3.1 General

CHAPTER

FUNICULAR

I I I

SHELLS

Funicular shells are a class of doubly curved

shells, the shape of which satisfies the desired state of

stress in its body for the given loading and boundary

conditions. The state of stress desired in an unreinforced

concrete thin shell will be pure compression unaccompanied

by shear and bending stresses. In funicular shells, the

shape of the shell is such that under a particular loading

condition, the shell is sUbjected to pure compression

unaccompanied by bending and shear stresses. Under other

conditions of loading, bending moments would develop and the

shell will no longer behave purely as a funicular element.

Analytically, it is possible to compute the funicular

surface of any ground plan for the given loading conditions.

The funicular shell is constructed as per the configuration

obtained from the analysis. By using simple techniques like

the use of a sagging fabric, funicular shells can be cast to

sat i s f Y s imp 1e loa din 9 and baundaryeo ndi t ion s (Ram aswamy

and Chetty, 1960).

Funicular shells are not limited by plan, shape or

size. It can be triangular, square, rectangular, circular

or elliptic of required dimensions. Models can be cast to

scale and its ordinates measured to be used for casting the

prototype. The shells can be cast either using the formula

for the ordinates of the shell from computations or by using

the simple sagging fabric technique developed by the

14

Structural Engineering Research Centre, Roorkee, India. For

many structures, funicular shells of different sizes and

shapes have been tried as roof element and many publications

on their use have been brought out. An important case was

that of a heavy duty platform built of precast funicular

shells for unloading cargo from the ships for the use of

Madras Port. A single funicular shell roof over an oval

ground plan was built as roofing for the Municipal

Corporation Conference Hall at Kanpur, India. Other notable

funicular shells designed by Structural Engineering Research

Centre were the roof for their testing laboratory at

Madras, brick shell roofs at the National Institute of

Design, Ahmedabad, the concrete shell roof of the Cathedral

at Lucknow, the roofs for the schools at Roorkee and

Chaibassa, the roofs for 600 units of houses for Bharat

Heavy Electricals, Thiruchirappalli, and the roofs for the

tenements of Tamil nadu Slum Clearance Board and the housing

project of Andhra Pradesh Police Housing Corporation. Some

work has been done in the field of funicular shells in the

United States of America at the U.S. Naval Civil Engineering

Laboratory, Port Hueneme, California. Allgood, Rail and

Chiu (1967) suggested that the sagging membrane technique

appeared to be a simple and satisfactory way of fabricating

shallow shells and that shallow compressive membranes

appeared to be suitable for floor systems in ordinary

buildings, bridges, docks and similar structures after

conducting tests on model shells of 48 x 48 cm and prototype

shells of 2.4m x 2.4m size. Taylor bLnd Allgood (1968)

15

reported the test results on funicular shells for various

thickness-to-halfspan ratios and rise-to-halfspan ratios on

model shells and prototype shells of the same size as in the

previous study. It was reported that the most probable mode

of failure of funicular shells of the type tested is

transitional buckling and as the rise and thickness are

increased, failure modes other than buckling become more

likely. Another finding of their study was that shallow

funicular shells required a relatively small amount of

reinforcement and have a low weight per unit area as

compared to flat floor slabs. Odello and Allgood (1970)

described the details of testing a funicular shell of 12.2m

x 9.15m size and 0.6 m rise. From the test results, it was

observed that shallow concrete funicular shells were capable

of carrying large uniform or concentrated load over

extensive spans and that the linear analysis provides

reasonable estimates of their behaviour only at low loads.

Odello and Allgood (1971) reported the results of a study on

the economy of shallow funicular shells over other types of

roofs. It is stated that funicular shells cost less than

comparable roofs for 15m to 25m span and floors of 15m to.

gOm span.

3.2 Oi fferences in Performance of Shell s as Roof and as

Foundation

The loads on a shell roof is mostly due to its

self weight only, whereas the shell foundation will be

sUbjected to the soil reaction due to all the structural

loads transmitted to it, thereby necessitating thicker

16

sections and depths for foundation elements compared to the

ones used as roof.

Unlike in the case of roof shells, buckling is

less likely in foundation shells, since the latter rests

directly on the soil and carries the backfill on top. In

foundation shells, the weight is directly transmitted to the

soil and therefore no significant stresses are induced in

the shells due to self weight.

Shell roofs require of scaffolding and formwork

which are costly and require skilled labour. When shells are

used as foundation elements, the difficulties in this regard

are significantly reduced. The shells can be cast over a

curved surface formed by suitably excavating the soil or on

a mound of earth formed to the desired shell shape.

The actual pressure distribution below a shell

foundation is complex due to shell-soil interaction, whereas

the distribution and the magnitude of loading on a roof

shell is predetermined.

Shells of all shapes and types adopted for roof

are not suitable for foundations because of various

limitations. Boloski (1959) mentioned that the downward

convex shells are also expedient from the point of view of

soil mechanics because of their favourable influence on the

distribution of soil stresses. He recommended use of shells

formed like elliptic paraboloid instead of mat foundations.

Tetior (1966) stressed that the shells, used in foundations

17

and supported on foundation soil by their whole surface,

take up distri~uted loads from the soil very efficiently in

comparison to planar elements.

3.3 Funicular Shell as a Foundation Element

Funicular shells of various lateral dimensions,

ri se and thi ckness, both rei forced and unrei nforced, have

been used as roof and floor elements. Single funicular

shell roof of the Kanpur Municipal Building (Ramaswamy,

Raman and Zachariah, 1962) is one of the largest single

funicular shells ever put to use. Smaller size, shallow

square and triangular funicular shells of 1 to 1.5m size in

plan have been in use as roofs and floors for many

structures. When these are used, suitable supporting beams

will have to be provided so that the shells with the

supporting beams form the roof or the floor. Such shells

usually do not have any reinforcement in the body and the

edge beams are of small dimensions with nominal

reinforcement of one or two 6 mm diameter mild steel bars

only. The thickness of the shells varies between 1.5cm to

3cm. It is seen that for shallow funicular shells, there is

considerabl~ savings in material and consequent reduction in

cost of roof or floor in comparison to conventional

reinforced cement concrete slabs. It is, therefore, logical

to use funicular shells instead of conventional reinforced

cement con~rete mat foundations so that the foundation costs

can be brought down and material saved. Much work on these

lines have not been taken up so far and there is complete

lack of literature on this topic. The difference in the

18

•.\1

functions of the roof and the foundation as outlined above

should be considered in detail so that it is possible to

uti 1i s e fun i cu1ar she 11s e f fie i e nt 1y as f 0 undat ion s • ~1 at

foundations are adopted to distribute the superimposed

loads on 1arge areas on soi 1s of poor beari ng capac i ty.

They have to be rigid enough to reduce differential

settlements due to the loading of large areas. The precast

funicular shells when assembled suitably with proper

interconnecting beams can form a foundation unit of

sufficient rigidity economically. compared to a reinforced

cement concrete mat. As the convex surface of the shell is

in contact with the soil. the shell is fully supported and

chances of buckling of the shell are remote. As stated

earlier. the convex contact surface tends to distribute the

superimposed loads on a wider area. thus reducing the

intensity of soil pressure. Studies by Szechy (l965) has

shown tha the influence of shape of the contact surface is

the biggest in the case of shallow foundations and there

will be no effect practically at depths greater than twice

the widt.h of the strip foundation. Funicular shells can be

made to form units of a centrally loaded footing with.

suitable connecting beams for columns. Possibilities exist

for the replacement of conventional reinforced cement

concrete footings by multiple funicular shell footing with

savings in material cost. Detailed studies in this regard

are also warranted before they can be adopted in the field.

A f 0 ur she 11 .,,' nit i side a11y sui ted for a squaref 0 0 tin g •

19

The rigidity of the unit can be suitably adjusted by

properly designing the connecting beams of the unit.

A strip foundation can also be composed of the

funicular shells with necessary connecting elements in the

form of reinforced concrete beams. The dimensions of the

shells may be adjusted such that the width of the unit can

be made up of two shells with the connecting element.

The performance of individual shells and the

factors that influence their behaviour as elements in rafts,

footings' and strip foundations requires detailed study.

Bearing in mind the potentialities of the funicular shell as

a foundation, studies on the subject commenced in the

College of Engineering, Trivandrum (Thomas, 1970) with which

the author was also closely associated. To bring out the

differences in the behaviour of the funicular shell as a

foundation and roof element, a series of laboratory and

fi el d tests were conducted on funi cul ar shell s of 60cm x

60cm size. A diagrammatic sketch of a typical shell is

sho'l,n in Fig.3.!. The dimensions of the edge beams of the

shell sand thei r rei nforcements and the concrete mi x used

for casting the shells were varied to study their effect on

the perform~nce of the shells. The details of shells tested

are given in Table 3.1.

The shells .."ere cast using the sagging fabric

technique with coarse aggregate passing 6mm sieve. The edge

beams were provided with 2 numbers of mild steel bars as

given in Table 3.1. The testing of the shells was done

20

•

after curing them for 28 days. Three numbers of cubes of

7.06cm size were cast from each mix of the concrete prepared

for casting the shells. They were tested for their

compressive strength on the day of testing the shells.

3.4 Testing of Shells as Foundation Elements

The shells were tested with their convex surface

in contact \'dtl1 the foundation soil and with equal loads

transmitted to the four corners of the shell on the edge

beams. The test set up was as given in Fig.3.2. The set up

consisted of a self-straining loading frame installed in

the test pit, and sand compacted to the desired

characteristics. The load was transmitted through a four

legged load transmitting frame by a hydraulic jack through

a proving ring and dial gauges were used to measure the

settlements of the shell at various points. The shell s

were tested on top of the sand bed. The foundation medium

consisted uniformly compacted river sand, the

characteristics of which were as given below:

1.

2 •

3.

.4.

5 •

6.

Dry density

Void ratio

Specific gravity

Angle of internal friction, p(from direct shear test)

Effective size

Uniformity coefficient

1. 7 gmfcc

0.58

2.68

36 0

0.2 mm

1.5

The dimensions of the test pit were large enough

so that boundary effect have little effect on the test

21

results. The loading arrangements were as shown in Fig.3.2.

The loading tests were conducted as per loS.No.1888. The

loads were app 1i ed in increments of 120kg at a time after

applying initially a ceating stress of 7ogm/cm 2 and

releasing it. After each increment of load. the dial gauge

readings were noted at intervals till the rate of

settlement \'/as less than O.02mm/hr. At th is stage. the

dial gauge readings were noted and were taken as the final

readings for that particular increment of load. The loads

corresponding to the first crack and at various stages of

development of cracks were noted for each test. The

loading was continued upto the failure of the shell and the

corresponding

were noted.

settlements of various points on the shell

3.5 Tests on Shells as Roof Elements

The arrangement for testing the shell as a roof

element was as shown in Fig.3.3. The testing was done using

a self-straining loading frame in the laboratory. The

loading \'Jas done by reaction jacking through 16 points on

the body of the shell and the procedure of the test was

almost the same as that for the testing of shells as

foundation element except for the fact that the readings for

each increment of load was almost immediate in these tests

unlike the former. Here again. the loads at first crack at

various stages of development of cracks and the failure

loads were noted and the corresponding deflections observed.

22

3.6 Results of Tests Conducted with Shells as Foundation

and as Roof Elements

The load-deflection/settlement graphs for the

typical shells L2. L4 and Ld2 tested as roof and foundation

elements are given in Fig. 3.4. 3.5 and 3.6 respectively.

The loads at first crack and the loads at failure for each

of the shells are given in Table 3.2

Based on the study. the following conclusions were

arrived at.

1. There \'Ias no appreciable improvement in the ultimate

load of the shell due to the increase in percentage of

reinforcement in the edge beams.

2. The failure mechanism developed through cracks which

started from the edge beams and proceeded to the shell

body.

3. By i ncreas i n9 the depth of the edge beams. a defi ni te

increase. in the load carrying capacity could be

attained.

4. The loads at first crack and ultimate loads were higher

for shells as foundation than the corresponding loads

for t est s \'1 i t h the she 11 a s roo fin 9 e 1em e nt • This i s

attributed to the fact that the pressures were more

uniformly distributed for the test with the shell as

foundation than in the latter.

23

5. The greater the rigidity of the edge beam, the greater

were the loads at first crack and failure when the

shells of 60cm size were tested as foundation.

6. It was possible for the funicular shells of thin

sections to carry higher loads in comparison with plain

elements of same thickness to the ground resulting in

general economy of foundation construction.

24

•

Table 3.1

------------------------------------------------------------Serial Shell Edge Beam 1,1 i x Reinfor- DesignationNumber tested Size cements of the

as shell------------------------------------------------------------1 Foundation 4cm x 4cm 1: 2: 4 2 mm FL2

2 , I I , , I 4 mm FL4

.3 , I I , , I 5 mm FL5

4 , I , I 1:1.5:3 2 mm FH2

5 , , , , , , 4 mm FH4

6 , , , , , I 5 mm FHS

7 , , 4cm x Bcm 1: 2: 4 2 mm FLd2

B , , I , I I 4 mm FLd4

9 , I I , , I 5 mm FLd5

10 Roof 4cm x 4cm I , 2 mm RL2

11 , , , , , , 4 mm RL4

12 I , , , , , 5 mm RL5

13 , , , , 1:1.5:3 2 mm RH2

14 , I , I , , 4 mm RH4

15 , , , , I I 4 mOl RH4

16 I , 4cm x Bern 1: 2: 4 2 mOl RLd2

17 , , I , , , 4 mm RLd4

18 , , , , I , 5 mm RLd5

------------------------------------------------------------* two bars were used in all cases

25

Table 3.2

---------------------------------------------------------Designation Load at first Ultimate load Testedof shell crack as

kg kg---------------------------------------------------------

FL2 537 575 Foundation

FL4 450 1830 , ,

FL5 665 1780 , ,

FH2 751 2480 , ,

FH4 720 2420 • •FH5 1140 2240 • •FLd2 1471 3220 ••FLd4 1450 3222 • •FLd5 1311 3490 • •RL2 494 1230 Roof

RL4 450 1255 ,.RL5 580 1306 • •RH2 775 1412 • •RH4 665 1360 • •RH5 990 2002 • •RLd2 775 1412 ••RLd4 775 2118 • •RLd5 1208 2002 ••

-----------~------------------------------------------ ---

26

•

2ern thie}< shell

rei nforeementSECTION ON XX

\c1ge beam'--+-----r----------:·

1f

•• . . A. );1 -f....Sen)". '.".-: ::-::.> ; : , , :. :0 ,.,"' :"' ..•.:A.::"':.· ..-·· ..·~ -6em

T

r~-----60cm--_.~

'-'--'-'-'-'--'- --xuato

PLAN

FIG. 3.1 DETAILS OF FUNICULAR SHELL MODEL

60cm x 60crn

27

•

I. -----4m

E

of loading

.. .....' . . : . ': ..

• •••• I' .'

o' • _'., .

. ', " ..." .· .' .. . " . .... .: .... ~',

· . ~ .' . '.

. . .. : . ',' . .'.' " ", .

.....· .', .

1y>:~i-"~~:n~" ~i1:lng

'. '.: ' ...... 1• .. ',... I· . '. ; .: . " .. ,... . '. I

:....: ',' ',' '.:-.: « ::'. ;; loading frame

'.•: :,-': :.'~. m:~;-:'-'"--'-"_:'-,.'_.'-':_.:'"tnl. ::.·..p:.:.·.·.:-I:.' . . .' . ,

.... , :::',:: ~~" ~<.-~ :. ".; .'1 compacted earth.. ' .' '.-' ; ,.' ..: ,: .. 1,· .' . .'.... ~ . ...

'. ,. . ..' .'.. ' • - .' ", I. ~ '. '. .: . ". . '. .' .' • I• • .'0 ' :. . ~ : •.•.•. I

· ~ .' ." ",•... ·0 : ".. :.. ".... .' '. ". ',' ., I." .: ,'. . ., ' ...... ' " :- :. :',-:', .....:/l,. "'.. : . ,........, ~ .-

EN

"1' 1"1/1' ~, - "" ~ •.•... ' •.I·,.'.·~.·.:-I " .I ~ -:

I·" ,"" '· . . ...I', ... : " . "

FIG.3.2DETAILS OF LOADING FRAME AND TEST BED FOR

FOUNDATION

28

load distribut ion setup

funicular shellsupport at corners of shell

!r----- Dial gaLge at centre of edge

beam----H---FUNICULAR SHE LL

-..-----tt-- Dial gauge at centre of shell._--+1---- Dial gauge at quarter span of

shellLOCATIONS OF DIAL GAUGES

loadi.ng f~ameproving ringhydraulic jack

-- -0 00

J •

I I..L 1

~ ---1L~ nII II- -

ELEVATION

load distribution setup

,'..,

'-'r;::;o.~-+--funicular shell------'"'---

tPoint through which, load IS

applied on the shellc:support at corners of shell

PLAN 0 position of jack

FIG33LOADING ARRANGEMENT FOR SHELLS TESTEDAS ROOF ELEMENTS SHOWING LOCATIONSOF DlA L GAUGES

29

"'-~I.~~

~~~~~h..,.~e~ ..~;::; .~~ ,_,-,- -......-.~.,... ---- ........0 __ .....-,e.-._'"

" D_"'O __O~'- ~'-..,t

.~ • - • .:::-- 0-- . -)( . -;(. _

f..~~, • ", ::-::-:--:=.:: }Test as foundation, ",• -, -IL..-

":: '1. ' -.

~ I'~' FL2~~~ \" \" ..." \

\ \

\~'< 1"-\1Test as roof RL2

]\J I I

LOAD, kg ~

0 0 300 600 900 1200 1500 1800 2100 2400 2700

1EE...22aI­u~3LLUJo

';::>4

~j~5l­I-

~60--0-0 QUa rter

)(_._. -x Central

span dial ga~e reading

dial gauge reading

.- - - - -II Reading of cJil1l 'gauge at centre of edge beam

FrG3.4 LOAD SETTLEMENT GRAPHS OF SHELLS TESTED

AS FOUNDATION AND ROOF E LEMENTS- FL2

AND RL2

30

2100 2400 27001500 18001200LOAD I Kg ~

600 900300~l·\o~~ ,~o ~~'t~ ~.'t, ... -':::-1-.

".~i" - _~~_.._._'x .... • a..."..: "- ~-,._ .

, '............... ,,0_ ° __._"'__" ~o a ........ -0_ ?-~-.-)(...- leo-

as fou rdatlonx .~ ......... J - 0- test" "-0 .~.. 0

" .~ I"x, .. ',._" .,~ --'-I FL4Jc" .,

• 0 '.

\c" '\'," 0 'I

)c, '\. '\, '0\. 'l\ test as ro of - R L4"\

\;

00

50---4,Qucrter span dial gauge reading~-._Cent ral dial gauge read ing"'---4 Dial gauge reading at the centre of edge beam '

FlGJ.5LOAD SETTLEMENT GRAPHS OF SHELLS TESTED AS

FOUNDATION AND ROOF ELEMENTS-FL4 -AND RL4

1

EE 2

1---

ZW

UJ~3

-> -.Jl-I-

~4

l

~

lJJN

LOAD1Kg _o 0 600 . 900 1200 1500 1800 21GB 2400 2700 3000 3300

E ~~ ~E \u"''"--~

... ,,', "" .,........... -Ul \' 't::-.h, "z 1 .... 'x,

f-_O "\. '" Jl~",'<JI...;: ·I~:-~~:--. __U ............... ~ ........'J< ., :::-x.......w ""- 11',- ....." .....---l 2 . '11 'n.... . -=,..--

til -"ox - K"'., '-.-.:.~~~ j Settlements of shell J:Ld-tJ.~ I"', I'll, , I, '-u.. -_~-. -~:::::-when tested as (.

Ul 3 "-, .... Il I -- -11__-~o::--I fou ndatlonI-- - x_ ._o::::--_...:..-__~_~~ "oDe flect ions of shel -11.. __~.=:~ /~ },.'" when tes ted as roof - --Q..- ,~-..,..._~ 4 ., RLd 2 . -_Jl~

r- I "II. ~ Ir-WUl

5 I I I I I I I ' I

o 0 Diar gauge at qucrte r spa nx-·_)C 11 )1 'I centrea----iS II IJ I) centre of edge beam

F1GJ.6 LOAD-DEFLECTION/SETTLEMET GRAPHS OF SHELL.S~

.~ND FOUNOTION ELEMENTS - FLd 2 AND RLd 2TES TEO AS ROOF