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THE ANALYSIS AND DESIGN
OF RAFT FOUNDATIONS USING
THE MOIRE TECHNIQUE.
A thesis submitted for the Degree o f Master
o f Science in the Faculty o f Engineering of
the University o f London.
by
MASOOP ALI ZAIDI!
B*Sc.(AIig)/ B.E .(C iv il) , D .C 0!« (B a tt.)
Battersea College of Technology,
London.
May 1966.
ProQuest Number: 10803840
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2
ABSTRACT
In structures carrying heavy superimposed loads involving closely
spaced columns Gr other supports, a raft foundation is normally adopted
especially when the soil bearing capacity is low. This type c f foundation
is generally economical i f the area under a group of independent footings
exceeds sixty percent o f the total covered area and offers the additional
advantage of simplifying the wafer proofing of basements. Furthermore,
a raft reduces the d ifferentia l settlement o f supports.
Two methods o f analysis are available for dealing w ith rafts0 In one
method the earth-pressure is assumed uniform and the raft considered as a
reversed floor slab or beam and slab system, the elastic properties of the
soil being ignored. Design methods on this basis are empirical and are
thought to be unrealistic.
An alternative approach treats the raff as a plate on elastic support,
the subgrade properties being assessed from tests on the so il. Methods in
this group are involved and only a lim ited number o f mathematical solutions
o f practical importance are ava ilab le .
The moird technique presents a means o f solving plate problems and
has been extensively used for dealing w ith plates of various shapes w ith
different boundary conditions and for a variety o f loading conditions.
The work reported in this thesis was directed towards applying the main/
technique for solving the particular problems relevant to the design o f rafts.
3
3 ABSTRACT (Contd.)
Rafts models were prepared from black Perspex and sponge rubber v/as
used to simulate an elastic subgrade. If v/as found that subgrade reaction
at low stress was a linear function of deflection, an assumption inherent in
the governing d ifferentia l equation. It was also established that results
obtained on a model can be applied to a prototype providing the model
dimensions and grade o f rubber are appropriately chosen.
Simple raffs o f square and rectangular configuration were tested and
theoretica lly analysed by the fin ite difference technique using a thirteen
point net. The comparison o f results enables conclusions to be established
regarding the u t i l i ty and also the lim itations of the experimental technique.
To my father and
to the memory o f
my mother.
5
. ACKNOWLEDGMENTS
I should like to express ray gratitude to my supervisor, M r.R .C .Vaughan,
for suggesting the problem, fo r his kind help and continuous guidance throughout
the work.
My gratitude is also due to Professor Z .S . Makowski, Dean o f the
Faculty of Engineering, for his encouragement and sustained interest in the work.
M y thanks are due to my fe llow research colleagues, staff of the
Computing Unit and Structures Laboratory, whose co-operation during my work
helped me in many ways.
I also wish to express my sincere gratitude to Pakistan Public Works
Department, M inistry of Works, Government o f Pakistan, Karachi, and
Pakistan High Commission, London, for the generous grant o f study leave.
F ina lly , I would like to thank Mrs. Renee Sousa for the excellent
typing and presentation o f this thesis.
6
CONTENTS
PageBrief survey of existing methods o f raft foundation design,
1*1,1 Introduction, 10
1 .1 .2 Brief history, 10
1 .1 .3 Necessity of adopting raft foundation. 13
1 .1 .4 . Types o f rafts. 14
... II
1.2.1 Categories on the basis o f analysis. 17
1.2 .2 Solid slab ra fts . -17
1 .2 .3 Broad assumptions and principles o f analysis. 18
1 .2 .4 Load on column strip . 19
1 .2 .5 Design. 21
1 .2 .6 Comments, 21
IIS
1.3.1 Beam and slab rafts. 23
1.3 .2 Analysis. 23
1 .3 .3 Design. 26
1.3 .4 Comments. 27
IV
1.4.1 Buoyancy rafts and basements. 28
1 .4 .2 Analysis and design. 29
1 .4 .3 Comments. 30
1 .4 .4 Hipped slab rafts. 32
V1.5.1 The soil line method, 33
1.5 .2 Soil L ine. 33
1*5,3 Plotting soil line , 35
1 .5 .4 General equilibrium o f a beam and slab ra ft. 38
7
CONTENTS (Contd.)
Page
1 .5 .5 Beam line , 39
1 .5 .6 General formula for the moment reduction factor. 40
1 .5 .7 Analysis, 41
1.5*8 Comments, 44
VI
1.6.1 Flat slab rafts on elastic foundation “-Allen & Severn method, 48
1.6 .2 Assumptions. 48
1 .6 .3 Conditions at external boundaries, 51
1 .6 .4 Loading. 57
1 .6 .5 Scope of the method, 60
1. 6 .6 Comments. 60
V II
1.7.1 Introduction and statement o f problem. 62
Chapter 2
2.1 The moirdi technique, 65
2.2 The moWapparatus. 73
2 .3 Model materials, 80
2 .4 Determination of D -the plate bending r ig id ity . 83
Chapter 3
3.1 Finite difference solution. 86
3 .2 General outline o f the method. 873 .3 Difference approximations to basic d ifferentia ls, g33.4 Difference equations relating to transverse and normal
deflections o f a plate o f uniform thickness subjected to
lateral loading. 90
3 .5 Procedure for analysis by fin ite difference, 91
3 .6 Boundary conditions. 91
CONTENTS (Contdo)
3 .7 Other than central nodes*
3 .8 Nodes on or near re-entrant corner,
3 .9 Column loads.
3.10 Accuracy of fin ite difference solution.
Chapter 4 Experimental Work,
4.1 Introduction.
4 .2 Model and prototype dimensional co-ordination.
4 .3 Transfer of model moments to real structure.
4 .4 Some assumptions.
4 .5 Load/deflection experiment for sponge rubber,
4 .6 Moire7test details.
4 .7 Some d ifficu lties associated w ith the testing o f raft
models on moirdL
4 .8 Photographic materials used.
Chapter 5I
5 .1 .1 Theoretical analysis by Finite D ifference.
5 .1 .2 Load vectors.
5 .1 .3 A note oh stiffness matrix.
5 ,1*4 Model N o, 1.
5 .1 .5 Consideration of load.
5 .1 .6 Equilibrium check.
5 .1 .7 Effect o f variation of k value.
5 .1 .8 Model N o ,2.
5 .1 .9 Model N o .3.
5 .1 .10 Model N o .4 .
9
CONTENTS (Conte!0)Page
I!
5.2*1 Reduction of experimental results. 160
5 .2 .2 Experimental results - Model N o . l , 165
5 .2 .3 11 " -M o d e l N o .2 . 167"
5 .2 .4 " " - Model N o .3 . 167
5 .2 .5 " " - Model N o .4 . 174
III
Comparison and discussion o f salient results.
5 .3 .1 Model N o . l . 174
5 .3 .2 Model N o .2 . 173
5 .3 .3 Model N o .3 . 133
5 .3 .4 Model N o .4 . 133
General conclusions. 190
Scope o f further research. 191
References 193
Appendix I 196
Appendix II . 201
10
CHAPTER 1
Brief Survey o f Existing Methods o f Raft Foundation Design.
1.1.1 INTRODUCTION
Foundations are generally divided into shallow or deep foundations-
the latter subdivided info those which are piled and those which are not!
Rafts may come in both the categories depending upon the sub-
structural configuration. Rafts, w ithout basements, laid c Ig s© to natural
ground level are necessarily shallow foundations, while those w ith base
ment and sub-basement floors laid at times even below the sub-soil water
leve l, belong to the latter category.
1 .1 .2 BRIEF HISTORY
Before the advent o f passenger lifts which became a practical proposi-
2tion in the 1870s, and the metal frame construction w ith light panel walls,
domestic and office buildings were generally restricted to five or six stories.
A ll these buildings had external load bearing w alls. Judged on the standards
of today, these comparatively small buildings o f traditional construction, the
equally traditional foundation types, such as brick footings, timber piles,
masonry or mass concrete rafts, were usually but not always adequate.
W ith passenger l i f t the biggest impediment in the way o f vertical
development o f buildings was removed. The vertica l development o f
buildings showed a ll round promise as can be inferred from the fo llow ing
fa c ts :-
(a) more office accommodation in business centres of restricted area*
(b) Without overstepping accepted limits of population density or o f
floor space ratios, i t is possible to greatly increase the open space on a given
site by concentrating the majority o f various accommodation requirements in
one or more high buildings, instead of covering the whole site by buildings of
moderate or low height,
(c) Consequently increased open space improves the day lighting of
the rooms, provides more fresh a ir , relieves the parking and tra ffic problems and
v irtua lly eliminates the hazard of spreading fires.
(d) Ultimate cost per square foot o f accommodation may be more
economical as services can be compactly located.
Thus it was seen that the demand for high buildings was no longer
encountered exclusively in the centres of great commercial c ities, but these were
increasingly required in many parts of the world w ith correspondingly varied
foundation conditions,
This trend had undoubtedly established itse lf by the late seventies o f the
last century, here and in the United States. The trend o f higher and yet higher
buildings has, even today, a great number o f advocates and buildings of vertica l
heights o f about two miles are now being conceived as feasible,^
In 1879, nine storey luxury flats in Kensington had been completed by
Norman Shaw and in 1882 the ten storey Montank Block o ffice building in
Chicago was eonstasefed by Burnham and Roof,
12
The foundations o f ta ll buildings have to carry heavy loads and the
traditional foundation types are inadequate to cater for the needs o f veiftieally
expanding buildings.
W .L .B . Jfenney for the first time in his nine Storey Home Insurance
Building in 1885 (in Chicago) and subsequently in 1890 in the sixteen storey
Manhattan Building, used metal frame w ith light panel walls and a ll steel framework
w ith light f i l le r panels.
A lternative types of foundations were being explored by engineers at this
tim e. The grillage was born in the 1880s. As grillages were perceptible to damage
due to deep adjoining excavations, these were discarded in favour of concrete pier
foundations. These were first used by Sooy-Smith in 1890 in the U .S .A . Concrete
pier foundations were then generally adopted in localities where good soil was
available w ith in 1001 below ground leve l. The piers gave rise subsequently to
precast concrete piles and steel p iles.
As early as 1785, the main principled for achieving the maximum horizontal
spread o f loads to the ground, including the buoyancy princip le o f deep raffs,
seem to have been realised. The buoyancy princip le expressed in modem terms
means that settlement is ch ie fly dependent on the net pressure on so il.
The earliest known raft laid in this country was a th ick mass concrete mat
for Westminster Penitentiary, bu ilt in 1817, but mass concrete rafts and the inverted
vault type did not become very popular. The former were required to be very th ick
to be e ffective . London Docks bu ilt in 1842 have been founded on 7 ft . th ick lime
13
concrete rafts and consequent!/ became heavy enough to offset the gain in the
bearing area* In the case of the latter construction costs were prohibitive*
W ith the advent o f reinforced concrete, the raft began to be w ide ly used
as the thickness needed for reinforced concrete raft can be very small as compared
to that of mass concrete, and the benefit o f i t being light in weight can be fru it
fu lly utilised* From 1914 onwards this type o f foundation became quite common,,
In short, the pressing need for ta ll and yet ta lle r buildings evolved the
follow ing three classes o f foundations:
(a) Spread footings o f reinforced concrete (simplest of the three) acid
steel grillages,
(b) Piles - (timber piles existing since early days) o f steel or reinforced
concrete, either precast or cast in situ, which transfer load as deep as
possible*
(c) Rafts o f reinforced concrete, which provide the maximum spread o f load.
It is this last class o f foundation which is discussed here in some deta il*
1 .1 ,3 NECESSITY OF ADOPTING RAFT FOUNDATION
Generally in a structure when the columns or other supports are
closely spaced, superimposed loads are heavy and soil bearing capacity
poor, a raft type of foundation is adopted. If any combination o f the
preceding three considerations render the area under a group o f independent
4footings more than sixty percent o f the total covered area, raft may also be
the most economical solution* Other alternatives besides being more
14
technica lly involved may also require costly building equipment and
specialist knowledge, both of which may adversely contribute to the cost
of the sub-structure.
A t building sites where the sub-soil water level is compara
tive ly high, such that the basement and sub-basement floors are below the
highest subsoil water level, the hazardous problem of water proofing and
breaking the hydrostatic head of subsoil water pressure is generally rendered
easier by adopting a ra ft. In such circumstances the raft slab, besides
fu lf i ll in g a primary structural requirement, may also serve the useful
constructional need o f rendering easy the task o f water-proofing the
bu ild ing.
1 ,1 ,4 TYPES OF RAFTS
The rafts met in common present-day practice are:
(a) Solid Rafts (with or w ithout thickening o f edges)
(b) Beam and Slab Rafts.
(c) Cellu lar Rafts
5O f more recent origin is the “Folded Slab R a f t , la id in Havana,
fo ra 3081 high bu ild ing .
The “Folded Slab Raft", may thus be considered a special type o f Raft.
The various types of rafts are shown in Figs, 1.1 and 1, 2 ,
15
tViG.L.,
WALLCOL-
G.L.
W A L L
SOLID SLAB R A F T
SOLID SLAB RAFT ( with thic kened edges)
T"
I *j X FI} *
t
H n
BEAM AND SLAB RAFT
CELLULAR RAF
F IG 1-1
16
4 Thick. Slab
FOLDED SLAB RAFT 2 '22
BUOYANCY RAFT
FIG 12
17IS
1.2.1 CATEGORIES ON THE BASIS OF ANALYSIS
Rafts on the basis o f analysis can be divided into two basic
categories,
(1) The earth pressure is assumed to be uniform under the ra ft, and the raft
acts as a reversed floor slab or a reversed beam and slab system, loaded
from below. The elastic properties of the soil are ignored,
(2) The earth pressure is assumed to vary depending upon the experimentalf
data o f the so il, and the raft may be assumed to be supported e las tica lly ,
Under category (1), various types of rafts are analysed and designed as
b rie fly described below.
1.2 .2 SOLID SLAB RAFTS
Loading Intensity - Dead and Live Loads that are to be
transferred to the ground are assessed. The load coming on each individual
column is calculated; a reduction, i f applicable, is allowed as per code
of practice? The total load coming on the columns, divided by the available
area, gives the intensity o f loading on the ra ft. W ith this intensity o f
loading, the raft is considered as a fla t slab and is analysed in accordance
w ith B*S, Code. (C .P , 114~Sec,3),
Since the code caters for only comparatively lig h tly loaded slabs
which are th in and fle x ib le , the method has its lim itations to single or twoSi
storied dwelling houses or buildings founded on sites having very poor
bearing capacity, where a raft o f about six inches thickness w i l l suffice.
IS •
O f the other emp? rica l methods resorted to in the analysis and
design o f such rafts, the one due to Punhan7is discussed h@re0
1 .2 .3 BROAD ASSUMPTIONS AND PRINCIPLES OF ANALYSIS
(1) Bays should approximately be square, o f aspect ratio not more than 1.2.
(2) Possibly a ll the columns should be evenly loaded, a condition that can
only be realised by adopting near-square panels and rigid planning of
partitions.
(3) Column pedestals, serving the same purpose as is served by drop panels
and column capitals in the case o f fla t slabs, should be provided„ The width
o f these pedestals is to be chosen approximately at one quarter o f the column
spacing. As the width o f the column strip is calculated from the pedestal
dimensions, fa ir sized pedestals are necessary , os column strip d w M
not be too narrow for obvious reasons.
(4) Effective depth o f slab assumed at one inch per foot o f clear span
between the pedestals for heavy loads, this may be taken less for loads
which are not very heavy.
(5) Width o f the column strip is taken as width of thb pedestal plus three to
four times the effective depth o f the ra ft. This is generally more than the
w idth determined by assuming 45° disper sion.
.4 LOAD O N COLUMN STRIP.
Supporting system of column strips under columns as shown in
Fig.* 1 <» 3(a).
1In the transverse direction the column strips are C^C^C^C ^
C - C X j C ' C X - C X * while in the longitudinal direction these0 S 2 © / I 2 /
w i l l be C X , C 7— - —, C X - C C . C 0C0— - - - , C ‘ C ‘ C 'T— — etc,5 6 7 7 3 1 J ' 4 2 2 5 6 7
The column strips intersecting perpendicularly w i l l thus form a grid system,,
The p rtio n o f the raft le ft out in the middle e .g . “cdfe1 is assumed to
be supported by the grid formed by the intersecting column strips 0 Load
coming on the slab portion “cdfe1 is transferred to the column strips
«—> and C^C^C^ ------ ■**, in the longitudinal direction and to
C X X X “ and C X X X ' in the transverse d irection .6 1 2 6 7 1 2 7
The suspended central slab panel ‘cdfe1 is assumed to transmit load to
the column strips as a two-way slab in accordance w ith the code. Having
assessed the loads being carried by the column strips, these being triangular
in pattern as shown in F ig . 1,3 .(c ) . , the column strips may be conveniently
analysed by moment distribution i f spans and loads vary or by any other method.
These may sometimes be designed as a series o f fixed beams.
The points of support are taken as knife edges, so the moments at the
supports may be reduced due to fin ite width of the supports. The modified
bending moment diagram is shown in Fig, 1.3(d).
20
SECTION XX
[a]
SOLID SLAB RAFT
[c]
FIG. 13
_ _ _ j 2
21
1 .2 .5 DESIGN
Sizes o f the elements o f raft are fixed as given in 1 .2 .3 . These are
checked for the modified bending moment 1 ,2 .4 .
Punching shear is checked at the f ace o f the pedestal in the usual
manner. Diagonal tension at a point 45° beyond the bottom o f the pedestal
is also kept w ith in the permissible lim it’s.
Reinforcement is provided for the requirements of the modified bending
moments. It is not usual to economise on the reinforcement and for a minimum
at least 50% o f the reinforcement is continued throughout at the top and
bottom, even in regions where it may not s tric tly appear to be necessary.
The general pattern of reinforcement and location of splices are in
accordance w ith the general practice; however, the planning of construction
joints may need special consideration.
1 .2 .6 COMMENTS
Thisssheme of design may turn out to be economical or wasteful depending
upon the fo llo w in g :-
(1) Spacing of columns
(2) Choice of sizes o f pedestals.
(3) Loads - the ir magnitude and nature.
(4) Concrete and reinforcement needed to constitute the grid system.
However, a judicious consideration of a ll the factors by an experienced
designer may produce quite a workable economic foundation even by this
method, which is de fin ite ly semi^empirical in its approach.
\ A t
-A- -A A 22
/
/ I \
/
/ I\\
/ I/ I
\\ \
\ \
n \ c , x L2X ” ” 1 X q X ! \in 1 / \ l \
\
ii /
' j /
\ i\ i
_ _N.
\V
\
X*/
COL
B E A M•SLAB
BEAM AND SLAB R A F T
FIG.V4
1.3.1 BEAM AN D SLAB RAFTS.
A typ ica l example of a beam and slab raft is given in F ig „1 .4 .
Sometimes when the bending moments to be resisted by column strips are
excessive and the thickness of slab given by the preceding method is
considered uneconomical, perhaps only on the consideration of economics
alone, than on anything else a beam and slab raft is adopted,, Detail
comments on this type of foundation appear in 1„3„4,
1.3 .2 ANALYSIS
Suitable dimensioned beams or walls are provided along each row of
columns in the longitudinal as w ell as in the transverse d irection. The
remaining area forming the central panels is provided w ith slabs supported on
these grid beams or w alls. When a supporting wall is not possible some of the
grid beams may turn out to be very heavy.
Assuming a uniform eariS* pressure, the centre of gravity of the column
loads is made to coincide w ith the geometric centre of the area o f the ra ft,
to elim inate any eccentric ity and thereby elim inating the non-uniform
distribution o f pressure. If, however, this is not possible every effort is made
to keep the eccent r ic ity to a minimum.
The grid beams and walls are analysed as flxed-end or continuous beams
v/ith triangular loading as in 1 ,2 .4 .
Slabs are designed one-way or two-way w ith fixed or continuous edges,
A typical example of an idealised beam and slab raft is given in F ig . l04 ,
The value o f the bearing pressure is assessed, approximate dead load
of the raft is estimated and the maximum effective upward load on each panel
is calculated.
From this load, or even sometimes as a percentage o f the total length
o f the raft beam, (5% to 6% of total length) the depth o f the beam is guessed,
the width may be taken as i depth. From the section so obtained the dead
load due to beams is computed for calculating the total dead load o f the raft,
i f the beams are splayed near the column supports, dead load due to splays is
also added. Dead load o f such panel w i l l therefore be the summation of the
fo llow ing; -
(1) Dead weight o f slab
(2) Dead weight o f beams
(3) Dead weight o f splays
This may be denoted by d „
If "A1 is the side dimension of square panel and fp^* be the assumed
bearing pressure, the maximum load a panel o f ground w i l l support w i l l be
equal to A ^ p ^ - d.
Now the load carried by each column is computed as the worst
combination of the fo llow ing;
(1) Dead Load.(2) Probable Live Load.0) Improbable Live Load.
V/ W WThis load may be indicated by ‘ col^, c o ^ / colg— — -e tc . From the
imposed loads on each column and allowable bearingload of pane^excess
bearing capacity or excess load per column is obtained depending whether the
expressior^.pb“ £a}=wcol Is negative or positive. Consider that fo r columns 1
and 3 this expression yields positive values w, and w ^, now w, and w i l l
indicate the excess load bearing capacity o f columns 1 and 3, while for
column 2, assume the expression to y ie ld a negative value wp, wnere w2
'w ill indicate the excess load the column 2, w i l l have to carry.
Effort is made that ~ w,+w3 where w , - w ^. Now the beam
spanning between columns 2 and 3 is made suffic iently strong to withstand a
central load W2, over the span between columns 2 and 3 . The excess load
on column 2 is thus transferred to neighbouring panels.
It is convenient in practice to draw a plan of the whole raft on which
the column loads and the bearing areas are marked. A second plan may then
be made, showing only the excess loads and excess bearing capacities. The
excess loads are then judiciously spread on longitudinal and transverse beams.
This being done, the excess load diagram of each individual raft beam Is drawn.
The bending moments for the design o f beams are then calculated as and
M y where
( 1) M j is the local moment due to spanning o f beam from column
WAto column, calculated as -y j- where W = Apb - £ W d .
(2) M „ Is the excess load moment being equal to v ^ A /2 and(3) Mg is the partial imposed load moment. A lower estimate of
this moment may be only 10% o f while a higher figure may be as much
as 75% of »
The beam is now designed to resist a bending moment equal to
Shears are computed from the loads and moments.
1.3 .3 DESIGN
A slab thickness o f 8 " to 10" Is commonly taken up to column loads of
200 to 250 tons, and a minimum thickness o f 6 " Is adopted for heavy column
loads. Slab steel Is generally restricted from 5 /8 “ to 1" diameter.
Rough approximate sections of beams are guessed, depth at 5% to 6%
of total length o f the beam, and width may be taken at 3- to 4 of depth,
Usual codal restrictions are followed for the reinforcement o f the beams,
For shear reinforcement a continuous stirrup system is adopted since at
any point shear o f e ither kind may occur due to uneven live loading. Though
8not a codal requirement, Manning suggests the fo llow ing relation between
the overall height and bar diameter o f a lin k ,
211 bar link should be not less than 4 1 overa ll,
5 /8 " " " " 5 5 overa ll.
3 " " " " 6 ' overal l .
In arranging the steel in the slab and beams general sagging deflection
o f the raft as a whole may cause high tension in steel, hence care is to be
exercised to provide adequate laps and end hooks.
27
1 .3 ,4 COMMENTS
( 1) In practice such idealised conditions may be rare to fin d . Regular
spacing o f columns and square grids throughout a building is d iff ic u lt to
maintain owing to irregular plots, positioning of lifts and stairhalls, and
other architectural or functional requirements.
(2) Most bye laws insist on the upper stories o f a building being stepped
back fo r consideration o f light and a ir . This is bound to result in the lig h tly
loaded outer bays and heavily loaded central bays of the bu ild ing . Hence
the basic assumption that a ll the columns are evenly or nearly evenly loaded
may not be possible.
(3) W hile assessing the bearing pressure to be adopted for design of
foundation, proper consideration must be given to the settlement that Is
like ly to occur due to the imposed load on the ground. O nly such a bearing
pressure can be taken which would keep the settlement w ith in reasonable
limits w ithout causing any distress To the super-structure. This may at times
reduce the allowable bearing pressure much lower than the safe bearing
pressure o f so il, hence resulting In an uneconomic foundation.
(4) A strong argument sometimes advanced in favour o f semi-empirical
methods o f design and analysis is that in precedence structures that have been
constructed on the basis of these, have withstood the test o f time very w e l l .
This argument when considered w ith reference to rafts may not be quite
tenable.
O nly about twenty to th irty years ago, stresses in concrete and steel
were taken at 600 and 16000 lb . per sq .m . and superimposed live loads for
various categories o f buildings were much higher, say 100 lb , per s q .ft. for
8office floors and the rafts then were designed for fu ll load conditions.
Judged on the basis o f the stresses of 1000 and 20,000 lb , per sq .in .
end superimposed live load for offices at 50 lb , per sq . f t . , these rafts had
a great margin o f safety to cater for reasonable variations due to partial
loadings and local variations in the bearing pressure.
Hence the rafts designed those days were at least 25% stronger than
calculations o f today would warrant and live load estimated for them was
50% too high. Therefore any raft designed today on precedence has to have
some additional allowance for conditions of imposed loading as wel l as for
provision of sections and reinforcement.
IV
,1 BUOYANCY RAFTS AND BASEMENTS 9
Buoyancy rafts and basements are also known as Box Foundations,
(F ig .1 ,2),
When the soil is poor and it is feared that it w i l l settle to dangerous
limits i f loaded, the princip le o f buoyancy is made use o f. The soil is exca
vated to comparatively greater depth and over-burden on it Is thus removed.
The weight o f structure is now balanced against the weight o f over-burden or
displaced earth. Theoretically in such a condition there should not be any
settlement as the net pressure to which the soil would be subjected at the sub
structure foundation depth w i l l be zero. This principle is sometimes made use
of for buildings on poor and highly compressible soils.
An over-burden o f 20' o f earth removed relieves the soil of about one ton
per sq .ft. A substructure would seldom weigh more than a quarter ton per
s q .f t . , and the additional loading of § ton per sq .ft. can be borne by the soli
at foundation level without any danger o f settlement. This additional pressure
can accommodate about six storeys of a normal office b lock,
A few examples o f this type of raft laid in practice a re : -
10(1) Foundations of a power house at Cossipur near Calcutta, 1949.11(2) Foundations at Pier 57, in New York Harbour,
(3) Basement fo r Shell Building, London .^1
(4) Government Building, W hitehall,London.
.2 ANALYSIS AND DESIGN
( 1) Walls are analysed as self-supporting cantilevers.
(2) Earth pressure on walls calculated by methods described in the
Institution of Structural Engineers Code of Practice fo r Earth Retaining
Structures.
(3) To cater for high bending moments at the junctions o f walls and
floors, substantial heels may be provided,
(4) Basement floor slabs are designed as rafts on the lines described
previously in sections 1 .2c4and 1 .3 ,2 .
For Sight or moderate bearing pressures, fla t slabs of uniform thickness
generally, or slabs w ith thickened edges in some cases may be suitable.
30
For high bearing pressures, which is rare in such cases, grids o f heavily
reinforced upstanding beams may be provided0 When settlement is not a con
sideration, a rarity in practice, ligh t floor slabs over independent column
bases may be provided,,
1,4*3 COMMENTS
A t first sight buoyancy rafts may appear to be an ideal solution for poor
soils, and so these have a greater appeal for a novice, yet there are many
practical d ifficu lties which lim it the ir use to a few idealised conditions only*
Some o f these are listed below*
(1) In p rac tica lly most of the soils at depths that are generally required for
buoyancy rafts, the subsoil water level is encountered* Keeping the water
table depressed during construction may ca ll fo r special precautions, tanking
and wafer proofing* A ll this may render the whole proposition of build ing on
such a site uneconomical.
(2) St is not possible to load the substructure as soon as if is constructed* In
water-bearing soils and especially in s ilty clays, t ilt in g of the substructure
may occur under u p lift pressures* To prevent this the fo llow ing remedial
measures have got to be taken;
(a) Ballasting the substructure*
(b) Keeping the water level depressed by continuous pumping*
(c) Anchoring the substructure*
A ll these methods, though they are certa in ly e ffective , substantially
increase the cost o f the substructure*
(3) Cell u lar buoyancy raffs sunk through a soft s ilty c lay suffer from the drag-
down effect on the walls i . e . the tendency of the soil in contact w ith walls o f
the substructure is to load the base slab rather than to relieve it o f load* These
drag-down forces are generally not very easy to account for in the analysis.
(4) As the over burden is removed, most soils swell* If is normal fo r London
clay to swell by 111 for every 20* o f over burden removed. The assumption that
no soil movement occurs is not s tric tly true. Special precautions have got to be
taken for minimising the effects of swelling,
(5) It is very d iff ic u lt , almost at times impossible, to construct deep basements
in bu ilt-up areas where the removal o f lateral pressure by excavation may cause
yield ing o f surrounding ground. This may generally ca ll for expensive under-
pi nning o f buildings on shallow foundations, close to deep excavations.
(6) Lowering o f water fable for basement construction may, in general, result
in the settlement o f the foundations of existing structures.
(7) To counteract u p lift pressure basement floors may turn out to be expensively
th ick .
(8) Water proofing such basements is often a hazard. Great care is to be
exercised in the process o f "tanking" i f concrete is being water-proofed, w ith
"patent additives" which have not stood the test o f tim e.
There can never be a greater nuisance than a leaking basement; i t is
almost impossible to remedy if,so every possible care has to be taken at the
time o f construction.
Mastic Asphalt for tanking has proved well in the past (B, S. 1097-1958),
o f late "polyvinyl chloride or polyethylene" is also being used as an impervious
membrane w ith success,
1.4,4 HIPPED SLAB RAFT
The rafts discussed in Sections IS to iV are the ones that are
considered to be more or less established in practice/ i f not established
in theory. For the foundations of a 308' high building in Havana/
5M artin and Ruiz have used a hipped plate raft shown in Fig..1 ,2 , This
non-conventional rather daring approach is claimed to have effected a
30% saving over the beam and slab ra ft.
Hipped or folded plates, as a structural form, seem to have made
the ir first appearance about th irty years ago and during the past fifteen
years these have received increasing attention in theory as w ell as in
actual practical construction. As hipped plate is a much more economica!
concrete span than a fla t slab, its choice in preference to other types of
structural forms is justifiab le .
As fla t slabs are used as rafts, a logical conclusion may be "‘why
not hipped slabs for ra fts?" Besides being economical, these may have
the added advantage o f spreading the load to an area greater than the
plan area of the build ing* Depending upon the site conditions, considera
tion o f formwork, placing of concrete, excavations e tc.,various arrange
ments of hipped slab rafts are possible.
1.5.1
1.5.2
Since this is a singular example of this type of ra ft, and since the
building has not yet been subjected to the designed intensity o f wind
loading, not much c ritica l appreciation o f this type of raft is possible.
The building is under observation and under fu ll design dead and live
loads, the observed results show excellent behaviour o f foundations,
U VTHE SOIL LINE METHOD
As pointed out in 1 .2 .1 , the second category of rafts does fake into
account the elastic properties o f so il, The soil line method of design was
( put forward by A .L .L . Baker in 1937. Though not mathematically exact,
i t is the first method in this line . The method is described here b rie fly ,
and is largely applicable to beam and slab rafts.
$081 LINE
ABCD, F ig , 1,5(1), may be considered a typical three spanned
beam w ith bearing areas shown shaded.
Resultant earth pressure at any point throughout the raft may be
denoted by we,q+we where S^e ' is constant and ’q 1 varies.
Let the total deflection o f the beam at any point bo *Y which w ill
be the algebraic sum of deflections suffered by the beam at the centre and
at the ends; *Y measured downward being considered positive.
The relation between 'Y 1 and ’q 1 for the bearing area o f a beam,
plotted along its span is defined as the SOIL-LINE of the beam* This
curve for convenience is assumed to be a straight line not fa lling beyond
certain lim its determined experimentally.
34
A
BEAM WITH BEARING AREA MARKED
(ii)
w L
i1 q-w<
LOAD DIAGRAM OF BEAM.
iii)
FIG 1-5
PLOTTING SOIL LINE
Fop soils generally the modulus of e lastic ity is defined as the load
required to compress a unit surface area by unit amount. This Is denoted
by k and can be represented as;
k =s Unit pressure__Total compression
On a homogeneous so il, i f a constant depth Is effective in taking up
compression on the so il, the value o f k may be constant throughout the area
however V may vary from point to po in t.
The deflection *Y' in terms of 'k * and the resultant earth pressure is
calculated in the fo llow ing way:
A bearing area w ith assumed unit pressures as (c f„F ig ,1 ,5 (ii))«
(a) w + q .w e at the ends,
(b) w ~ q w 0 at the centre,
(c) Modulus o f e lastic ity o f soil = k at ends,
(d) " 11 " 51 = nk at centre - n being always
greater than un ity ,
(@) Assuming straight line variation for the pressure from the centre
to the ends.
Considering a particular raft beam w ith general settlement 'S’
and deflection 'Y 1, we may write down the fo llow ing relations, as Is
obvious from F ig , 105 ( ii) .
(~)Y - Wp+qWp w«“ q .w p . . .— ------ --------j~— — eq„1 -5,1 (")css deflection downwards is
positive,
Eq« 1*5*1 can be written in the form:
Y = - (1+i) + we/i i x eq.1.5*2.k n — 1: ♦ — /
K n
(-) sign before - 2. ( 1- —) is introduced since *k' and "nk1 can be k n
interchanged.
Equ* 1 #5.2 is known as the equation o f soil line for a particular area,
wThe gradient o f the so il- line is given by the expression
nwhich w il l be maximum when n = 1, and k has a minimum value of kmj n,
and a minimum value when n and k have maximum values. These maximum
and minimum values o f gradient are w ith in the lim its 2we and w@O T M M M M M O M
kmin kmaxrespectively. For a ll practical purposes n =4 gives a low enough value of
the gradient, so the practical maximum value of 'n 1 may be w ritten as
kj i ___k *> fmin
The Intercept o f the soli line Is given by the expression + we j
— p£ -(l~ "jj1)- which w il l have the fo llow ing two lim iting values:
max
w~ i w ■ i— ( t * -------- ) a n d - r ~ - ( I * * --------- )
Kmin nmax kmin nmaxw
for n = o c these values reduce to equal to + _J=L—mi n
n-1Also from e qu ,1 „5 .2 for condition o f zero deflection q
So the flattest soil line w ill have a valu« o f intercept given hy ine
j w _ , ,
expression + e \nmaxk -max^ nm_. +1 max
Therefore the lim its o f so il- line are given by the fo llow ing equations:
V - „ 2q * w 0 + W e . 1 .Y - - p J — is - — ( ] ~ — — • ) eq*1 *50o 0kmin kmin n.max
Y = ' - ^ (nm ax-1) i 5 4k k (n -hi) e q .la 0 *4 .Kmax Kmax ' nmax '
The majority o f the cases of soils, barring those which suffer from sudden
changes o f safe bearing pressure from spot to spot, are covered by
We/ k ^ V^O i.e . the maximum gradient o f the soil line and the
I 1maximum intercept /3 0 .
Examining equations 1*5*3 and 1*5.4 , i t ?s inferred that the soil
line lim its depend upon
we * n(max)/ kmax' anCl kmin *
The values kmax/ kn1j n and nmax are assessed for a particular site from
experimental data. Pressure tests, settlement records of existing buildings
on sim ilar soils, bore hole logs etc* are a ll made use o f to arrive at these
values*
The value of we is taken as one-quarter to one-third more than the
average intensity o f loading over the raft computed on the basis o f load
coming over the columns and load bearing w alls.
/
GENERAL EQUILIBRIUM OF A BEAM AND SLAB RAFT*
Beam and slab raft may be considered as a horizontal frame kept in
equilibrium by vertica l forces acting on it* Column loads act ve rtica lly
downwards and these may be indicated by
Earth pressure varying as we + qw^ and imparting reactions to
columns, which are considered as points o f support.
fIt is convenient to sp lit up the vertica l faces into component forces
Fa ' Fb and Fc*
Fa - being the force acting at the columns as points o f support due
to uniformly distributed loads o f intensity w0/ considering the raft to be a
reversed floo r system. The column reactions due to wQ loading may be
represented by k ^ ,k 2,k ^ — ———k .
F[j - the balance forces C ^ -k j, ^2 ^2' positive
fo ra heavily loaded column and negative for lig h tly loaded columns.
If eccentric ity , considering the whole ra ft, is zero it is easy to see
Z * b = o.
Fc - fo r varying earth pressure (variation in pressure from the
average pressure we, being given by + qw0, where *qr varies from point
to point) the reactions at the columns are calculated and form the system
of forces F * The force Fc must be distributed throughout the raft so as
to form a balanced force system* Forces Fa,Ff3 and Fc w ill impart
moments and deflections which are denominated as Ma,M j:),M c and
Yq/ Yb and Yc respectively*
It can be seen that Ma and Ya can be calculated in the same way as the
moments and deflections for a beam and floor slab system.
For calculation o f M b and Yb, since the supports w ill not stay at the
same level under forces Fb/ the same methods as employed for the solution of
indeterminate frames may be resorted to*
Forces Fc involve consideration of soil properties and can best be
computed as described hereunder.
Fc - is further considered to be equal to three component forces where Fc ]
depends upon. ^ properties o f the soil of the particular site* fc>) the shape and size o f the bearing area,
and consists o f the variation in pressure throughout the raft from the average w@/
which would occur i f the raft were perfectly rig id .
FC2 " consists o f the variation in the pressure throughout the raft from the average
we/ due to deflection only*
Fc3 “ consists o f the variation in pressure throughout the raft from the average we
due to local variation in the yie ld ing capacity o f soil i . e . due to local hard or
soft patches o f soi I .
BEAM LINE
The line expressing the relation between the variation of the distribution
of soil pressure and the total deflection o f the beam.
For a particular beam load diagram o f which is given in F ig .1 05 ( iii) , thew j_4
deflection Y is given by Y=0*0035q -gy- , where E, I, W and L convey
their usual meanings.
S»nc<£> the intensity of loading may bo shown to approximate the load
pattern given in F ig , 1,5 ( iii) , the deflection Y , due to forces F may bec c
assumedY = 0.0035q. wL4
El
As the deflection Y , under force Fa, w ill be very small as compared
to deflection Yb, under force the total deflection Y o f the raft ignoring
wL.4small quantity Ya, may be written as Y = Yb+0.0Q35q *gj~ eq, 1 .5 *5 .
Equation 1 *5 ,5 . when represented graphically gives the beam line*
GENERAL FORMULA FOR THE MOMENT REDUCTION FACTOR
Equ.l ,5 ,5c gives the general equation of beam lin e . If V 1
be the average slab reaction, assume the maximum value M bx, = tw l^
o f M b, at a certain point be* along the span o f the beam.
If Yb « is the maximum deflection o f a simply supported beam carrying a
maximum moment M bxt, due to a uniform load w, then
Yb , = 5/4 8 ^ = 0.104 twi i
A Yb = 0 .104 r^ ( a s Y b ^ r Y b«) e q . l . 5 ,6
If the moment diagram is a parabola t " l , otherwise i t w il l have to be
m ultip lied by a suitable constant.
The general soil line equation may be written in the form
2 ,a .w
The value of Y from equations 1 .5 .6 and 1 .5 .7 elim inating 'q ' is
given by 2w * + w l4 ,1A, eA 0.0035 /E l
2weYb /k + 0.0035 C
Also Yb( 2we + 0.0035 w l4 )( 1<~ _______ El )
2 we Yb + 0.0035 w l4 ' k “El x C
w l4 k____________ El x v/q____________
1+ .00175 wl/ 1 x k x C x ^ L j — ,4
If Y u / -7 wL4
t wLTwc
y = U , J-= —r — and V = CkE.we twe
1+ OC’ /SZequation 1,5*8 can be represented as r =........"q~|y\/~~
Now for various values o f V 1, a relation between V and *2* can be
plotted on a graph F ig, 1,6 (A fter A *L,L» Baker),
ANALYSIS
A fte r having described soil line , the beam line and the moment
reduction ratio V , the analysis o f raft by this method is done in the
fo llow ing stages,
(D Planning the general arrangement o f beams and slabs forming the
raft grid to suit the architectural drawings and making adjustments that
would fac ilita te the general calculation o f foundation design*
The arrangement o f raff beams may be arrived at in the fo llow ing manner,
(a) Decision about a suitable bearing pressure o f the so il.
42
9
a
7\ V
6
5
3
2
0 X1000
. 0 001 75 Z+ 1" 0 0168 + 1
7 \N 6 KI ' E w c
» _ C K
FIG.V6
( b ) A r e a of foundation around each column may be marked by
squares such th a t the sides o f squares are parallel to column
diagonals.
(c) Beams are best provided from comer to corner o f bearing area
squares*
A fte r the arrangement o f beams has been decided upon, if may be
checked that (i) The total area x safe bearing pressure -
unit dead weight o f raff * Sum o f column loads*
( ii) If the raft is assumed to fracture along the bearing areas, the
isolated footings so formed should continue to support the super
structure w ithout much distress,
( i i i ) Generally, as far as possible and especially on c lay, the centre
o f gravity o f column loads should coincide w ith the centroid o f the
raft*
(2) Computation o f Column Loads,
(3) Considering the general equilibrium o f the raft as a horizontal
frame, determining the forces Fa and for various conditions of
live and dead loads.
(4) Plotting moment diagrams for individual grid beams v iz * M
diagram and the worst condition Mfo diagram.
(5) Adopting beam sections which may be adequate for shear, and
deciding upon the splays o f the beams, such that optimum fle x ib il ity
o f raft beams is made use of,(Beams may be splayed horizonta lly in
<? A
view o f f le x ib il ity ) .
(6) By applying the general formula for V* across the soil line lim its,
p lo tting the worst (M^-hM ) diagrams for each beam.
(7) Having computed the worst pattern for moments and shears, deciding
upon the best possible arrangement o f reinforcement.
An idealised raft beam arrangement as given by Baker is shown in
F ig .1,7(1),
1 ,5 .8 COMMENTS
(1) The experimental investigation involved is Song (much more than
warranted by conventional methods), and its application to a particu la r
case may not be very easy.
(2) The analyst, often used to the concept that moments in a raff depend
to a greater extent on column loads and bearing power o f soil may find to
his dismay that the basic soil constants here are km| n, and nmaxs more
on which the resultant moments on a raft may depend, than on the former
tw o. , ' ■
(3) Since the raff beams are allowed to deflect, these deflections may
develop moments in the beams o f the frame o f superstructure, which may2
be o f the magnitude o f - g y * where is the load per foot run on the
superstructure frame beam. This magnitude of the beam moment Is based
on the assumption that the raff beam is a hundred times as s tiff as the
frame beam. The correction to be applied to the frame beam moments, due
45
! :
4 6
to raft beam deflections not being easy, the designer is prone to adopt
sections and reinforcement for the superstructure members, which are on
the safe side, hence uneconomical. The saving effected in foundations
may thus be offset*
(4) Generally in buildings w ith rafts, columns are very heavy due to
high imposed loads and so very s tiff . These may also be highly stressed
in direct compression. Due to f ix ity o f columns some re lie f o f moment
occurs for the raft beams, while the feet o f the columns w ill carry
additional moments. If these have not been designed for the additional
moments, columns at feet may be stressed at times beyond the allowable
lim its,
(5) The method takes account o f the variation in pressure which is
assumed to fo llow a straight line law w ith in certain lim its, depending upon
the experimental data. The actual variation of pressure may not be linear,
(6) While computing loads on foundations, a general tendency is to err
.'•onthe higher side, and so the load computed may be excessive. In the
case o f such an estimation o f live loads, a convex-upwards deflection
w ill occur, when a ll the dead loads and part of the live load acts on the
foundation. In an unframed building superstructure, this w ill produce
horizontal tension in the walls, which may produce ugly cracks in the
masonry and finishes. If the reverse is the case the walls w ill be in com
pression, which is not damaging, henca erring in estimating the live loads
on a lower side may be better.
(7) Deflection calculations take into account the E ’ value of concrete
a variable facto r. A ctua lly that value o f I: should be taken into account,
which gives results that agree w ith deflection iests, A simple beam tort
may be carried out for this purpose. Such that the same !E‘ value is
maintained, the quality o f concrete during the whole laying process of
foundation should be carefu lly controlled,
(8) In th is method, the designer has to be very meticulous in providing
sections o f beams which are only as rig id as needed,. Sometimes here, as
well as in other cases, where too rig id a foundation has been provided
assuming uniform earth pressure in a certain region, fa ilure may occur due
to lack o f f le x ib il ity o f certain members especially the raft beams. An
example may c la rify th is. Suppose ‘S1 in F ig . l ,7 ( ii) is the soil line o f the
bearing area o f a beam, the beam line for which is given by , The yd -
ordinate o f the point where the beam line intersects the soil line w ill give
the resultant deflection of the beam, For beam let this be y^S30 o15,lc
Evidently a ll the beam lines o f beams having same span as B., w il l pass
through k, the point where B„ intersects q = 0 , Now it is possible to have$
a beam, the beam line for which passes through k and intersects the y axis
at 0 .7 5 Ilc This beam w il l obviously be three times as flex ib le as the first
beam. The deflection for this beam having for beam line is only 0 ,20 ” .
Assuming same column loads for both the beams, since the first beam
is three times as s tiff as the second beam, the ratio o f moments in the beam
1.6.1
1.6.2
4 8
w ill be as
moment in 3,- 3 x 0 .1 5 = 2.25
moment In "gT2“
Nov/ I f has been reinforced for a deflection due to uniform intensity of
loading i . e . def = 0 ,25 " and only nominal steel has been provided at the
top as hanger bars, i t is like ly to fa i l . This beam would not have fa iled
i f i t were designed as flex ib le as Bg, as then it would have to carry much
less moment.
Y L
INTRODUCTION
Flat slab raffs supported on elastic foundations by A llan and Severn
method. The method suggests a relaxation technique for the solution of
resulting simultaneous equations when the fourth order biharmonic plate
equation from theory o f e las tic ity is broken info two second order simul
taneous equations. Finite difference approximation is used fo r obtaining
the relationship between load intensity, soil reaction and def feet Ion .
ASSUMPTIONS
(1) The raff is assumed to act as a slab supported on elastic foundations.
(2) The imposed weight o f the build ing Is transferred.to the foundation
e ither through columns or basement walls, or through a combination o f both,
(3) The basement walls are su ffic iently s tiff along the ir lengths and any
particular w a ll settles uniformly along its entire length,
4 9
(4) k - *he coefficient of e lastic ity o f soil may vary from point to point
and this variation can be accounted for in calculations; however, calcula*
tions become much simpler i f a uniform value for ’k ’ is considered*
The well-known relations from the mathematical theory o f elastic
plates dsemade use o f . The system o f axes adopted and sign convention for
moments and shears is given in F ig ,1 ,8 ,
Bending moments p6r unit length In terms of deflection {measured
positive downwards) are given by:/ ^ c* \
M = -D ( d- v r J 1 6 1^ t'; x " I" N
% = - ° C ^ r + ! -6 ’ 2 ‘
The tw isting moment per unit lengths acx» \
M = + D ( • - ' V ~ 'xy D x 3 y /
where v - is the Poisson’s ratio e
Shears, measured per unit length are given by
<9 M ^ J?',Q = ~ ~ = **■ q*— Vl ) 1 ,6 .3
x 7>k d x \ , ..x
q = in.= . L f D P w ) where vY + -~r,y dy oy x ■ S ” csxV
(Laplace operator in two dimensions)
If D be the flexural r ig id ity o f a slab o f thickness fh,* and Young's
3modulus E, then D = Eh» where v is the Poisson's ratio*
-v&;Again i f q {x , y) be the intensity of loading at a point (x ,y ) measured
positive downwards and k be the modulus o f e las tic ity o f the foundation,
the downwards deflection at (x, y) must satisfy the equation
50
X
+ Q
( i i i )
•• 0 ”" Mxy
FIG. 18
rt i O a
4w = q~kw 1 .6 ,5 .
If w is replaced by w 1 such that w ‘ - Dw and k by k 1 such that k —
equation 11 .6 ,5 can be w ritten as
kV w ^ q - k ’w 1 1*6. 6 .
(which is a biharmonic o f the fourth order»)
2 1A function M foy) -W "w is now introduced to break the fourth
order biharmonic into two simultaneous second order equations such that
V 2w ! = -M
T 7 2M = - (q -k 'w 1) 1 ,6 ,7 ,
Equations 1 ,6 ,7 are now solved for w 1 and M . Having known w*
and M , the stress resultants at any point can be found out by means of
above equations. However i t is necessary that such conditions exist that
equations 1 .6 .7 yie ld a unique solution,
1.6.3 CONDITIONS AT EXTERNAL BOUNDARIES OF THE SLAB WHICHREKlDERlf POSSIBLE TO OWXlNXUNlQlJE SdTOTTON OFIQBsJ
(1) O nly straight boundaries are assumed to exist.
(2) A long the straight boundary parallel in the direction of an axis, say
for instance y-axis, bending moment along x-axis would be zero,
i . e . M = - s - v i i i i ' l = 0 . 1.6 .7(a)
Also i f the w all has not been provided along the boundary
q r 0 , i .e . Q — — >—x D u x
Also d ifferentia ting ( v *
' ' i.OgO.
(3) If an external w all exists it may be assumed to influence the foundation
slab in the fo llow ing manner,
(i) the w all may settle uniformly along its edge,
( i i) the w all imposes no bending moment at the edge of the slab i.e-.
w*~ is constant along the length o f the w all v iz w* " W (where W h
constant) 1 .6 .9 .- Ii. y
d soi ~ i ^ = 0 . 1. 6 . 10.c) ZL*"
RELAXATION METHOD
The method as developed by Southwell and his co-workers is quite powerful
in solving some physical and engineering problems. As if is not possible to
discuss more mathematical aspects o f the relaxation method such as conver
gence o f the relaxation process, the error caused by the replacement o f the
derivatives by the ir fin ite difference approximations, suffice if to say that
the method works w ell in problems involving partial d ifferentia l equations,
where instead o f one, two variables are to be found.
In our case the governing equations are:JL
V w 1 = -M
VZm = -(q -k V )
and the boundary conditions are given by the equations 1 ,6(8 ,9 , 10),
The fin ite difference approximation to the governing equation at a typical
nodal point *0‘ are (c f. f ig , 1.9)
w*« + w b + w b + w ‘ ,~4w‘ + h^ M “ 0 .1 2 3 4 o o
and M^+M2+M g+M ^-4M0 + h ^qg -h^kw = 0 ,
53
5f
6 f
r
4 tY
(i)
2 f
“ 1 "
X
1 3
5 f
t - W a l l
-Wall
I1
1
1
- W a l l 1
0 h tI1
1
I
4 *8 f
( i i )
<I
- W a l l
---------------**X
3 0 1
8 f
( iv )
FI G. 19.
where v /1 , w w w — '-and M M g M — -e tc . are the values of
variables w ‘ and M at the nodal points 1 ,2 , 3, 4 - “ -=— etc .
Equations 1*6.11 w ill hold when the nodal point !o ‘ does not lie on e
boundary, and the mesh is square with its sides* parallel to the axes* It is
diagramatically represented by F ig&U 9 0( i) 0
BOUNDARY CONDITIONS
When the nodal point 50 ' lies on a boundary some points may lie outside
the slab; these points are termed fic titious points, for instance I f is
fic titious as shown in F ig . l „9 ( ii) , and points I f and 2 f, are fic titious in
F ig. 1 a9(iiI)o The values o f w 1 and M w ill evidently be fic titious at the
fic titious points. St is for the elim ination of these fic titious points that the
boundary conditions are made use o f.
If condition 1.6.7(a) and 1 .6 .8 hold, the fin ite difference approximation
to 1 .6.7(a) at the nodal point '0 ! in F ig . 1.9(H) is given by
,' 1f+w,3- 2w ,0+ v (v^ -tw '4 - 2w '0) = 0w
where from w 1 2w * q - w w ‘^ ) 1. 6 , 12.
Frorrs 1.6.12 the fic titious value of w ', can be elim inated in terms of other real1
values o f w !$. This w il l render the first o f equations 1.6.11 as
w * + w ' - 2 w * + !r— M n = 0 . 1 .6 .13 .A 4 u !=v 0
Equation 1.6*13 is used in numerical computation.
S im ilarly fo r the 2nd of equations 1.6*7
where w ^ w 1 end w 'g are fic titious . w '^ is eliminated by equation ?*6* 12
and Equations sim ilar to T.6J2for w 1 and w* work our to be2 ^
w '5 = 2w ,2 -w '6+ — - M 2
, 2 1 .6 .15 .and w ' = 2w , ,-w ,_+ ^5—— Mo
8 4 7 l - v *
Eliminating fic titious values from 1*6.15 we get
M = M^+v(M2+M ^) -2 (1 +v)Mq+ — ^ w^ - w^ - w^ ) 1.6.16*h
Substituting 1 .6 .16 in the second o f eqs.1.6.11 we get
(1 +v)(M2+M^)+2M ^“ (6+2v )M0+ ^ y ^ ’k w '^ fi^qO ~ Q*
h ’ 1 .6 .17 .Equation 1 .6 .17 can now conveniently be used for numeiical computation.
EXTERNAL WALL
When an external w all exists the boundary conditions are given by
eqs. 1 ,6 .9 and 1 .6 .10 .
Att the nodal point ‘O' in R g a1.9 (iv ), making use o f eqs. 1 ,6 .9 and 1o6„10
■ j W,n = w >and M =n ) 1.6,18." ‘0 )
If *W' the uniform settlassent o f w all is obtained eqs.l .6.11 may be dispensed
w ith, in such a case.
A t the corner such as shown in F ig . l .9 ( i i i) , when a boundary w all does
exist eqs. 1 .6 .18 w ill hold, while in the absence o f the w a ll, however, the
fic titious values may be elim inated by making use of 1, 6.12 and 1.6.14
which give conditions for boundary parallel to y~axis0 Similar conditions
for boundary p<zra!!ol to x-axis a re o b ta in ed as
w x ~ 2w fQ * * w v ( w ‘ j +w*2“ 2w Sq) 1 6 , 1 9
and 1_°v_ (w '^+w ‘^ -2w '^ -w 1 y=w*^+ 2w '^) 1, 6.20
h 2
From equations 1*6,12, 1,6.14 and 1. 6 , 19, 1.6020,
w ‘ l = 2w 'o "w ,31.6,21
w '2 = 2w 'o 'w,4
Substituting from eqs.1.6,21 in the difference equation
w’^ - h ^ w ^ h 2^ 0 =
it is seen th*»t M =0 (as h '^ -0) . 1. 6.22
while in the other difference equation, substituting the value of
from eqs,1,6*16 and making use of relations 1, 6,21 and 1, 6*22,
eq. 1 .6 .23 is obtained as
2(M^+M^) + (4w,2+4wI -8w,Q+2w, ~2wl^ )~ h \w , * I /“q = 0 . ,1 r.6v23‘ h
As there w ill be no shear force acting at the boundary at the corner In the
absence o f a w all
^ f 1 = 0 and w ‘ =w5, ~w ‘ -I-w r. 9d x . 2\ y 5 . 0 - 7 8
Substituting for w*^ and w *0 from the equations
vh ^w * ~~ 2w * “*w1 *t’ ■ AAw 6 3 7 1-v 3
and w b = 2w l i!-w ,,+. vh ^8 4 6 ■=--------/VL
I ~v 2
fic titio n w !r Is elim inated and the fo llow ing relationship is obtaineds
2-w ’o +W,3 +w,4 - w ,7+ f l j - j - - (M 3 +M 4 ) + g jf_ v )(h 2 q0 ~ h '2 kw '0)=0 1 ,6 ,2 4 ,
In computation 1*6.23 and 1,6,24 are used*
BOUNDARY AT THE FOOT OF INTERNAL WALL
If "O' is the nodal point, as deflection along the wall w il l be constant,
w,2=V/,g:=w 4 " ^ (Const) 1 ,6 ,25 .
The first difference equation w ill then be
(w1 +w,^«2wIq)+ h = 0 .
i *e, M^= 2w ,q -w ,^ -w , j1.6„.26<
h 2
Computation is done using eqs.l ,6 ,25 and 1 ,6 .26 .
1,6.4 LOADING
As already defined, ‘q 1 Is the intensity o f superimposed loading
on the ra ft. Except when load bearing walls exist, the load Is transferred
to the raft through the feet o f columns.
A t a nodal point *0* this Intensity o f pressure has been termed as
qQ , in which form it appeals in equations. Depending upon the location
o f nodal point four different cases arise, and q computed as shown in
F ig J r lG ,
COMPUTATION OF CONSTANT DEFLECTION 'W ? A LO N G A WALL
The wa lls are assumed to settle by the same amount *w* throughout
the ir lengths. This is computed by considering the equilibrium o f the
whole ra ft.
58
2
h q = Q
4CO
INTERNAL NODE
2f
If
hq = ^ ° o
hq = 2Q
1f
(ii)BOUNDARY NODE
2
4
CORNER NODE RE-E N TRANT CORNER
FIG.NO.
59
Total downwards load acting on the raft must equal the total upthrust
due to compressed foundation „
Total downwards loads = ^ .Q Q
✓ 2Total upthrust ” £ ^ t h k ’w 1
where = 1/ for nocss not on the boundary
= \ t for nodes on a boundary, but not on a corner
= i , fo r nodes on a comer
- I , for nodes on a re-entrant corner
= 0, where deflection is negative u e „ raft rises loca lly*
The constant deflection W, under a walli
where implies a ll nodes whereW = l Q o ~ ^ 2 0<w,<
columns and not walls are located and , takes into account a il nodes
at the foot o f w all *
The raft may have to be split up into component parts for competing
the equilibrium equations for deflection under each w a ll, i f the walls do
not form a continuous system, fo r computing the deflection under each
w all separately,,
RELAXATION CALCULATIONS
A fte r having obtained the fin ite difference approximation relations
for a ll the nodal points, the relaxation calculations are performed in the
usual manner* The two variables w 1 and M may be relaxed either
separately or simultaneously.
60
1.6.5 SCOPE OF THE METHOD
(1) Boundaries o f the raft are regular.
(2) Rafis may have complete externa! wails or internal waifs, or a
combination of both,,
(3) Rafts may have no walls.
(4) The right-angled re-entrant corner at which no rigid external w all
of the building is provided may be taken info account.
(5) The method may also account for any number of wall systems, and
tilt in g of w all can be provided fo r.
1 .6 .6 COMMENTS
(1) The technique is important as if attempts a precise method of
calculation o f flex ib le raff stresses, when the assumptions made are correct
(2) Since the introduction of moment distribution by Prof*Hardy Cross,
numerical technique of successive approximations in engineering mathemati
has found favour w ith most o f the engineers. This is c numerical technique
and has the added advantage that i t can be programmed on a computer.
(3) ft demonstrates that d if f ic u lt raft problems could be tackled by
practising engineers whose general tendency hitherto has been to overcome
such problems by resorting to alternative types of foundations, wh*ch may
generally not be the most economic types.
(4) The method may yield results that are good enough for practical
purposes, although it may become very involved, when interaction
between the walls and the raft is to be accounted fo r.
(5) In the case where beam and slab raft is an obvious choice, the
method cannot be applied.
(6) Relaxation technique becomes very cumbersome when k varies
from point to point,,
(7) Relaxation technique is also very tedious i f thirteen point fin ite
difference pattern is used, hence factorization of plate equation is
essential.
(8) If re-entrant corners are encountered which are not at right angles
the technique may not work,
(9) Tomlinson in his recent book (first published in 1963} has the
fo llow ing to say about this method:
"A recent analysis of stresses by A llen and Severn utilises relaxa
tion method to calculate the transverse deflections and hence the moments
and forces in a raft assumed to act as a slab supported by an elastic
foundation. However in view o f the va ria b ility o f natural soil conditions
and complex stresses in rafts due to unequal loading, unequal settlement,
temperature and moisture movements, i f would seem to the author 9
(Tomlinson) that such treatments have no place in practical foundation
design and the engineer w il l be w ell advised to adopt simpler methods as
those described by Manning"
The method is certa in ly o f practical importance and does not deserve such
a derogatory comment.
VJ l_
INTRODUCTION AND STATEMENT OF PROBLEM
O f a ll the various types o f rafts, the two most common are fla t slab\
rafts and beam and slab rafts. The latter, though at one time quite popular,
as these were economical, are perhaps now no longer economical besides
having the fo llow ing defects*
(1) I f the beams are positioned below the slab to keep the floor
unobstructed, the excavation o f the beams has to be below the raff slab
leve l. This makes the general excavation badly criss-crossed by trenches,
and as the soil is disturbed, its bearing capacity is impaired. The worst
e ffect is that even on a site where the safe bearing capacity can be
assessed as fa ir ly uniform over the whole area, once excavation for beams
is done this would vary from point to po in t. There being no satisfactory
method to find out this varia tion, the designer is le ft to the only alternative
o f taking a low safe value for bearing capacity, which besides being un
realistic may also be uneconomical. If beams are placed above the slab,
usefulness of basement becomes very lim ited, almost entire ly defeating its
very purpose.
(2) The shuttering for the beams is a costly item o f work and may
generally offset the saving contemplated in using a lesser volume of concrete
(3) From the proceeding comments (re f, 1 ,3 .4 ), i f w il l be clear
that even to form an economical grid system of beams the designer is faced
w ith great limitations, and more or less the architectural planning has to be
in accordance w ith the most economical foundation pattern of this type;
this o f course is rarely possible w ith the present trends o f flex ib le
architecture.
Thus i t is seen that the beam and slab raft is not a very satisfactory
type o f raff generally. The fla t slab raff is not subjecled to such lim itations.
It can accommodate flex ib le architectural planning, and the bearing
capacity o f the site once assessed can be relied upon m the soil is not
disturbed by odd excavations.
The existing methods o f analysis and design of solid slab rafts fa ll
into two categories: (1) Semi-empirical methods which do not fake into
account the elastic properties o f so il, (2) Methods considering the elastic
soil properties, for example relaxational method due to A llen Bi Severn-,
involve lengthy calculations and may be out o f the scope of a design office
not having computer fa c ilit ie s .
The purpose o f this research is to find an experimental technique
which can be used for the solution o f rafts and is good enough for
practical purposes.
The M o ir^ method has been used v/ith success for finding out stress
distribution in la tera lly loaded plates. The present effort is to modify
this technique, such that it can be used for the analysis and design of
rafts, which can be treated as plates supported on elastic foundations.
To check the va lid ity o f experimental results obtained i t is proposed
to check these theoretica lly and to establish some useful conclusions.
CHAPTER 2
THE MOIRtf TECHNIQUE 16~17~18
When alternate solid and open regions in two patterns are mads to
interfere e ither mechanically or op tica lly , distinct solid and open region
pattern is observed, due to the phenomenon known as moird effect,,
F igs.2, l ( i ) and ( i i) represent sim ilar patterns formed by alternate
white and black lines. If b 'and '^a re superimposed the resulting pattern
* c* w il l be seen which w ill consist o f a uniform dark fie ld , as the black
lines o f pattern V lie over the white lines o f pattern “a*. In any other
position the resulting pattern, say fd ‘, w il l show partial- dark and lightA l t
fie lds, forming w haty iknown as moird fringes.
C learly, a ligh t zone w ill appear whenever 'b 1 superimposed on *a*
is displaced by a distance equal to the pitch o f the rulings on the patterns.
So in this resulting moire pattern the spacing o f the centres o f the dark and
ligh t fie lds w il l correspond to a displacement equal to the spacing of the
rulings on the orig inal patterns ‘a 1 and ‘b*.
When a set o f alternate black and white lined patterns represented
by lines k and lines I in F ig ,2 .1 (H ) are superimposed and rotated s lightly
w ith respect to one another, a distinct pattern of alternate white and black
lines in a transverse direction appears. This w ill again be a moire pattern.
Considering the sets o f lines k and I, the points where k, and l „
U-2 and I2- - ™ - intersect, the intensity o f ligh t is weaker than in the
F ORM ATI ON OF MOIRE P A T T E R N S .
immediate neighbourhood o f these points of intersections, to the le ft or to the
right o f these points o f intersections. Consequently, a fringe representing the
locus o f the points of intersections o f the k and I lines o f the same denomina-
tion , would appear.
Immediately to the right o f this fringe after some interval the line k.,8
would intersect w ith the line ^ w ith 1^, w ith 1 and so on0 These
intersections again would give rise to another fringe. Subsequent fringes to
the right w il l represent points o f intersection o f lines k j & 1^, k^ & L ,
k - & L e tc ., and k . & I ,, k^ & l „ , k0 & L and so on,3 5 1 4 2 5 3 6
St is thus seen that between two successive fringes the loc ii o f in ter
section shift from a certain set o f intersection points o f kn & ln, k &
I .I# k , , and I on the right, to the points o f intersections o f k and n+17 n+1 n+2 ^ ' r n
I k - and I ~ on the le ft, i . e . two successive fringes d iffe r frorn one n-1 n-1 n-2
another by a constant proportional to the pitch of the grid lines.
Although patterns used in moird method o f analysis may not necessarily
be alternate black and white lines these, depending on the specific problems,
may be c ircu la r grids and moire fringes obtained as an effect o f d iffraction,
refraction or re flection , the discussion here w il l be restricted to straight
ruling grids and the fringes obtained by reflection interference only, the
latter being relevant to plate problems.
In the moird' apparatus subsequently described in 2 .2 , the plate model
is executed as a reflecting surface and the screen is a pattern o f alternate
white and black lines.
68
Under in it ia l load conditions the reflected grid lines are in it ia lly almost
straight and p a ra lle l le d Fig *2 ,2 ). The same model when loaded and
deformed, distorted pattern o f grid lines is seen F ig*2*3 . A superposition o f
both these grid lines as shown in F ig ,2 ,4 exhibits moire fringes. These
fringes as described above are contours o f equal displacements of the grsd
lines in a direction at right angles to the rulings* Choosing a particular
shape o f the screen, these fringes may also be made to represent the contours
o f equal slope. In analysis by moire method they form the in itia l record on
which subsequent analysis is based for the calculations o f stress resultants.
QUANTITATIVE ELUCIDATION OF MOIRE FRINGES
With reference to F ig ,2 .5 , M M j represents the reflecting model
under in it ia l load condition shown by firm line and under fina l load, show?;
? deformed by dotted lin e . Any point P w il l assume a position P1 after the
model has deformed under load. It may be seen that since PP1 is small as
compared w ith distance o f the model from the screen, it can be neglected.
Under in it ia l load condition let S be the image o f a certain point Q
on the screen when reflected from the model surface at point P. Under load
from a deformed model, S w ill be the image o f certain other specific point P.
on the screen when reflected from the model at point P' (P & P1 may be treated
as co incident.) If the model under load has rotated through a small angle
?0 ,. then i t may be shown that ,2
QR = 2 a 0 (1+ /<j2 ) 2 .1 .
as 0 =tan 0 , i f 0 is small.
Model under in it ia l load, (1st exposure)
. F ig02 ,2 „
Model under fina l load. (2nd exposure)
F ig .2 .3 .
71
Model showing moire fringe pattern when both
exposures are taken on the same negative.
F ig. 2 .4 .
20
72
M O D E L
C amer a
R uiedscreen
M O I R E " A P P A R A T U S
CORNER
S QUA RE PLATE P F IG .g j.
Relationship 2.1 is true i f the screen is f la t . By choosing the screen
in the form o f a straight cylindrica l surface (Radius of cylinder R ” 3,5a),
the term fo r can be ignored without much loss of accuracy, and the
simpler relationshipQR = 2a 0 2 .2
is obtained.
THE MOIRE APPARATUS (F ig .2 .6 )
The apparatus essentially consists o f the fo llow ing; ~
(D Model loading frame.
(2) Screen* (F ig ,2 ,7 ).
(3) Cam era,(F ig.2 .8 ).
(4) A r tif ic ia l L ighting.
loading Frame It is a frame on which it is possible to f ix the model
conveniently in a vertica l position. The verticals o f the frame ate slotted
5 section's and the horissontals are slotted channels which a llow for a reasonable
adjustment in the positioning o f model w ith respect to screen and camera.
Loading device consists o f simple right angled levers mounted on
horizontal tubes. The loads are applied ve rtica lly on the levers and are
transferred as horizontal thrusts to the model surface or to the block-board
backing, depending on whether the model is to be applied w ith a point or
uniformly distributed load.
It is essential that the fric tion at the bearings stays a minimum* This
can easily be ensured by a simple experiment which checks the resultant
thrust against an applied load,
moire' apparatus.
Fig. 2 .6 .
Screen of the mo ire'apparatus.
Fig. 2 .7 .
r.
Camera of the moire‘*apparatus.
Fig. 2 .8 .
A few sets o f spirit levels o f "suspender1 or c ircular disc type are used
to check that the load applied is transferred to the model or blackboard
backing in a direction tru ly normal to the surface. The accuracy o f the
results, in ter a lia , depends greatly upon the accuracy of the loading device.
It is experienced that i f the model is not of a big size and is to be
applied w ith loads at more than five or six points, such that the clearance
between the two consecutive loading points is only three to four inches, the
levers when placed near to each other fo u l. In cases like this it is quite
d iff ic u lt to apply the load in this conventional method. W ith the models on
which experimentation was done during this work, it was found that loading
the models by levers was not possible. Small pulleys were subsequently
employed which too did not prove quite satisfactory,
However, it maybe realised that the model can be conveniently loaded
i f i f could be placed in a horizontal position. The d if f ic u lty o f clogging and
fouling levers is thus obviated, and the loads on the models can be easily
suspended ve rtica lly ,
A horizonta lly placed model has got to be photographed through a
mirror, the qua lify o f the photographs then is expected to deteriorate.
Screen The screen consisfe of a circular disc curved in the form of
straight, c ircu lar cylinder w ith a radius equal to three and a ha lf times the
distance o f the screen from the reflecting surface o f the model* The error
introduced due to neglecting second order terms in the formula 2 01 is then
only
The screen disc is adjustable. St is capable o f movement in horizontal
and vertica l planes, besides being able to move along its own axis. The
adjustment in the vertica l plane is necessary to maintain the desired distance
between the model and the screen, in the horizontal plane to have a
properly orientated photograph o f the model, and the adjustment along the
axis o f the screen enables the grid to be orientated along the axes o f co
ordinates of the model.
On the curved screen any pattern of grid can be neatly pasted*
Rulings of Screen For the present work very coarse rulings of
alternate white and black lines, fo r which the word ‘g rid5 is normally used,
were found satisfactory. These may be produced by drawing or scribing
on physical surfaces or may be printed photographically on sensitized
surfaces by optical projection from small negatives. Fine grids called
'gratings1 having up to 70,000 lines /in . have been produced by Sir T 0Merton
at the N .F . Laboratories (1950). The range of interest for moip^ experi
mentation is covered up to 1,000 pnes/in . which are easily available
commercially. About eleven rulings per inch of grid give w ell defined
moir^ fringes for plate problems.
Since the moird fringes are the loc ii o f point?, where slope is the same,
and to know this slope Q R (cf.eq ,2 .2 ) is measured, which is representative
o f the pitch o f the grid and forms the basic unit o f measurement, i t becomes
obvious that for accurate measurement, the rulings should be o f uniform thickness, straight and accurate.
79
Testing Accuracy o f Screen Rulings Photographically by .Moire
An accurate reflecting surface is obtained and photograph o f the screen
is exposed on a negative keeping the reflecting surface tru ly ve rtica l« A
second exposure o f the screen is taken on the same negative by rotating
the screen through a sma S3 angle (3 °to 5°), care being taken that the
reflecting surface is not disturbed during the two exposures* If the resulting
photograph exhibits straight uniformly spaced m oirl fringes, the accuracy of
the screen grid rulings is established*
CAMERA
In the centre of the c ircular screen is a hole which houses the objective
lens o f a camera* An ordinary ZEISS IKO N MAXIMAR camera for photo
graphic plates o f size 9 x 12cm* w ith Tessar 1 :4 .5 , F -5 ,3 (objective)
supplied w ith the apparatus, was found quite satisfactory for obtaining the
molr^ photographs for this work*
The objective o f the camera is fixed in position and focussing is done
by moving the ground glass screen by means o f knurled screws* This is essentia!
as the distance between the objective o f camera and the model is to remain
fixed once the screen has been adjusted*
LIGHTING
Photographs are taken w ith a rtific ia l ligh ting , W ith the apparatus are
supplied four lamps attached to each corner o f the loading frame, throwing
uniform ligh t on the grid screen. The light from these lamps was. not found
30
19very satisfactory in photographing some o f the models by Gupta, who also
found that the intensity o f ligh t was also not adequate. The qua lity o f uniform
19lighting o f the screen was markedly improved by Gupta who used two Colortran
Line Lights dispensing w ith the use o f the lamps supplied w ith the apparatus.
Each set o f lights consists o f six 200 watt bulbs backed by reflectorsc The
position o f line lights, w ith reference to the screen, can be changed at w ill
t i l l an adequate uniform intensity o f ligh t is obtained, as can be adjudged by
means o f an exposuremeter.
It may be mentioned that very good results are obtained by keeping the
lights at the ir maximum intensity and by keeping them as far from the screen
as is possible, even w ithout the use of diffusion screens on the line lights*
MODEL MATERIALS .
For the purpose of obtaining moird' fringes, any material which has a
surface that would reflect the illuminated screen w ithout much loss o f light
and pronounced mirror effect would be good. From this consideration on ly,
perhaps external surface mirrored glass would be the best. However, besides
needing a reflecting surface, there are the fo llow ing other conditions that
require to be fu lfille d by a model m aterial.
(1) The material should be easy to machine and work w ith in the range of
equipment o f a small workshop.
(2) The elastic constants should be known and be o f such magnitude that
the required deformations are easily produced under laboratory loading equipment.
(3) Behaviour o f the material under load w ith regard to creep, fatigue and
temperature should be known.
From the foregoing considerations the two categories of materials
v iz . Plastics and Metals, have the fo llow ing comparative merits and demerits*
(1) For plastics, the flexural stiffness for a plate o f giver? thickness is
less than for a corresponding metal plate . Therefore for any loading it
is possible to use a s ligh tly th icker plate, and hence allow greater
deflections (without significant membrane action), which consequently
produces a better fringe pattern,
(2) Plastics are easy to work and i f unusual model configurations are
used, manufacture o f model is much simpler.
(3) Metals have a constant value o f modulus of e lastic ity which is
generally independent o f temperature and humidity and also creep may
not be very important. W ith plastics these effects have to be accounted
for during experimentation.
(4) Metals can be machined to almost exact thicknesses, hence error
due to variation o f thickness o f plate can be more or less controlied.
However, it is very d iff ic u lt to obtain a suffic iently smooth meted
surface to give a good defin ition in the fringe pattern. The same is
also true about silvered plastics, silvering the plastic being quire a
specialised technique.
82
Opaque black "Perspex" acry lic sheets, although not as good a
reflecting medium as is silvered plastie or polished bras% is found to be
quite a suitable mode! material* It takes good polish, shows no surface
blemishes, has a mirroring effect and is obtainable in a suitable range of
thicknesses, A fte r checking the thickness o f the sheet at a number o f spots,
it is possible to find a portion o f the sheet which is suffic iently big for making
a model and w ith in which the thickness variation is 2% to 3% , During the
whole o f this work only black "Perspex" has been used as model material c
20-21ELASTIC CONSTANTS
Poisson's ratio o f Perspex is 0 ,35 . The value remains essentially
o oconstant over the temperature range - 25 € to +50 C , A t temperatures
O ' -above 100 C large deformations occur and Poisson's ratio approaches 0 ,5 ,
FLEXURAL MODULUS
The modulus o f E lasticity o f Perspex depends upon temperature, creep
o f the material and the rate o f loading, In temperatures ranging from 0°C
o »
to 80 C, the value o f E is given by the empirical relationship
5E = (5*26-,042 T) x 10 (where T is temperature in degrees).
Assuming an average temperature o f 78°F (25,55°C) the value o f v. m
4 ,1 9 x 1 0 can be read o ff the l 0C , l , graphs.
Since just a fter loading, before the photograph is taken, 5 minutes are
generally allowed for creep and an exposure time of about 4 minutes was
given, creep is calculated at (5 + ^ /2 ) 7 minutes. This assumes that
03
during exposure of 4 minutes i t is the intermediate state of the reflected grid
a t two minutes which is photographed.
When exposure time is comparatively large, as has been during this
work, i f may be pointed out that i f during exposure the creep continues, the
qua lity o f the photograph obtained w il l be poor, in the sense that the negative
w ill show the same defect, which is observed when the object moves with
respect to the camera.
It is possible on the moire apparatus to have an idea o f creep after the
load is applied. The model is loaded and exposed say for a period o f five
minutes, and the single exposure negative is developed, I? Hie grid lines
are sharp, distinct and w ell defined during exposure, there has been no creep.
If, however, the grid lines tend to split out and not much contrast is
d iscernib le, creep during exposure is indicated.
Such a test photograph before starting the actual fringe photography
serves to check the fact that loading device and the model do reach a stable
state *A t temperature and fo ra creep time o f 7 mins,, the value of E is
5taken as 4 02 5 x 10 (p *s , ia)
DETERMINATION OF D0
pjL 3The bending r ig id ity enters the moment equations. Since
f f l “ V )i f can be computed i f the elastic constants of the model, material are known
accurately, and "h ," the thickness o f the plate can o f course be measured
accurately, which i f not constant average value may be taken in some cases*
Experimental determination o f ‘D 1 is however possible on moire
apparatus. Especially suitable for this purpose are the two theoretica lly
known cases;
(a) A square plate loaded at two comer points along a diagonal
and supported on the remaining corners along the other diagonal*
(b) A plate o f the shape o f an equilateral triangle, simply supported
along the edges and la te ra lly loaded by a uniformly distributed load
throughout.
In case (a), a square proving plate is made out o f the model material,
care being taken that the thickness over the area o f the plate stays constant
or nearly constant. The corners o f this plate are finished as shown in F ig ,2^(1
which enables supporting and loading at the geometric points where the corner
forms.
If the axes are chosen along the diagonals and the grid lines are kept
along one diagonal, moir^ fringes w ill be straight, equidistant and parallel
to one o f the diagonals.
If 'S1 be the distance between any two consecutive fringes, D is
found out by the relationship, other symbols carrying the ir usual sense*
rs = r P /o • ** •/ 1-v d
Using a square plate for the determination o f 3 D! has the advantage of
serving a quick check on the su itab ility o f model material, as any variations
in thickness o f the plate and *E1 tend to give rise to imperfect fringes, which
may be wavy and k inky, and even unevenly spaced.
The only disadvantage o f this test is that even w ith small loads, i f the
model material is th in , deflections obtained may be excessive. When this is
the case, i t is better to resort to equilateral triangle p la te.
Sometimes it may even be possible to find out the 'D 1 value through a
simple equilibrium check on a model under test.
• CHAPTER 3,
FINITE DIFFERENCE SOLUTION
A raft foundation as previously indicated may be treated mathematically
as a la te ra lly loaded plate resting on elastic foundations* The simplest
assumption that the intensity o f subgrade reaction is proportional to deflection\
is va lid ly made, as the deflection is expected to be generally small and the
linear relationship over this small range is justified.
There is a widespread impression among engineers that the elastic theory
o f plates is far too d iff ic u lt as a design procedure. In terms o f classical series
solution perhaps this is justifiable when less simple shapes and d iffic u lt
boundary conditions are encountered the theory breaks down.
Now the advancement in the use of computers and before that the
development of relaxational methods has to a great extent brought the use of
the elastic theory o f plates w ith in the scope of most design offices,provided
the solution o f the governing equation is sought by replacing i t in its fin ite
difference form. The method of fin ite difference is also sometimes known as
"the elastic web technique" or the method o f "individual displacements"*
23According to Wood, so much work has been done by Marcus, Nielson
and Westergaard for the solution o f some standard p la te by this method, that
engineers indentify fiin ite difference methods solely w ith the bending of slabs,
the method in itse lf beingm^ch more universal in its application* In
engineering i f is perhaps used because o f its great convenience ond power for
obtaining deflections, stresses, c ritica l loads and natural frequences o f anV t
elastic system,In.fact where in a mathematical relation derivatives 0 l'/ ; ,r a x
’7\~ * / (c' / £, , , etc* i* e , slope, curvature, twist e tc . appear, the method
/ d x *
may be applicable, hence it is as general in its application as are the
p reced ing parameters.
GENERAL OUTLINE OF THE METHOD
Over an elastic system, deformations of the system in terms of
individual displacements o f a number o f chosen nodal points (also denominated
as nodal stations, grid points, or simply as nodes or stations) are expressed to
equal the intensity o f externally applied load. Since the externally applied
loads are not uniquely defined, the method may be viewed as an inverse
approach. The loads corresponding to any hypothetical set o f displacements,
each in terms o f a few local displacements, can be w ritten dov/n establishing
what may be called “stiffness equations". The stiffness equations once solved
by any convenient means e .g . computer or relaxation method, flex ib i l ity
equations can be arrived a t. From the f le x ib il ity equations the deflections
o f the system in terms o f any conceivable set o f loads would constitute solution
to any static problem*
In structural problems, the intensity o f loading being known, i t is the
object o f the analyst to find out the displacements to obtain the stress
resultants. In fin ite differences the displacement pattern is assumed and the
intensity o f loading is investigated. The displacements are set up as a set of
linear algebraic simultaneous equations in individual local nodal displace
ments as variables* The right hand side of these equations represent the
intensity o f loading,
DIFFERENCE APPROXIMATIONS TO BASIC DIFFERENTIALS
Assumptions;
(1) The grid chosen is fine
(2) The individual nodal points are close together and the deflections
"w " on the nodal points define the deformed surface of the plate*
The choice o f axes and the numerical denomination of nodes is given in
F ig „3 .1 ,
St is easy to show that to a first order approximation, i f h is the pitch
o f the grid, for the nodal point 'O':
a better approximation for the slope being
( ^ y ) 0 = a r (2w4+3wo-6w2~W ll)
The second derivatives or curvatures are given by?
/ 0 == 1
/ D W . „ s \ = 1
yA
w hile the twist and fourth deiatives by:
( v)w/ ^ 0 hT ( wr 2V"3>
= “ r ( w2- 2wo ^ 4 )
Opera t ing
Mxy =-Myx
M xMxyOx
J f V M x . - ^ d x )W / Qx7 ( Q x t | ^ L d x )
axM ( M y | y * - d y )
■S f e i f f ) f f i f f ! f f l f i f l f f l f f i f f i f i i f f 1 f f l
/ h-------------- r / ^ x - o n --------<( M x y ^ ^ y / Q y ^ d y )
K hQ h
(iii)
T Y P I C A L I N T E RNAL N O D ES
FIG. 3-1
DIFFERENTIAL EQUATIONS relating to transverse and norma I daf Sections of a plate of uniform thickness subjected to lateral locidmg»(SmalI deflection and th in p la te).
Elemental plate o f dimensions dx.dy, showing the various moments and
shears per unit width is shown in F ig „3 .1 ( ii) . The figure also explains the
sign conventions and axes o f coordinates* ‘q* being the intensify o f loading
measured positive downwards w ith reference to Fig«3«1 (i' 1), the fo llow ing
relations may be obtained?
ju ^ Q y - i+ — ■*- = - q 3.1x 0 y
d x Q x
d M y - 3 Mxy _ ^3.2
Mx = ) 3 .4
M y - -D (™ ~ + v 3 .5
M xy = -M yx = D(1 -v ) 3 .6
— - + A " - ~ = q/D 3 .73x-<' dx*<>?~ <>y*
5 V t o = q /D .
For e lastica lly supported,, la te ra lly loaded plates, the load 4cw is added to
the lateral load ‘q 1 to cater fo r the subgrade reaction^ where the plate rises
loca lly *k‘ is zero.
PROCEDURE FOR ANALYSIS BY FINITE DIFFERENCES
The plate is divided into a number of suitable squares (only square grids
have been used throughout the work). The greater the number of squares the
greater w ill be the accuracy and so w ill be the number of stiffness equations.
The accuracy o f the method is subsequently discussed in 3 ,10 .
The comers o f each square w ill form a nodal po in t. Since the problem o f a
plate reduces to the integration o f equation 3„7 , and for a particular case
the solution so obtained, i f i t satisfies'the boundary conditions, stress
resultants can be easily found. It is the solution of this equation which is
attempted in fin ite difference form such that the various boundary conditions
are accounted fo r.
BOUNDARY CONDITIONS
A typ ica l internal node is shown In F ig .3.1 ( II I) . The local nodes
are marked as already shown In F ig .3 0l( i) * Transient subnodes a ,b ,c and d
are also marked in F ig .3 . 1(0 as these are needed for derivation o f the fin ite
difference approximation, subsequently disappearing in the fina l result^
The fin ite difference approximation o f a frypidc# interned %!@decan! be
evaluated e ither by direct substitution for difference equivalents o f the basic
d ifferentia ls in equation 3 .F . or by choosing intermediate stations. The
latter are especially useful when the difference relations at nodal points at
the re-entrant corner and nodes adjacent to it are to be evaluated.
92
Equation 3 ,7 can be written as:
d j? 5x2 zT2 T y 2 5127ay2 a x2. dy2” ^Expanding in the forms as given in 3 .3
20wo-8(w j‘1’w2+w3+w4^ + 2(w5+w6+w7+w8) + w9+v/10'fw l ] ’hvr ^
= h^qg/D 3 .9
i f lD ‘ is not constant at a il the nodal points, various D*s w ith respect to
nodal points v iz , D ^ ,D j,D 2~~ e tc . can be accounted fo r , Equation 3 ,8
M A J +~ V b ( ^ / d x . b y ) k j “ ^ F L ^C l^TsA i c. Oyi }
_?Vl
If *h* is reasonably small i . e . when a fine grid Is chosen and the plate
expected to be o f a uniform thickness: — — = D, and when the fin ite difference approximations
for the derivatives are substituted in equation 3 .10 , equation 3 ,3 is obtained
as a special case.
It may be seen that equations 3 ,9 and 3o l0 do not involve PoIsson‘s ratio: V
except in *Dr«
Equation 3 ,9 can be used for any node which is not less than two grid pitches
away from a boundary, a ll such nodes are termed as central stations.
For square and rectangular plates, the fo llow ing cases arise when a ll
the nodes of grid are considered as is necessary to analyse any p la te .
(1) Nodes lying on a free boundary at least two grid pitches away from a
corner.
(2) Nodes on a free boundary one grid pitch away from a corner.
(3) Corner nodes on a free boundary,
(4) Nodes one grid pitch away from a free boundary and at least two grid
pitches away from the corner,
(5) Nodes one grid pitch away from a free boundary nearest to a corner.
The above five cases can be solved considering:
(a) Bending Moment across a free edge is zero,
(b) Nodes lying beyond the free edges of plate may be considered as
situated on a plate of zero stiffness.
(c) The rig id ity o f the plate in a direction parallel to a boundary for a
boundary station may be considered as one half o f the actual plate
rig id ity in a direction not along the boundary.
OTHER THAN CENTRAL NODES
Consideration o f case (1) in paragraph 3*6, starting w ith the equilibrium
equation 3 ,3 , i t can be seen that besides the transient subnodes !a ! and 'b 1
four fic titious nodes v iz , 1 ,5 ,6 , and 10 are obtained.
Considering !b ' above
( M x ) q = ( M x ) 1 = ( M y ) , = ( M x y ) a = ( M x y ) b = 0
and taking rig id ity o f plate along the free boundary as one half o f the rig id ity
o f plate at a central point, fina l relation for the node is obtained.
Nodes fa llin g in category (2) y ie ld 2 ,6 ,7 ,1 0 and 11 and transient
subnodes a ,b & c as fictitious,from considerations (a), (b) and (s j in 3 ,6 . ,
the fina l relationship is easily obtained.
For corner nodes (3) above, seven fic titious nodes v iz , 1 ,2 ,5 ,6 ,7 , 10
and 11 are obtained besides the three transient subnodes a ,b and c a Starling
w ith the equilibrium equation and from consideration (a):
(Mx)0 = (My)0 also from 'b '
<Mx>. ~ (»^y)2 = (^xy^a = (^xy )b = (^x y )c ~
Considering the r ig id ity o f plate along the boundary as ha lf the normal
r ig id ity , the fina l relationship fo ra corner node is obtained.
For nodes ‘4 1 above, only one fic titious node v iz . 10 is obtained, which
from consideration 'a ' is elim inated yield ing w ^ = w ^-2w ^“ v(w ,-2w ^+w ^).
This value o f w„~ when substituted in the relationship obtained for a central 10 r
node, the fina l relationship for nodes (4) is obtained.
For nodes of category ‘5 1, only two fic titious nodes are 10 and 11.
From the three considerations, the fina l equation for these nodes can be
worked ou t,
NODES O N OR NEAR RE-ENTRANT CORNER
The fo llow ing three cases arise to enable the analysis of the model w ith a
re-entrant corner which has been Included in this work.
(1) Node on a re-entrant corner,
(2) Node on a free boundary one grid pitch away from the re-entrant corner
and at least two grid pitches away from a comer.
(3) Node on a free boundary one grid pitch away from a comer and a re
entrant comer,
In case (1) above only one fic titious node 'd ' is obtained besides the fic titious
transient subnode V .
Starting w ith the equilibrium equation 3 „3and making use o f the consi
derations 'a 1 and *b * in 3 .6 , the fina l relationship for this node is worked ou t.
Consideration ‘c ’ above Is not va lid for re-entrant comer. The rig id ity
at the re-entrant corner w ill obviously be more than one half and less than
one in both x and y directions. It is assumed to be .75 o f the fu ll r ig id ity
o f the plate and w ith this r ig id ity the fina l relationship is obtained.
Node *6' fic titious as i t Is, appears tw ice in the fin ite difference
approximation o f the equilibrium equation and is elim inated once assuming
(Mx ^2 ~ 0 and again (M y)j = 0 , This though not absolutely necessary would
be obvious i f the symmetrical nature of the relationship at the re-entrant
comer is to be maintained* which is essential i f the matrix representing the
stiffness equations is to be symmetrical.
In case (2) above, nodes 4 ,5 & 9 are fic titious besides the transient
subnodes 'a 1 and 'd 1.
Starting w ith the equilibrium equation and taking appropriate rig id ities
o f the plate at various boundary nodes, the fina l relationship for the node is
obtained.
In case (3) above nodes 1 ,5 ,9 & 10 are fic titious besides the transient
subnodes ‘a 1 and 'b 1. These are eliminated as in case (2) above.
The fin ite difference relationships for various nodes including nodes at the re
entrant or near re-entrant comer are given in F igs,3.2 to 3 ,9 ,
It may be mentioned that the central node does not involve any
‘v1 except in D, the rig id ity o f the plate, hence independent o f Poisson's
ra tio .
A ll other boundary stations involve ‘v 1., where the nodal points
appear one grid pitch away from the boundary V appears only in single
power, while for nodes on the boundary v^ is also involved.
The value of V being fa ir ly constant for black Perspex used during
th is work for models, in F igs.3,2 to 3 ,9 , the difference relationships have
also been given w ith the value of "v1 substituted, the fina l form in which
these have been used in the analysis.
The coefficients of the nodes o f the lefthand side of a ll re lation
ships when added algebraically adds up to zero. This serves as an e q u ili
brium check on the working, firs tly when the boundary relationships are being
evaluated and secondly when the stiffness equations for the analysis o f a
model have been completed,
COLUMN LOADS
In the plate equation 13 W = q , q is the intensity o f down
wards loading. For plates supported on elastic foundations this becomes
4 . kwV O > = y D - , v/here kw is the subgrade reaction *
In fin ite difference form the righthand side o f this equation w ith
A l4reference to a particular node ‘o ' becomes q ^ i ' -kQw0 JL In rafts
the column loads are considered, and i f Q q is the column load for a central
97
12
1.1
2
►
Xb
Xd
0Xa
10
KEY D IA G R A M
OPERATING ON tu
2- u -6 +u 2-u -Z /y77S ///~yVA //7V /A //V //S
fOPERATING ON uj
1*657 7 7 7 7 7 7 /7 7 7 A / / / V / A / / 7 / 7
Q oh“IT K I L uj,D
Qoh Kh UUD
Nodes one grid pitch away f rom boundary and at leasttwo pitch as away from corner.
FIG. 3'2
98
+b
12+•a
1 10
8
KEY D IA G R A M
Operating on u i 0
f _ ' f fr n ------------ j i
- 4 + 2 u t2 y2 , - 4 t2 u t2 u 2
•5-5u^ I 8-4u - J u2 i 'V /V / S ? r / / A / / / / / % / / / ? / A / / / / S 7 7 7
Operat ing on luq
2-u -6>-2u
T
2 - u
Qoh2 K h 4 U4,2D
t r — \ L
' 43 *75 - 3 0 5 5 6 *2 3 2 5 -3 '0 5 5 *43875 7 7 7 / / J 7 / 7 / \ / / / / > / /7 7 7 7 / / / / A / 7 / / /
2 4Q-oh K h qj
D 2D
Nodes on boundary at least two grid pitch qwqyfrom corner
99
11
7 2 6
fc +b
2 3
fi
l +
TD
0+a
1 10
' 4 5
KEY D I A G R A M
O p e r a t i n g on tuT fI
f y r f1 2 T o * 2 - 4+ 2 u+2 u J - 3 * 2 u + i r
' 5 - 5 U 2 1 7*5^4M“2 '5 u 2 I
O p e r a t in g on at
7 V 7 / / ~77V7/ } V / / / / /
- 6* 2u
Qo h Kh4
i fr f
i I «'43575 -3 '055 5 *79 35 -2 1775
/ 7 / 7 / A / / / / / A -i f/ 7 7 7 / / / / A / / / / /
1*65
Nodes on boundary one grid pitch away from corner
D 2DUJ
Qoh K h at2D
12
c+
b
04- +d a
8 4
10
100
O p e r a t i n g on
5 - 5 “
KEY D I A G R A M
- 3+2 u+u3 - 2 u- u
V / / 7 7 Y / / / r / / A ! f “
2 -2 u - 3+2U+u^
'B - ^ u 2
Ope r a t i ng onuj n
fi
Qoh Kh uu
iii
f j------------
! !I 1 *
A3 875 —2 1775 2 1 775 -2 1775 4 - •»V /7 '7 7 7 r'
I______ I13 -21775
‘438751 Nodes on c orners
D 4D
Q0h2 Kh4 a.D 4D
FIG. 3'5
101
+c
\
12 3+d
f0
+a
1 10
KEY D IA G R A M
O p era t in g on ai
2 - u-8
O p e r a t in g onui
1*65- 5 ' 3 1*3
-5*3
1*65- 8
Nodes one pitch away from c orne rs
Q*h Kh_D
uu
Qoh Kh* — — w |
D P
FIG. 3-6
102
Ope r a t in q
Ope r a t i nq
9
KEY D IA G R A M
/
onuJ0 '/2-U Yj
' * 5 -5 u 2 —
-4 *5+5 u+u?
- . -7*! +. cn c
i •
0* 5 - 5 u z „ , 2 4
Qoh 3 h1 i
15-uV/7//7/7
-45+51
77T77 / / / /7
J4U 2
D 4D
2 -7 '5 ^ 5 u 2-u
I I
on uu
r 65
-7 *325 Q 0h 3Kh
-4 ' 2 05
1*65
Nodes on re -en tran t corners
103
1'
7 2 6
1
4C +bh
12 34d
04a
1 10
8 4 5
KEY D I A G R A M
O p e r a t i n g on w .
2 - 2 5 u
71
2- u
5-5u^
-4 5 + ’5u+u2; / / / / / / / / / / /|/
875-4 u-2'5u2
•75U 7 -
O p e r a t i n q on w
I 9125
654375
420251
2625
A3 875V l ' I II*-3 05 5
Qoh2 Kh^ UJ “ D 2D 0
J- J*Qoh K h ii|D
. / / / / / / / / / / / / / / / / /
N o des on bou n d a r y one gr id pitch a w a y f ro m
r e - e n t r a n t o r n e r s
2D
104
11
12
*+c
■+d
+a
8
10
KEY D IA G R A M
*75O p e r a t in q on w
*75 u
-6+2 u
'75tO p e r a t i n g on w
202 5 2625
- 5 ‘3 6 1 0 5
i-2‘17751 65
u 2 u*Q oh Kh2D
w
Qo h2 Kh4 wD 2D
Nodes on b o u n d a r y between a corner and
r e - e n t ra n t corn er
105
n od e 'o 1
Q q = qQh2, so fo r a central node in terms of column load the
righthand side o f the stiffness equation becomes
Q c h2 - W 4D D
For other cases if w ill have the fo llow ing value:
"T T " ~ "— 2D °F a noc e on boundary away from corner,
O h2 k w h4 ' o „ o o" D * ” ” ""’45""”— or a comer node .
Q „h 2 3k w„h4° " ° ° fo r a node at the re-entrant corner,D 4D
The ahoye are the values used for computing the righthand side of the stiffness
equations during analysis,
3,10 ACCURACY OF FINITE DIFFERENCE SOLUTION
The error in fin ite difference solution can be made small in two/A
extreme ways advocated in the past: (1) by taking a fin® grid (2) by including
suffic ien tly high order differences, A compromise between the two alternatives
25is suggested by A llen & Windle by systematically incorporating the fourth
order difference approximations on a grid which need not be excessively fin e .
The thirteen point square grid used here does employ fourth order differences.
As the plate is divided info a grid to give nodes on which the
deflections 'w V are assumed* i f the value o f these deflections are such that
the deflected shape o f the plate is exactly defined* the method should yield
106
nearly exact results. This is only possible i f nodes are in fin ite ly close
together (a practical impossibility) yield ing in fin ite number o f simultaneous
equations.
W ith a coarse grid o f 4 x 4 for a simply supported sauare plate*26
Timoshenko has shown that maximum central deflection is obtained w ith 1%
error only* while moments at the central point are obtained w ith about 4i%
error fo r a symmetrical loading case. For asymmetric loading a 6 x 6 grid
27has been used by tivesley to obtain reasonably accurate results,
23According to Wood* for rectangular plates i f the short side is
divided into six portions keeping to a square grid* reasonably accurate
results are obtained, A rectangular grid* besides other regular geometric
shapes* may also be used i f suitably adjusted to produce the required
boundary conditions. Grids other than square grids fend to give less accurate
results and do reduce the number o f resulting equations.
Consequently during this work in the theoretical analysis by fin ite
differences only square grids have been used. The grids adopted ares
(1) For a square raft w ith one central column a grid o f 8 x 8,
(2) For square raff w ith foir symmetrically located columns
a grid of 6 x 6,
(3) For a rectangular raff w ith six columns a grid o f 6 x 10 has
been adopted, the shorter side being divided into six,
(4) For the ‘I 1 shaped raft w ith fourteen columns* a square grid
is adopted* the longer side being divided into nine segments and
the shorter info s ix . These are the same number of divisions as has1§ . been taken by A llen & Severn in analysing the ir representative raft
107
chosen for the Illustration of tho ir relaxation technique.
The results near point loads are notoriously inaccurate„ If the
column dimensions are suffic iently large to compare w ell w ith the grid
size, comparatively good results adjacent to the columns may be obtained.
This a t times w il l increase the number of equations greatiy, which may be
overcome by having f in e r grid sub-divisions near the columns. However,
i f the column load may be made to be uniformly distributed over one grid
size, reasonable results near columns may be obtained, which may be
concluded as a corollary . to paragraph 3 .9 .
108
CHAPTER 4
EXPERIMENTAL WORK
4.1 INTRODUCTION
Structural models may be used w ith advantage in the fields o f
education, research, development, design and construction,, In addition
to providing data on general pattern o f structural behaviour, and providing
means for checking experimentally the results o f analytical procedures,
model studies are generally being established as acceptable methods for
the direct design of structures. The present experimental work has been
done w ith the latter concept in v iew .
A fte r a model has been constructed and experiments on i t concluded,
i f is essential to convert the results obtained on model to predict the
behaviour o f fu ll scale structure or prototype i ,e , the scaling factors by
which model quantities be m ultiplied to obtain the corresponding prototype
quantities, must be known,
4.2 MODEL AND PROTOTYPE DIMENSIONAL CO-ORDINATIO N
V
To render the governing biharmonic equation dimension I ess, cox
ordinates as shown in F ig ,4 ,3 (i & i i) are chosen so that X = L , Y~ L
W /and W = / L , where L is a representative length, which means any
convenient length in the raft slab system,
Kl Dw 3LW 3 WNow V x - ^ r * " “g r
109
and - ^ - = 1 s lH .L ' S X X
-\4S im ilarly a w ■_ 1 / c- W
& x4 ^8 ^ X 4 '
Hence the bi harmonic in a dimension less form can be written as?
if ~ ^a * \ k+ t " )
T 1 d x /y c>x '• d y l q / 4
As applicable to rafts, the relationship w ill be
* 0 £ L + qi.3 KL4c ) x 4 W l B y 4 ~ - g ~ * W 4 . 1 .
Now for dimensionless s im ilarity between the model and the prototype the
righthand side o f equation 4.1 • for the model and the prototype must be
identical i . e . the parameters q l^ and kL.4 for modef and the proto-T T "
type must be the same,
If subscripts'm1 and 'p s refer to model and prototype respectively,
assuming geometrical s im ilarity in plan alone
qml m3 q p., Li 3
>m ... , Dp
and kmL.m4 ^ kptp4
Dm ' De
4 .2
4 .3
from equation 4 .3
4 .4
4 ,4 w ill give scaling factors from the prototype to the model. Once
-rn aJ ^p^m
scaling factor Lo /. has been chosen appropriately, it w ill automatically fsx *7 # ,
value for % A im/ as can seen ^rom E q .4 .2 , It is thus seen that the
sea ling ~down factor to the model from the prototype automatically
fixes the scaling factors for the intensity o f loading.
A fte r the model has been made and tested qm is known, hence qp
can be known. Once stress resultants are known for a specific qp , stress
resultants for any other intensify o f loading can be found out, the law of
superposition being applicable w ith in the range of small deflections „
Now it remains to be examined that the rafts generally encountered
in practice should be able to be represented as models, w ithin the range of
mo I refexperi mentation.
The various variables to be considered are as follows:
km - may vary from 1 to 5 Ib /in ^ (k - values of various easily
available sponge rubbers).
0 ^ » may vary from 9.854 lb , in . to 630.634 lb , in , for Perspex sheets
o f 1 /16” to i " thickness.
3 3kp “ may be assumed to vary from 50 lb /in to 500 lb /in since
only on comparatively poor soils raft foundations are la id ,
6 PDp - may be assumed to vary from 441.944 x 10 to 119,538 x 10 lb in
as rafts generally designed seldom have a thickness o f less than 1281 and more
than 3 6 ",
Varying km in steps of f l ‘ from *1* to '5 ', Dm in five steps from 1/16"p
to i " , Dp In five steps from 441,944 x 10° to 119,325 x 10 , and k in five
steps from 50 to 500, a table has been computed to give the scaling factors
o r mo^e s/ anc corresponding for load intensity scaling
factor. It is seen by examining the table that an appropriate scaling
factor can always be chosen for making a raft model which Is o f convenient
dimensions for moire' tests, i f the approximate linear dimensions o f the pro~
totype along w ith other relevant data are known.
For the purpose of illustration an instance of a very simple case may
be taken,
A square raft 10'xlO1 carrying a single central column subjected to
a concentric load o f 100,000 lb .
Other data for this footing:
t= 18n, E = 3 x l0 ^ lb /sq .In . v = 0.15 k = 100 Ib /ln ^
Dd = 148,935 x 107 lb . in . The scaling down factor for a model
o f 1 /8 " thickFerspex sheet (Dm = 78,829 ib .in ) w ill be 20,8 , i .e . a
model of 5,769" square w ill be required.
Stress resultants obtained on this model for an intensity of loading
giving satisfactory moire fringes can be made to represent stress resultants
■ On/on the prototype for t l# specific Intensity o f loading, such that ‘ ‘ / q .^5=2085
as seen from the tab le .
An extract from the ^ A m and ‘^ / fable as applicable to
1 /8 " th ick perspex models is reproduced In Appendix I.
TRANSFER OF MODEL MOMENTS TO REAL STRUCTURE
The moments obtained on the moire/ apparafus can be worked out
in dimensionfess form. These moments w ill then represent moments on the
prototype fo r the specific intensity of loading as may be seen from the tab le .
Actual moments on the prototype fo r the actual intensity o f loading can then
be easily worked out.
As a requirement o f sim ilitude the value o f Poisson*s ratio for the
model and the prototype must be the same. Since this requirement cannot be
satisfied, the value of v being 0.15 for concrete and 0,35 for Perspex, the
model moments involving V* say at boundaries w ill not yield true prototype
moments.
These w ill however be on the safe side. A t central points where 'v 1
exists only in ‘D* in plate equation, exact values of moments o f prototype
can be obtained by taking the V* value of the prototype material in the
moment equation along w ith the Devalue of the model,
SOME ASSUMPTIONS
For the partial d ifferentia l equation to be applicable to plates
supported on elastic foundations, the fo llow ing salient assumptions are made,
(1) Relationship between the maximum relative deflection of any two
points o f the plate and the thickness of the plate should be such that the
ordinary theory o f bending o f thin plates can be applied.
(2) The subgrade be uniform in character and provide continuous support
for the raft slab and p la te . The subgrade reaction at any point may be given
by Hr «w where k is the modulus of subgrade reaction determined experimentally
and w is the deflection ,
(3) The material be homogeneous and isotropic.
(4) The c ritica l stresses remain w ith in the elastic limits of both the slab
and the subgrade.
(5) "Hie subgrade may be considered as o f in fin ite thickness.
In the models tested, the above assumptions need to be satisfied as
nearly as possible w ith in practical lim its* Per3pex,being the model material,
satisfies to a great extent the assumption of isotropy and is homogeneous 0
Relative deflections can be maintained w ith in reasonable limits by checking
these by means of ordinary dia l gauges before moire*experimentation is started
For the models sponge rubber has been used as subgrade. The sub-
grade reaction in the case o f sponge rubber can be considered as proportional
to deflection w ith in the range o f small intensities of loading generally
encountered in the moire experimentation. To test the su itab ility o f sponge
rubber for the v a lid ity of assumption (2) above, a simple experiment was
devised. Various grades of sponge rubber obtained from manufacturers were
tested.
LOAD/DEFLECT IO N EXPERIMENT FOR SPONGE RUBBER
Two hard blackboard pieces o f 6 ,,x6 ,b4 11 th ick were taken, A
6!,x6" test rubber piece was placed between these two wooden pieces, *ihe
upper blackboard was subjected to a uniform intensity o f loading by
suspending loads through a cable terminating in a brass disc at the upper
end and in a loading hook on the lower, The upper blockboard carried four
corner targets which could be sighted through the telescope of a cathetometer
The load-deflection diagram o f sponge test piece was drawn for
each test piece. It was found that the load-deflection diagram over a wide
range o f intensity o f loading is not a straight lin e , (F ig .4 ,1 ). Due to creep
effects the two readings taken once when the load was increasing and the
other when it was decreasing are not the same* However, i f the rubber is
in it ia lly subjected to an intensity o f loading of about 0 ,8 lb /s q .in . for a few
hours and the load is then removed and the rubber is tested again for load/
deflection, no hysteresis effects are then observable.
In a ll the tests carried out on moir<^ the subgrade rubbers were first
subjected to compression for a few hours then the load was removed for an hour
before the moire tests were started.
Over the range of intensities of loadings from 0.1 lb /sq0in . to A
0,35 lb /sq .in , the load deflection curve may be reasonably taken as a
straight lin e .
For the rubbers tested, the k~vaSues were found to d iffe r from 1 to
36 ,3 lb /in . The apparatus used is shown in F ig .4 .2 .
MOIRE TEST DETAILS
The details o f moire experiments conducted on four models w ith various
loading cases are as follows:
Model N o , 1
The dimensioned diagram o f the model is given in F ig,4.3p
Choice o f dimensions - A simple case o f a raft w ith one central column w ith
a prototype in mind was tested. Since if is conventional to provide a pedestal
in a slab footing, the model also carries a pedestal as shown in the figure.
CA
TH
FT
OM
FT
FR
R
EA
DIN
GS
IN
cm
s.
115d f
53 ‘9 0 0 - ( 0 *000)
83' 600*11 81")
-JZo LOAD ASCENDING
LOAD DESCENDING
no-rAM Size of rubber 6^ 6('511 fiO ” Initial loud 1*0 l b .
94Load intensity138
LOAD IN lb.LOAD-DEFLEGTION DIAGRAM OF SPONGE RUBBER If
SUBJECTED TO VERY HIGH INTENSITY OF LOADING
FIG . A1
116
Apparatus for determination of k value o f rubber.
F ig. 4 .2 .
117
V l --- -
D im e n s io n !e s s form
M O D E L NO. 1
Dimensional form
Pedestal
5-769
LD
K 1- -i
V/
P L A N
2TXT
5-769
S E C T I O N
DIM E N SIO NED S KETCH.
118
Various aspects o f load distribution under the pedestal w il l be
considered for theoretical analysis*
Moire Details
Loaded by suspending loads model kept horizontal*
Photographed through a mirror at 45° fixed to the loading frame,
Plate (1)*In itia l Load i lb*
Final Load 6 lb .
Total 6 i lb ,oAngle of screen 0 *
1st - exposure time 5 min.
2nd - " " 5 m in.
Time for creep 5 min.
Subgrade - Sponge rubber SPjo
Temperature 75 F.
Plate 2*Same details as in Plate 1 except
Screen at 45° *
Plate 3*In itia l Load i lb .
Final Load 8 1b.
Total 8 i lb .
Screen at 0 ° ,
1st exposure time 4 min.30 secs*
2nd " 11 4 m in ,30 secs,
Time for creep 5 rnins.
Temperature 75°F,
119
Plate 4
Same details as in Plate 3 except Screen - 45°.
Plate 5
Same details as in Plate 1 except subgrade rubber used was W-2610.
oSame details as in Plate 5 except Screen - 45 ,
Plate 7
In itia l Load £ lb .
Final Load 6 lb .
Total 6 i lb .
Rubber W - 2623
Screen - 0°
1st exposure time 4 m in .30 secs.
2nd “ " 4 m in .30 secs.
Time fo r creep 4 m in,30 secs.
Temperature 77°F.
Plate 8
Same details as for Plate 7 except Screen - 45°.
Plate 9
In itia l Loading £ lb .
Final Loading 6 lb .
Total 6£ lb .
Subgrade Rubber N o.-W -2 6 3 2 ,
Screen - 0 ° .
120
1st exposure time 4min„30secs.
2nd 11 11 4mih.30secs.
Creep time 4min,30$ecs«
Temperature 78°F.
Plate 10
Same details as for Plate 9 except Screen - 45 °.
Plate 11
Same details as for Plate 9 except rubber used No#-W “ 2 6 l4 ,
Plate 12
Same details as for Plate 11 except Screen - 45°»
Plate 13
In itia l toad \ lb .
Final Load 6 lb .
Total 65 !b.
Subgrade Rubber - SPj - 9 “ x 9" (extended subgrade)
Screen “ 0 ° ,
1st exposure time 4min*30secs.
2nd “ " 4min,30secs,
Creep time 4rnin,30secs.
Temperature 78°F,
Plate 14
Same details as for Plate 13 except Screen ~ 45°,
Model N o ,2
A square raft model 9 ,,x9 l,x l / 8 n th ick w ith four columns symmetrically
placed was chosen for the second model. The centre to centre distance between
121
the colurnn&'vere kept 6" and these were placed at a distance o f l i "
from the boundary. Dimensioned sketch is given in F ig ,4 ,4 „
M o if^ tes t details are as follows, other details being the same as
for Mode I I ,
Plate 1 (Subgrade rubber used SP-j)
In itia l Load i lb , on each co l.
Final Load 7.\ lb ,o n each co l.
Total Load 10+2 lb .
1st exposure time 3min,30secs<»
2nd " " 4m in.l5$ecs,
Time for Creep 4rnin.
Screen 0 ° .
Temperature 75
2
o
Same details as for Plate 1 except Screen 90°
1
Same details as for Plate 1 except Screen-O
Plate 4
In itia l Load i lb , on each co l.
Final Load 1st coj.* l i l b ,
2nd c o l. 3 i lb ,
■■■. 3rd co l. 3 ilb „
4th c o l, l i t b .
Total Load (lG+2)Sb,
1st exposure time 3min»30secs«2nd 11 11 4min.15secs,Time for creep 4m in.Screen 0° Temperature 75 F,
1-54
—i
122
MODEL NO. 2
to
*in
\ i
5 dig, col.
-5 dig, col. -5dia. col.
\
|— 1 5 — H
inCM m
1 - 5 HPLAN
inEi/Td / V V '/ T ?"7 77 r v Z 7 / V V 7 > v / 71 y / ' / / v r 7" y " ? '> ^ > y y y r / r - r i
SECTION
D IM EN SIO N ED SKETCHFIG. 4*4
123
Plate 5
Same details as for Plate 4 except Screen - 90 .
Plate 6
Same details as for Plate 4 except screen - 45 .
Plate 7
In itia l Load - \ lb , on each co l.
Final Load1st Column “ 1'JSb.
2nd C ol, ~ 3 ilb .
3rd C o l. ' - l i l b .
4th C ol. ~3-gSb,
Total Load (!0+2)!b.
Screen - o°
1 st exposure time 3mIn• 15secs•
2nd " 11 4min.30secs.
Time for Creep 4m in.
Temperature 76°F. ■
Plate 8
\ oSame details as for Plate 7 except screen 90 .
Plate 9
Same details as for Plate 7 except Screen 45°.
Plate IQ
In itia l Load i lb , on each column
Final Load , . , nu1st column 21b*2nd c o l, 31b.3rd co l. 41b„4th col „ 1 lb .
Total (10+2)lb, Screen - 0 ° ,
124
1st exposure time 3min.45secs.
2nd 11 11 4min*30secs.
Time for Creep 4m in.
Temperature 76 F.
Plate 11
Same details as for Plate 10 except screen - 90°
Plate 12
oSame details as fo r Plate 10 except screen - 45 *
Model No* 3
A rectangular raft model 13*19" x 7 ,91 " x 1 /8 " th ick w ith six
columns symmetrically placed was chosen for the third model. The centre
to centre distance between the columns was kept 5 ,276", while the
distance between the centre of columns to the edge was kept as 1*318",
Dimensioned sketch is given in F ig ,4 ,5 .
Moird test details are as follows, other details being the same as for Model No*1
Plate 1 (Subgrade rubber used SPj)
In itia l Load i lb * on each co l.Final Load
1st c o l, 2 ilb „2nd co l, 3 f lb ,3rd co l* 2j?l b«4th co l, 2j>!b«5th co l, 35tb*6th co l. 2*2lb*
Total (17*5+3)Ib .1st exposure time 4rnin.30secs.2nd " 11 4min.30secs»Time for Creep 4min,30secs«Temperature 78 F. Screen - 0 ° ,
125
MODEL NO. 3
• 5*dia. cols.
. 5 cl i a c o l s . 5 d i a cols.-
Q —*|
10 a = 13-19 inches. PLAN
• 5 ' <125*
j i*-a—pi* 4 a 4 a- a n
L .SE C TIO N
*5*•125''
h-a—& 4 a- a —i
T. SECTION.
DIMENSIONED SKETCH.
FIG. 4*5
Same details as for Plate 1, except screen
Plate 3.
In itia l load i lb . on each co l.
Final Load
1st <col. S^lbt
2nd n 4 ilb .
3rd ti l£ lb .
4th it l i l b .
5th ii 4 \b ,
6th ii 3 i!b
Total Load (19+3)1 b.
1st exposure time 4mtn«30sec.
2nd 11 l! 4min«30sec.
Time for Creep 4rnin.30sec.o
Temperature 78 F ,
Screen 0 ° .
Same details as for Plate 3 except screen
Plate 5
In itia l Load ilb » on each co l.
Final Load 2 i lb . on each co l.
Total Load (15+3)lb,
1st exposure time [email protected] 11 n 4min.30secseTime for Creep 4min.30secs.Temperature 78°F.Screen 0 ° ,
127
Plate 6
Same details as for Plate 5 except screen - 90°.
Plate 7
in itia l Load i l b . on each co l.
Final Load
1st co l. 2 j lb .
2nd co l. 4 i lb 0
3rd c o l. 2 i!b ,
4th co l. 2 i!b .
5th co l. 4 i ! b .
6th c o l. 2 j lb ,
Total Load (18*6+3)lb.
1st exposure time 4min.30secs.
2nd " " 4min.30secs.
Time for Creep 4min„30secs.
Temperature 78°F„
Screen 0° ,
Plate 8
Same details as for Plate 7 except screen - 90°0
Plate 9
In itia l Load i lb . on each column.
Final loadI 1st co l, 1 lb .j 2nd co l, 5 lb .| 3rd c o l, 2 lb ,I 4th co l. 3 lb .I 5th c o l« 6 lb .j 6 th co l. 2 lb .
| Total Load (19+3)lb.!
128
1st exposure time 4min,30secs.
2nd 11 " 4min.30secs,
Time for Creep 4min»30secs,
Temperature 79°F,
Screen 0 ° .
Plate 10
Same details as for Plate 9 except screen - 90°.
Plate 11
Same details as for Plate 10 except that first exposure was done
at fina l load and the second at in itia l load position*
Plate 12
Same details as for Plate 9 except screen 26.6^
Plate 13
Same details as for Plate 9 except screen *=* 116,6°
Model N o „4 .
A typ ica l L-shaped model raft w ith fourteen columns was chosen
for the fourth model, the dimensioned sketch of which is given in F ig .4 ,6 .
Moiraf Details (Subgrade Rubber SP,)
In itia l Load l / l 0 lb , on each o f the columns*
Final Load:1st col - .4 lb .2nd ii ,84 lb .3rd 91 ,671b.4th l» ,671b.5th II 1.341b.6th II 1.771b.7th II 1 ,.68lb„8th 1! . 8 1b.
MODEL NO. 4 129
5- 6*' - j----------- 4-Ofr -3 4 2 £
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D— ..4
03
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in
I5 6
' --17
'8
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a- ---------------------- 7 . $ " ----------
* -3 75"^d ia . c o lu m n s .
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0 a
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5- 64
I-4.06- 3-425
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•125
3* ----------- 4- 3 '/ ! <*'/ mjm 3L . S E C T I O N
Left hand side T. S E C T I O N .L.125
•125 R ight hand side T. S E C T IO N
DIME N SI 0 NED S K E T C H
FIG. 4*6
2 81
25---
---4*
—2-
8125
130
9th co l. 1.34 lb .10th ii 2.2 lb .11th it .84 lb.12th ii .4 lb.13th ii 1.1 lb.14th .67 lb.
Total Load (14.72+1.4 )ib .
1st exposure time 4min.0sec.
2nd " 11 4min.0sec<,
Creep time 4min,0sec,
Temperature 76°F.
Screen 0 ° ,
Plate 2
Same details as for Plate 1 except screen - 90 .
Plate 3
Same details as for Plate 1 except screen - 24°*
Plate 4
oSame details as for Plate 1 except screen - 114 *
Plate 5
In itia l Load 1/10 lb . on each of the fourteen columns.
Final Load1st co l. 21b.2nd " 1.0 lb3rd " 1.0 lb4th " 0,51b5th " 2.01b6th 11 3.01b7th w 2.01b8th " l.QIb9th " 2«0lb
10th M 3.01b11th " 1.01b
Plate 5 (Contd,)
12th c o l, 0 ,' 13th co l. 0.51b.
14th co l. 1
Total Load (18.4+1 #4 )lb .
1st exposure time 4min .SOsecs.
2nd "
Time for Creep
Temperature 72°F,
Screen ~ 0 ° .
Plate 6
4min.30secs(
4min.30secs,
Same details as for Plate 5 except screen - 90°,
Plate 7
In itia l Load 0,1 lb . on each o f the fourteen columns.
1st co l. a2 lb .2nd " 1 lb.3rd " 1 lb.4th ,5 lb .5th " 1 lb,6th 11 1 lb.7th " 1 lb .8th " 1 lb.9th 11 1 lb.
10th n 1 lb .11th " 1 lb .12th " .2 lb .13th " 1 lb .14th " 0.5 lb .
Total Load (1 1 .4 + 1.4)lb,
1st exposure time 2ndCreep time
4m?n.30secs.4min.30secs«4min.30secs3
Temperature 72°F. Screen 0°,
132
Same details as fo r Plate 7 except screen 90 .
9
In itia l Load 0,1 lb . on each of the fourteen columns.
Final LoadC ol, 1 - .1 lb „ C o l,8 - .4 ib „ C o iJ 2 - ,1 lb
0 ,6 lb , on the remaining eleven columns.
1st exposure timeA o Total (7,2+1,4)lbo 4min,30secs*
2nd 4min,30secs,
Creep time 4min,30secs.
Temperature 76°F,
Screen 0 ° ,
Plate 10
Same details as for Plate 9 except screen - 90°,
Plate 11
In itia l Load 0,1 on each of the fourteen columns.
Final Load1st co l. 0,1 lb .2nd " 2 ,6 lb .3rd ' 2 .6 lb .4th ' 0 .6 lb .5th ’ 0 ,6 lb .6th 1 2 .6 lb .7th ' 2 ,6 1b.8th ' 0 .4 lb .9th 0q6 lb .
10th 1 2 .6 lb .11th ' 2 .6 lb .12th ' 0 .1 1 b .13th * 2 ,6 lb .14th 11 0 ,6 lb .
Total Load(2 1 ,2 + 1 ,4 ) lb .
133
Plate 11 (Contd.)
1st exposure time 4min.30secs.
2nd 11 " 4min.30secst,
Creep time 4min.3Qsecs,
Temperature 76°F.
Screen - 0 ° .
Plate 12
Same details as for Plate 11 except screen 90°.Typical M o ir^ fringe patterns are given in Appendix I I ,
Some D ifficu lties associated w ith the testing of Raft Models on the M oir£ Apparatus.
As already mentioned in 2 .2 , the loading frame o f the moir</
apparatus is suitable for loading models horizonta lly, when the model
itse lf is kept in a vertica l position.
In the beginning it was tried to load models by means o f loaded
strings passing over pulleys, as loading by means of levers was not possible*
The device was found workable i f the loading points were less than six,
but otherwise the pulleys and attachments fouled. As the loading points
were expected to be much more than this, the device o f loading the
models through pulleys was abandoned.
If the model is placed horizontally it can very easily be loaded
by suspending the loads through nylon chords, an easy and accurate device
as the fric tion of the pulley bearings and lever attachments then does not
come into the picture at a l l . It was therefore decided to photograph the
134
models through a mirror,
A small slotted angle stool was made on which the models could
be placed and levelled in position, such that the virtua l image of the
model is located opposite to the central portion o f the screen.
A special 24"x18" mirror (silvered on the outside to cut out the
mirroring effect) inclined at an angle o f 45° to the loading frame o f the
apparatus, was used for photographing the models. The angle of incline
o f the mirror could be accurately adjusted at 45° by using ordinary
longitudinal and c ircu la r s p ir it levels.
PROBLEM OF LIGHT
In it ia l photographs through the mirror did not give very good
results. The fringes showed lit t le contrast and the intensity o f tight
over the whole negative seemed to vary. Although every effort was made
to test that the screen was uniformly lighted, as was tested by mean: of a
very sensitive exposunemeter, (S.E. 1 -Exposure photometer-Salford
E lectrical Inst,Ltd,London).
Subsequently a fter much tr ia l and error, a solution was found in
completely shrouding the mirror and the model such that no other ligh t
except from the screen could possibly reach the model on the mirror.
Normally for photographing the model d irec tly , an exposure time
o f 1 to 22 minutes is adequate. By photographing the model in the manner
described above, the exposure time had to be increased to 3 to 5 minutes
to get good fringes. Sometimes the exposure time for the first and
second exposure generally kept the same, was also varied to obtain
better contrast.
In the end, it was seen that excellent fringes could be obtained
i f proper care and precautions were taken * It is always helpful to take
a few tr ia l photographs prior to actual tests.
Deflections
Before testing the models, dial gauges were fixed at a few
strategic points to assess the maximum relative deflection. Under the
in it ia l load, the gauges were set to zero reading. The fina l readings
were taken under the fina l load condition after the creep tim e. The
difference between the two final readings of the gauges gave the
maximum adjudged deflection.
In some o f the tests these deflections were deliberately not kept
w ith in i the thickness o f the p la te .
PHOTOGRAPHIC MATERIAL USED
For photographing the models Gevaert (Ortho“ 05“ 9x12cm.
A n ti-H a lo ) plates were used and found to be satisfactory. These plates
are o f extra hard qua lity to give maximum contrast, which makes uniform
lighting of screen imperative. As astigmatic effects cannot be tolerated
the smallest aperture was used while faking photographs,,
136
Ilford N -50 (Thin Film Half 7 one Backed) plates were also tried , but the
qua lity o f the negatives, although satisfactory, was not considered equal
to the Gevaert plates*
Developing
Gevaert G~5 developer was used for developing the plates,,
Generally 4 i to 6 minutes was found adequate developing tim e.
ST OP BATH - 1% A cetic acid solution was used as stop bath. The
plates were put in stopbath for 20 to 30 seconds.
Fixing
Ilford IF“ 2 (Acid Hypo Fixing salt) was us<t.$ for fix in g . Five
minutes of fix ing time was found satisfactory.
A fte r fix ing , the plates were washed in running water for about
ha lf an hour to ensure that the quality o f negatives does not deteriorate
w ith the passage o f time*
137
CHAPTER 5*
Theoretical Analysts by Finite D ifference.
I
5^1.1 The raft is divided to form a suitable g rid . The number of grid
stations is restricted by practical considerations, compatible w ith the
computing fa c ility at hand and the accuracy aimed at* The accuracy
o f the solution has already been discussed in Chape3,Sec*104
A fte r the grid has been decided upon, the grid stations are
denominated in order, giving proper regard to the symmetry arising out o f
the loads applied at various columns and the geometry o f the raft* If the
banded nature of the stiffness (coefficient) matrix is desired, it is necessary
to denominate the grid stations in such a manner that adjacent stations
above and below an internal station d iffe r by the smallest whole number
possible, while the adjacent stations to the le ft or right d iffe r by one or
vice versa.
Using appropriate operators on each o f the grid stations, the
stiffness matrix can be w ritten down or computed* Since the number of
equations involved is generally large enough to render hand computation
almost impossible, an electronic computer is necessary.
The theoretical analysis associated w ith this work was done on the
. . . - 28 Ferranti Sirius Computer, The programmes were written in autocode and
29sub-routines were used fo r solving the resulting linear algebraic
simultaneous equations*
138
5.1*2 LOAD VECTORS
In framing the load vectors It may be pointed out that the column
loads need not be m ultip lied by any factors v iz , a factor o f 2 - at the
boundaries, 4 , at the couriers and 4 /3 at the re-entrant comer (Ref,3 ,9 )
15as is necessary in A llen St Severn's Method (Ref, 1 *6 .4).
This is because while working out the operators a continuous
plate is contemplated, the stiffness o f which abruptly changes at the
boundaries. As discussed in 3 .6 , the plate rig id ities at the boundary are
suitably modified and no m ultip lication factors for the column loads are
required.
5 .1 .3 A note about the stiffness matrix - The diagonal elements of the stiffness
m a lm also contain the kw^ terms. These terms are comparatively small
in magnitude and are responsible to render this matrix non-singular and
obtain a unique solution to the resulting simultaneous equations.
The accuracy In computing the diagonal elements is o f paramount
importance. The author can recall an instance when a change o f about
7.9% (diagonal element o f 6.76703961 was reduced by .5329) in the
fifth diagonal o f a stiffness matrix o f 15x15 yielded completely non
sensical results.
Resulting deflections are not so sensitive due to such a change in
off-d iagonal elements. Although these too have a considerable effect
when the fina l values o f moments are considered.
139
While carrying out the analysis o f Model No .4 , having a stiffness
matrix o f 60x60, i t was seen that a change o f 1% in an element la tera lly
situated three columns away from the diagonal, gave values o f deflections,
on the basis o f which when superimposed load was computed for the
equilibrium check, the values o f loads obtained were almost tw ice the
values of applied loads. The moments obtained from these deflections were
absolutely non-sensical.
Numerous such instances may be quoted. A ll these singularly
point to the fact that when computation is aimed at by this method,
scrupulous care is needed to see that the equations are framed accurately
and precisely to obtain worthwhile results.
The d iff ic u lty can o f course be overcome by having a few b u ilt-
in periodic checks in the computer programmes and fin a lly checking the
accuracy o f deflections obtained from considerations o f equilibrium .
The theoretical analysis o f the four models experimentally studied
is b rie fly discussed here.
5 ,1 ,4 Model N o , 1
The model was divided into a fine grid o f 8x8, As symmetry exists
along the centre-lines parallel to the axes o f co-ordinates and also along
the diagonals the 81 grid stations yield only fifteen unknown deflections.
Fifteen fin ite difference stiffness equations were computed using
appropriate operators, k value o f subgrade was experimentally determined
140
to be = 2.459,166 Ib /in ^ .
5 .1 .5 CONSIDERATION OF LOAD
In the model the load was applied through the I'b c l'b c i” high
pedestal (R ef.F ig .4 ,3 ). The pedestal was rig id ly fixed to the raft slab
to simulate the effect o f m onolith ic ity in concrete footings. The app li
cation of load to the raft slab may be considered in the fo llow ing three
possible ways.
(a) The load may be considered to be applied as a point load at
grid point 1, o f the key diagram in F ig .5 .1 (a practical Impossibility,
30but usually considered when such footings are designed by fin ite difference ) 9
(b) The load may be considered as uniformly distributed over the whole
1 " x l 11 area of pedestal and transmitted to the footing slab as point load
acting at the C .G . o f the quarter portion of the pedestal,
(c) The load transmitted to the footing slab as shear along the
periphery of the pedestal. Also the footing was analysed for a uniform
distribution of reactive load, assuming the footing slab to be supported
in e lastica lly , when the superimposed load was considered in manner (a),
(b) and (c) above.
To study the effect o f the variation of k-value of rubber on the
3stress resultants, the k-value was varied from 0.9559166 Ib /in to
3 35,7959166 Ib /in in steps o f 0.25 Ib /in , during the theoretical
analysis.
141
Deflections and moments when the superimposed load was
considered in manner (a) and the footing supported e lastica lly are given
In F ig .5 ,1 .
Deflections and moments when the superimposed load was
considered in manner (b) and footing supported elastica Hy are given in
F ig .5 ,2 .
Deflections and moments when the superimposed load was con
sidered in manner (c) and footing supported e lastica lly are given in
F ig .5 *3 .
The deflections and moments for uniform distribution o f reactive
load corresponding to the above three cases are given in Figs.5 ,4 ,5 .5
and 5 .6 .
5 .1 .6 EQUILIBRIUM CHECK
As at times the fin ite difference equations can be notoriously i l l -
conditioned, the deflections obtained after the solutions o f equations,
invariably in a ll cases, were checked by a simple equilibrium check.
Reactions being assumed to be linearly proportional to deflection,
—■— oIf is easy to see that JL.^ k w h must be equal to the externally applied
load, where k Is the subgrade reaction, w the deflection, h is the mesh
size, and o< a factor being1 - at central nodes, i - at the boundaries, i -a t the corners.J - at the re-entrant comers,
142
\
MODEL NO. 1
/
\ /\ / 13
\ / 10 11\ / 6 7 8/ 1 \ 2 3 4
/ \/ \
/ \
15
14129_5
D e f lec t io ns in inches.
/
KEY DIAGRAM Showing axes of symmetry
0*0000690 *0679 *0609 *0564oopo
000o
rooIT)•0806 *0785
000•1740o
CNCO OO<30CO
0000353•5051uoouo CO
COCOCNo
uouoCOCD
0911 *1845Bending m o m e n t s
1756
Mx
THEORETICAL ANALYSIS [ a s s u m in g central p oint lo a d )
\
FIG.51
143
M O D E L NO.1 (ELASTIC SUPPORT]
/
\\ / 13
\ / 10 11\ / 6 7 8/ K 2 3 4
/ \/ \
\
/15
14129 _5
\\
D e f le c t io n s in inches
'0 694 0683 •0 6 5 5 0617
KEY DIAGRAM S h o w in g axes of symmetry
*0575 8 0 000o
oto
0701 0 660000
C7>COCO•0801 -0785 0747
0001712
uor-toCN
cocotoco
totouo
coCO
0000381988‘516 6
COcoCO
uooCOCOco
•o 0*000 a in lb . in . per in
Mx
*03262074B end in q m oments
636314 03
THEORETICAL ANALYSI S (assuming u.d. load on pedestal)
FIG. 5-2
144
/
MODEL NO.1\ /
\ / 13\ / 10 11
\ / 6 7 8/ ! \ 2 3 4
/ \/ \
/ \
15
14129
D e f l e c t io n in inches
/ '
KEY DIAGRAM Sh owing a xes of s vm m e trv
\
06210696 06580685 0 00 00581Oo
o
inCM
0744 *0733 0702 0 66 3000
inoco
079 3*0 7840001706
o
CD■MrCDCOCO
CD CMCN*084 9 *0829
00003 97*5299 2020cr»cmin
inoCMco
/c o // // co/*0870 67 000
in l b . i n per in.
Mx
03676424
B e n d i n g m o m e n t
THEORETICAL ANALYSIS (assuming s h e a f d is t r ib u t io n ]
FIG. 5-3
MODEL NO. 1
( SUPP O R TED NON E L A S T I C A L L Y . UN IFOR M D IS T R IB U T IO N
OF LO AD ]
\ /\ / 13
\ / 10 11\ / 6 7 8
/ 1 \ 2 3 4/ \
/ \/ \
145
15
14129 _5
Def lec t io ns in in c h e s
KEY DIAGRAM S h o w in g a x e s of sym m etry
- • 0 276 ° 0 0 00° ao'CD
- 0 3 2 0
"-j*CNLOO000
LOo-0 2 7 1
0510
iocoLO
LO
co CD*0109 - 0 1 2 7•1917 0504 000
totoCN
O- 4*OCO
lO
aCOLO
5377 2134 *000
toLOCN
LOCOCO
CO
CO
0*00 1*79 60 >199 *2067
Bend ing m o m e n t s
0328 0000in lb. in per in
M x
THEORETICAL ANALYSIS ( a s s u m i n g central point load 1
FIG.5-4
146
\ /
\MODEL N 0,1
1S U P P 0 R T E D NON E L A S T IC A L L Y , U N IF O R M D IS T R IB U T IO N
OF L O A D l
/
\ /\ / 13
\ / 10 11
\ / 6 7 8/ l \ 2 3 4
/ \/ \
/ \
15
14129_5
/ '
\\
\
D e f le c t io n s in in c h e s
KEY D IAG RA M ( S h o w i n g a x e s of s y m m e t r y )
- •0194 -*0205 - 0 2 3 6 —02 77 -*0322 o 0*000
toto
- 0231-0 1 8 70*000
COo toLO
CO0*000•1870
CNCNLOCN
COt"00CO
toLO
0*0000463•2187
CNoncooo
CN
CDOCOLO
0*000 1 0 7 9 0 0*00023156799
Bending m o m e n t s ?n lb in pc r inMx
THEORETICAL ANALYSIS (a s s u m i n g u .d . lo a d on pedesta l )
FIG. 5-5
\\
MODEL NO. 1
[ S U P P O R T E D NON E L A S T IC ALLY, U N IF O R M D I S T R IB U T IO N
OF LOAD]
\ /\ / 13
\ / 10 11\ / 6 7 8/ K 2 3 4
/ \/ \
/ \
/
147
'(s
14129_5
\
/\
Defl e c t i o n s in inches
KEY D IA G R A M ( S h o w in g a x e s of sym m e t r y )
- 0 1 8 3 -0 1 9 4 - 0 3 0 8 o 0*000-------7/ o / •/ /CD
to- 0 2 1 9
0*000
ocoto00- 0 0 7 5 - 0 090 -0129 LO
•1863 0*000
o<NCO(N
LOLO00CO
04COLOLO
695586 22 05 0*000
CO coLO
LOCD
CO
9111 672 6 B
2360Be no1 i nq mom ent s
0446 0*000 in lb in per in*
THEORETICAL ANALYSIS (a s s u m i n g shear d istr ibut ion)
FIG. 56
148
In a ll the cases solved for Model N o . l , the equilibrium check gave
satisfactory results, as w il l be seen from the fo llow ing few values
chosen at randum.6,00019. 6.0018, 6,0014, 5.998518,5.998518,5.9979
as against a value o f 6 lb . - the externally applied load. The e q u ili
brium check gave values o f external loads which are w ell w ith in 0 ,1% .
EFFECT GF VARIATION OF k-VALUE
When the k-value increases, the deflections and consequently the
moments also decrease. It is interesting to know that in the lower limits
3o f k up to 3 lb /in deflections decrease rapidly w ith the increase in
3k-va lue . While in the higher lim its of k above 3 ib /in , the decrease is
less pronounced. A graph for deflections along the centre line o f footing for
various values of k has been given In F ig ,5 .7 ,
W ith the moments, maximum and other values, this does not appear
to be the case. The moments more or less decrease linearly w ith in the
range o f these k-values, as can be seen from the graph in F ig ,S ,8 -a very
important finding i f the trend is established for other cases o f rafts. Having
proper regard to the scaling factor, perhaps most o f the soils encountered
in practice, on which rafts are generally la id , are covered by the k-value
3o f 1 to 5 Ib /in , If this is the case, only one experiment w ith any suitable
k-value would enable one to predict the moments for a whole range of
k-values and consequently for various types of soils.
DE
FLE
CTI
ON
S
IN IN
CH
ES
■ v v ' * »;vA \ r * 'r r K * .A T ' V '«A'.'-T-'J ) ^ 'Jv.-'. ' A .'v-1' A.'!,.1 v.AA* AA'VV'.VvA 1
1 J F
61b.
FOOTING DETAILS
1.Sponge rubbeF-
Block board
\\ /
\ /\ // 1 \ 2 3 L.
/ \/ 4 \
/ \/ KEY D I A G R A M
.2 5 -
K » 0.9559 Ib /n
*2059
1*7059
1*9559
•05
0-01 532
D E F L E C T I O N S ALONG X X' FOR V A RY IN G VALUES
OF V
I 150
•9000
.8 9 0 0
KEY DIAGRAM
• 8 8 0 0
8 6 0 0 1 -
B E N D IN G , \ M O M E N T
8500 £
DE FLE C T IO N
•06 .8400
• 8 3 0 0
1-95G ' 2.956K in lb per in
M A X I M U M M O M E N TS AN D DEFLECTIONS FOR VARIOUS
956
VA LUES OF K
FIG. 5-8
151
5 ,1*8 Model N o ,2
The model was divided into a grid of 6x6, yield ing 49 nodal
points. !n geometry, symmetry exists along the centre Sines parallel to
the axes of co-ordinates and also along the diagonals. If the same load
is applied on a ll the four columns, 49 nodal stations y ie ld only 10
unknowns. When the same load is applied on the two columns along
any diagonal, the number o f unknown is sixteen, while when symmetry of
load exists along a centre line parallel to one of the axes of co-ordinates,
the number of unknowns is 28, When different loads are applied to a ll
the four columns, the total number o f unknowns w ill evidently be 49.
The load was applied at the centre o f d la , columns, i " high,
rig id ly fixed to the raft p la te . Since the column area is about one ninth
o f the area of one grid square, considering the load to be applied as a
point load over the grid station is like ly to y ie ld higher moments loca liy ,
the moments at other points are expected to be fa ir ly free of this local
varia tion .
Results o f the fo llow ing three cases are given:
(1) 2 i lb . on each column in F ig .5 .9 .
(2) l i l b . on one set o f diagonally opposite columns.S^lb. on the other set o f diagonally opposite columns in F ig .5 .10 .
(3) J ilb , on one set o f columns on one side o f the centre lineparallel to x-axis and 3 ilb . on the other set o f columns,Fig,5.1!.
The value o f k was experimentally determined as
1.979269 lb / in ^ .
152
MODEL NO. 2
>»2
Mx
\
/K E Y i D IA G R A M \
Showing ax<?s of sym m e tr y )
D e f lec t io ns in inches
• 0602 • 0659 • 0784 •0850
oo
•0559 • 0613 -0738 ? 0-000o
ir>LDoCM
•000
COCDCOCM
COCM
-000•17 22
cr>cr»o.CM
ocoCDCO
CDCO
cnco
0-000• 0570-.1509
Bending m om ent in I b . i n . pe r i n .
T H E O R E T IC A L ANALY SIS [ f inite d i f f e r e n c e ]
FIG . 59
153
MODEL NO. 2
DEFLEC! TIONS
MOMEN TS MOMENJS
KEY DIAGRAM
DEFLECTION TN INCHESShowing symmetry
0602 082 5 1257
O'OOOO K. O \ po
0*000oo
0559 0737 *0979 o
oo
o•0276 •1175 •3855LO o
o
oooo
oo
_*U67 :2170-*3021 o
COCNI00
00
LOo
CO
cnCsl
cocoCO
oo
oooo
COCOCO-*3716 o -3997oo
MOMENTS IN LB.IN. PER IN.X2 My
THEORETICAL ANALYSIS | Fin ite difference]
154
MODEL NO.2
DEFLECTIONS IN IN
■0352 -0424 -047 5
DIAGRAM \/ Symmetry 1 along 'XX' \
I•0602 -0844 -1144 -1347
.0307 •0390 •0438 •0559 •0788 •1085
»0228 .0308 .03 74 •04S1 •0684 •0918
•0191 •0272 •0344 •0459 •0637 •0846
•1261
■1091
1013
MOMENTS IN LB. IN. PER IN.
>N2
O’OOO *0732 --2626oooo
lOcn
o -OO•n-
cooLDMx cy
I
-•3932 — 2011 ■3332 O’OOO................ c yoo
o
0 * 0 0 0
oo
^ o^ QiOCO
•U64
oot>- o OO o l o o52464
oor o 2 ao fc>CDo-3636
8o
CN OCOro-2183
oo
r o o o. o roN
*47 7 8
0 0 0 0
l o
LO
•0272
oLO
-V9i9
r o- j -CD
-2863
CNLOCN
-16 54
o7?•1648
O'OOO
r->
- j ’
-0025
CNCOOJ9V-1761
C?LOCN
-•2543
•N-CD•N-
. 7
-1524
LO(NOO
*07 90
CD
$Csl
cr oo o
LD
7?O'OOO
(NCDCDLO
‘b'OOO
THEORETICAL ANALYSIS [F in i te d i f fe re nce )
155
The equilibrium check yields the fo llow ing values o f externally applied
loads*10.000 324010.000 270010.000 3160
as against an externally applied load of 10 lb*
5 .1 *9 Model N o .3.
Dimensioned sketch o f the model is given in F ig .4 *5 . The
shorter side was divided into six and the longer into ten* giving rise to
seventyseven grid stations* Considering geometry o f model, w ith
respect to chosen grid, it is easy to see that symmetry exists along the
centre line parallel to axes o f co-ordinates* If symmetry of loading
exists along the centre lines, the seventyseven grid stations yield only
twentyfour unknown deflections. When symmetry o f loading exists along
the centre line parallel to *y‘ axis, the number of unknowns w ill be
fortyfour.
The load was applied at the centre o f the i " d !a0, i " high
columns, rig id ly fixed to the raft slab. The column loads wore considered
as point loads over the grid stations. Error due to this approximation may
not be considerable except loca lly , as already stated in the case of
Model N o .2 .
The model was analysed for various loading cases, three of which
are given here.
(1) Deflections and moments for equal loads o f 2 i lb . on each column are given in F ig .5 ,12 .
156
ODEL NO. 39 2(3 21 22 23 24
■ 15 16 17 18
8 9 10 11 12
2 3 4 5 6
KEY DIAGRAM
■12 70 1225 1136 -1098 .1137 -1193
■1198 •1169 -1078 •10*1 ■1078
•1085 1055 • 0995 0966 •0987
•1034 •1007 -0958 • 0933 •0947
1K3
1019
0967
D E F L E C T IO N IN IN C H E S
0 0 0 0 -1806 - 2 0 5 3 - 3 0 8 0 - *0639 *U12
>s2
CD
CO
LDOOCM
CDosf
oooo
CD
0 0 0 0 m
ooo »
CM
*3696 ?
oooO S fo-•2046 o
ooo9 £
CD-•3102 o
‘00
00
"O CO CO •550
3
CM
■ o o o o 1
CMCDID
0367 c? 1
-=2180 “ 1
00
- 2 8 8 0 *5
o
•=1281 § *11
inoo
• 0 0 0 0 " f
COro
- 0 5 0 0 °?________ i_<25
- 2 2 7 7 <?-
i
00ID
-2815 £ -
cn
-'1565 S
Mx M OM ENTS IN L B . I N . PE R IN .
oooo
•7138
■1695
•0183
/VTHEORETICAL ANALYSIS (Finite d i f f e r e n c e )
FIG.5'12
(2) Deflections and moments for equal loads of 2^1 b* on external
columns and 3 f lb . on central columns are given in F ig .5 .13 .
(3) Deflections and .moments when symmetry o f load exists along the
centre line parallel to axis of *x* for loads of 3^\hf 4 i\b and l i l b
respectively are given in F ig .5 ,14 .
~ 3The value c f k was experimentally determined as 1,57964 lb / in .
The equilibrium check yields the values of 15,0004370,
17,5004750 and 19,0004710 against externally applied loads of 15,
17,5 and 19 lb . respectively, adequately confirming the accuracy of
theoretica lly determined deflections.
,1 0 . Model N o .4
Dimensioned sketch o f the model is given in Fig„4„6« The
raff is L-shaped. D ividing the longitudinal side info nine equal divisions
and transverse side info six equal divisions gives rise to sixty nodal
stations. As no geometric symmetry exists along any o f the axes, sixty
grid stations give rise to s ix ty unknowns.
The model is w ith one re-entrant corner. The re-entrant comer,
and for stations adjacent to i t , require the use of special operators as
given in Chapter 3, Figs,3 ,7 ,3 ,8 and 3 .9 .
AH the columns are 3 /8 “ d ia , i " high, r ig id ly fixed to the
raft slab. Boundary and comer columns are so placed that the raft
boundaries are tangential to them. Column at the re-entrant corner is so
situated that the produced boundary lines o f the raft are tangential to i t .
19 20 21 22 23 2*!MODEL NO. 3
13
7
1
1*. , 16 17 18,
8 9 10 11 12
2 3 * 5 6
KE i DIAGRAM
.1297 -1169 *0993 *0866 *0 8 2 5 -0840
•12 38 •1126 • 0952 •0833 •0797
•1133 •1023 •088* • 0780 •073 8
•1085 •0979 •0852 •0755 •0711
0819
0740
’ 0705
D E F L E C T IO N IN IN C H E S
*0 0 0 0 *1924 -1 9 7 6 - 3 4 5 3 - 2 1 5 9 *J123
>s2
roOoo
CDCDCMCM
’ I
oCOCO
ooo , o 00 . o•0000 « •
° so ° o o o 4 0 o ^
. CM °
•37 6 9 ? -*2 013 o .
go 10
34,5 5 o
Oooo• ^
-’220 0 m *
o o o o -22
03 v r cn
LD CD
•0C01 r* - 2 2 0 2 S 1 ! *1
LO
-326 2 ?*1
O
-2*73 Si
00CO
•oooo ??
1 1 1
i<2 CMcn o
- ’0483 7 ,-*2323 *> ■
1
COCP
-•320 2 ?
T
COID
-•2637 ??
oooo
*3006
- 0 5 6 5
-15*8
Mx MOMENTS IN L B . I N . P E R IN .
THEORETIC At ANALYSIS. I Fini t e d i f fe r e n c e )
159
MODEL NO. 32*, , 25 26 27 28 29 30 31 32 33
13 14 15 16 17 18 19 20 21 22
2 3 4 5 6 7 8 9 10 11
K E Y DIAGRAM
•1578 -17 62 -1579 -1457 -1427 -1404 -1221 -1017 *0865. -0756
•1778 •I 683 •1498 -1379 •1347 •1338 •1146 •0952 -0808 •0706
•1620 •1520 •1385 •12 81 -1233 •1186 -1041 •0876 •0739 •0624 •
•I 54 9 -1459 •1335 -1237 •1183 •1124 •0995 •0843 •0708 -0590
0628
•0571
•04 92
D E F L E C T IO N IN I N C H E S
*0000 *2681 -2440 -3651 - 0 2 8 5 *6 3 8 6 ‘0830 - 2 0 6 2 - 1 7 5 8 *0788 “0000
° - J oo co o ' < j) O O
•0000 >* *5322*
° ^ o in o o-2441 *•
o c-. o cn o ^13677*
° “>'S t-o C"* 0 CN O CD O 'J '
-0467' ’9618 ^
o ^§ s° o*0600 *i
o ^ o o O o-2103 T
CD
CN-1906 *i
O v jo cn-1690 ~
_ m oo co O o1982 *
ro! cn
S 3ro n°
*0000*. I *0644'*
in55CN
-33 97 '
ocn•o-CN
-106 5 1
CNooroCN
•3051*1
oCDCOCN
-006 7'
CNt**00
-1672
CNCDC".
*0050^CNLO
00 00 ^
c£cn
-‘05 78*
c—CNCO
-2812^
55
-3314*'
roCNin
-1406 ^
CNCNCDro
* 1 2 2 2 ’ •
oororo
-0C38 1
00CD
CN-0441 *1
Vj-roroCN
t".ID
ro‘iC-*inIDin
ooo’000 0
*0000
0000
z
MxM O M E N T S in lb . in . p e r in .
T H E O R E T IC A L A N A L Y S IS ( Finite d i f fe rence )
FIG. 5*14
S tric tly speaking, none of the columns is located such that its centre is
coincident w ith the grid station. A ll the columns, except three in the
second row from the rlghthand side (c f,F ig .4 .6 ) have fo r the purpose
o f theoretical analysis been assumed to impart point loads at the grid
stations nearest to which these are located.
The three remaining columns are assumed to impart proportionate
load to the contigua&s arid points. This is like ly to introduce some error.
Since in practice such a case is very like ly to occur, the configuration of
the columns in this form has been deliberately chosen.
3The k-value was experimentally determined as 2.3996913 lb /in .
The results o f the two theoretical cases are reproduced here in
Figs.5.15 and 5 .1 6 . The equilibrium check yields a total externally
applied load o f 11.4187880 and 7.1937590 as against loads o f 11A & 7.2,j j l
1 REDUCTION OF EXPERIMENTAL RESULTS
As already discussed In Chapter II, mosr/fringes represent the
contours o f and . To be able to calculate the momentsBy
* —— and be known besides the values of D and v .
This requires differentiation o f the first derivative values obtained
from the fringes w ith respect to x and y by any suitable device,
graphical or ana ly tica l. The method used throughout this work is as
b rie fly described as fo llows.
and twist , i t is needed that the numerical values ofx mY . xy
MODEL NO.4.
.0665 .0431 .0290 .0258 . 0296 .0296 •0317 •0 3 2 5 .0310 .0319
.05*8 •0371 .0256 .0222 .0240 .0 2 5 0 .0269 •0281 •0284 •0297
.0(98 .0334 .0231 .0202 .0222 .0233 •0254 .0 2 7 0 • 0278 •0 304
.0(60 .0319 .0227 .0203 • 02 24 .0 2 4 0 .0 2 61 .0 2 7 6 • 0280 .0294
ooooft
LOLOOCM
.0501 .0347 .0256 ,0241 .0 2 7 5 .0279 • 0303 .0315 .0302 •03121
.0559 .0401 • 0310 .0307 •0362
cor>oi
COoCM
•0690 .0481 •0377 •0398 •0515 -1
DEFLECTION IN INCHES
- 2 8
00cncoloO
LO
cmCM'Ioooo
.oooogoo
.0000 -
CO CM LO
.0000
•vfCMLO
.0000 ^
LOCO
.0000
0000
erfCOCO
*7
,0000
o
oooo
T H E O R E T IC A L AN ALYSI S ( F in i t e dif
MODEL NO. 4.
.0673 •0455 •0347 .0369 .0471 • 0490 • 0 5 3 8 0576 0592 • 0665
.0626 •0441 .0337 .0335 •0394 • 0428 •0473 •0519 0567 •0651
•064 0 •0440 •0331 .0325 •0381 • 0411 .0459 •0511 •0571 •0685
.0611 .0437 •0340 .0 340 .0398 .0432 .0477 .0522 •0570 • 0653
•0644 •0464 •0378 .0402 .04 91 • 0505 .0548 •0584 •0598 • 0670
.060 •049 3 .0435 .0498 .0641
.0710 •0541 .0501 • 0629 *0888
D E F L E C T I O N IN IN C H E S
oooo
mLOoo
co
cnCMcn
LOoocji*
oooo
T H E O R E T IC A L A N ALYSIS [ F in i t
163
For instance in Model N o . l , i f if is desired to obtain moments at
p t.3 , (c f.F ig .5.1 -key diagram) the fo llow ing procedure is fo llow ed.
An enlargement of the negative o f the fringes is made to exact
size o f the model over which the appropriate grid is marked.
For obtaining the value of at '3* centres o f fringes where "v'Jt i.x
these cut the grid line 12-11-10-7-3-10-11-12 are marked on the grid line .
G rid line 12-11-10“ 7~3-1Q~11-12 is now considered as the base line . A
suitable vertical scale is chosen. Fringes are designated numerically from
le ft to right;where the first fringe cuts the base line a point is marked.
Successive points are marked by stepping up or down by one unit o f
vertica l scale, the next fringe section w ith the base line . Proper regard
should always be given to the sign. Sections of the same fringe w ith the
base line more than once maintain the same elevation o f the marked
points i .e . no stepping is done for various section points of the same
fringe,
A smooth curve is then drawn to pass through each one o f the
points so obtained. The choice of the vertical scale should be such that
this is possible. This would then be the first derivative curve. Opposite
point ‘3* on the first derivative curve a tangent is drawn by d ivider method;
the angle which the tangent to the curve makes w ith the base is accurately
measured.
The tangent of this angle, multi plied by the scaling factor involving
the vertica l scale chosen and the constant o f the moire'apparatus
164
w ill then give the value of at point ‘3*,d x 2
For obtaining JSLSb . , a sim ilar curve is drawn w ith respect& y z -
to grid line 3-7-10-11-12, Having obtained the second derivatives, the
moments Mx and My are easily found out by mere substitutions in the
basic equations, assuming the values o f constants D and v are known.
It may be pointed out that the zero o f the first derivative is not
known; this, however, does not affect the determination of the magnitude
o f the second derivative. The signs of the slopes may also not be given
e x p lic it ly in the contour lines. This can be fixed from mechanical
considerations as at least at some points in the model the sign o f the
second derivatives can easily be guessed.
16Although quite a number of methods v iz , Slaby's method for
d ifferentiation/the paper scale method and the mirror method for drawing
normal to the curve may be used for obtaining the value o f second
derivatives, and each may have its merits and demerits . In the author*s
opinion, i f accuracy is the main consideration, best results are obtainable
i f actual tangents are drawn to the first derivative curve at the required
points. The method, o f course, tends to be laborious i f a great number
o f points are involved. Working from enlarged fu ll size prints, though it
may be convenient, has its defects. The worst o f these perhaps is that in
drying the photographic paper does not retain its exact size. This
variation in size may be possible to account for in calculations, entails
unnecessary labour, especially because the coeffic ient o f expansion of
photographic paper does not appear to be linear nor the same in two
orthogonal directions.
When fringes are not w ell formed or are splayed, marking the
centre o f the fringe may become d iff ic u lt on the p rin t. By judging the
intensity o f light the centre o f the fringe can better be located when
working d irectly from the enlarger.
EXPERIMENTAL RESULTS - MODEL N Q J .
The pedestal is devoid of any fringes/land it can be assumed that
under load if did not undergo any deformation at a l l .
The fringes are sparsely located in the neighbourhood of nodal
points 5 ,11 ,13 ,14 and 15, and to obtain the values o f • andd «* o. . ■\
*■ C) LO. i f the *— “ ** and —1——■ curves are extrapolated except in
■dyz d yvery especial cases, the results obtained may not be at a ll re liab le .
The experimental moments are given in F ig05 .1 7 a St may be
mentioned that the result at p t . l is marked w ith an asterisk, showing
that s tric tly speaking it is not possible to obtain the values o f moments
at this po in t. However, tG obtain the values of moments in as close a
neighbourhood of this point as is possible for the and JzLZx—CK- 3 V
curves, the fringe sections w ith the base--line at the boundary of
pedestal were projected on the section line 5 -4 -3 -2 -1 , which evidently
w il l give an approximate assessment of the moments.
.77
08
*'
.660
7 -U
77
-331
8 -2
757
166
/
M O D EL NO. 1
[ M O I R E ]
\ 5 S 12 K 1,\ 4 8 11 n
\ 3 7\ 2
\/ \
/ \/ \
/ \\
KEY DIAGRAM Showing axes of symmetry
oooCO
• 0 95 8 *0836 cn M
IT)OOC"JCN
OCO
.0501• 0177 .0228
cn
COco •1638
ooCOuo
• 6439 . £608
COoCO
Mx
BENDING MOMENTS BY MOIRE ' TECHNIQUE
FIG. 517
167
5 .2 *3 EXPERIMENTAL RESULTS -MODEL N O .2
Case 1. It is not possible to obtain the values o f at points 4 ,7 ,' r 11 ' " <rL
9 and 10 and the value o f — S^Lat point 10. Hence the results at3 y x
pts. 4 ,7 and 9 are bound to be unreliable, while at p t.10 it is not
possible to obtain any experimental results.
The experimental results are given in F ig .5 .18 .
Case 2 . St is not possible to obtain the values o f --CL-Cri»T-' at pts, 1 ,2 ,3 ,4 ,, ^ £ .
"N , ^
5 ,6 and 7 and the values o f — a* P^s. 1 ,2 ,7 and 8,
The experimental results are given in F ig ,5 .19 ,-2- C* - r • x
Case 3 . It is not possible to obtain the values o f at pts. 1 ,7 ,8 ,' — I d?
14, 15,16,21,22 and 28 and the values o f at p ts .1 ,2 ,3 ,4 ,5 ,6 , & 7.
The experimental results are given in F ig .5,20,,
5 .2 *4 EXPERIMENTAL RESULTS-MODEL N O .3- - ^ ^ ?
Case 1. It is not possible to obtain the values o f at pts. 13 & 19~ ... a oc2
and the value of at pts, 19,20,21,22,23 and 24 being the boundaryS y '2'
points.
The experimental results are given in F ig .5 .2 1 ,,
Case 2 . It is not possible to obtain the values o f at 13 & 19c > x .^
and the values o f ■bhCSii-'* at pts. 19,20 and 24.ci y
The experimental results are given in F ig .5 .22 .- v 2 -
Case 3 . It is not possible to obtain the values o f at p ts,23,34,43
"ST , 5and 44 and the values o f at p ts .34 and 44.
The experimental results are given in F ig ,5 .23 .
168
/
MODEL NO. 2
K E Y - D IA G R A M \| Showing axes of symmetrys|
*0713
MxCDCDO
CDCD CD
•1990 -0857
CDO£"•
CD
aOCN
- *2 517 -.2051 - ‘ 121 9
Bend inq moment in Ib/m. p er in.
BENDING MOMENTS BY MOIRE^
FI G. 5*18
KEY DLAGR AM Showing svm m e tr v
— 2820 • 3839CDLOO
COLDCO
CDOCOO
•2 0 6 9 -3385CDO-s/° sscn
-3135 -1916'
COCO LOO
cnCNCN
LOoCDCN
- 2 9 0 5
B e n d in g m om c nts in I b . i n . per in .
BENDJNG MOMENTS BY MOIRE"
-26
32
-1
57
3
06
21
0
*00
0
MODEL NO. 2
22X -------- 27 28
X
S ym m e try la long^ XX
0*000
0 *0 2 1 7
-0 * 0 9 1 3
• 1257o
O
LOO
0 *0 5 5 1 v
•158/;
-0566
CO CO LO CM' i - ‘0 U 6
-2632
CMCDO
oCOCM
LOOcoCM
CO-sfCOo
-1686
-2554
-2062
-4 2 9 5COoLO
LOO
OCOCM*1
COoCDCM
-3694
-1830
coo■ i CO
o
- 3 4 5 7
-3113
CO
OCO
CO-rC~~'CO
-1368
-3005
-307 6
_l203S 00
o
0*000
CM
CO
COcnLOCM
3301
1247
CMCO
•7
oopo
ocncoCM*1
CDCO
0*0601
- 0*1011
LO
-0 7 65 -0M 799
B e n d i n g m o m e n t s in tb. in .per in
Mx
BENDING MOMENTS BY MOIRE FIG. 5‘20
\/
171
MODEL NO. 3 1113
7
1
20 21 22 23 2 4
14 15 16 17 18
8 9 10 11 12
2 3 4 5 6
KE Y | DIAGRAM ( Showing a x e s of s y m m e t r y )
H
on*0402
*12 57
COinCM 3120
-2029
oooCOo
coCM
°+-2190
- 3 2 Uco« oicn
coLO o'I +•'2963
- • 0 6 9 5sfCMs
LOI**
*39 88
1 +-XJ261
CDcnro
cnLO jt-*lM
’6456
¥CM
•0912
cnroco>rl
cnCM '0292
MOro•I m o i
LO
j £ f 3198
M00CMCM +1071
cncn
•1257
-0287
roroin
__L_- 0 5 5 8
»cncn 1-2318
rocnCDro 72 998
cnCD
Y P2467
ocno•sC F0874
JMxBending m o m e n t in. in. lb. per in,
B END IN G MOMENTS BY MOIKE'
i
MODEL NO. 320 2 22 23 2 4
, 15 16 17 1®,
8 9 10 11 12
2 3 4 5 6
( Showin gK E Y DIAGRAM
a x e s of sym m etry )
l - l *!203 - 1 6 9 5 - '4295 - ‘3890 - 0 4 1 0
00o00
cnoo'roCN
t-"o00"-r
COo
*0633 °
CN
9 CN cn CO
'2540 ° -
COcnin
9 co 1 o
CO
•1802
rcoo
17 So
-3531
NCOCO
TcnN
•1938 ™
CNCN
-•0509 9 -0224 ?
CJ)in
-2067 ^CN
-3630 ? -
N
•1612 S -........... i
in
-0389 S
1
cn
-0698 £
CN00
-2574 7
1
o
-3147 7
i
'N-inin
^2286 7
co
o*r
*1063
- ‘0796
-1209
*1 MxBending m om ents in lb . in .per in
BENDING MOMENTS BY MOIRE"
173
MODEL NO. 3.27 2R
- X
KEY D IA G R A M
( - )
Cnin oo p
1474 -24*49
CO
LOo
COCnCO
CO
cocnCNo
-•4*788 -0831 *5194
innto
co
'1289 -
o o
7 ~3318 - 2884 ( - ) H
COCOCO
•0672*
in8o
•266 0®
cn•n -CN
•2 9 99 ’ -
...... ocnmo
•*43<7 -
trr-o.oin
•0662*
2
•5 9 7 7
CnCOo-
•0 4 0 2 '-
V"CO
O
•2 6 22 -
cOoino
-1986*
cnCN
•0 9 3 5 °CN
CO
0 3 5 9 -
o*CD(p
•0 6 0 6 *-
Cncn<o
2 9 2 3 -
CNv *COCO
•4509*-
00oCNCN
•2188 1
mCN
• 2 0 7 9 -
COCDCNCN
• 0 1 7 8 -
cnoNCN
•27n‘‘-
cnin
•23 28 -
CNcnCN
•0011 *•
OS 70 - •1231 • •4101 - •4541 - •2810 • 0174 -0 7 5 0 - •2541 - •2362 - •0618 •+
•2140
1157
•0889
cnLO
coi
cncncoco
cnoocnin
incoin
in3i
co
' I
COCOcnco coco
*1
cncoI
ocoCN
incoinCN ‘ i
MxBending m o m e n t s in I b . i n . p e r i n
BENDING MOMENTS BY MOIRE
FIG. 523
5 .2 .5
5 .3 .1
174
EXPERIMENTAL RESULTS-MQDEL N O .41 ■ ■■■ 1 1 - — - i - - r i i . - — - - - - - - - - -. r r . — ^
Case 1 * It is not possible to obtain the values o f at p ts .l to 5, 19,dJC 'L
31,32 and 54 to 60 and the values of at pts«5/ 6/ 10,15,20,21,25,^ y 2•
32, 45 to 48, 53,54 end 57 to 60. ■ . •
The experimental results are given in F ig ,5 .24 .
Case 2 . In the regions of p ts .l to 32, the fringes are very sparsely
distributed. This is because o f the small moments to which this region is
subjected for this particular loading case.
Values of moments are calculated from points 33 to 60. It is not
-possible to obtain the values o f at p ts.54 to 60 and the values of; \ X %"
at p ts.33,34,40,41,42,47,43,-54.and.60.d y x
The experimental results are given in F ig ,5 .2 5 ,
JILCOMPARISON AMD DISCUSSION OF SALIENT RESULTS
Model No.1
Considering the footing as non “ e lastica lly supported w ith uniform
reaction, the moments obtained are generally higher than when the footing
is considered e lastica lly supported.
The shear type of distribution of load gives the least value of
maximum moment.
In F ig ,5 .26 the bending moments along the centre line of the footing
are compared for the shear type distribution o f load. The continuous curve
is drawn through the points obtained by fin ite difference analysis,
considering the footing to be supported e las tica lly .
175
MODEL NO. 454 47 40 33 26 21 16 11 6 11-----
55 48 41 34 27 22 17 12 7
56 49 42 35 28 23 18 13 8
57 50 43 36 29 24 19 14 9
58 51 44 37 30 25 20 15 10 ■
59 52 45 38 31
60 53 46 39 32
—*2 694
KEY D IA G R A M .
•3369 - 1 5 3 9 - 3 2 7 3 --1042 - 0 2 4 7 '0414 -2 3 6 9 |
( - )
COm T
-•215 2 -coinc-o
CDCm101
CDCOino
3043 - 1 7 2 7 i -
in10coo
0620 1 ..-1619 1 -•1159 ' - - 0 1 6 2 --2336
inocmo
inci
CMS5o*4
M “
Oocm
(JO10in
CO
RinM fc-*i
CMCMcm
CMCOin
- •2931 - .*2850 *—*1872 - ‘0835 -*1723 1 - 0 7 5 9 ‘
&in
0953 "
•vj-
Cl 9 3 i ^CMCDCDO*1
CO-s fin
7
r *ininCMCOo
COincm
i
inooo
- •1905 - *3182 4--1864 1 «04fc9 - 0 9 2 6 -: 0 5 78 -
CM CO
,0585V^2213
•j*CO00CM
l - U
m TCD R
CM
I
oinoo
coCDCO
—2 8 6 5 ' - ' 1 0 0 9 ‘ —2 20 3 1
■sjin
*1
inID
coCm10
'23 61 1 - * 2 070 -0103
CO
CM
•1695 *». -2509
( - )
inCMCM
o<M00CM
I
CMCMin
*' -3110 ‘ -.-4 0 33 --2798inooCMcm
c m
M
COCmCMCM
I- •3 6 9 4 - 5 3 23 -*4210
inoco
CM
oCO
inCDCM
COinco
-1054
( - )
incmCmo*1
<n COCO CDo ino o‘i *+
OoCM00o
5»2 Mx
—FT<n -j-Cm ICM *~*
' b e n d i n g M OM EN T IN tb. in per in
-•0516
•1009
■ 0217
( - )
B E N D I N G M O M E N T S BY M O IR E " FIG.52A
176
MODEL NO.4& l i l D 33 26 21 16 11 5 1
55 £8 £1 3 L 27 22 17 12 7 2
56 LS 12 35 28 23 18 13 8 3
57 50 L3 36 29 2 L 19 \L 9 L
58 51 U 37 30 25 20 15 to 5
59 52 LS 38 31
60 53 LG 39 32
K E Y D IA G R A M
( - ) - 2 0 3 6 -'4186 - 1 6 9 5CO£p
.0613
inCO
•U 3 0 ‘
roCDino
-2 2 5 8 •1529oino
.0291 ' •
o£o
-178 6 ‘ l
inCOin-1921 -•2062
o CD cn
—16
81•
CNf -COo
r'1709 '
CDinro
-2 2 6 8 V . -19L3
CNCN
.0365 7
COoCO
-•1839 ’’ ,
e'enin
-2461 V -1752
CDinin
‘0451 ’■ -
e'enco
-»2517 •'
vf00in
-3157 V . ■•1750 t
>s2
MxCOP I
( - ) T -
00
in
-♦2798 ' •
oinCN
• U 0 2 V -2 9 1 9
urr-’ I
roooo* «
'vi-oo
CO.
ov
CD.00CM
COCOCD
3LO
CD- s fO CN
BENDING M O M E N T IN l b . in per in
B EN D IN G MOMENTS BY MOIRE FIG. 5 25
\ /\ /
\ /\ // \ 2 3
/ \/ \
/ \K E Y | D IA G R A M S
—(i i) elastic support
X - MOIRE M O M E N T S
M O M E N T BY F I N I TE D IF F E R E N C E
< r - ( i ) assuming uniform distribution of load (non-e las t ic support)
COM PARIS IQ N OF D ISTRIBUT IO N OF BENDING MOMEN T ALO NG THE C E N T R E LINE OF F O O T IN G
FIG. 5-26
178
It is easy to see that the moments obtained experimentally are
in quite close agreement, except at the boundary point where It is not
, S'f'njkalways possible to obtain the values of J l i i^ a n c l — ——• from the
hDC2' 7>y Xfirst derivative curves,,
Experimental Conclusions
(1) Near the point load and boundaries the results are unreltable„
They can also be unreliable where the fringes are sparsely distributed,
(2) Extending the subgrade rubber, i .e . a rubber providing a much
greater area than the area o f the mode I,did not a lte r the fringes
perceptibly as compared to the fringes obtained w ith subgrade rubber
equal in area to the model area. This may be explained as due to low
Poisson's ratio of the sponge rubber,
(3) Over the various grades o f rubbers used, the fringe pattern
remained the same and unaltered, confirming thereby that the characteristics
o f the reaction exerted by the subgrade did remain the same,
(4) The moments obtained by moir^ method can for a ll practical
purposes be used for the design o f such a footing, as the points where ~
from a designer's point o f view these are important, the results are quite
reliable,
5 .3 .2 Model N o .2
Case No.1 The experimental and the theoretical moments are compared
In Fig,5 ,2 7 , In the FIg.X|X| is a mid-section and may be said to
correspond to a midstrip in a fla t slab structure while X2X2 corresponds
BE
ND
ING
M
OM
EN
T IN
L B
.IN
. PE
R IN
.
179
MODEL NO. 2X
— XX
K E Y I D IA G R A M( S h o w in q a x e s of s y m m e t r y )
M>
C E N T R E U N EOF T F O O T I N G .
!L
Mx
—2
F IG . 5-27
CO M P A R ISO N OF D ISTRIBUTION OF MOM ENTS ALONG X ^ AND ) y
180
to the column strip.
The maximum negative moments obtained are in quite close
agreement, while the maximum positlive moments at the feet of the
columns experimentaliy obtained, should generally be less than the
theoretical moments. This is understandable as in theoretical analysis
the column loads have been treated as point loads while actually the load
is transmitted to the raft slab distributed over a fin ite area. In this case,
hov/ever, there is quite close agreement in the maximum positive moment
obtained.
Case N o .2 The maximum relative deflection between any two grid
stations Is .0798" (63,8% of the thickness of the p la te,)
Along the column strip the moire results are generally lower and
so are those at the middle strip. Yet it may appear that the moments by
m oir/are not much out, for the purpose of actual design. It also appears
that w ithin this range of deflection, the membrane stresses do not appear to
be of much consequence 0
The experimental and theoretical results for this case are compared
in Fig.5,28o
Case N o .3 The maximum relative deflection between any two grid
stations is ,1156 (92.5% of the thickness of the plate) and the co-ordination
of experimental and theoretical results is not as good as in the case of
loading cases 1 and 2.
The experimental and theoretical results for this case are compared in F ig .5,29.
181
MODEL NO. 2
91
—XX —
KEY [ D IAG RA M ( Showjjng sym m e t r y)
X X
Mx
1016 U10
- . 6 x,x ,
COMPARISON OF D IS T R I B U T IO N OF MOMENTFIG. 5-28
182
MODEL NO. 2
/
x 215
22
M 2
8
X;23
3 U 5 6 1//
2k
L£L
17
l i
18
21
U X21
28 X.
/ KEY 1 DIAGRAM \( S y m m e t r y a lo n g a x is of xN )
ci
r£
/ / i \ V " \ 7
i ki *
M y -
k /
\ '' v •1
8 D \
\ '
10*
11 12 / f /
13i
\k
Mxi—
A H E O R E T IC A LMOIRE"
Mx- 2
COMPARISON OF D IS T R IB U T IO N OF M O M EN T ALONG X_X1 AND X X■I I _. " ......... - -------- 1-1 Z 2 FIG. 5-29
5 ,3 *3
5^3*4
183
Model N o ,3
In the first two loading cases, the theoretical and experimental
results are compared along sections corresponding to transverse column
strip, mid-column strip in Figs.5*30 and 5 = 31 and along the longitudinal
column strip and mid-strip in Figs.5 ,32, 5„o3.
The theoretical and experimental results along these sections
are in good agreement except at the boundaries and at column points.
In the third case, the agreement of results is not good* This may be
partly attributed to the large deflection (slightly over one thickness of
the plate) to which the plate is subjected in this case.
Model N o .4
Case 1, The experimental and theoretical moments are compared along
sections $ ]S ] (typical longitudinal column strip) and52S;9 (typical
transverse middle strip).
A t points where the moments are small In magnitude the comparison
is not good while in the regions o f maximum negative moment the experi
mental and theoretical results are quite close. Near the columns once
again the theoretical results are higher than the experimental. Comparison
o f moments is done in Figs.5 .34 .
Case 2 . Theoretical moments at points 1 to 32 are very small and
experimental results are not given for these* Experimental and theoretical
moments are compared along sections 5-jS^, $2^2 anc ^3^3 ' n ^ ‘9«5.35,
The rnoire^moments are generally lower than the theoretical moments*
BE
ND
ING
M
OM
EN
TS
in
lb.
in per
in
184
MODEL NO.3.
^ ^ T H E O R E T I C A L* MOIRE'
22 23 21
KEY DIAGRAM ( Showing ^ s y m m e t r y)
M x (s
Mx (s
Mx(
M x f s ^
c. i . of columns.
C O M P A R IS O N OF D I S T R I B U T I O N OF MOMENTSFIG. 5*30
BE
ND
ING
M
OM
EN
T
in lb
. i n
per
in
MODEL NO. 3
2
S.
19 20
J ?
7
1
2J 22 23 2 4
15 16 17 18,
8 9 10 11 12
2 3 4 5 6
THEORETICAL
-S.
—s.
KEY D IA G R A M( Showing axes of > s y m m e t r y )
$ MOIRE
colurr ns-columns
My (
- 2
- 3
▲COMPARISON OF DISTRIBUTION OF MOMENTS
FIG. 5-31
Ben
din
g m
om
ent
in lb
.in.
per
in.
186
M_0 DEL NO. 3
GRAM
Mx
- 2My (
Mx(s
C.L OF COLUMNS
-^—Theoretical
Moire
COMPARISON OF DISTRIBUTION OF MOMENTS
FIG. 5-32
MODEL NO. 319 20
13
s _ .
21 22 23 2!4
K E Y DIAGRAM
187
1*. > i s 16 17 1 8 ,8 9 10 11 12
----- w-----
2 3 C 5 6
i ----- (1-----
- s
M J S
- 2
My [S.
C.L
columns.
——*MTheoret i c a l r M o i r e '
C.L
,p^icolumns
C O M P A R IS O N OF D I S T R I B U T I O N O F M O M E N T S
FIG. 5*33
Bend
ing
mo
men
t in
lb.i
n.. p
er in
.
188
COMODEL NO. L
C O M P A R IS O N OF PI STRIBUTION OF
MOMENTS
r—
a
m
,5 6 £ 9 L2 3 5 2 8 2 3 18, 13
L3
U
LS
■ LS
KEY D IA G R A MTheoretical
± M o i re ''c.l. of colft.c .1 of cols. c.l ■ of col s . cl._pf I col s.
L2
Mx
ALONG S«Sto
- 2
A L O N G S S 2- 3
M x
-•5
nr,
BE
ND
ING
M
OM
EN
T
IN lb
.in
per
in.
3
£ 0
139
MODEL NO.i .
COMPARISON OF D I S T R IB U T IO N OF MOMENTS
Theoret icd
c . l .o f c o t s .#
» —a
—•— — «
56 £9 £.7 35 28 23 18. 13 8 , 3
57 50 13 36
U
CS
• £6 .. 4
-S .
KE Y DIAGRAM.
c.l .of
2 8
- 2M x
- 3ALONG S,S
Mx- 3
ALONG
cols .
I
I!
GENERAL CONCLUSIONS
From the experimental work reported in this thesis the follow ing
general conclusions regarding the application o f the moire technique to the
design o f raft foundations may be drawn.
( ! ) W ith the exception o f values at boundaries and under loads, the
agreement between the moir^ technique and the theoretical treatment was
generally found to be acceptable. Taking into account the normal design
approach o f providing substantial reinforcement, even on faces remote from
tensile stress, the agreement justifies the use of the moir^ technique for
dealing w ith rafts treated as plates on an elastic foundation.
(2) The technique may be used w ith great advantage to select the most
economical design when alternative raft designs are being considered« At a
glance the fringes give an indication o f the stress distribution in the raft slab
so that the design leading to the most uniform distribution may be selected,
(3) In the case o f one of the models tested in which a range of k values
was examined, the maximum moment was found to increase almost linearly w ith
decrease in k va lue. This finding requires further investigation in order to
examine more elaborate load patterns and various column combinations and
geometric shapes o f rafts. The approach offers the possibility o f analysing a
model experimentally so that approximate moments for various k values of
the soil can be predicted.
191
GENERAL CONCLUSIONS (Contd.)
(4) The moire technique is quick, cheap and straightforward, and does
not involve complex analysis to produce a workable design for a ra ft.
Scope o f further research
It must be pointed out at the outset that very lit t le work has been under-
taken in this f ie ld . In particular, there is no record of the moires technique
being applied to the solution of plates on elastic foundation and therefore there
is considerable need for further work. The fo llow ing aspects are considered to
be important.
(1) The development o f alternative methods of simulating subgrade
support so that complex plate problems o f this type can be solved.
(2) A study o f the stress distribution around cut-outs and holes in plates
supported e las tica lly .
<3) An investigation o f the su itab ility o f the moird technique for other
than simply supported condition studied in the present work. However, the
inherent lim itations which exist when the mo ire* technique is applied to laterally
loaded plates w ill apply in this case also.
(4) An investigation of the bending stresses in rafts formed from e lastica lly
supported folded slabs,
(5) W ith the advent of computers- rigorous mathematical analysis, at
times leading to complex numerical computation is no longer p roh ib itive . The
moir£ technique may serve as a useful check for assessing the va lid ity o f
simplifying assumptions made in the theoretical treatment.
192
GENERAL CONCLUSIONS (Contd.)
(6) The moire technique may be used for studying the effect in stress
distribution due to local lack o f subgrade support, a condition which may be
encountered at site, and may have a major effect on the behaviour o f a ra ftQ
REFERENCES
(1) Lift!©, A .I . Foundations (Edward Amold(PublIshers)Ltd, London, 196K(2) Skempton A .W . Foundations for High Buildings (Engineer V - l98 N~51£0,
• ' D ec .1954).
(3) Frischmann W.VV. Tail Buildings (Science Journal, October 1965).
(4) Baker A .L .L . Reinforced Concrete (Concrete Publications Ltd,London.Ed.1956).
(5) M artin , 1 and Ruiz,S. Folded Plate Raft Foundations for 24“ Storey Building.(Am.Cone.Inst.Journal V o l.3 1 ,N o .2 ‘1959).
(6) Scott; W .L .,G ia n v ille ,H „W ,and Thomas F .G . Explanatory Handbook on the B,S0Ccd® of Practice
for reinforced concrete. (Concrete Publications Ltd.1957).
(7) Dunham W.Clarence Foundations of Structures (McGraw H ill Book CompanyInc.London,1950.)
(8) Manning G .P , Design and Construction of Foundations (ConcretePublications Ltd, London-Ed. 1961)
(9) Tomlinson M .J . Foundation “ Design and Construction(Pitman)Edc 1963.
(10) Eng. News Record. A pril 14,1949.
(11) Eng. News Record. A pril 11, 1952.'
(12) W illiam s, G .M .J . Design of Foundations of Shell Building, London.
Proc.4th In t.C on f. Soil M echanics'4 Foundation Engineering,V°l. 1,1957 -Butterworth Scientific PubS icat ions-Londan.
^13) -P ike ,C .F . "New G ovt. O ffices" W hitehall Gardens.,The Structural Engineer-Vci.XXV! N o A p r i l 1948.1
(14) Baker A .L .L . Raft Foundations? The Soil Line Method.(Concrete Publications Ltd, London,Ed. 1956.)
(15) A llen - D .N .d e G . andSevern R.T. Stresses in Foundation Rafts ~
(i) (Inst, o f C iv il Engineers^'Proceedings-v-lS-Jan. 1960-paper 6416)„
( i i) (Inst* o f C iv il Engineers~Proceedings“ V” 20-O c t, 19^0-pape r 6532).
194
REFERENCES (Contd0)
(III) Inst, of Civil Engineers “Froceudings- Vol. 25 July 1963, paper No.6628,
(16) Ligtenberg F „K . The moire method - A new experimental method forthe determination of moments in small slab models,(Society for Experimental Stress Analysis; VoLXIi,
■ N o. 2, A pril 1955).
(17) Zienkiewicz OjC, andHolister G.S, Stress Analysis “Recent Developments in Numerical
and Experimental Methods (John Wiley and Sons Ltd, 1965),
(IB) He enyi M . Handbook of Experimental Stress Analysis,(Editor) John Wiley & Sons Inc,, London 1960 Ed.
(19) Gupta, K.K, ’’Distribution of Elastic Moments in Flat Slabs withparticular reference to Lift Slab Structures”,Ph.D. Thesis - London University, 1965.
(20) Benjamin, B.S* "Folded Plate Structures in Plastics".Ph.D. Thesis - London University, 1965#
(21) S. C.l .Ltd, Plastics Diyision Welwyn Garden C ity , Herts.Information Service Note No.896. Engineering Design Data -for "Perspex" Acrylic Sheet.
(22) l.C .I.L td ,P las tics Division,Welwyn Garden City,Herts. "Perspex" acrylicmaterials.
(23) Wood, ReH. Plastic and Elastic design of slabs and plates - withparticular reference to concrete floor slabs,Thames and Hudson “London, 1961.
(24) Williams, D. An introduction to the Theory of Aircraft Structures"(Edwa rd A rno I d( Pu b I i she rs) Ltd, 1960.
(25) ^ Ile n iu.N . de G j{ne fin ite Difference Approach “Contributing authors-ana Windle D .W . ^ j y • .
(26) Timoshenko, 5, andWoinowsky-Krieger, S# Theory of Plates and She I Is ( 'A c Graw Hi 11-2nd Edition)
(27) Llvesley R .K . and Analysis of a loaded cantilever plate by finite differenceBirchall P.C. methods. June 1956, RuA.E.(Farnborough,Hants.)
Tech,Note M.S.26. M in . o f Supply,London,W„C.2.
(28) Description of the Autocode, Ferranti Ltd. Computer Department.(List C.B.302 - Nov.1961),
195
REFERENCES (Contd.)
(29) Nonstandard Autocode Subroutines ~ Ferranti Ltd, Computer Department,(List CS 335C. July 1963).
(30) Teng, C . Wayne Foundation Design.(Prentice Hal! Inc. Englewood C lif f . New Jersey-
Eda1962.
196
APPENDIX
MODEL AND PROTOTYPE DIMENSIONAL CO-ORDINATION
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TABLE 5
Plate 1Sponge Rubber SPj
Plate 2Sponge Rubber SP.
MODEL NO, 1
202
Plate 9 . Sponge'Rubber
(W-2632)
Plate 10 Sponge Rubber
(W-2632)
89999999
4349
203
MODEL N0 .1 .
Plate 13.Sponge Rubber SP]
(9nx9ri - extended subgrade)
204
MODEL N O .2.
205
MODEL NO.2
Wig ™ .,-1 11..
Plate 5
SPl
^
^
'206
MODEL NO.2
Plate 7
SP,
illlW IIIIH
Plate 8
SP,
12191
207
MODEL N O .3
Plate 1
S?1
Plate 2
SPl
^
^^
^
208
MODEL N O .3
Plate 3
SP,
43
58
m o d e l h o
6889 97
37
^
^ 2 £ £ l n o .4210
Plate 7
SPj
"mmn
I k i
Plate 8
spi
45
^97350
2 1 1
Plate 10
SP,
03