cm,
d = Thickness of the raft or beam in cm,
R = Radius of the raft in cm
C-2.1.1 For K > 0.5, the foundation may be considered as rigid
C-3

Determination of Critical Column Spacing

C-3.1 Evaluation of the characteristics h is made as follows:

where

k = Modulus of sub-grade reaction in kg/cm3 for footing of width B in cm

B = Width of raft in cm,
E, = Modulus of elasticity of concrete in kgf/cm2
1 =Moment of inertia of the raft in cm4
Modulus of compressibility of the soil is the additional property required in this particular case.
5.5.2 ACI Committee, 436

Suggested design procedure for combined footings and mats - American Concrete Institute Journal, October,
196614
Relevant extracts from this paper are given below:

I1
a

Ig


29

Footings supportingjield structures

Continuous strip footings supporting structures which because of their rigidity will not allow the individual
columns to settle differentially should be designed as rigid footings with a linear distribution of soil pressure.
This distribution can be determined on the basis of simple statics.
To determine the approximate rigidity of the structure, an analysis must be made comparing the combined
stiffness of the footings, super-structure framing members, and shear walls with the stiffness of the soil. The
relative stiffness will determine whether the footing should be considered rigid or flexible.
The following formulas may be used in this analysis :

where

E = Modulus of elasticity of the materials used in the structure, kips per sq.ft (metric tons per sq.m)
I, =Moment of inertia of the structure per unit length, ft3 (m3)

IF = Moment of inertia of the footing per unit length, ft3(m3)

Es=
Modulus of elasticity of the soil, kips per sq.ft (metric tons per sq.m)
b =Width of footings, ft (m)

per
An approximate value of ElIC unit length of building can bedetermined by summing the flexural rigidity
of the footing (E'L,) the flexural rigidity of the each framed member (FIB)and the flexural rigidity of any
shear walls (F3112) where a and h are the thickness and height of the wall, respectively.
Computations indicates that as the relative stiffness K, increases, the differential settlement decreases
rapidly.
For K , = 0 ,the ratio of differential to total settlement is 0.5 for long footing and 0.35 for a square one.
For K , = 0.5 , the ratio of differential to total settlement is about 0.1.
If the analysis of the relative stiffness of the footing yields a value above 0.5, the footing can be considered
rigid and the variation of soil pressure determined on the basis of simple statistics.
If the relative stiffness factor is found to be less than 0.5, the footing shall be designed as a flexible member
using the foundation modulus approach as described under section 6.4 of the report.
Columns Spacing

The column spacing on continuous footings is important in determining the variation in soil pressure
distribution. If the average of two adjacent spans in a continuous strip having adjacent loads and column
spacings that vary by not more than 20 per cent of the greater value or is less than 1.75/h, the footing can be
considered rigid and the variation of soil pressure determined on the basis of simple statics.
- If the average of two adjacent span, as limited above, is greater than 1.75/h, the design of the footing shall
be governed by subgrade modulus theories.
For general cases falling outside the limitation stated above, the critical spacing i t which the subgrade
modulus theory becomes effective has to be determined individually.
Evaluation of the factor can be made on the basis of the following formulae:
'

30


K, = SR,r

Where

K, = Coefficient of vertical subgrade reaction, Kips per cu ft (metric tons per cu m)

K, basic value of coefficient of vertical subgrade reaction for a square area with width b = 1 ft (0.3 m).
'=
Kips per cu ft (metric tons per cu m)

b = Width of footings, ft (m)
S = Size or shape factor for a footing on a particular type of soil
E, = Modulus of elasticity of concrete, Kips per sq ft (metric tons per sq m)

I = Moment of inertia of footings ft4 (m4
For sandy soils the size factor S can be determined from the following formula:

with a limiting value of 0.25 for large footings.
As for clay soils, the shape factor S can be determined from the following formula:

When n is the ratio of the longer side to the shorter side of the footing. As for extremely long footings,
where n approaches infinity, S can be assumed as 0.67.
can
Values for Kt,, be determined from the results of field tests performed on the subgrade of the proposed
structure or can be estimated on the basis of empirical values in "Evaluation of coefficients of Subgrade
Reaction" by Terzaghi.

5.5.3 Hetenyi's Criteria
From theory of beams on elastic foundation, Hetenyi proposed rigidity criteria on the basis of hLterm which
considers width, length and elastic properties of the media. This term is

hL =

(K,.L ~ ) " ~
4 El

where

K, = KB = Modulus of sub grade reaction X Width of footing - units of psf.
L = Total length of foundation member

E = Modulus of elasticity of footing material
I = Moment of inertia of footing
If 1 c W4 footing can be considered as rigid. For value between W4 and l semi rigid, and elastic, if
,
3
> I ~ o w l e sfound this criteria of very limited application.
7
'~
5.6

Modulus of Sub-Grade Reaction

One of the important terms required in analysing foundation on the basis of flexible footings is value of modulus
of sub-grade reaction also called coefficient of sub-grade reaction for the particular soil in the foundation of
the buildings. Mathematically, this can be axpressed as intensity of soil pressure required to create a unit

31


3
I

deflection. Theoretically, it can be determined by performing a plate load test and plotting a curve of soil
pressure versus deflection. In actual practice, however, many other factors enter and actual value in field is
different from what can be determined by a simple plate load'test. Major problems associated are:
(a) Soil is not perfectly elastic and results are effected by the magnitudes of soil pressure and deflection
(b) Footing size affects the value
(c) Footing shape also affects
(d) Depth at which footing is located also affects
(e) Soil stratificationand other changes with depth which may not show when testing with a small plate
(f) In methods where soil modulus is determined in laboratory, site condition can not be exactly
duplicated in field laboratory
(g) Various authors have suggested different factors to take these problems into account
On the other hand, certain authors have suggested very simple values for modulus of sub-grade reaction
which can be determined from bearing capacity factors used in Terzaghi bearing capacity equation.
5.6.1 Recommended by ~ o w l e s ' ~

Has related value of modulus of sub-grade reaction with safe bearing capacity by the relation Ks = 36 qa where
qa is the allowable bearing capacity in Kips per sq ft. A slightly improved values are also suggested by the
equation.

where c is cohesion, Nc and N q are bearing capacity factors, Sc and Sq are shape factors for particular soil in
foot units .
Moreover:

Sc = I + -

NcxB
NcxL

General values suggested by Bowles are given below:
Soil

Range of Ks. Kef

~ o o s sand
e

30 - 100

Medium sand

60 - 500

~ e n s sand
e

100-800

clayey sand (Medium)

200 - 500

Silty sand (Medium)

150 - 300

Clayey soil :

qu 5 4 Ksf
4<qu18Ksf

75 - 150
150 - 300

32

RAFT FOLINDATIONS-DESIGN AND ANALYSIS

5.6.2 IS :
2950 P r I Indian Standard Code of Practice for Design and Construction of RafC Foundation
at
2950 1981'

-

Provision relating to determination of modulus of sub-grade reaction are included in Appendix B. This is
reproduced below. Figures given in bracket in Tables I and I1 are in Kipdc ft. units.
B-1

General

- 1 The modulus of subgrade reaction ( k ) as applicable to the case of load through a plate of size 30 x 30
cm or between 30 cm wide on the soil is given in Table 1 for cohesionless soils and in Table 2 for cohesive
soils. Unless more specific determination of K is done (see B-2 and B-3) these values may be used for design
of rafl foundation in cases where the depth of the soil affected by the width of the footing may be considered
isotropic and the extrapolation of plate load test results is valid.
Table I Modulus of Subgrade Reaction (K) CohesionlessSoils
for
Modulus OfSubgrade Reactions (K)in kg/cm3.

Soil Characteristic
Relative Density

(1)

For dry or moist state

Standard Penetration test
value ( N )

For submerged state

(2)

0.9(57)

1 5 to 4.7
.

.
0.9to 2 9

(95to 300)

(57to 185)

4.7 to 18.0

2 9 to 10.8
.

(300to 1146)

Dense

1 5 (95)
.

10 to 30

Medium

(4)

< 10

Loose

(3)

(185 to 687)

30 and over

* The above values apply to a square plate 30 X 30 cm or beams 30 cm wide
for
Table II Modulus of Subgrade Reaction (K) Cohesive Soils
Soil Characteristic

Modulus of Subgrade Reaction (K)in
K ~ / C ~ ~

Consistency

Unconfined compressive strength,
kg/cm2

(1)

(2)

(3)

Stiff

l to 2

2 7 (1
. 72)

Very stiff

2 to4

2 7 to 5.4 (172to 344)
.

4 and over

5.4to 10.8 (344to 688)

Hard

* The values apply to a square plate 30 x 30 cm. The above values are bared on the assumption that the average loading
intensity does not exceed half the ultimate bearing capacity.


B-2

33

Field Determination

In cases where the depth of the soil affected by the width of the footing may be considered as isotropic the
value of K may be determined in accordance with IS :9214 - 1979". The test shall be carried out with a plate
of size not less than 30 cm.
B-2.2 The average value of K shall be based on a number of plate load tests carried out over the area, the
number and location of the tests depending upon the extent and importance of the structure.
NOTE IS:9214 - 1979 lays down that Ks can be determined as slope of the secant drawn between the points
corresponding to zero settlement and point corresponding to 1.25 mm settlement of a load settlement curve
obtained from a plate load test on the soil using a 75 cm dia plate or smaller dia with corrections for size of
the plate used.
B-3

Laboratory Determination

B-3.1 For stratifed deposits or deposits with lenses of different materials, evaluation of Kfrom plate load
will be unrealistic and its determination shall be based on laboratory tests (see IS: 2720 (Part XI)- 1972" and
IS: 2720 (Part X1I)- 1 9 8 1 ) ~ ~
B-3.2 In carrying out the test, the continuing cell pressure may be so selected as to be-representative of the
depth of average stress influence zone (about 0.5 B to B)
B-3.3 The value of K shall be determined from the following relationship

where

Es= Modulus of elasticity of soil (see Appendix A)
E = Young's modulus of foundation material
p = Poisson's ratio of soil ( see Appendix A) and
I = Moment of inertia of structure if determined or of the foundation
B-3.4 In the absence of laboratory test data, appropriate values of Esand p may be determined in accordance
with Appendix A and used in B-3.3 for evaluation of K.
8-4

Calculations

B-4. I When the structure is rigid (see Appendix C) the average modulus of sub grade reaction may also be
determined asfollows:

Ks =

Average contact pressure
Average settlement of the raft

Appendix C lays down the method of determining the rigidity of superstructure and foundation and has
+n dealt with in para 5.5 above.
Appendix A lays down mettrod of determination of modulus of elasticity of soil by field tests or laboratory
tests.
~ w
'~
Equation in B-3.3 above is based on work carried out by ~ e s i c~ o~ . l e shas observed that the 12th root
of any value will be close to 1 and equation can be considered to be equivalent to

Raft foundations _design_and_analysis_with_a_practical_approach

Raft foundations _design_and_analysis_with_a_practical_approach

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