raft foundation

1. Raft
Foundations
Design and Analysis with a
Practical Approach
SHARAT CHANDRA CUPTA
Advisor, Indian Buildings Congress,
Former Chief Engineer
Central Public Works Department
PUBLISHINGFOR ONE WORLD
NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS
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2. Copyright O 1997 New Age International (P) Limited, Publishers
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3. CONTENTS
Preface
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I
1. INTRODUCTION
2. NEED OFRAFT FOUNDATION
3. TYPES OFRAFT FOUNDATION
4. SURVEY OFAVAILABLE LITERATURE
I 1 Foundation Engineeringby Peck, Hansen and Thornburn
I 4.2 Foundation Design and Practice by Elwyn.E.S. Seelye
1 4.3 FoundationDesign by Teng
4.4 Foundation of Structuresby Dunham
i
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
FoundationAnalysis 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 DesignersHandbookby CharlesE. Reynolds and
JamesC. Steedman -9th Edition 1981
IS 2950 (PartI) 1981-Code for Design and Construction of Raft Foundation Part I
~ e s i ~ n
Eleventh InternationalConferenceof Soil Mechanics and Foundation Engineering
San Francisco, August 12- 16,1985
Foundation Design and Construction by M.J. Tomlinson,5th Edition, 1986

4. CONTENTS
i
Handbook of ConcreteEngineering-Mark Fintel - 2nd Edition, 1986
ReinforcedConcreteDesigner Handbook by Charles E. Reynolds and James
Steedman, 10thEdition, 1988
Building Code Requirementsin ReinforcedConcrete-ACI -318- 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
FoundationAnalysis and Design by Bowles, 4th Edition, 1988
Proceedingsof Indian Geo-Technical Conference 1992,Calcutta,December, 1992
Designs of Foundation Systems-Principles and Practices by Nainan P. Kwian, 1992
13thInternational Conferenceon Soil Mechanics and Foundation Engineering,
New Delhi, January, 1994
Soil StructureInter-action -The Real Behaviour of Structures,published by the
Institution of StructuralEngineers, U.K. The Institution of Civil Engineers,
U.K. International Association for Bridge and StructuralEngineeringin March, 1989
5. DESIGN APPROACH AND CONSIDERATIONS
5.1 Rigid Approach
5.2 FlexibleApproach
5.3 Parametersfor Raft Design
5.4 Pressure Distribution Under the Raft
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 Soilsin Field
5.6.4. IS 8009- Part I - 1978.Code of Practice for Calculationof Settlementsof
Foundations - Part I -Shallow Foundations. Subjectedto Sy_mmetrical
Static Vertical Load.
5.6.5 Recommendation by Alpan and Prof. Alarn Singh
5.6.6 Summary
6. STRUCTURALDESIGNERSDILEMMA
7. STUDIESCARRIEDOUTONEFFECTOFVARIOUSPARAMETERSONDESIGNOFRAFT 38
7.1 Study 1 40
7.1.1 Examples Selected 41
7.1.2 Raft Size 41

5. CONTENTS
5 7.1.3 Soil Investigation
!
7.1.4 Load Considered in Study
7.1.5 Analysis
: 7.1.6 ' Discussions of Results
7.1.7 Conclusions
7.2 Study 2 -Effect of Horizontal Loads
7.2.1 Example Selected
7.2.2 Discussion of Results
7.2.3 Conclusion
7.3 Study 3: Comparison with Conventional Rigid Methods
7.3.1 Details of ConventionalMethod: Combined Footing Approach
7.3.2 Examples Selected
7.3.3 Discussion of Results
7.3.4 Inverted Floor Method
7.3.5 Conclusions
1
7.4 Study 4. AnotherOffice Building
7.4.1 Example Details
7.4.2 Comparison of Results
tt
1
7.4.3 Discussions of Results
! 7.4.4 Conclusions
I
8. STUDIESCARRIEDOUT ON ANALYSIS AND DESIGN OFPILED RAFTS
1
t
i
8.1 Design Procedures being Used
! 8.2 Example Selected
8.3 Soil Data
8.4 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 RafPAnalysisBased on FiniteElement Approach
8.5 Study of Parameters Influencing the Raft Behaviour
: 8.5.1 Effect of Raft Stiffnesson the PileLoads and Raft Moments
; 8.5.2 Effect of Superstructureand RetainingWalls on Foundation Stiffness
I 8.5.3 Effect of EarthquakeLoads and Moments
I 8.5.4 Effect of End Bearing and Friction Piles
8.5.5 Summary of Results
I 8.6 Discussions
8.7 Conclusions
9. JOINTS IN RAFl'S
10. SUMMARYOF STUDIES

6. 11. FACTORSAFFECTING CHOICEOFMET,HODOFANALYSIS
12. GUIDELINES
APENDM -ILLUSTRATIVEEXAMPLES
A.l ConventionalRigid Method -Combined footing approach
A.2 Flexible Raft -Beam on elasticfoundation
A.3 Piled Raft-Plateon elastic foundation
CONTENTS

7. INTRODUCTION
I
i
t
In 1957,when the authorwas a studentof 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 commonsense is that sense which is quite uncommon."
After 34 years of experience in Civil Engineeringconstruction and design, the author only wonders how
true the statement of his Professor was and how much more it is true in case of foundationengineering.
1.1 Foundation engineering has been practised asan art, without help of science,since time immemorialupto
1920when it had achieved a considerableamount of refinement. It was in the earlier 1920sthat a concerted
!effort was made to study and undentand the physical laks governing the behaviour of sub surface materials,
i.e.. soil from which foundationsderived 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 1919when Karl Terzaghi, popularly
known as 'father of soil mechanics', made successfulattempt to explain the phenomenon of settlementoti a
scientificbasis. Though study of soil mechanicshas provided us with new techniquesfor selectingappropriate
type of foundation and predicting the behaviour of completed structures,it has not been able to decreasethe
importanceof the accumulated experienceof 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 scienceand computer application,
structural design is still defined as:I5
a creation of a structuralfonn to satisfy a number of requirements.It is a combinationof art and science.
As a rule, there is no direct procedure leading to the solution of a specificproblem. An engineer uses all
his resources of knowledge experience and imagination toproduce a trial scheme. He then constructs a
mathematicalmodel of suchasolutiontoassessitsadequacyand ifnecessary, modifiestheoriginalconcept
inthe lightofanalytical results. Theprocess isrepeated until thedesignerissatisfied withthefinalproduct,
taking into accountnot 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 toarrive at decisionsbased on incompletedata Intuitive decisionswhich have to
be taken,appear to be diametrically opposite to the logical nature of ...'

8. 2 RAFT FOUNDATIONS-DESIGN AND ANALYSIS
Foundation design and analysisis, 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. Adesigner, when facedwith a task of designinga 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
adequatesafetyand,aboveall,acceptanceof the design by the clientandthe professionalcommunity 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 variousapproachesas availablein engineering literatureat a particular time, comparetheir merits
and demeritsand select the most suitablefor his purpose. He, therefore, perforce selectsa 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 solutionmay
not be as satisfactory as he feels.
An efforthas been made in the followingchaptersto explain the variousapproachessuggestedin 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
comparativeidea of availableapproacheswith 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.

9. NEED OF RAFT FOUNDATION.
Raft or Mat foundation is a combined footingthat 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 supportedon piles
as well.
Raft foundationis 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 compressiblelenses 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 differentialsettlement.
(c) When structuresand equipment to be supported are very sensitiveto differential settlement.
(d) Where structures naturally lend themselvesfor 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-structureis 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 differentialrotation and differentialsettlementof individual
footingscausing 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-foldadvantage:
(a) Ultimate bearing capacity increases with increasing width of the foundation bringing deeper soil
layers in the effective zone.
(b) Settlementdecreases with increased depth.
(c) Raft foundationequalises the differential settlement and bridges over the cavities.Every structure
has a limiting differential settlement which it can undergo without damage. The amount of
differentialsettlement between various parts of a structuresupported on a mat foundation is much
lower than that if the sarne.structurewas supported on individual footings and had undergone the
same amount of maximum settlement.With these considerations,maximum total settlementwhich

10. RAFT FOUNDATIONS-DESIGN AND ANALYSIS
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 shally
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 Somedesigners work on the rule that if more than 50%of the area of the structureis 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 excessivedeflection 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 I
footings on both sides. It is also stated that these effects are considerably reduced as length over breadth ratio
I
of the footings approaches unity. There are practically no such effects in the case of punching shear failure. 1
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 somespecial circumstan-
11
ces .
I
2.3 Situations exist in practice w h p a soil stratum contains compressiblelenses or the soilshave a formation
where individual layersof soil are neither parallel nor can be reasonably stratified intodifferent 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
I
be tolerated by the structure.
I
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 structureis 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,

11. NEED OF RAFT FOUNDATION 5
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
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 floorconstruction. It is inconvenientand often impossible to providetemporary 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 footingsare subjected to very large eccentricloadings, and one is faced
with the possibility of excessive footing rotation, excessive differential settlement or possibility of exceeding
the allowablebearing capacity of the soil at somelocation. 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 hisjudgement 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.

12. TYPES OF RAFT FOUNDATION
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 structureof 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
suitablewhere the column loads aremoderate and the column spacingfairly small and uniform. Pedestals are
utilised to distribute the load on a bigger area in case of heavy column loads.
3.3 Slaband beam raft is used as afoundationforheavy buildings where stiffnessis the principalrequirement
to avoid excessive distortion of the super structureas a-resultof variation in the load distribution over the raft
orthe compressibilityof the supportingsoil.Theserafts, however,havemany obviousdifficulties.If the beams
aredeep, ribs placed below the basement floor or raft, the bottom of the excavation becomes badly cut up with
trenches, impairingthe bearing valueof the soilbecauseof itsdisturbance.Water proofingin caseof 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 floorslabcan be designed,but it createsconsiderableamountof uncertainties, difficulty of construction
and quite often even in such cases, raft is designedas a slab of uniform rhickncss.

13. TYPES OF RAFT FOUNDATION 7
Raft,as aslabof uniform thickness, hasan additional advantageofprovidingbetterwater-proofingtreatment
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
RAFT SUPPORTED ON PILE R A F T SUPPORTED ON SOIL BUOYANCY RAFT
------------- FLAT PLATE RAFT
--------------------- ---------------------- ---------------
i
' FLAT PLATE WITH PEDESTALS BEAM AND SLAB RAFT
: ------------------------- ------------------
Fig. 3.1 Various types of rafts
FRAMED RAFT
-----------

14. SURVEY OF AVAILABLE
LITERATURE
Testbooks and design manuals by various authors suggest varying approachesto analysis and design of raft
foundation. Differences of opinion existin the method of analysisproposed to be adopted while determining
moments,shear forcesforthedesignof raft. Oncethe bending moments and shearforcesareknown, structural
designdoes not present anydifficultyandthereexistsno differenceof opinioninthisrespectexceptveryminor
differencerelating to desired thickness of slab and the effectivenessof the shearreinforcement
Methods suggested by different authors are summarised below. Thesehave been arrangedchronologically
with reference to date of publication of the testbooktdesign handbook.
4.1 Foundation Engineering by Peck, Hansenand hornb burn^
Raft is usually regarded and designed as an inverted continuqusflat slabfloor 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 specificationsfor the design of flat slab floors. On account of erratic variation in
compressibilityin almostevery soildeposit, therearelikelyto be correspondinglyerraticdeviationsof the soil
pressure from the averagevalue. Sincethe moment and the shears are determined on the basis of the average
pressure, it is consideredgoodpractice toprovidethis slabwith more than theoreticalamountof reinforcement
and to use the same percentage of steel at top and bottom. This method has been widely used, often with
completesuccess.On the otherhand,ithas alsosometimesled to structuralfailurenot only of the slabbut also
of the super structure. Therefore,its limitations must be clearly understood. The analogy follows only if the
differentialsettlement between columnswill be small and if the pattern of the differential settlementwill 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 slabsresulting in unconservativedesign.
Rafts are sometimes designed as if they rested on a bed of closely and equally spaced elastic springs of
equal stiffness.The contactpressurebeneath any small area is then proportional to the deflectionof the spring

15. SURVEY OF AVAILABLE LITERATURE 9
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 stressdeformationcharacteristicsof the soilbut alsoin acomplexmanneron the shape
and sizeof the loaded area and the magnitude and positionof nearby loaded areas.Evaluationof 'K'fordesign
is difficult and fraught with uncertainty. Whatever method may be adopted for design, there is no guarantee
that the deflectionsof the raft will actually be unimportant.In case,the structurecovers a fairly large area with
possibilities of differentialsettlements,itis not enoughto provide great strengthin the slab.It is also necessary
to provide sufficientstiffness.However, a stiff foundation is likely to be subjectedto bending moments far in
excess of those corresponding to the flat slabsubgrade modulus analysis.
There appears to.be no furtheredition of this book after 1954.
4.2 Foundation Design and Practice by Elwyn. E.S. seelye9
According to Seelye after determiningthe soil pressuresat 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 furtheredition 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 followsthe planner distribution.The mat is analysed as awholein each of twoperpendicular
directions. Thus the total shear forces acting on any section cutting across the entire mat is equal to the
arithmeticsum of all forces and reactions(bearing pressure)to the left orrightof the section.Thetotal 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 momentscan be determined by the principlesof simplestatics,the distribution
along this section is a problem of highly indeterminate nature, the average moment not being indicativeof the
sign and the magnitude of the bending moments in the individual strip in either direction. In order to obtain
someidea as to the upper limit of these values,each strip bounded on central line of the column bays, may be
analysedas independentcontinuousor combinedfootings.If thecolumn loadsare 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 calculationsis not alwaysjustified or practicable in case of raft.foundationsbecause
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 soconstant that plastic soil will
compressand adjust itself so thateach 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 averagepressure

16. 10 RAFT FOUNDATIONS-DESIGN AND ANALYSIS
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 adjacentcolumns should not differ very much and the bays
ineitherdirectionshouldbereasonably,equal inlength,thelargerspacingnotexceeding 1.2time, thesmaller
one and the columnsshould be arringed in reasonably straightrows.
Fig. 4.1 Plan of assumed columns strips and distribution of loads
Onemethod of making a preliminary analysisof sucha mat is onthe basis of an assumedsupportingsystem
of columns strips that constitute a grid of beam along the column rows in each direction.The portion of the
slabsin the central areas is taken up tobe supportedby this grid. The effectivewidth of these stripsor shallow
beams has to be assumedand it is normal to take it slightly more than, what is determinedby 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 advisableto reinforce both sides equally because
any yielding of end restraint will increasethe,tension in the top of the mat above the computed value. Each
column stripmay be analysed by moment dishbution if the variation of loading or spansmake thisdesirable,
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 furtheredition of this book after 1962.
4.5 IndianStandard CodeofPracticeforDesignandConstructionof RaftFoundation-IS2950-1965'
There aretwoapproachesfor design-conventionalmethod andtheelasticmethod. In theconventionalmethod,
thefoundation 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 coincideswith the line
of action of theresulting forcesof alltheloadsactingon thefoundation. Themethod is normally used in design
because of its simplicity .A generousamount of reinforcementis provided to safeguard uncertainties caused

17. I SURVEY OFAVAILABLELITERATURE 11
I by differential settlement. The raft is anabjsed as a whole in each of the two perpendicular directions. Thus,
total shear forcesand total bending moments acting on any section cutting across the entireraft is equal to the
arithmeticsum of all forces and reactions/moments to the leftor right of the section.The actual reinforcement
I 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.
I In the other, the soil is assumed as a continuous elastic medium obeying Hook's Law. These methods are
applicablein case the foundation is comparatively flexibleand the loads tend to concentrateover small areas.
The actual reinforcement can be one-and-a-half times that required theoretically.Thefamous 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.
i
(2) apply only to fairly uniform soil conditions and for fairly horizontal planes of separation of layer
below.
I (3) foundations in seismic area and/or to vibratingload shall be given special considerations.
i This code has been revised in 1973.Kindly see para 4.7.
I 4.6 RafL Foundation- The Soil Line Method of Design by A.L.L. ~ a k e q
! Accordingto Mr. Baker,the design of raft as areversed floorisdangerous.Engineersbeing awareof this, who.
' therefore,normally adopt the second method in which earth pressure is assumed to be uniform throughoutand
moments are obtained at any section by statics. He, however, feels that in the secondmethod also high values
'
of moments are obtained, which may or may not bepresent, 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 furtheredition of this book after 1969.
4.7 Indian StandardCode of Practicefor Designand Constructionof Raft Foundation
1.S :2950 (Part-I)1973~
In the revised version of the code, followingmethods of analysis have been proposed:
(a) Assumption of linearly varying contact pressure
(b) Perfectly rigid structures
(c) Perfectly flexible structures
(d) Structuresstiffened along one axis
(e) Structuresstiffened along both the axis
(f) General methods:
(i) Based on modulus of subgradereaction, 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 applicablewhen deformations of raft under loads are small as compared to
the mean settlementof the structure.

18. 12 RAFTFOUNDATIONS-DESIGN AND ANALYSIS
Method (c) is applicable for structures which have relatively less stiffening members specially resting on
very stiff foundation soil. In this case, the deflectionsof the raft are same as the settlementsof 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)arenot applicable.Detailsgiven in the codedo not provideenoughguidance
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 settlementsas 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 authorsof chapters on spread footings,combined and special
footingsand mat foundationrespectively.Chapter on floatingfoundation has been writtenby 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 assumesthat:
(a) Mat is extremelyrigid 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 straightline or a plain surface such that
the centroidof the contactpressure coincideswith the line of action of the resultant forceof all the
loadsactingon themat.When matfoundationissupportedon piles,pilesareassumedto be perfectly
elastic. Raft is considered to be rigid when the column spacing is less than 1.751hor when the mat
is supportinga rigid super-structure.his sameas defined by Heteny.The mat is analysedas a whole
in each of the two perpehdicular directions. The mat is divided into perpendicular bands of width
between centre lines of adjacentcolumnrows. Each band is assumed to act as an independentbeam
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
springseach of which is not affected by others.Thisfoundationmodel is alsoreferred to asWinklerfoundation.
Analysis procedures have also been developed for the beams on the simplifiedelastic foundation concept.
The mat is considered as a plate and the effect of each column load is consideredin area surrounding the load.
Using the method of super-imposition,effectof allthe column loads within the zoneof influence iscalculated.
Among computer-orientedmethodssuggestedis finitedifferencemethod,based on the assumption that the
sub-gradecan be substituted by a bed of uniformly distributedelastic 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 expressedby a differential equation in terms of deflection at the adjacentpoints
to the right, left, top and bottom. These simultaneous equations are solved with an electroniccomputer and
deflection at all the points are determined. Once deflectionsare known, the bending moment at any point in
each direction is determined from theory of elasticity.
The finite element method transformsthe problem of plates on elastic foundationinto a computer-oriented
procedure of matrix structuralanalysis.The mat is idealised as a mesh of finiteelementsinter-connected only

19. SURVEY OFAVAILABLE LITERATURE 13
at the comers and the soil may be modeled as a set of isolated springs or as an elastic isotropichalf 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 economymay 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
directionsat 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 irrespectiveof the method used foi
analysis.
Second edition of this book is published in 1991. Please see para 4.21.
4.9 FoundationAnalysis and Design by Joseph.E. ~owels'
"
The mat may be designed as rigid structuresthereby soil pressureare computed as Q = V/A in the case where
the resultant of the forcescoincide with the centre of the mat area. If resultant has eccentricitywith 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 appropriatecolumn linesas shownin Fig. 4.2. For the seriesof beams, shear and moment
diagram may be establishedusing eithercombined footinglanalysisor 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

20. 14 RAFT FOUNDATIONS-DESIGN AND ANALYSIS
When the soil bearing pressure is low say 0.5 ~i~slft2
(25 K N I ~ ~ )
or less and if the deformationof the mat
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 methodshave not been widely used in the
past, are ease of making conventional solution, which have been generally satisfactoryand 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 straightforward conversion to a value of modulus of sub-grade reaction exists.
Various methods for elastic analysislike finiteelementand finitedifferenceshave alsobeen 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
7
)
1
8
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
requirementsof this code.
15.10.2 TheDirect Design Method of Chapter13 shall not be usedfor design of combinedfootings and mats.
15.10.3 Distribution of soil pressure under combinedfootings and mats shall be consistent withpropemees
of the soil and the structureand 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 FoundationDesign and Constructionby M.J. ~omlinson'~
Mr.Tomlinsonstatesthatitis wrongin principalto assumethat araftactsasaninvertedfloorslabon 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
fordeflectionunderthe most favourablecombinationofdeadandliveloadand variation in soilcompressibility.
Guidance is required from the soil mechanics engineer on the estimatedtotal and differential settlementfor
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.Basementrafts carrying heavy building on weak soils are often foundedon 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 bedrock or other relatively incompressiblestrata.The piles also
have the effect of stiffeningthe raft and reducing or eliminatingre-consolidation of ground heave, thereby
reducing differential settlementor tilting. In such cases, considerableheave takes place with further upward
movement caused by displacement due to pile driving. After completionof piling, the swelled soil should be
trimmedofftothe finishedlevel.Thebasementwallsshouldgenerally be designedasself-supportingcantilever
retaining wallseven though they may eventuallybe supported by thefloorconstruction and additionalstability
against overturninggiven by super-structureloading on top of the wall.The basementfloor slabsmust be able

21. SURVEY OFAVAILABLE LITERATURE 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
t bases with only a light slab between them, there would be likelihood of failureof the slab from the pressure
of the underlying soil.
g
Fifth edition of this book has been out in 1986.Please see para 4.17.
4.12 Design of Combined Footings and Mats ACI Committee 33614
The committeeobserves that no authentic method has been devisedthat 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 factorsshould be considered while examining any problem:
(1) Soil type immediately below the footing
(2) Soil type at the greater depth
(3) Size of footing
i (4) Shape of footing
(5) Eccentricity of loading
(6) Rigidity of footing
(7) Rigidity of the super-structure
(8) Modulus of sub-gradereaction
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 suggestedfor footings which can be consideredrigid to the extent that only very small relative
deformationsresult from the loading. The rigidity may result from the spacingof the columns on the footing
from the rigidity of the footingitself 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 pressureby means of sub-gradereaction has been suggestedwhere sub-soils areof such
characterthat the deformationsare localisedin 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
a
is suggested that uniform or lineardistributionof soil pressure can be assumed andthedesignbased 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
1 detailed design procedure based on plate theories has been recommended.
4.13 Pile FoundationAnalysis and Design by H.G.Poulos and E.H.Davis 1 9 8 0 ~ ~
: In this book,Chapter 10deals with piled raft systems.Theauthor saysthat,"in designof 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 illogicalto 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 introducedsolely for the purpose of reducing the settlement

22. 16 RAFT FOUNDATIONS-DESIGNAND ANALYSIS
designquestionbecomesnot"how many piles arerequired to carry the weightof thestructure"but "how many
piles are required to reduce the settlementto an acceptancelevel".
However, in Chapter 5, the settlement behaviour of a free standingpile is obtained from the elastic-based
analysis.The pile isdividedinto number of elementsand theexpressionsforvertical settlementof the pile and
the soil at eachelementin termsof unknownstresseson thepilesareobtained andsolved,imposingthevertical
displacement compatibilitycondition, to arriveat the settlementbehaviour of the pile. As a further extension,
the unit consisting of a singlepile with an attached cap resting on the soil surfaceis 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 thecap. The book givescharts indicating interaction factor
between the raft and thepile for various values of length of the piles, diameter of the pile, poisson ratioof soil,
height of soil layer over the rigid stratum and the cap diameter.The method is further extended to group of
piles upto about 40numbers.Curves are drawn which are applicableonly for rigid rafts or perfectly flexible
rafts. The entire emphasis is to work out the ratio of the load carried out by thepiles and the raft soil system.
No details are given on &e method to determine the bending moment and shear forces in theraft. 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 ConcreteDesigners Handbook by Charles E. ReynoldsandJamesC. Steedman-
9th Edition 1981"
This book suggests the analysis of a raft foundation supportinga seriesof 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 symmetricallyarranged,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 impracticableowing to the extent of the raft being limited on one or more sides,the plan of the raft
shouldbe made so that theeccentricityof the total loadingisa minimum,though this may produce a raftwhich
is not rectangular in plan.
4.15 IS 2950 (Part I) 1981-Codefor Designand Constructionof Raft Foundation PartI ~ 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 assumingplanner distribution of contactpressure is applicableto
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 stiffnessfactor K >0.5 or columns spacing less
than 1.75A. Methodsof determiningvalue of K and hare giveninthecode.Conventionalmethodisapplicable
when either of the two conditions are satisfied. The value of K depends upon the flexural rigidity of the
super-structure,modulusofthecompressibilityof thefoundationsoil,thicknessof theraft,lengthof thesection
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 r
a
f
tis analysed as a whole in each of the two perpendicular
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

23. SURVEY OF AVAILABLE I-ITERATURE 17
the sub-grade conslstsof an infinitearray of individualelastic springs each of which is not affected by others.
This method is more or less same as the famous soil linemethod.
When conditions,asmentionedabove, for flexiblefoundationsare not satisfied ,a method based on closed
form of solutionof elastic plate theory has been suggested.The distributionof deflectionand 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.
i
: The code also lays down that:
(a) Sizeand shapeof the foundationadopted affectsthe magnitudeof subgrademoduluswhich should
be taken into consideration.
(b) Considerationmust be given to the increasedcontact pressure developedalong the edges of the raft
on cohesivesoilsand the oppositeeffect on granular soils.
(c) Expansionjoint should be provided when the structuresupported by the raftconsistsof severalparts
with varying heights and loads or there is a change in the direction of the raft.
(d) Thiscodedoesnot explicitlyprovideany guidanceas tohow factorsemphasisedin (a)and (b)above
should be allowed for.The secondpart of the coderelating to constructionaspect is still not printed.
There has not been any furtherrevision and this code was reaffirmed in 1987.
4.16 EleventhIntenationul Conferenceof Soil Mechanics a d Foundation Engineering San Francisco,
August 12- 16,1985~~
In the conferencewhile two papers werepresented on instrumentation of pile raft foundation and cap pile soil
interaction,there was no recommendation or paper on design of raft foundation.
4.17 FoundationDesign and Construction by M.J. Tomiinson, 5th Edition, 1986"
There is no significantchange 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 ConcreteDesigner Handbook by CharlesE. Reynoldsand James Steedman,
10thEdition, 1988~'
There is no change in recommendationsfrom what was done in the earlier edition published in 1981
4.20 BuildingCode Requirements in ReinforcedConcrete -ACI -318 -1989~'
Building code requirements since their second edition in 1977have 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 referencehas been made to 'design procedure for combined footings and mat
i
sper reportprepared by ACIcommittee336'and alsotoa paper 'simplified design offootingsby' Kramrisch,
Fritz and RpgersPaul published in American Societyof CivilEngineersProceeding,V. 87,NOSM 5,October
1961,p. 19.

24. 18 RAFT FOUNDKTIONS-DESIGN AND ANALYSIS
4.21 Foundation Engineering Handbook by Hsai-Yang-Fang2nd Edition, 1 9 9 1 ~ ~
This edition has omitted the chapter on mat foundation which was originally'includedin first edition.
4.22 Design of CombinedFootings and Mats -ACI committee336 2R -88Publishedin 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 suggeststhat:
(a) Maximum unfactored design contact pressure should not exceed the available soil pressure deter-
mined 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) Combinedfootings and mats are sensitiveto time dependent sub surfaceresponse. 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 areato the node, effectivedepth, and change of modulus with depth. The use of uncoupled
springs in the model is a simplified approximation. The time dependent characteristicsof 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 misleadingresults.
(c) Caution should be exercised when using finite element analysis for soils. Without good empirical
results, soil springsderived form values of subgradereaction 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 subgrademoduli to obtain an adequatedesign.
(d) The response of a footing is a complex interaction of the footing itself, the superstructureabove,
and the soil. That interaction may continue for a long time until final equilibrium is established
between the superimpos&lloadsand the supportingsoilreactions. Moments, shears,anddeflections
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 accuratedeterminationof the contact pressures
and associated subgraderesponse.
(f) For mat foundationsmodulus of subgradereaction cannotbereliably estimated on the basis of field
plate load tests becausethe scale effectsare too severe. I
(g) Mats may be designed and analysed as either rigid bodies or as flexible plates supportedby elastic I
foundation. A combinationanalysis is common in current practice. An exact theoretical design of
mat as plate on an elastic foundationcan be made. However a number of factors like, difficulty in 1
I
projecting subgraderesponses,variation in soil properties both horizontal and vertical, mat shape, * ;
variety of superstructureloads and assumption in their development and effect of superstructure
stiffnesson mat rapidly reduceexactnessto a combinationof approximations.The design is further
affected by excavationheave.
(h) After propottioningthe 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.

25. i SURVEY OFAVAILABLE LITERATURE 19
(i) In casecolumn spacing is less than 1.75divided by hor themat is very thick and variation of column
loadsand spacingis not over2096, mat may be designed by treating it asarigid body and considering
I strips both ways. These strips are analysed as combined footingswith 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 criteriais not met with an approximateanalysiscan be made using the method suggested
by ACI Committee 336 in 1966.
(k) Computeraidedfinitedifferences,finitegridorfiniteelementmethodscan be used wherecomputers
are available. The report gives details of these3 methods. In any of these 3 methods node pressure
should not exceed the safe bearing pressure value recommended by the geotechnicalengineer.
(1) A mat analysisis only as good as the soil parameters. Since it is very difficult for the geotechnlcal
engineer to provide accurate vdues of moGulus of subgradereaction, the structural designer may
do the parametric study, varying the value of K over range of one half the furnished value to 5 or
10times the furnished value.
(m) The analysisand design of combinedfootingsandmats isa soil-structureinteractioneffortin which
there is no uniquemethod to determine mat deflection. Thedeterminationof mat deflectionextends
far beyond the analysis of a beam or finite element model to the prediction of subgrade response.
Theprediction of subgraderesponse,though partofthestructuralanalysisof themat, ismore elusive
than designerswish to admit.Experience with extensivemeasurementsof both foundationloadings
and subgraderesponse are needed to develop a high degree of confidence in the method selected.
A very close working relationshipmust exist between the geotechnical and structural engineersto
properly analyse comb~ned
footings and mats.
4.23 Foundation Analysis and Design by Bowles, 4th Edition, 1 9 8 8 ~ ~
In this editionanalysisof mat foundationhasfurther been elaboratedconsiderably.Among thedesign 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
computerprogrammes which are relatively easy to use and mat being generally too expensive and important
not to use most refined analyticalmethod available.
The approximate flexible procedure suggested by ACI Committee 436 (1966) has been retained and
elaborated.Further details have been given for finitedifferencemethod, finiteelement method and finite grid
method applicablewith computer.
4.24 Proceedingsof IndianGeo-TechnicalConference 1992,Calcutta, December, 1 9 9 2 ~ ~
This conferencedoes not have papers relating to design and analysisof raft foundation.
4.25 Designs of Foundation Systems -Principles and Prrictices by Nainan P. Kurian, 1 9 9 2 ~ ~
Thebook detailsconventional approachto raft designasa flatslaband beam andslabraft, followingthe Indian
Standard Code of Practice, more on the inverted floor approach. The book only mentions that an integrated
analysisof the beam and slab on the computerby 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

26. 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 13thInternationalConferenceon Soil MechanicsandFoundationEngineering,New Delhi
January, 1994~~
II
A paper by M.F. Randolph was presented as a special lecture on design methods for Pile Groups and Piled i
Rafts.
The paper recalls thatinmajority of thecases where pilesform part of thefoundationfora building or other
structures, the primary reason for inclusion of the piles is to reduce settlements. However, once the decision
has been made thatpiles arerequiredthetraditional design approachhas been to ensurethatthe total structural
load can be carried out by the piles, with adequate factor of safety against bearing failure. However, there is
elasticinteraction'betweenthe raft and soil below, between piles and piles asthe performanceof a pile within
i
agroupisaffectedby thepresence of otherpiles. Thekey question thatarisesinthedesign of pilerafts concerns
I
the relative proportion of load carried out by raft and the piles and the effect of additional pile support on ,
absoluteand differentialsettlements.,Thepaper suggeststhatthisdistributionof load betweentheraftand piles 1
be taken into account. The paper also gives methods by which this proportion of load between the two
components are carriedout.
I
4.27 Soil StructureInter-action -The RealBehaviourof Structures,publishedby the Institution of
StructureEngineers,U.K. The Institution of Civil Engineers,U.K. InternationalAssociation
forBridge and Structural Engineeringin March, 1 9 ~ 9 ~ ~
The above institutions constituted a joint committee under Dr. Sam Thornborn which prepared this report.
Pointing out that,
(i) Red behaviourof structuresin contactwith ground involvesan inter-activeprocess beginning with
the construction phase and ending with a state of balance after a period of adjustmentof stresses
and strains within the structureand within the ground influenced by the structure.
(ii) Actual behaviourof thestructurerelatesto the inherentspatial variations in the ground and it should
be appreciated that these variations are not always readily identifiable by occasional and local
boring, sampling andtesting.
Thereport dealswith thequestionof soilstructureinteraction in 2parts. PariIrelatesto structuressupported
by ground and Part I1for ground supported by structures.
(a) Under structures supported by ground, the report points out that engineers could estimate the
settlementsfor aperfectly flexibleload or they could estimatethe avenge settlementof a rigid load
but in between these limits, the engineers could say nothing.
(b) Analytical methods have been developingso rapidly over the last few years that it is now possible
to obtain solution to many complexproblemswhich a few years ago would have been quite out of
reach. If used sensibly and with discernment,these powerful analyticalmethods can be of consid-
erable 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 understandingof the idealisationsthat 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

27. SURVEY OFAVAILABLE LITERATURE
i
structural load is applied short term settlements take place, the part of the structure in existence
t distorts and the overall stiffnessgradually increases.The cladding is then added and may substan-
tially 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.
(d) The report has an appendix which has reviewed currently availabletechniques for the analysis of
the total soil structure system. More readily available computerpackages that utilise these techni-
ques, have been listed in the appendix.
(e) 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 structuralanalysis. It is a poor physical model. Theresultsof analysisbased on use
of this model may be excessively sensitiveto the pattern of applied load.
(f) 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 extensionof this method where elastic theory is used for strainsonly and
then stressesare calculatedusing the variousdeformationmoduliof the soil is better approximation.
In a further improvementof a layered coniinuum theexact stressesand strains in a layered soil mass
are calculated.
(g) Super structure stiffness has a marked influence on the behaviour of the raft and should not be
ignored although the quantitativeassessment 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
stiffnessof the building.Sincethe raft is in intimatecontactwith the supportingsoil,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 ignoredeformation 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.

28. DESIGNAPPROACH AND
CONSIDERATIONS
Summary of methods suggested by various authors discussed in Chapter 4 would indicate that basically two
approacheshave been suggested for analysing the behaviour of raft foundation:
A. Rigid foundation approach
B. Flexiblefoundation approach
5.1 RigidApproach
In rigid foundation approach, it is presumed that raft is rigid enough to bridge over non-uniformitiesof soil
structure. Pressure distribution is considered to be either uniform or varying linearly. Design of rigid raft
follows convkntional methods where again following two approacheshave been suggested:
(a) Inverted floor system
(b) Combined footing approach
In rigid rafts, differentialsettlementsarecomparatively low but bendingmoment and shearforces to which
raft is subjected are considerably high.
5.2 Flexible Approach
In flexiblefoundation approach,raft is considered to distribute load in the area immediately surroundingthe
column depending upon the soil characteristics. In this approach differential settlements are comparatively
larger but bending moments and shearforces to which the raft is subjected are comparatively low. Analysisis
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 inde-
pendently, i.e., Winklers's foundation
Based on these two basic approaches, methods suggested include simplified methods subject to certain
limitationswhich can be carried out by manual computation.Also now availableare computerbased methods

29. DESIGN APPROACH AND CONSIDERATIONS 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 elementmethod 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.
Sufficientlyaccurate 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 charac-
teristics 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
s is the secondary
compression settlement. The immediate component is that portion of the settlement which occurs simul-

30. 24 RAFTFOUNDATIONS-DESIGN AND ANALYSIS
taneouslywiththe load application,primarilyas aresultofdistortionwithinthefoundationsoils.Thesettlement
is generally not elastic although it is calculated using elastic theory. The remainingcomponents result from
the gradual expulsion of water from the void and corresponding compression of the soil skeleton. The
distinction between the consolidation and secondary compressionsettlementis 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 timerate of settlementiscontrolledby therate at which watercan be expelledhorn
the void spacesin the soil. The secondary compressionsettlement,the speedof settlementis 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 granularsoil that
the expulsion of water from the pores for all practical purposes is instantaneousand thus foundation settles
almost simultaneously with the application of load. In cohesive soil, it takes considerabletime for water to
escapeand thussettlementin cohesivesoilscontinuemuch longer.In fact, it has been reportedthat thepressure
under a mat foundation on clay may vary from time to time.
It is usual to assumethat the soil below the foundation is an isotropichomogeneousmaterial 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 stratumisrelativelythin ignoringtheeffect of layering,itmay 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
decreasesthe settlementconsiderably. 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 stratumlike 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 encounteringa 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 significantdepth. Moreover it is
to be remembered that properties of soil constitutingeach layer which determine the shear strength charac-
teristicsand settlementcharacteristicsof the soil becomemoreimportantasraftsaregenerally adopted in areas
where soils of poorer types are'~ncounteredand which some years ago might have not been taken up for
constructionat 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
coefficientof the soilisbelow 0.1mm per second,soiliscohesiveandprobability of surfacewateraccumulated
against basement walls exist'. In such situations,it may be necessary to design raft foundationsof basement
for water uplift also.
The conventional analysisof 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
elasticityorcoefficientof sub-grademodulus),settlementof rigid footingwillbe uniform andthatfor aflexible
footing the settlement would be non-uniform and but if this is the case then how can the contact pressure be
uniform(underarigid footing).Inreality wehave asoil snuctureinteractionproblem andthereisanon-uniform
soil pressure and differentialsettlementswithin thefootings.It has been suggestedthat in case of squarefooting
resting on clay on averagecontactpressure of 0.6PIAwithadditional0.1PIAalong edgeswould be reasonable

31. . DESIGNAPPROACH AND CONSIDERATIONS 25
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.
1 For footingson sands a pressure distribution of uniform soil pressure is reasonable.
I Rigidity of foundation getsmodified by the rigidity of super-structure.Arigid super-structurewill not allow
1 differential settlementto take place in foundation.Situationcan arisewhen aparticular column of the building
: may be hanging from the super-structureand 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 structuremeans less differentialsettlement, large variation of contact pressure
i-
i and high bending and shear stress in foundation members. A flexible foundationwith 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 shearfailurein the soil,the contactpressure 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
1 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 bedifferent.Many times it may take several years to achieve final settlement.
There are situations in engineeringpractice where footings areplaced so closeto each other that their zones
of influenceoverlap. Studieshave shown that effect of adjacentfootings may vary considerablywith angle of
shearing resistance. For low values they are negligible. For higher values they appear to be significant
particularlyif footingis surrounded by otherson all sides.Therearepractically no effects in case of punching
shear failure. It is generally recommended that interferenceeffect 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 distributionunder a raft

32. 26 RAFT FOUNDATIONS-DESIGN AND ANALYSIS
Fig. 5.1 (a) is applicable when the mat is supportedon hard rock and column loads are transmitted to the
rock on areas of relatively small sizedirectly under the columns.If the raftrestson a stiff densesoil,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 commonlyjustified 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 settlementsof structure.
A flexible foundation with a flexible super-structuresand a comparatively rigid soil below will behave as
a flexible foundation and would result in large differentialsettlementsand low stresses.Thus:
(i) Arigid member ischaracterisedby high bendingmoments and relatively small,uniform deflections.
Over all differential settlementsare 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 settlementwould be of higher
orders.
Rigidity criteria proposed by various authoritiesare discussedbelow:
5.5.1 Proposed by IS :
2950 (PartI) 19813
Appendix C of this standard gives the method of deciding rigidity of super-structureand foundation. This is
reproduced below:
Rigidity of Superstructure and Foundation
C-1 Determination of the Rigidity of the Structure
C-1.1 TheflexuralrigidityEl of the structure of anysection may be estimatcdaccotding to the relationgiven
below (see also Fig. 5.2):

33. DESIGNAPPROACH AND CONSIDERATIONS
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.
J
H =total~height
of the infill In cm,
E, =modulus of elasticity of frame material in kg/cm2
Ib =moment of inertia of the beam in cm4
where
/
. 1 = Spacing of columns in cm,
h, =Length of upper column in cm,
hl =Length of lower column in cm,
4
I,, =Moment of inertia of upper column in cm ,
Il = Moment of inertia of lower column in cm4
If =hioment of inertia of foundation beam or raft in cm4,

34. 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
C-2 Relative Sti#hess Factor K:
C-2.1 Whethera structure behave as rigid orflexible depends on the relative stimss ofthe structure and
thefoundation soil. This relation is expressed by the relative stimessfactor K given below:
(a) For the whole structure
(b) For rectangularrafts or beams
(c) For circular rafts
where
El =Flexible rigidity of the structure overthe length (a) in kg/cm2
E, =Modulus of compressibilityof the foundation soil in kg/cm2
b =Length of the sectionin the bending axis in cm,
a =Length perpendicularto 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, thefoundation may be considered as rigid
C-3 Determination of CriticalColumnSpacing
C-3.1 Evaluation of the characteristics his made asfollows:
where
k =Modulus of sub-grade reaction in kg/cm3for 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 compressibilityof the soil is the additional property required in this particular case.
5.5.2 ACI Committee,436
Suggesteddesign procedure for combinedfootings and mats -American ConcreteInstitute Journal,October,
196614
Relevant extractsfrom this paper are given below:

35. I DESIGNAPPROACHAND CONSIDERATIONS
29
a
1 Footings supportingjield structures
Continuous strip footings supporting structures which because of their rigidity will not allow the individual
I columns to settledifferentiallyshould be designed as rigid footingswith a linear distribution of soil pressure.
g This distribution can be determinedon the basis of simple statics.
To determinethe approximaterigidity of the structure,an analysismust be made comparing the combined
stiffness of the footings, super-structureframing 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 formulasmay be used in this analysis :
where
E =Modulus of elasticity of the materials used in the structure,kips per sq.ft (metrictons 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 (metrictons per sq.m)
b =Width of footings,ft (m)
An approximatevalue of ElIC
per unit length of buildingcan bedeterminedby summingthe flexuralrigidity
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 settlementis 0.5 for long footing and 0.35 for a squareone.
For K, =0.5 ,the ratio of differentialto total settlementis about0.1.
If the analysisof the relative stiffnessof the footingyields a value above0.5, the footingcan be considered
rigid and the variation of soil pressure determined on the basis of simple statistics.
If the relativestiffnessfactor is found to belessthan 0.5,the footingshall 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
consideredrigid and the variationof soil pressure determinedon the basis of simple statics.
- If the averageof two adjacent span,as limited above,is greater than 1.75/h, the design of the footingshall
be governed by subgrademodulus theories.
For general cases falling outside the limitation stated above, the critical spacing it which the subgrade
modulus theory becomes effective has to be determined individually.
' Evaluation of the factor can be made on the basis of the followingformulae:

36. 30 RAFTFOUNDATIONS-DESIGN AND ANALYSIS
Where K,=SR,r
K, =Coefficient of vertical subgrade reaction, Kips per cu ft (metrictons per cu m)
K
'
,
= basic value of coefficientof vertical subgrade reaction for a square area with width b = 1ft (0.3m).
Kips per cu ft (metrictons 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 elasticityof concrete, Kips per sq ft (metric tons per sq m)
I = Moment of inertiaof footings ft4(m4
For sandy soils the size factor S can be determinedfrom the following formula:
with a limiting value of 0.25 for large footings.
As for clay soils, the shapefactor Scan be determinedfrom 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.
Values for Kt,,
can 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
(K,.L ~ ) " ~
hL=
4 El
where
K,= KB = Modulus of sub grade reaction X Width of footing- units of psf.
L =Total length of foundationmember
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
3 semi rigid, and elastic, if
>I
7~ o w l e s ' ~
found this criteria of very limited application.
5.6 Modulusof Sub-GradeReaction
Oneof theimportanttermsrequiredin analysingfoundationon thebasisof flexible footingsisvalueof 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

37. DESIGNAPPROACH AND CONSIDERATIONS 31
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
differentfrom what can be determined by a simple plate load'test.Major problems associated are:
(a) Soil is not perfectly elasticand results are effectedby the magnitudes of soilpressureanddeflection
3 (b) Footing size affects the value
(c) Footing shape also affects
(d) Depth at which footing is located also affects
(e) Soilstratificationand other changeswithdepthwhich may not show when testing with a smallplate
I (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 differentfactorsto take these problems into account
On the other hand, certain authors have suggested very simple values for modulus of sub-gradereaction
which can be determined from bearing capacity factors used in Terzaghibearing capacity equation.
5.6.1 Recommended by ~ o w l e s ' ~
Hasrelated valueof modulusof sub-gradereaction with safebearingcapacityby therelationKs =36qa 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 Nq are bearing capacity factors,Sc and Sq are shape factorsfor particular soil in
foot units .
Moreover:
N c x B
Sc = I+-
N c x L
General values suggestedby Bowles are given below:
Soil Range of Ks. Kef
~ o o s e
sand 30 - 100
Medium sand 60 - 500
~ e n s e
sand 100-800
clayey sand (Medium) 200 - 500
Silty sand (Medium) 150 - 300
Clayey soil :
qu 5 4 Ksf 75 - 150
4 < q u 1 8 K s f 150 - 300

38. 32 RAFT FOLINDATIONS-DESIGN AND ANALYSIS
5.6.2 IS :
2950 P
a
r
t I Indian Standard Codeof Practicefor Design and Constructionof RafC Foundation
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 modulusof subgrade reaction (k)as applicable to the case of load througha plate of size 30 x 30
cm or between 30 cm wide on the soil i
s given in Table 1for cohesionless soils and in Table 2for cohesive
soils. Unlessmore specific determinationof K is done (seeB-2 and B-3)these values may be usedfor design
of raflfoundation in cases where the depth of the soil affected by the width of thefooting may be considered
isotropicand the extrapolationof plate load test results is valid.
TableI Modulus of SubgradeReaction (K)
for CohesionlessSoils
* Theabove values apply to a square plate 30X 30cm or beams 30cm wide
TableII Modulus of Subgrade Reaction (K)
for Cohesive Soils
Soil Characteristic
* Thevalues apply to a square plate 30x 30cm. Theabove values are bared on the assumption that the average loading
intensity doesnot exceed half the ultimate bearing capacity.
Relative Density
(1)
Loose
Medium
Dense
Modulus OfSubgrade Reactions (K)inkg/cm3.
Standard Penetrationtest
value (N)
(2)
< 10
10to30
30 and over
For dry ormoist state
(3)
1
.
5(95)
1
.
5to4.7
(95to300)
4.7to 18.0
(300to 1146)
Modulus of SubgradeReaction (K)in
K ~ / C ~ ~
(3)
2
.
7 (1
72)
2
.
7 to 5.4(172to 344)
5.4to 10.8 (344to 688)
Soil Characteristic
For submergedstate
(4)
0.9(57)
0.9to2
.
9
(57to 185)
2
.
9to 10.8
(185to687)
Consistency
(1)
Stiff
Verystiff
Hard
Unconfinedcompressivestrength,
kg/cm2
(2)
l to 2
2 to4
4and over

39. DESIGNAPPROACHAND CONSIDERATIONS 33
B-2 Field Determination
In cases where the depth of the soil affected by the width of thefooting may be considered as isotropic the
value of K may be determined in accordancewith 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 locationof the tests depending upon the extent and importance of the structure.
NOTE IS:9214- 1979lays down that Ks can be determined as slope of the secant drawn between the points
correspondingto zero settlement and point correspondingto 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 unrealisticand itsdetermination shall be based on laboratory tests (seeIS: 2720 (PartXI)-1972" and
IS: 2720 (PartX1I)- 1 9 8 1 ) ~ ~
B-3.2 In carrying out the test, the continuing cell pressure may be so selected asto be-representativeof the
depth of average stress influence zone (about 0.5 B to B)
B-3.3 The value of K shall be determinedfrom thefollowing relationship
where
Es=Modulus of elasticityof soil (seeAppendix 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 theabsenceof laboratory testdata,appropriate values of Esand p may be determinedinaccordance
with Appendix A and used in B-3.3for evaluationof K.
8-4 Calculations
B-4.I Whenthe structure is rigid (seeAppendix C)the average modulus of sub grade reaction may also be
determined asfollows:
Average contactpressure
Ks =
Average settlementof the raft
Appendix C lays down the method of determining the rigidity of superstructureand foundation and has
+n dealt with in para 5.5 above.
Appendix A lays down mettrod of determination of modulus of elasticityof soil by field tests or laboratory
tests.
Equation in B-3.3 above is based on work carried outby ~ e s i c ~ ~ .
~ o w l e s ' ~
has observed that the 12throot
of any value will be close to 1 and equation can be considered to be equivalentto