May, 1958 165

Analysis of Cylindrical Tanks with Flat Bases by Moment Distribution Methods

by Amin Ghali,

T 1. SyIlopSis

HE moment distribution method is used to find the moments and the ring tension in the walls

and the bases of the following two types of cylindrical tank :

(a) Cylindrical tank on a rigid flat foundation. (b) Cylindrical tank with flat base supported on a

For the first type a trial and error method is used to determine the width of the ring-shaped part of the base which will lift up from the foundation surface.

In the second type the variation of the thickness in the overhanging part of the base is taken into account. Graphs are presented in the appendix to facilitate the solution.

The design of each type is illustrated by a numerical example.

cylindrical shaft of smaller diameter.

M.Sc., Ph.D.

In order to apply the method we need to compute the fixed-end moments, the stiffness and the carry-over factors for each element.

This method of moment distribution was used by Mtirkus, Gy.l* in Hungary and by Lavery, J. H.2 in Australia for the analysis of certain types of cylindrical tanks. Some of the data given by MBrkus will be used in this study.

3. Stiffness, Carry-over Factors and Fixed-end Moments in Cylindrical Walls

(a) Stiffness The moment which causes unit rotation at a hinged

end of a cylindrical wall varies according to the different conditions of support at the far end. In the following the stiffness factors are given for the three cases shown in Fig. 1. It is convenient to express the stiffness S

Fig.

2. Introduction The usual procedure of the moment distribution

method of Hardy Cross could be used to take account of the continuity of the walls of cylindrical tanks with their roofs or bases. A vertical element of the wall is considered together with a radial element of the roof or the base. The method involves the calculation of moments at the ends of the elements under artificial conditions of restraint, then a distribution of un- balanced moments by arithmetical proportion when the artificial restraints are removed. The fixed-end moments per unit length developed at the edge of the cylindrical wall due to the liquid pressure, and those developed at the edge of the circular plate are deter- mined, the unbalanced moment is distributed between the connecting elements in proportion to their stiffness. The term “ stiffness ” here means the moment needed at the end of the cylindrical wall or the plate to produce unit rotation of this end. Also, if a moment is distri- buted to one end of the cylinder (or the plate) while the other end is held fixed, a fraction of the distributed moment is carried over to the fixed end of the cylinder (or the plate). * The index numbers refer to the items in the list of references

at the end of the article

1

in terms of E where E is the modulus of 12 (1 - p ) ’

elasticity and p Poisson’s ratio. In other words the moment which causes a unit rotation at an edge of

E stiffness S is equal to S. 12 (1 - p ) ’

For the three cases of Fig. 1 we have :

case a (Fig. la)

sinh PH cosh PH -sin PH cos PH S = 2pt3 ( cos2 PH + cosh2 PH

case b (Fig. 1 b)

sin (3H cos PH -sinh pH cosh @H sin2 P H -sinh2 PH S = 2pt3

case c (Fig. IC)

sin2 PH + sinh2 pH S = 2pt3 sinh PH cosh PH -sin @H cos PH

166 The Stmctural Engineer

t is the wall thickness and H its height. is a dimensionless factor.

PH =

where R is the wall radius.

The term QH

. . (4)

For cylindrical walls having big values of PH ( QH> x ) , the terms between brackets in equations ( l ) , ( 2 ) and (3) tend to unity. The stiffness a t one end will be the same whatever the conditions at the other end, and will be equal to

S = 2pt3 . . . . (5)

Most elevated tanks as well as some grounded tanks have dimensions which give values of pH> x , and the stiffness of the walls can be easily calculated by equation (5).

(b) Carry-over Factor If a moment is applied at the edge of a cylinder

while the other edge is fixed (Fig. lb), a fraction of this moment will be carried-over to the fixed edge. The ratio between the applied moment and the moment developed at the far fixed edge is the " carry-over factor." This factor depends upon the value PH, it may be a fraction with positive or negative sign. In Table 1 below the carry-over factors are given for values of PH between 1 and 6.

Table 1.-Carry-over Factors for Cylindrical Walls

For long cylinders ( p H > x ) , the carry-over factor is very small, which means that a moment applied at one edge dies before it reaches the other end.

(c) Fixed-end Moments The fixing moment at the bottom of a cylindrical

wall having its top edge free, and filled with liquid of specific gravity y, could be expressed by the relation :

There exist tables and curves 1 , 3 , 4, 5 and 6 which give the moment a s well as the ring tension in cylin- drical walls fixed at their bases and subjected to triangular loading, with the top edge under various conditions.

4. Stiffness, Carry-over Factors and Fixed-end Moments in Circular Plates

The bending of a circular plate loaded symmetrically with respect to its centre has been exhaustively treated by many authors (see for example references 2, 7, 8 and 9. With the usual assumptions cmsidered in the elastic theory of plates, it could be shown that the deflection of the plate is governed by the differential

** The term 1.3068 = 43(1 -pa), in which is taken equal to + in which p. is taken equal

4 -

1 l * The term -- 3.4156 - 22/3 (1 -p:) to 4.

equation in the x , y , z system of co-ordinates

where W is the deflection ; q the intensity of loading ; D is the flexural rigidity

Ed3 D = 12 ( l - $ ) . . . (8)

in which d is the plate thickness.

This differential equation (7) can be expressed in a polar system of co-ordinates in which the centre of the plate coincides with the origin of the system. In the case of circular symmetrical loading the differential equation will be,

d4w 2 d3w l d2w 1 dw ~ + ~ ~ - ~ d , z + $ d , = D ' '

- (9)

w in this equation indicates the deflection of all points which lie on a circle of radius r. The solution of the differential equation (9) in its general form is

A I , A z , A3 and A4 are the integration constants which are to be determined from the edge conditions of the plate.

The principal moments will be acting in a radial direction M r , and in a tangential direction Mt. Their values per unit length are :

andMt=-D(- -+ 1 dw p-) d2w Y dr

When a circular plate is built continuous with a cylindrical wall or with another ring-shaped slab, the radial moments in the wall and the slab per unit length of their common edge must be equal. Also,

the rotation in the radial direction (g) of the edge

of the plate must be equal to the rotation at the edge of the cylinder.

Similar to what was considered when dealing with cylindrical walls, the stiffness at the edge of a circular plate is equal to the radial moment applied at this edge to let it rotate a rotation cquals unity.

If a radial moment (Mr) l is applied at edge 1 of a ring shaped slab of radii a and b (Fig. 2 ) , while the

P

I l

Fig. 2

other edge 2 being fixed, a fraction of this moment (Mr2) will be carried over to the fixed edge. The

May, 1958 167

ratio between the applied radial moment per unit length and the corresponding value of the moment developed at the far edge is the carry-over factor f1-2. The carry-over factor depends upon the ratio between the two radii, and may be a value less or more than unity.

The radial moments at a fixed edge due to a loading on the slab are called the fixed-end moments. Tables and graphs are available, 2 and 8, which give the fixed-end moments, stiffness and carry-over factors in circular and ring-shaped slabs for various loadings and edge conditions. These were obtained from the basic equations (lo), (1 1) by choosing the integration constants which suit the edge conditions. The values given in table 2 below were calculated by Mhrkus (S), they are given here since they will be used later in this discussion.

and the base may cause a ring-shaped part of the base near its edge to bend as shown in Fig. 3, whereas the inside circular part of the base may remain flat.

This is a nonlinear problem as regarding the bending of the circular plate, the conditions at the edges of the deformed part of the base are changing with the deflection, and the deformation of the plate will not be proportional to the load applied on it.

The stiffness of the deflected ring-shaped part of the base depends upon the dimension b (Fig. 3), and this depends upon the unknown moments which are developed at the edge of the base. A trial and error method will be used here ; first the radius b will be assumed and then corrected to satisfy the conditions of the problem. These conditions are that at a circle of radius b, the deflection, the moment and the slope of the deflected surface are zero. The deflected

Table 2.-Stiffness, Carry-over Factors and Fixed-end Moments in Ring-shaped Slabs of constant thickness (Fig. 2)

Outer radius Inner radius

Stiffness at edge 1 E l d3 E

a - - _

b I T

1 .o 1.1 l .2 1.3 1.4 1.5 1.6 1.7 1 .8 1.9 2.0

0 0.0254 0.0518 0.0789 0.1065 0.1347 0.1632 0.1922 0.2214 0.2510 0.2808

T Carry-over factor fi-2

0.5000 0.5374 0.5676 0.5996 0.6308 0.6604 0.6894 0.7174 0.7446 0.771 1 0.7969

-- a b - -

1.0 1.1 l .2 1.3 l .4 1.5 1.6 1.7 1.8 1.9 2.0

Fixed-end moments due to a distributed load Q I MI

i n terms of q b2

0 -0.0007 -0.0033 -0.0072 -0.0126 -0.0199 -0.0277 -0.0373 -0.0484 -0.0608 -0.0746

b a -

1 .o 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

M2 in terms of

4 a2

0 -0.0008 4 . 0 0 3 5 -0.0081 -0.0150 -0.0247 -0.0380 -0.0570 -0.0862 -0.1433

5. Sign Convention for Moments Any system of signs could be followed, similar

to those used in the usual moment distribution analysis of beams or frames. In this discussion, because of symmetry, the analysis is carried out on one half of the structure, namely the left half. An external moment applied at the end of an element is positive when it tends to rotate this end in the clockwise direction.

6. Variation of the Moments away from the Edges The moment distribution serves to calculate the

continuity moments at the intersection of circular slabs with cylindrical walls or between elements of ring-shaped slabs supported along annular rings. Starting from these moments the variation of stresses throughout the cylinder or the slab can be easily calculated. Data is available, 1 and 4, which give the moments and ring tension at different heights in cylindrical walls subjected to triangular loading as well as for radial moments applied at the edge. Also, there exist tables and curves, 2 and S, for the radial and the tangential moments at different radii of a ring-shaped slab due to a distributed load on the slab or due to radial moments applied at the edges. The final ring tension and moments in the wall or the slab can be obtained by superposition.

7. Cylindrical Tank on Rigid Flat Foundation When a cylindrical tank is constructed on an

absolutely rigid foundation-such as solid rock- or if the tank is constructed on a thick stiff plain concrete footing, and the wall is built monolithic with the base, the continuity moment between the wall

+&4 -4 Fig. 3.-Cylindrical tank on a rigid foundation

part of the base can be considered as if totally fixed at the circle of radius b, but loaded in a manner that the radial moment at the fixation is zero. This may be seen by summing the two moment diagrams for the two loadings shown in Figs. 4a and b. A right assumption of the distance b should satisfy the con- dition that the moment at radius b is zero. This can be quickly checked and the assumption modified until the right value of b is reached. Not more than three trials should be necessary. The procedure of calcu- lation is fully explained during the solution of the following numerical example.

ExampLe 1 :

(a) Given Data Tank diameter 2R = 40' ; tank height. H..= 16' ;

floor thickness d = 10" ; wall thickness t = 10" ;

1 6 8 The Structwal Engineer

(d) Carry-over Factor This factor will be used to find the radial moment

carried over to the inner edge of the ring slab (at radius b) when a moment is applied at the outer edge

from Table 2 for - = 1.25 f1-2 = 0.584 a b

Fig. 4.-Radial bending moment diagram on a circular ring slab.

weight of liquid = 62.5 lb./ft.3 ; weight of concrete 1 = 150 lb./ft.3 ; Poisson’s ratio p =- 6

The tank is assumed to be placed on an absolutely rigid flat foundation.

(b) Loads The bent part of the base plate will be loaded

downwards by the weight of the liquid plus its own weight

10 Q 62.5 X 16 + 150 X 12 = 1125 lb./ft.’

(c) Stiffness Factors 1.3068H 1.3068 x 16.4

wall : @H = - - 4Ft 1/20 x 0.83 = 5.25 > x

5 25 16.4

.*. sz-1 = 2Pt3 = 2 X L..-- X 0.833 = 0.368

base : In order to estimate the stiffness of the ring (part 2 -3, Fig. 5), the dimension b must be assumed.

I

(e) Fixed-end Moments wall : By substituting in equation (6) we get

62.5 x 20 x 0.83 x 16 M2-l = 3.4156 ( I - & )

= 3930 Ib.ft./ft.

base : By interpolation from Table 2 we get a b - = 1.25 and - = 0.8 b a

M 2 - 3 = - 1125 X 162 X 0.0053 = - 1525 lb.ft./ft. M3-2 = + 1125 X 202 X 0.0035 = + 1575 lb.ft./ft.

The distribution procedure is shown in Fig. 6a. The moment obtained at radius b = + 733 lb.ft./ft., which should be zero if the assumption of the radius b was correct.

For the second trial we take b = 18’. A similar calculation will give the moments shown in Fig. 6b, with the moment at radius b = - 1110 lb.ft./ft. Hence the right value of b, which gives zero moment at radius b, must lie between 16‘ and 18’.

A reasonable value to be assumed for the next trial may be taken by making linear interpolation between the two previous trials, this gives ;

733 733 + 1110 b = 16 + (18-16) = 16.8’.

The moment distribution for the third trial with b = 16.8 is done in Fig. 6c, the corresponding value of the radial moment at b = 10 lb.ft./ft., which is very small, and no more trials need to be considered.

The variation of ring tension and moments are obtained by the help of graphs from references l and 8,

As a first trial we take b = 0.8a = 16’. a 20 b 16 - -- - - - 1.25

By interpolation from Table 2 we get

Y

Fig. 5

the values are tabulated below. Diagrams for the ring tension and the bending moments are shown in Fig. 7.

It may be seen from this example that the base is subjected to radial moments only near its outer edge, and they diminish very quickly away from the edge. Hence it is possible to construct the middle part of the base with a reduced thickness. We can also see

May, 1958

Table 3.-Ring Tension and Bending Moments in the Walls

1 69

I distance from top edge

due to liquid pressure on a cylindrical .: B

wall hinged at the bottom

a the bottom 2 due to restraining moment-2910 lb. at

total (lb./ft.)

,,, due to liquid pressure on a cylindrical wall hinged at the bottom

H r"

the bottom W

due to restraining moment-2910 lb. a t

.- a total (lb.ft./ft.)

0

- 300

+ 70

- 230

0

0

0

that the tangential moments in the base are small and need not be considered.

8. Cylindrical Tank with Flat Base Supported on Cylindrical Shaft of Smaller Diameter

Water towers of medium capacities are often made of the type shown in Fig. 8. The flat base may be

fi

-%F I

fb) second +,h / b (8' I

"h I

I (bzeio)

Fig. 6.-Yament distribution : Solution of Example 1.

' 0.2H 1 0.4H 0.6H

+ 4300 + 9000

+ 170 0

+ 4470 + 9000

- 50 - 20

+ 30 + 85 -

- 20 + 65 l

+ 13300

- 1300

+ 12000

+ 530

+ 205

+ 735

+ 12500

- 3710

+ 8790

+ 1480

-

+ 955

preferred to other types because of the simplicity of its shuttering and construction. A considerable reduc- tion of the stresses in the base is achieved by taking the diameter of the supporting shaft less than that of the tank wall. A suitable choice of the overhanging length is necessary to obtain values of the positive and negative radial moments which require the minimum base thickness.

In the following this type of tank will be analysed by a moment distribution method. The joint between the tank wall and the base (joint 2, Fig. 8) can rotate and also can move downwards. A direct moment distribution calculation could be applied if the stiffness, the carry-over factor and the fixed-end moments of the overhanging part are evaluated taking into account that the outer edge can move downwards but is not free to rotate.

For the sake of economy as well as good appearance the cantilever part of the base slab is usually made tapered with the greater thickness at the inner edge. The stiffness of this part is greatly affected by the variation of the thickness and consequently the bending moments and the ring tension in the tank wall will also be affected. A method for the estimation of the stiffness, the carry-over factors, and the fixed-end moments in ring-shaped slabs of varying thickness is presented in the appendix. These values are calcu- lated by the Author and plotted in curves (Figs. 13, 14 and 15) for slabs of various ratios of inner to outer diameters, also for various ratios of the thickness at the inner and outer radii. The fixed end moments are given due to a concentrated load P per unit length on the outer edge (which represents the load from the wall and the roof), and for uniformly distributed load Q per unit area, (which represents the self-weight of the slab base and the weight of the liquid above it).

The analysis of this type of tank by moment distri- bution will be explained while solving a numerical example.

Example 2 : The concrete water tower shown in Fig. 8 of 120,000

gallons capacity is supported on a cylindrical shaft 28' diameter, cast monolithic with the tank base. The thickness of the shaft wall near the top is 8". The tank is supposed to be covered by a roof weighing 40 lb./sq.ft., simply supported on the tank walls. It is required to find the bending moments in the slab base, and the moments and ring tension on the walls of the tank and the shaft. Weight of the contained

170 The Structural Engimir

Fig. 7.-Ring tension and bending moment diagram for a tank on a rigid foundation (Example 1)

liquid 62.5 lb./cu.ft. ; weight of concrete material 1 150 lb./cu.ft. Poisson's ratio p = - 6

(a) Loads (see Fig. 9a) Weight of roof, acting vertically on the top edge of the wall

17 2 P = 40 X -- = 340 lb./ft. = 340 lb./ft.

8 I t Own weight of wall = 150 x -;- x 22 = 2200 lb./ft.

Total vertical load at the outer edge of the base slab

Distributed load on the overhanging part of the base (part 2 -3, Fig. 9a) including its own weight =

= 2540 Ib./ft.

(b) Stiffness Factors The stiffness of all elements will be estimated in

E terms of 12(1 -p").

41 = 62.5 x 22 + E) X 150 = 1590 lb./ft.2

Distributed Ioad on part 3 -3 including own weight

2 x 12

- - 0 795 - 260

.I i 26

12

t - q2 = 62.5 X 22 + - X 150 = 1700 lb./ft.2

068P I I0.5Sf Q5021

0 4 I

\ S -

0 (h) 2 k b u / ; o n and carry-over fucfors

MP- 4

r c o m

/Y2- 3 M3- 3

f 3650 + f7980

- 5250 - t 6'6'20 . 3930 - + U950 h2680 - - 3380

- 9uo c- t //90 + 640 - - 8f0

- P30 - + 480 + (50 - f90

- 50 c- f 70 + U 0 - - a0

c 20

- JZUO f 26690

l

M3- 3 '

- 38700 FE f f s f 7080

f. 1700

+ 040

t /oo

c fO

Fig. 8

,May, l958 171

(i) Tank wall (element 2 -1) : 1.3068 H 1.3068 x 22 - 8.52 p H = dR.t-J17 X

- ~ _ _ _ - 8

3 8.52 22 s2-1 = 2 p 13 = (;) x 2 x 7 = 0.229

(ii) Base slab (element 2 -2) : thickness at outer radius d = 9’’ Thickness at inner radius

= d + d . d ’ = 9 + 1 7 = 2 6 ”

d’ 2 - = 1.89 17 9

Ratio of inner to outer radii = c = - = 0.795 27 34

From the curves of Fig. 13 or from Table 5 (see the appendix) we get

l s2-3 = (&)3 X 3% X 4.07 = 0.491

3 1 s3-2 = ( X X 5.10 = 0.615

(iii) Base slab (element 3 -3’) : The radial moment which causes unit rotation at the edge of a circular plate of radius c . R

E d3 - (1 + P)*

d3

c . K where s = - (1 + p).

3 1 :. s3 = (S> x i3.5 x 1 .l667 = 0.880

(iv) Shaft wall (element 3 -4) : 1 .3068 3

s3 = 2 pt3 = 2 x x (A) = 0.258 $3.5 x

The distribution factors are calculated in the usual way and given in Fig. 9b.

(c) Carry-over Factors The carry-over factors in the tank and shaft walls

are zero. The carry-over factors in the base from 2 to 3 and from 3 to 2 are

f2-3 = -1.26 andfs--2 = -0.795 (taken from Table 5 in the appendix).

(d) Fixed-end Moments (i) Tank wall (element 2 -1)

(ii)

(iii)

Base slab (element 2 -3) :

From the table or the graphs given in the appendix we get

M2-3 ‘ due to distributed loadql ;

‘ due to concentrated load P ; M = 0.270 X 2540 X 3.5 = 2,400

M = 0.064 X 1590 X 3.52 = 1,250

l l

3,650 Ib.ft./ft.

due to concentrated load P ;

due to distributed load q1 ; M = 0.920 X 2540 X 3.5 = 8,180

M = 0.502 X 1590 X 3.52 = 9,800 k13-2

17,980 lb.ft./ft.

Base slab (element 3 -3’) :

The radial fixing moment at the edge of a circular slab of radius = c R, uniformly loaded by a load q 2 is equal to ;

M = q 2 (?)2. . . . . . . (13)”

M3-3’= 1700 X- 13.5* - -38,700 lb.ft./ft. 8

-

The distribution process is carried out in the usual way and is shown in Fig. 9c. Because of the relatively large values of the carry-over factors, the convergence is comparatively slow, but this does not add much difficulty to the problem. After the moments ,at the joints are obtained, the variation of thle fing tension and the moments in the tank wall is obtained by superimposing the effect of the liquid pressure on a wall fixed at the base plus the effect of the relaxed moment (4050 -3240 = 810 lb.ft./ft.) during thc dis- tribution process. The moment and ring tension along the shaft wall are obtained by considering the effect of a moment (= 2700 lb.ft./ft.) acting at the top. The variation of, the ring tension and the moment in a long cylindrical wall ( p H > x ) due to a radial moment M O applied at the edge (Fig. 10) is given by the equa- tions

= + 4050 lb.ft./ft. . . . (6) * This can be easily proved by the theory of bending of circular

plates, see Timoshenko, Strength of Materials, Part 11, p.132. * See Timoshenko, Strength of Materials, Part 11; p.140.

Fig. 10

172 The Structural Engineer

Fig. 11.-Moments and ring tension diagrams for a water tower (Example 2)

T=-- 3.4156 e-pxsirt px, . . . . t (14)

a n d M = MO e-pxcos px . . . . . (15)

The variation of the radial and tangential moments in the base slab 3 -3' is obtained by adding the moments due to a distributed load on a simply sup- ported circular slab plus the effect of the restraining moment along the support (-29390 Ib.ft./ft.). A radial moment M O applied on the edge of a circular slab will cause constant radial and tangential moments equal to MO. The radial and tangential moments at different radii of a simply supported slab loaded by a uniform load Q are given in Table 4.

In the overhanging part of the slab the tangential moments are usually small and need not be considered. The reinforcement bars used as distributors to the main steel in the radial direction are normally sufficient to resist the tangential moments.

I

(p=;)

Table 4.-Radial and Tangential Moments in a uniformly loaded, simply supported Circular Slab

y i R 1.0 0.8 0.6 0.4 0.2 0

Radial moments in terms of q . H * 0 0.0712 0.1267 0.1662 0.1900 0.1979

Tangential moments in terms of q.Rs 0.1979 0.1942 0.1042 0.1379 0.1642 0.1830

- ~ ~ _ _ _ ~ ~

-------

Of the bending moments and the ring moments for circular rings tapered towards the edge tension in the tank Of the above are shown are needed when calculating the stresses in a cylindrical in Fig. 11. tank with a flat base supported on an inner circular

support, as the tank of Fig. 8.

9. Appendix : Stiffness, Carry-over Factors and Fixed-end Moments in Circular Ring-shaped

Slabs of Variable Thickness

Circular plates of variable thickness require even for the simple cases very tedious calculations. The stiffness and carry-over factors and the fixed-end

* The accuracy of this assumption is checked at the end of the appendix.

Because the inner and outer radii of the ring (Fig. 12a) are usually nearly equal, i.e. c is some factor round unity, the tangential moments in the ring are small and the only important moments are in the radial direction. Hence, such a ring-shaped slab is more or less acting as over-hanging radial beams'. The variation of the thickness of the ring has an effect on its statical behaviour which is assumed for simplicity as the same effect on a radial beam with the same variation of the thickness.

May, 1958 173

l ------l

Fig. 12b

(a) Stiffness Consider an element of the ring slab of Fig. 12b

between two vertical radial sections with a width unity at the outer edge. The thickness of this element varies linearly between d at the outer edge and (d + d d ’ ) at the inner edge. The stiffening effect of the adjacent elements of the ring-shaped slab on the bending deformation of the elemental radial beams can be taken into account by increasing the moment of

inertia of each beam in the ratio ~

The flex-

ural rigidity of the elemental beam at edge A

1 11 -P2>’

which is also equal to the flexural rigidity of the slab at the outer edge. The length of the elemental beam I is equal to the difference between the two ring radii

I = R (1 -C) = c’R

The thickness of the elemental beam at any point ( X = X . I , see Fig. 12b) between A and B = d + dd’x, and the breadth at the same point = 1 -(l -c). x = 1 --G‘%. The flexural rigidity at x is

E EIx = (1 -xc‘) (d + d d ’ ~ ) 3 . (17) 12 (1 - p )

The stiffness of the elemental beam at edge A is the moment which causes a unit rotation at A . It could be proved (10) that this moment is equal to

SA= - D A 1

1::: (1 +’x) (1 + d‘43 d X . (18)

which is the stiffness per unit Iength of the outer edge of the slab. The stiffness at edge B is the moment required for a unit rotation at edge B, while edge A is held against rotation (but free to move in the vertical direction). The value of the stiffness per unit length of edge B is

The values of the integral in equation (18) were calcu- lated for different values of d’ and c’ (= 1 -c) , and then the stiffnesses S, and S, were calculated, they are given in Table 5 and plotted in graphs (Fig. 13). The values given are for ring slabs having the ratio of the inner to the outer radii c = 0.6, 0.7, 0.8, 0.9 and 1.00, and values of d’ between zero and 2.

Fig. 13.-Stiffness SA and S , at edges A and B of ring-shaped slabs of variable thickness

l74 The Structural Engineer

(b) Carry-over - . Factors . . - . - . .

.A moment applied at edge A of the elemental beam Fig. 12b will be carried over to edge B with its full value. If a moment S , per unit width is applied a t A , th,e moment per unit width will be developed at B

i s equal to - S,. With the sign convention used,

the carry-over factor from A to H

l C

1 fA-* = - - . . . . (20)

C

By a similar way, the carry-over factor from R to A

fB+ = -c . . . . (21 )

(c) Fixed-end Moments The F.E.Ms. are calculated for the two loadings :

(i) A concentrated load P per unit length on the slab. (ii) A distributed load q per unit area of the slab. For both loadings the edge A is supposed to be re- strained in direction but free to settle downwards. By considering a radial element of the slab it could be proved (10) that the F.E.Ms. per unit length of the edges A and B are as follows : (i) Due to a concentrated load P per unit length on

edge A (Fig. 14)

l - I

1

M,= PI [ f 1

0

l M, = - (PZ-M,) . . . . . . C (23)

(ii) Due to a distributed load q per unit area (Fig. 15) M A =

. . (24)

The values of the fixed-end moments M , and M , are calculated by equations (22) , (23), (24) and (25) and given in Table 5 and plotted in graphs Figs. 14 and 15. They are expressed in terms of PI for a concentrated load P per unit length on the edge A , and in terms of qZz for a distributed load g per unit area. The slabs considered have the ratio c between 0.6 and 1.00 and d’ between zero and 2.00.

Fig. 14.-Fixed-end moments M A and MB at edges A Fig. 15.-Fixed-end moments M A and M B at edges and B of a ring-shaped slab of variable thickness due A and B of a ring-shaped slab of variable thickness due to a concentrated load P per unit length of edge A. to a distributed load of q unit area.

May;.1958 . 175

. . . . Table 5.--Stiffness, carry-over factors and fixed end radial moments in ring-shaped slab of variable thickness*

T I F.E.Ms. due to Stiffness factor Carry-over factor I ' - ! P/unit length

on outer edge distributed load

qlunit area ~-

f B - A from

B to A

~- SA

DA -

in terms of

l

I .oooo I .6335 2.3143 3.0248 3.7552 4.5000

-~ fA-B from

A to B

- 1 .oooo

,>

,,

- 1.1111

, ?

$ 9

,,

- 1.2500

, ,, ,

,*

- 1.4286

B

t

- 1.6667

t

t

t

--- AfB

in terms of P.1

---

0.5000 0.5833 0.6429 0.6875 0.722 1 0.7498

0.5459 0.6388 0.7058 0.7561 0.7953 0.827 1 ~-

0.60 l9 0.7065 0.7827 0.8404 0.8858 0.922 l

0.6720 0.7923 0.8803 0.9476 I .0004 1.0427 ___-

0.7628 0.9040 1.0080 1 .Os75 1.1512 1.2022

direction.

_-____ n!A

4 . 1 2

in terms of

0.1667 0.1269 0.1009 0.0826 0.069 l 0.0590

0.1668 0.1278 0.1019 0.0837 0.0704 0.0596

0.1671 0.1289 0.1034 0.085 1 0.0718 0.0606

0.1680 0.1305 0.1049 0.0868 0.0731 0.0616

~-

0.1692 0.1324 0.1072 0:089 1 0.0754 0.0632

SB

- D A

in terms of

1

l .oooo 1.6335 2.3143 3.0248 3.7552 4.5000

M A in terms

of P.1

0.5000 0.4167 0.357 l 0.3 125 0.2779 0.2502

-. -

0.5087 0.425 1 0.3648 0.3195 0.2842 0.2556

____-

0.5185 0.4348 0.3739 0.3277 0.29 14 0.2623

0.5296 0.4454 0.3838 0.3367 0.2997 0.2701

0.5423 0.4576 0.3952 0.3475 0.3093 0.2787

M,

4.12

in terms of

0.3333 0.373 1 0.399 l 0.4 174 0.4309 0.4409

0.3517 0.3950 0.4238 0.4440 0.4588 0.4708

0.3744 0.4222 0.4540 0.4765 0.4935 0.5075

0 4029 0.4565 0.4930 0.5189 . 0.5385 0.5549

0.4402 0.5015 0.5435 l .5737 0.5965 0.61 69

c'= 1 -c d'

-~

0 0.40 0.80 l .20 1.60 2.00

0 0.40 0.80 1.20 l .60 2.00

- l .oooo

- 0.9000

,

,

1.0546 1.7378 2.4774 3.2537 4.0537 4.87 1 1

~-

l . 1204 1 B642 2.6766 3.5341 4.4201 5.3305

0.949 l 1.5640 2.2297 2.9283 3.6483 4.3840 -- 0.8963 1.4914 2.1413 2.8273 3.5361 4.2644 ---

0.8441 1.4152 2.0479 2.7196 3.4 176 4.1356

0.7830 1.3342 1 .g486 2.6048 3.2906 3.9984

. .

- 0.8000

, S

, p

0 0.40 0.80 1.20 1.60 2.00

0.20 , p

,,

--

0.30 ,, ,. 8 ,

$ 9

,

0.40

. ,.I

$ 8

. .

1.2016 2.0217 2.9256 3.885 1 4.8823 5.9080 -- l .3050 2.2237 3.2477 4.3413 5.4843 6.6640

0 0.40 0.80 1.20 1.60 2.00

0 0.40 0.80 1.20 l .60 2.00

- 0.6000

,)

* Thc outer edge is free to move downwards but restrained ir

1 76 The Structural Engineer

(d) Check on Original Assumption The accuracy of the analysis of ring-shaped slabs

by the consideration of elements taken as separate radial beams will be checked here by the comparison of the values obtained by this analysis with those obtained by an exact slab analysis for the case of a slab of constant thickness. Consider the slab shown in Fig. 16, it is required to find the stiffness a t end B,

or the radial moment per unit length of the circum- ference at B which makes a unit rotation at this end while the outer edge is restraint against rotation.

In this case there is no loading on the slab. The equation which defines the deflected surface of the ring is :

W = A1 + A z l o g r + A 4 r 2 1 0 g r (see equation 10). The constants AI, A2, A3 and A4 are to be determined from the following edge conditions

a t e d g e B , r = c R ; O = l a n d w = O , ]

at edge A , r = R ; 0 = 0 andQ = 0, where 8 is the rotation and Q is the shear.

The constants A1 to A4 which satisfy these conditions are

1 *

A s = - 2 R (1 -C’)

and A 4 = O J The radial moment

The stiffness S , is equal to the moment at edge B which causes unit rotation,

l? 1 -c

Substituting for R by - we get

D (1 + c2) -(l -c2) . . (30) S , = - c C ! l + C )

The moment at the outer edge will be D 2 c D 2 c

(Mr)r= R R (1 4 2 ) I (1 + c) =--=--.- (31)

The carry-over factor from B to A

Consider the case of a ring slab having c = 0.6 and p = 6. Substituting these values in equations (30) and (32) we get

and 2 x 0.6

fB-A = (1 + 0.6)’ 1.3055 - - 0.5745

The corresponding values obtained by consideration of an elemental radial beam are (see Table 5 for c = 0.6 and d’ = 0) :

and

Comparison between the values obtained by the two methods shows the degree of accuracy in the original assumption adopted for the determination of the values given in Table 5 and the graphs of Figs. 13, 14 and 15.

fB-* = 0.6000

Acknowledgment This investigation was carried out at the Department of Civil

Engineering, Leeds University, partly in the course of an analysis concerning the Structural Behaviour of Concrete Tanks. The writer would like to express his sincere thanks to Professor R. H. Evans, D.Sc., Ph.D., M.I.Struct.E., M.I.C.E., M.I.Mech.E., and to Dr. E. Lightfoot,M.Sc .,Ph.D.,A.M.I.Struct.E., A.M.I.C.E. whose guidance and careful supervision enabled this paper to be written.

1.

2.

3.

4.

5 .

6 7.

8.

9.

10.

References MBrkus, Gy., “ Analysis of Circular Storage Tanks with Plane Covers and Floor Plates by the Moment Distribution Method,” Viziigyi Kozlkmenyek (Hydraulic Proceedings), 1953, II, Budapest. (In Hungarian with German and English abstracts). Lavery, J. H., “ Continuity in Elevated Cylindrical Tank Structures,” The Journal of the Institution of Engineers, Australia, Vol. 20, 1948, October and November. Salter, G., “ Design of Circular Concrete Tanks,” Trans- actions of the American Society of Civil Engineers, Vol. 105, 1940. “ Reinforced and Prestressed Concrete Tanks,” published by the Concrete Association of India, Bombay, 1953. (A reprint of the PortlaEd Cement Association Chicago). Gray, W. S., Reinforced Concrete Reservoirs and Tanks,” Concrete Publications Ltd., London. North, J . C. “ Cylindrical Reinforced Concrete Surface Tanks,” New Zealand Engineering, December, 1952. Timoshenko, S., “ Theory of Plates and Shells,” McGraw Hill Book Co., London, 1940. Markus Gy., “ Analysis of Circular Plates by the Moment Distribution Method,” Viziigyi Kozlkmenyek (Hydraulic Proceedings), 1952, I, Budapest. (In Hungarian with German and English ,+bstracts). Oravas, G., Analysis of Collar Slabs for Shells of Revolu- tions,” Proceedings of the American Society of Civil Engineers, Vol. 82, March, 1956. Ghali, A., Ph.D. Thesis, “ The Structural Analysis of Circular and Rectanplar Concrete Tanks,” University 01 Leeds, 1957.