Pro! G. S. Ramaswamy, Director, Strnctural Engineering Research Center, Roorkee, India

1110 Funicular Coneopt

The normal sequenee followed in designing shell roofs demands Ihal lhe firsl slep is lhe selection of lhe shape or geomelry of lhe shell surface. Then , lhe slresses in lhe shell surface under loads are delermined. In selecling shell shapes, no deliherate effort is made to ensure a favourable state Df

slress in lhe malerial selecled for building lhe shell . Malhe­matical convenience seems to be the only consideration influencing the choice af shapes. There are many advantages to be gained by reversing this traditional sequence ; ar, in other words, we may assume a desired state cf stress in the material after carefully examining its strong and weak points and then proceed lo find a shape in wlUch lhe chosen slress slale will prevai! under a specified loading condition. Shapes so found are called funicular shapes.

We may ilIuslrale Ihis concepl wilh reference lo brick masonry. Our efforl in building wilh brick masonry will naturally be direcled lowards fully exploiling ils slrong poinl , its compressive strength. Because tensile strength is its weak paiot , it is necessary to ensure, at the same time , that the structure develops little or no tcosiao. These two desirable objeclives are simullaneously realized by choosing a slale of pure compression unaccompanied by shear in the shell roof. An elemenl taken oul of such a shell and ils plan projeclion are shown in Figure 30-1. The following well-known relalions exist between the real stress resultants NX1 Ny , and N xy and lhe pseudo slress resultanls Nx, Ny, and N xy and belween lhe real loads F x' F y' and F z and the pseudo loads X, Y, and Z.

N ~ 1+ q2 x 1+ p2

(30-1 a)

(30-lb)

251

(30-1 c)

and

X= Fx~ 1+ p2 + q2 (30-2a)

Y F;.J 1+ p2 + q2 (30-2b)

Z Fx~ 1+ p2 + q2 (30-2c)

where :

8z 8z p =- andq= -

8x 8y

We may wrile down lhe equalion of equi!ibrium for lhe element in the z direction as:

pX + qY - Z (30-3)

Most shell roofs in practice carry only vertical loads. For Ihis condilion of loading X = Y = O, and Equalion 30-3 lakes lhe formo

2N xy 8x8y

- Z (30-4)

LeI lhe vertical load acling on lhe shell be g per unil area. Inserting llUs value in equalion 30-4 , we gel:

252 Designing, Engineering, and Constructing with Masonry Products

2 2N ~+

xy 6x6y

(30-5)

In practice, shallow shells are general1y favoured for roofing applications. For such shell roofs. 1'2 and q2 are small in comparison with unity. and Qne may write:

2 N 62z +

xy 6x6y (30-6)

At this stage let us specify the desired stress state. As already explained . it i5 desirable to have a pUfe compression state in the masonry unaccompanied by shear. Shear stresses should be obviated as they cause tension. For 0l'timum .!!.tilization of the compressive strength of the material Nx and Ny must be equal. These conditions may be formulated as follows:

(30-7)

where N is the magnitude of the desired compression in the masonry . and:

(30-8)

~----------------x

y ,LF./# __ Fy N,

Ny

J--x F---N, y

Figure 30-1.

Inserting these conditions in Equation 30-6, we get ,

(30-9)

Equation 30-9 dennes the desired shape of the funicular surface. This relation may be generalized to make it inde­pendent of the coordinate system employed by recasting Equalion 30-9 in vectorial form as:

(30-10)

It must be distincUy understood that a shell surface can be made funicular only for one specined loading condition. In our case, vertical loading was selected, because the dominant load to which a sheli roof is normally subjected is vertical. However, experience gained with numcrous funicular shells a1ready built shows that , a1though their shape is selected for optimum performance under vertical loads, Ihey behave very well under other loads.

AnaIytical Methods for Finding Funicular Shapes

Equation 30-10 is the weU-known Poisson's differential equation of mathematical physics. It may also be noted that it is the equation which governs the height Z of the PrandUI membrane in the Membrane Analogy for Torsion . Many available results relating to torsion af prisma ti c bars afvarious cross sections2 may, therefore , be drawn on to find funicular shell shapes corresponding to various ground plans. Figure 30-2 gives equations af funicular surfaces over some af the usual graund plans. The ordinates of a funicular surface over a given groulld plan can also be computed accurately at desired intervals by manual methods of computation such as Relax­ation. Alternatively , by employing numerical methods, the Poisson 's equatian can be solved on an electronic digital computer to arrive at ordinates of lhe surface accurately.

Experimental Methods of Generating Funicular Shapes

A simple method of generating funicular shapes successfully is to make a wooden mould whose ground plan is similar to that of the structure which is to be roofed over by a funicular shell. A flexible fabric , stretched taut across lhe mould , is loaded with wet plaster of paris. The fabric is now allowed to sag and the p]aster to set. On inversion, the shape obtained is the required funicular surface. The ordinates or the prototype roof are easily obtained by multiplying the ordinates of the model by the scale factor. The modol of a funicular shell roof over an oval ground plan, generated by tltis process, is shown in Figure 30-3.

A more sophisticated method is to blow up a rubber membrane over the desired ground plan and measure its ordinates.

It is also well-known that the Poisson's equation involved can be solved by the Electrical Analogy Method.

Funicular Brick Shell Roofs for Industrial Buildings 253

In building large funicular shell roofs , the author's experi­ence dictates that ordinates necd to be at 2' intervals for accurate fabrication af forms.

Bending Analysis of Furneular Sh.lIs

Equation 30-10 dermes the funicular surface only ir a membrane stress state prevails . Secondary bending stresses do arise in lhe shell in Ule neighbourhood of the edge members. A study was reeently made of a funicular shell using the shallow shell bending theory of Vlasov3 . It was found that even after bending stresses are taken into aceount . lhe sheJl is more OI

less in a state of low compression throughout. The small tensile stresses that develop are well witltin the limits that unreinforced brick masonry can withstand without cracking.

Examples of Praetical Application

The author has sueeessfully built two types of funicular shells using brick masonry. The details of the first type are given in Figure 304. Several shells of this type have been built for the National Design Institute, Ahmedabad, India. Figure 30-5 shows the finished view of one of the shells. The shells measuring 41' x 41' are supported 00 reinforced concrete edge beams }'-2" x 3'-6" which, in tum, rest on reinforced concrete colurnns. The roof is 4-1/2" thiek and is built ofbricks laid on edge. The joints wltieh are 1/2" and 1-1/2" wide are filled with mortar. The brickwork is lightly reinforced by the provision of 3/8"-<liameter reinforeement at 10-1/2" in both direetions. The joint width is entirely governed by the need to provide adequate concrete coveI to the reinforcement to prateet it fram corrosion caused by emorescence. The handmade bricks had an average compressive strength of 800 p.s.i. Th.is quaJity of brick was considered quite adequa!e for the purpose as the

o Groul'Id plau [\Ultion \0 \ .. ,urra" ~ no ' ~o.+c. -t

z. ti,o.~ ~ , ,

$ ' -(-o)

)lI'[ n- l ,3 ,5 • . .' i

(i co~h W ) n", n ff b c.os --

c.osh -za- "

m -

z • 16!.~ 2:= .-, ~(_\)T .. n

1 x n _ l, 3, 5, .

• ( , - c.O$h lI1tY) ---rã""" c.o~ n"

co!.h 'fi;- -,,-

$ g a1. b1

(1. 2 ) l X X . -- • + Y I

2N ai "b2 ar b'"-

~

(f)-• x , . ...!..-(x2 +y2_ a1) .. >-- . ----l

w- z.~ [1 (x,2 +y2)_-h()I,5 - 3xi)-z~ az] , • %

~~ ~ Figure 30-2. Equations of funicular surfaces over some of the usual ground plans.

Figure 30-3. Model of a funiculor shell roof over an oval ground plano

254 Designing, Engineering, and Constructing with Masonry Products

I ~ tnncr face of edge beam

PlAN SHQWING REINFORCEMENT

, EdO' b,em

• 3/8 Dia . bors @

4 Nos #. CIC both ways

J /4 dtO, bors

3/e" Dia. stirrups

4 Nos. 3/4· dia. bars

_ 4 Nos." dia. bors

14·-+ SECTION X X

Figure 30-4. Funicular shell in reinforced brick wark.

Figure 30-5. Finished view af funicular shell.

~Inner face of edge beam

>-t-_ __ 3/8~ Dio. bars ot 101ft clc both woys

DETAll AT A

Funicular Brick She/l Roofs for Industrial Buildings 255

r f-' f-' f-' --I '~ I '" I 9 5/;

I / - "\ I

I A ) I "-r- / I

i I

I i I I I---Inner foce of edge beam

PLAN SHOWING THE ARRANGEMENT OF BlOCKS

~' boom

{ Hollow cloy blocks

4 Nos. vt dia . bars

llá Dia stirrups AfOC,,- 4 Nos. 5/0' dia. ba r!>

'2' +-=-+ SECTION X X

Figure 30-6. Unreinforced funicular shell built with hollow c/ay blocks.

fac~ of edge beom

ISOMETRJC VIEW DF A HOLLOW ClAY 8l0CK

DETAIL AT A

Figure 30- 7. Finished view of roof soffit.

256 Designing, Engineering, and Constructing with Masonry Products

compressive stresses that develop in lhe shell under a vertical load of 52.5 p.s.f. were of the order of only 175 p.s.i.

A different technique was employed for a masonry shell roof recently built at Roorkee , India. Measuring 22' x 20', it is built of extruded hollow clay blocks of the type shown in Figure 30-6. The 1:4 cement morta r joints between the blocks are only 5/8" wide. The shell is supported on reinforced concrete edge beams which rest on colurnos. Details Df this roof are given in Figure 30-6. A finished view of the roof soffit cao be seeo in Figure 30-7.

Conclusion

Brick funicular shell roofs of the type described in this chapter offer ao economical means af roofing large cúlurnn­free spaces required for factory buildings. They are especially suitable for wide application in the developing countries af the world where there is a pressing need to conserve lhe small supply of cement and steel. The brick funicular shell roofs developed by lhe author are ao outgrowth af lhe extensive work carried out by him aver the past ten years 011 reinforced concrete funicular shells.4 ,S

x,y,z z=f(x,y)

p

q =

r =

s =

t =

Nomenclature

co-ordinates defining lhe surface of the shell Equation of the surface 8z 8x 8z 8y

82z 8x2

82z 8x8y

8 2z

8y2

x y Z ,,2 g N

Stress-resultant in the shell in the x direction Stress resultant in the shell in the y direction Shear slress resultant Psuedo slress resultant in lhe x direction Psuedo stress-resultant in the y direction Psuedo shear stress resultant Externai force acting on the shell elemen t in the x direction ExternaI force acting on shell element in y direction Externai force acting on shell element in z direction Psuedo force in x direction Psuedo force in y direction Psuedo force in z dircction Laplacian operator Verticalload acting on the sheU per unit area Desired compressive stress in the shell

References

1. Timoshenko, S" and Goudier, J. N. , "Theory of Elasticity," Ellgineering Societies Monographs, McGraw-Hill Book Co. , 1951, pp. 268-269.

2. Timoshenko, S., and Goodier, J. N., "Theory of Elasticity," Engineering Societies MOllographs, McGraw-Hill Book Co., 195 I , pp. 259-285.

3. Vlasov, V. Z. , "Generce Theory of Shells and its applicatian in Engineering," TTF-99 , NASA, 1964.

4. Ramaswamy, G. S., "Analysis , Design and Construction af a New Shell of Double Curvature ," Proceedillgs of the Symposium 011 Shell Research, Delft , Augus! 30-September 2 , 1961.

5. Ramaswamy , G. S., Raman , N. V. , and George , Zacharia, "A Funicular Shell for the Conference Hall of the Kanpur Municipal Corporation - Design and Construction ," lndian Concrete/ouma/, Vol. 36 , No:9, September, 1962.