analysis of cylindrical tanks with flat bases by moment distribution method

7/24/2019 Analysis of Cylindrical Tanks With Flat Bases by Moment Distribution Method

1/12

May,

1958 165

Analysis of Cylindrical Tanks with Flat Bases

by Moment Distribution Methods

by AminGhali,

T

1. SyIlopSis

HE momentdistributionmethod is used to find

the moments and he ring ension in the walls

and the bases of the following two types of cylindrical

tank

:

(a)Cylindrical tankona rigid lat oundation.

(b)Cylindrical tan k 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 foundationsurface.

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 t o 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 t o apply the method we need

to

compute

the fixed-end moments, the stiffness and the carry-over

factors for each element.

Thismethod of momentdistribution 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-overFactorsandFixed-end

Moments in Cylindrical Walls

(a)

Stiffness

The moment which causes unit rotation a t a hinged

end of a cylindrical wall varies according o thedifferent

conditions

of

support at the far end.

In

the following

the stiffness factors are given for the threecases shown

in Fig.

1. It

isconvenient to express the stiffness S

Fig.

2. Introduction

Theusualprocedure of the momentdistribution

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 he base. Themethod nvolves the calculation

of

moments at the ends of the elements under artificial

conditions of restraint, hen a distribution of un-

balancedmomentsbyarithmeticalproportion when

the artificial restraintsare removed.The fixed-end

moments per unit length developed a t 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 o their stiffness.

The term stiffness here means the moment needed

at

the end of the cylindrical wall or the pla te toroduce

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 eld fixed, a fraction of the distributed

moment is carried over o the ixed 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

inerms of

E

where

E

ishe modulus of

12 (1

- p )

elasticity and p Poissons ratio. Inother words the

moment which causesaunitrotation a t 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. 1b)

sin 3H cos

PH

-sinh

pH

cosh @H

sin2

P

H

-sinh2

PH

= 2pt3

case

c

(Fig.

IC

sin2

PH +

sinh2

pH

S = 2pt3

sinh P cosh

PH sin @H

os

PH


2/12

166

The Stmctural Engineer

t is the wall thickness and H its height.

isadimensionless actor.

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 )

nd

( 3 )

tend o unity.The stiffness a t one end will be the

samewhatever heconditions at he oth er end, and

will be equal to

S

=

2pt3 .

.

. . (5)

Most elevated tanks as well as some grounded tanks

have dimensions which give values of

p H >

x , and the

stiffness of the walls can be easily alculated y

equation (5).

(b) Carry-overFactor

If amoment is applied at he edge of a cylinder

while the other edge is fixed (Fig.

lb),

a fraction of this

moment will be carried-over to he fixed edge. The

ratio between theappliedmomentand he moment

developed at hefar fixed edge is the carry-over

factor.Thisactor epends upon the value

PH,

it may be

a

fractionwithpositive or negative sign.

In Table 1 below the carry-over factors are given for

values of

PH

between 1 and 6.

Table1.-Carry-overFactors 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-endMoments

The fixing moment at he bo ttom of acylindrical

wall having i ts op edge free, and filled with iquid

of specific gravity y, could be expressedby the elation :

There exist tables and curves 1 ,

3 ,

4, 5 and

6

which

give the moment a s well a s t he ring tension in cylin-

drical walls fixed at their bases and subjected to

triangular loading,with the op edge undervarious

conditions.

4. Stiffness, Carry-overFactorsandFixed-end

Moments in Circular Plates

The bendingof a circular plate loaded symmetrically

withespect tots centre as been exhaustively

treated by many authors (see for example references

2, 7, 8 and 9. With he usualassumptionscmsidered

in

the elastic theory of plates, i t could be shown th at

th e deflection of the plate s governedby thedifferential

**

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:)

t o

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 ymmetrical 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.

Theprincipalmoments will be acting n a radial

direction M r , and in

a

tangential direction Mt Their

values per unit length are

:

a ndMt=- D( - - + 1

dw

p-)

2w

Y dr

When

a

circularplate sbuiltcontinuouswitha

cylindrical wall or with nother 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 t he 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 sequal o heradial moment applied at this

edge to let it rotate a rotation cquals unity.

If

a radial moment

( M r ) l

is applied a t 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

( M r 2 )

will be arriedover to he fixed edge. The


3/12

May,

1958

167

ratio between the applied adial moment per unit

lengthand he corresponding value of the moment

developed at the far edge is the carry-over factor f1-2.

The carry-over factor depends upon the rat io 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

andgraphsare available, 2 and 8, which give the

fixed-end moments, stiffness and carry-over actors

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 par t of the base may remain flat.

This is a nonlinear problem as regarding the bending

of the circularplate, the conditions

at

the edges of

the deformed part of the base are changing with the

deflection, and the deformation of the pla te 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

willbe

assumed and then corrected to satisfy he conditions

of the problem. These conditionsre 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 FactorsandFixed-endMoments

in

Ring-shapedSlabs of constant thickness (Fig. 2)

Outer radius

Inner radius

Stiffness at edge 1

l d3 E

a

-

_

b

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

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

M 2

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 theusual moment distribution analysis

of beamsor frames. In his discussion, because of

symmetry, theanalysis is carriedout on one half of the

structure, namely he left half. An external moment

applied at the end of an element is positive when it

tends torotate hisend n he clockwise direction.

6.

Variation of the Momentsawayfromthe Edges

The moment distribution serves to calculate the

continuitymoments at the intersection of circular

slabswith cylindrical walls or between elements of

ring-shaped slabsupported along annular rings.

Starting from these moments the variation

of

stresses

throughout he cylinder or the slab anbe easily

calculated. Data is available, 1 and

4,

which give the

moments anding tension a t different heightsn

cylindrical walls subjected to triangular loading as

well as for radial moments applied at the edge. Also,

the re exist tables and curves, 2 and S, for the radial

and he angential moments at different radii of a

ring-shaped slabdue to adistributed 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. CylindricalTank on RigidFlatFoundation

When

a

cylindrical tank isonstructed

on

an

absolutely rigid foundation-such as solid rock-

or if the ank isconstructed on a thick stiff plain

concrete footing, and the wall is built monolithic with

the base, thecontinuity moment between the wall

+ 4

-4

Fig.3.-Cylindrical ank

on

a rigidfoundation

part

of

the base can be considered as

if

totally fixed

at the circle of radius

b,

but loaded in a manner that

the radia l moment at the fixation s 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 hould be necessary. The procedure of calcu-

lation s ullyexplainedduring he solution of the

following numerical example.

ExampLe

1

:

(a)

GivenData

Tank diameter 2 R = 40 ; tank height.

H..=

16'

;

floor thickness d =

10

; wall thickness t = 10 ;


4/12

1 6 8 The StructwalEngineer

(d)

Carry-overFactor

This factor will be used to find the radial moment

carried over to he inner edge of the ing slab (at

radius b ) when

a

moment

is

applied at the outer edge

fromTable 2 for -

=

1.25 f1-2

=

0.584

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

;

Poissons ratio p =-

6

The tank is assumed to be placed on an absolutely

rigid flat foundation.

(b)

Loads

Thebentpart 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 )

StiffnessFactors

1.3068H.3068

x

16.4

wall : @ H=

-

4Ft 1/20 x 0.83

= 5.25 > x

5

25

16.4

*. z-1 =

2Pt3 = 2 X L. X 0.833

=

0.368

base : In order

to

estimate hestiffness of the ring

(part 2 -3 Fig. 5), the dimension

b

must be assumed.

I

(e) Fixed-endMoments

wall : By subst ituting 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 nterpolation from Table 2 we get

a

b

- =

1.25 and

-=

0.8

b

a

M 2 - 3 =-

125 X 162 X 0.0053 = 525

lb.ft./ft.

M 3 - 2 =

+

1125 X

202

X 0.0035 = + 1575 lb.ft./ft.

Thedistributionprocedures 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 he second trial we take b = 18. A similar

calculation will give the moments shown in Fig. 6b,

with the moment at radius b = - 110 lb.ft./ft. Hence

the igh t value of b , which gives zero moment at

radius

b,

must lie between 16 and 18.

A reasonablevalue to be assumed for thenext

trialmaybe akenby making inear nterpolation

between the two previous trials, this gives

;

733

733 + 1110

=

16

+

(18-16)

= 16.8.

Themomentdistribution or he hird rial with

b

= 16.8 is done n Fig. 6c, the corresponding value

of

the adial moment at b

= 10

lb.ft./ft., which is

very small, and no more trials need to be considered.

Thevariation of ring tension and moments re

obtained by thehelp of graphs from references l and 8,

As a first trial we take

b

= 0.8a = 16.

a

20

b 16 -

-

.25

By

interpolation from Table 2 we get

Y

Fig.

5

the values are abulated below. Diagramsor the

ring tension a nd 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


5/12

May,

1958

Table3.-RingTension and BendingMoments in the

Walls

69

I

distance from to p edge

due to liquid pressure on a cylindrical

.

B

wall hinged at the bottom

a

the bottom

2

due to restrain ing moment-2910 lb. at

total (lb./ft.)

,

due to liquidpressure on a cylindrical

wall hinged

at

the bottom

H

r

the bottom

due to restraining moment-2910 lb. at

.-

a

total (lb.ft./ft.)

0

-

00

+

70

-

30

0

0

0

that he tangentialmoments n he base are mall

and need not be considered.

8.

Cylindrical

Tank

with

Flat

Base

Supported

on

Cylindrical

Shaft of SmallerDiameter

Water towers of medium capacities are often made

of the ype shown inFig. 8. The latbase may be

fi

- F I

fb

second + ,h /

b

(8

h

I

bzeio)

Fig. 6.-Yament distribution : Solution of

Example

1.

0.2H 0.4H

0 . 6 H

+

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 ank wall.

A

suitable choice of the overhanging

length snecessary to obtainvalues of the positive

and negativeadialmoments which require the

minimum base thickness.

In the ollowing this type of tank will be analysed by

amomentdistributionmethod.The oint between

the tank wall and the base (joint 2, Fig.

8)

can rotate

and also an move downwards. A directmoment

distribution calculation could e applied if the stiffness,

the carry-over factor and the fixed-end moments of the

overhanging partareevaluated aking ntoaccount

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 s usually made

tapered with the greater thickness a t the inner edge.

The stiffness of thispart sgreatly affected by the

variation of the thickness and consequently the

bendingmoments and he ring ension n the ank

wall will also be affected.

A

method for the estimation

of the stiffness, the carry-over factors,and the ixed-end

moments in ring-shaped slabs of varying thickness is

presented n the appendix.Thesevaluesarecalcu-

lated by he Author and plotted n 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 andouter radii.The fixed endmoments

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 i t) .

The analysis of this type of tank by moment distri-

bution

wll

beexplained while solving anumerical

example.

Example 2 :

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

gallonscapacity ssupported on acylindrical shaft

28 diameter, astmonolithicwithheank base.

The thickness of the shaft wall near the top is 8 . The

tank issupposed to becovered bya roof weighing

40

lb./sq.ft.,simplysupportedon the ank walls.

It

is required to find the bending moments n the slab

base, and the moments and ring tension on the walls

of the ankand heshaft. Weight of the contained


6/12

170

The Structural

Engimir

Fig.7.-Ring ensionandbendingmomentdiagram ora ankonarigid

foundation Example 1)

liquid 62.5 lb./cu.ft. ; weight of concretematerial

1

150 lb./cu.ft. Poisson's ratio

p

= -

6

(a)

Loads

(see Fig. 9a)

Weight of roof, acting vertically on the op edge of

the wall

17

2

=

40

X

--

=

340 lb./ft. = 340 lb./ft.

8

I t

wn weight of wall = 150

; 22

=

2200

lb./ft.

Total vertical

load

at the outer edge

of

the base

slab

Distributed loadon theoverhangingpart

of

the

base (par t 2 -3, Fig. 9a) including its own weight

=

=

2540 Ib./ft.

(b) Sti f fnessFactors

The stiffness of allelements will beestimated n

E

terms of

12(1

-p ).

41

=

62.5

x

22

+

E

150

=

1590

lb./ft.2

Distributed Ioad on part

3

-3 including own weight

2

x

12

-

-

0

95

-

260

.I

i

6

12

t

-

q2 =

62.5

X

22

+

-

150 = 1700 lb./ft.2

068P

I0.5Sf Q5021

4

\

-

(h) 2kbu ; on

and ca r r y -ove r fucfors

MP- 4

r c o m

/Y2- 3 M3- 3

f 3650 +

f 7 9 8 0

-

5250 -

6 6 20

.

3930

-

U 9 5 0

h 2 6 8 0

- 3 3 8 0

-

9uo

c

//90

640

-

- 8 f0

-

P30 -

480

+

(50

- f 9 0

-

50

c

f

70

+ U 0 - a0

c

20

-

JZUO 26690

l

M3-

3

-

38700 F f f s

7080

f

1700

+ 040

t /oo

c

fO

Fig. 8


7/12

,May, l958

171

i )

Tank wall (element 2

-1)

:

1.3068H 1.3068 x 22 .52

p H =

dR.t-J17 X

~ _ _ _

8

3

8.52

22

2-1

=

2

p 13 =

x 2

x

0.229

(ii) Base slab (element

2

-2) :

thickness a t outer radius d = 9

Thickness at inner radius

= d + d . d = 9 + 1 7 = 2 6

d

=

1.89

7

9

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

7

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 a t

the edge of a circular plate of radius

c

. R

d3

1 + P)*

d3

c . K

here s = 1 +

p).

3 1

: s3

=

S>

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 istributionactorsrealculated in the

usual way and given in Fig. 9b.

(c) Carry-overFactors

The carry-over factors in the tank and shaft walls

are zero. The carry-over factors in thebase 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-endMoments

i) Tank wall (element 2 -1)

ii)

(iii)

Base slab (element

2 -3)

:

From the table or the graphsiven in the appendix

we get

M 2 3

due to distributed loadql

;

due to concentrated oad 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 a t 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

-

Thedistribution process is carriedout 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 his does notadd much

difficulty to the problem. After the moments ,a t he

joints are obtained, he variation of thle fing tension

and he moments in the ank wall is obtainedby

superimposing the effect of the liquidpressure on a

wall fixed at th e base plus the effect of the relaxed

moment

(4050 -3240

=

810

lb.ft./ft.) during thc dis-

tribution process. The moment anding 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 > )

due to a radial oment

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.

*

SeeTimoshenko,Strength

of

Materials, Part 11;

p.140.

Fig.

10


8/12

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 - p x cos px

. . . .

. (15)

Thevariationof heradialand angentialmoments

in he base slab 3

-3'

is obtainedbyadding he

moments due to a distributed load on a simply sup-

ported circular slab plus the effect of the restraining

momentlongheupport

(-29390

Ib.ft./ft.). A

radial moment M O pplied on the edge of a circular

slab will cause constant radial and tangentialmoments

equal to

MO.

Theradialand angentialmoments 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 notbe considered.

The reinforcement bars used asdistributors o he

main steel n the radia l direction are normally sufficient

to resist the tangen tial moments.

I

p=;)

Table4.-Radialand Tangential Moments in auniformly

loaded,simplysupported Circular Slab

y i

R

1.0.8 0.6

.4

.2

Radial moments

in terms

of q . H

0

.0712.1267

.1662

.1900.1979

Tangential

moments in terms

of q.Rs

0.1979 0.1942

0.1042

.1379

.1642

.1830

~

Of

the bendingmoments

and

the ring moments for circular rings tapered owards he edge

tension

in theank Of the above are

shown

are neededwhen 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-overFactorsand

Fixed-endMoments nCircularRing-shaped

SlabsofVariableThickness

Circular plates of variable thickness requireeven

for

the simple cases very edious calculations. The

stiffness and carry-overactors andhe 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, .e.

c

is some factor round

unity , he angent ial moments in the ring are small

and heonly mportant moments are in the adial

direction. Hence,sucharing-shaped slab

is

more or

less actings over-hangingadial beams'. The

variation of th e thickness

of

theringhasaneffect

on its sta tical ehaviour which is assumed for simplicity

as he same effecton aradialbeamwith hesame

variation of the thickness.


9/12

May, 1958

173

l

------l

Fig. 12b

(a) St i f f ness

Consider an element of the ringslab of

Fig.

12b

between twovertical adial sections with a width

unity

at

the outer edge. The thickness

of

this element

varies linearly etween d at the outer dge and d+ d d )

at t he inner edge. The stiffening effect

of

the adjacent

elements of the ring-shaped slab n the bending

deformation of the elemental adialbeams an be

takennto account by increasing the moment

of

inertia of each beam in the ratio

The flex-

ural rigidity of the elemental beam a t 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 elementalbeam at any point

( X = X .

,

see Fig. 12b) between A and B

=

d + ddx ,

and hebreadthat he samepoint

=

1

- l -c).

x = 1

-G . The flexural rigidity at

x

is

E

E I x

= (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= -A 1

1:::

1

+x)

(1 + d43

d X . 18)

which is the stiffness perunit 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 ree to move in thevertical

direction). The value of the stiffness per unit ength

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 adii c

=

0.6,

0.7 .8,

.9 and

1.00, and values of d between zero and

2.

Fig. 13.-Stiffness SA and

S ,

atedges A

and

B

of

ring-shaped slabs of variable

thickness


10/12

l74

The Engineer

(b)

Carry-o ver Factors

- .

.A moment applied at edge A of the elemental beam

Fig. 12b will becarried over to edge B with its full

value. If a moment

S ,

per unit width is applied a t

A ,

th ,e

momentperunitwidth will bedeveloped at

B

i s

equalo - S,. With he sign conventionused,

the carry-over factor from A to

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.

arecalculated for the t w o loadings :

i )

A

concentrated load

P

per unit length on the slab.

i i) A

distributed load

q

per unit area

of

the slab.

Forboth loadings the edge A issupposed to be re-

strained n direction but free to ettle downwards.

By

considering

a

radialelement of the slab t could

be

proved

(10)

hat 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 ength on

edge A (Fig. 14)

l

-

I

1

M,= I [ f

0

l

M= -

PZ-M,) . .

. . .

.

C

(23)

(ii) Due to a distributed load

q

per unit area (Fig.

15)

M A =

. .

2 4 )

The values of the fixed-end moments

M ,

and M , are

calculated

by

equations

( 2 2 ) ,

(23),

( 2 4 )

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 adistributed load g per unitarea.Theslabs

considered have the ratio 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-endmoments

M A

and

M B

atedges

and

B of a ring-shaped slab of variable thickness due

A and B

of

a ring-shaped slab of variable thickness due

toaconcentrated oad

P

per unit ength of edge

A.

to a distributed oad of q unit area.


11/12

May;.1958

. 175

. .

. .

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

F.E.Ms. due to

Stiffness factor Carry-over factor

I '

-

/unit length

on out er 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

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

0.8271

~-

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

-

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

0.4251

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

0.3731

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

1B642

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 bu t restrained ir


12/12

176 The

Structural

Engineer

(d)

Check

on

Original Assumption

Theaccuracy of the analysis

of

ring-shaped slabs

by he consideration of elements akenasseparate

radia l beams will be checked here by the comparison

of the valuesobtainedby hisanalysiswith hose

obtained by an exact slab analysis for the case of a

slab of constant thickness. Consider theslab shown

in Fig.

16,

it is required to find the stiffness at end B ,

or the radial moment per unit ength of the circum-

ference at B which makes a unit rotation at this end

while theouter edge is restraintagainst otation.

In his case there is no oading on he 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 A I , A 2 , A 3and 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

*

A s = -

2R

(1 -C)

and A 4 = O

J

The radial moment

The stiffness

S is

equal o he moment at edge B

which causes unit rotation,

l

1

-c

ubstituting for R by e get

D

1 + c 2 ) -(l -c2)

.

. (30)

S = -

C l + C )

The moment at the outeredge will be

D

2 c D 2 c

(Mr)r= R R (1 4 2 ) 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 hese values in equations (30)

and (32) we get

and

2 x 0.6

fB-A = (1

+

0.6) 1.3055

-

.5745

Thecorrespondingvaluesobtained by consideration

of an elemental radial beam aresee Table 5 for c = 0 6

and

d =

0)

and

Comparison between the values obtained by he wo

methods shows the degree of accuracy in the original

assumptiondopted for the etermination 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 th e Dep artm ent of Civil

Engineering, Leeds University, part ly in the ourse

of

an analysis

concerning theStructuralBehav iour of Concrete Tanks .The

writer would like o express his sincere thankso Professor

R.

H.

Evans,D.Sc.,h.D.,M.I.Struct.E.,M.I.C.E.,M.I.Mech.E.,

and toDr. E. Lightfoot,M.Sc

.,Ph.D.,A.M.I.Struct.E.,

.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 CircularStorage Tankswith

Plane Covers and Floor Plates by the Moment Distribution

Method,

Vizi igyi Kozlkmenyek Hydraulicroceedings),

1953 II, udapest.

(InHungarianwithGermannd

English abstracts).

Lavery, J. H., Continuity nElevatedCylindricalTank

Structures,

TheJournal of the Institution of Engineers,

Australia,

Vol.

20 1948

October and November.

Salter, G.,

Design of CircularConcrete Tanks,

Trans-

actions of the American Society of Civil Engineers,

Vol.

105

1940.

Reinforced and P restre ssed Concrete Tanks , published by

the Concrete Associat ion of India, Bombay, 1953. ( A reprint

of the Portl aEd Cement Association Chicago).

Gray, W. S., ReinforcedConcreteReservoirs andTanks,

Concrete Publications Ltd., London.

North,

J .

C.

CylindricalReinforcedConcrete urface

Tanks,

New Zealand Engineering,

December, 1952.

Timoshenko, S.,

Theory of Platesand Shells,

McGraw

Hill Book Co ., London,

1940.

MarkusGy.,

Analysis of Circular Plates by the Moment

DistributionMethod,

ViziigyiozlkmenyekHydraulic

Proceedings), 1952 I Budapest.

(In Hungarian with German

and English ,+bstracts).

Oravas, G., Analysis of CollarSlabs orShells of Revolu-

tions,

Proceedings

of

the American Society

of

Civil Engineers,

Vol. 82, March, 1956.

Ghali, A., Ph .D. Thesis,

The Struc tural Analysis of Circular

and Rec tan plar Concrete Tanks,

University 1 eeds,

1957.

analysis of cylindrical tanks with flat bases by moment distribution method

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