Computer-aided design of circular liquid retaining structures in accordance with BS5337

Computer-aided design of circular liquid retaining structures in accordance with BS5337 A. R. L L O Y D and W. S. D O Y L E

Department of Civil Engineering, University of Cape Town, Rondebosch 7700, South Africa

A computer program (DART) for the analysis and design of all types of circular liquid containment structures and pressure vessels is presented. Meridionally curved thin shell finite elements are used for the linear elastic static analysis of variable thickness, branched thin shells of revolution. General external mechanical loading is permitted, and the stress variation through the thickness can be determined. Displacements and stresses due to thermal loading can be determined, and a temperature gradient through the shell thickness can be taken into account. All load cases may be axisymmetric or asymmetric. Prestressed concrete and reinforced concrete structures are designed in accordance with the British Standard Code of Practice BS 5337 (1976), 'The Structural Use of Concrete for Retaining Aqueous Liquids'. Many of the design facilities of the program are optional, and these include optimization of structural size and shape, an elastic foundation analysis for floor slabs, and a bill of quantities.

sections, namely analysis and design, linked as shown in the flow chart below:

ROUTINE

t

IANALYSIS ROUTINES

] NO

:~

I INTRODUCTION A wide variety of circular shells exist in everyday life. In reinforced and prestressed concrete there are various pipes and pressure vessels, there are reservoirs in a range of sizes, there are elevated water towers, and there are silos and hyperbolic cooling towers to name but a few examples. Similar structures are also being built with other materials. Approximate methods exist for the analysis of these structures, and no doubt they are adequate for some of them. Now with the availability of the electronic digital computer, sophisticated and accurate analyses can be performed rapidly and inexpensively. In the computer program discussed in this paper, the further facility is incorporated to perform the structural design automatically, and to determine an optimum size of structure together with a bill of quantities should this be required. The program DART has been written to conserve the maximum amount of computer storage space and to design structures rapidly and efficiently.

NO

YES

~

(END ~

L_/ DART can be used for the analysis of any axisymmetric thin shell, with any number of material properties. The design section with the optional optimization facility, however, has been written specifically for concrete thin shells. The program is written in standard ASCII Fortran for running on a Univac 1100/81 computer, but can be used on most other systems with minor modifications. A detailed acount of the program capabilities and special features appears later. MERIDIONALLY CURVED AXISYMMETRIC SHELL E L E M E N T

THE DART P R O G R A M DART stands for the Design of Axisymmetric Reservoirs and elevated water Towers, but also caters for sludge digester tanks, circular chimneys and towers, arid hyperbolic cooling towers. The program comprises two main

0141-1195/81/010035-0752.00

© 1981 CML Publications

The finite element analysis section of the program is a modified version of the SABOR4 program developed by Witmer et al. 2. The shell of revolution is idealized as an assembly of discrete thin shell ring elements which may be curved meridionally if desired. A typical element is

Adv. Eng. Software, 1981, Vol. 3, No. 1 35

w

xs0 = 2/1"(- O2w/OsO0 + sin O/r Ow/O0 + cos q~ Ov/Os-

v

(sin q) cos q~/r)v-Oq~/Os Ou/Os)

nodep/ r ~ " ' - ' ~ ' ~

//~

xo = llr(llr(OZw/002) - c o s ~olr(Ov/O0) +

l-,. ' !

.

.

.

[OwlOs + uOglOs]sin q~)

.

where e and x are the axial and bending strains, i" is the radial coordinate, and tp is the meridional slope of the midsurface. The shell is assumed to be homogeneous and isotropic and in a condition of plane stress. The stress-strain relations are thus given by:

/node(p+1) Figure 1.

Discrete-element geometry and nomenclature

described in Fig. 1. The meridional slope tp of the shell midsurface is given by:

°'s

trs°

E

=~

tr° ~o(s)=ao + a l s +a2s 2

(1)

where the constants a t are to be determined, and s is the meridional distance along the element (see Fig. 1). The circumferential variation with 0 of the midsurface displacements are represented by Fourier series harmonics j as:

u(s,O)= u(°)(s)+ ~ uU)(s) cos jO + ~ fiU)(s) sin jO j=l

j=l

I! ° i]r 2

es°

0

~ e°

(5)

where E is Young's modulus, and v is Poisson's ratio. The individual element stiffness matrices are assembled into the system matrix, and a consistent load vector is assembled for each Fourier series harmonic. A solution is determined for each harmonic, and these are summed to give the solution for the full load condition. Only one harmonic (the zeroth) is required for axisymmetric loads, whereas typically eight to ten harmonics are required for non-symmetrical loads. DESIGN TO BS 5337

u(s,O)= v(°'(s)+ ~, vU)(s) s i n jO + ~ 5U'(s) c o s jO j=l

1- v

j=l

w(s,O)= w(°)(s)+ ~ wU)(s) c o s jO + ~ ~,U)(s) sin jO j=l

(2)

j=1

where u is the meridional displacement, v is the circumferential displacement, and w is the radial displacement. The meridional variation of the midsurface displacements for each Fourier series component is given by: u0 (J) (s)=~1o3 +~2(J) s VoU)(s) = ~°3")+ ot~)s w oO~ts j _

,~0) ±

,~0") ~ ..L

(3)

,,0%2 .a_ ,,O')~3

where ~ are the generalized co-ordinates. In addition, the meridional rotation is given by:

p~ = OwU)/Os+ uU~.Oq~/Os

(4)

The shell is assumed to be thin and to obey the Kirchoff assumptions that plane sections before bending remain plane after bending. The strain displacement relations are based on the general theory of Novozhilov 3 and are: (~)o = Ou/Os- wOq~/Os

The design section of the DART program is based entirely upon the limit state philosophy in CP 110 (1972): 'The Structural Use of Concrete '4, with particular reference to BS 53371. The limit state of cracking has been found to be the most important consideration in this type of design 1a, and is usually the governing criterion when compared with the ultimate limit states of flexure and shear. The program is therefore structured in the following order. (1) Serviceability limit state of cracking (SLS): direct tension in immature concrete; flexural tension in mature concrete; direct tension in mature concrete. (2) Ultimate limit state checks (ULS): flexure; shear; local bond and anchorage bond. BS 5337 provides for an alternative method of design similar to that of its predecessor CP 2007 s, but this is not included in the program since limit state design generally results in a more economical solution 6. Direct tension crackin 9 in immature concrete t The quantity of reinforcement required to distribute cracks due to restrained shrinkage and early thermal movement can be determined using Appendix B of BS 5337. Two formulae are given, involving the maximum crack width Wma~and maximum crack spacing Sm~e s.,~=~of,/fb.2 p and W,.a~=Sm~(T~ + T2).ot/2

(6)

These two equations are combined in the program to determine the reinforcement ratio p by:

(e~o)o = 1/r(rOv/O0- v sin q~+ Ou/O0) p = (f./L)(T1 + T,)~q~/4 w,,,,,

(7)

(eo)o = 1/r(Ov/O0 + u sin ~o+ w cos q~) x~ = - (O2w/Os 2 + uO2 ~o/Os2 + Ou/Os &o/Os)

36

Adv. Eng. Software, 1981, Vol. 3, No. 1

where ~ is the coefficient of thermal expansion of mature concrete, ~ois the bar diameter, T1 is the fall in temperature

Table 1. Allowable steel stresses in direct or flexural tension for serviceability limit states Allowable stress (N/mm 2) Class of exposure A B

Plain bars

Deformed bars

85 115

I00 130

between the hydration peak and ambient, T2 is the seasonal variation in temperature, f , is the direct tensile strength of immature concrete, fb is the average bond strength between concrete and reinforcement, p is the tension steel/concrete ratio AJbd. The value of Wm,xis limited to 0.1 mm, 0.2 mm and 0.3 mm for exposure class A, B and C respectively. The three classes of exposure are defined as follows. Class A: exposed to a moist or corrosive atmosphere or subject to alternate wetting and drying. Class B: exposed to continuous or almost continuous contact with liquid. Class C: not exposed to liquid nor to moist or corrosive conditions. The minimum steel ratio required to distribute cracking is determined as (Appendix B in BS 5337):

p¢,~,=L,/f,,

(8)

wherefy is the characteristic strength of the reinforcement. The minimum steel ratio may be reduced to 2/3 flcrit depending on the method of construction chosen from Table 6 in BS 5337, in which different methods of crack control are presented. This facility is incorporated in the program. The user may also state whether the section is restrained against shrinkage or not. An unrestrained section will contain only the minimum steel ratio Petit,and not that given by equation (7).

x = d [ - ae(p + p') + x/0t~(p + p,)Z + 2ae(p + p'd'/d] (11) Equations (9), (10) and (11) form the basis of the flexural cracking check, and are solved in the program using an iterative procedure, since the equations are interdependent. The program determines the reinforcement p required to limit the crack widths, and modifies p where necessary. Generally this cracking criterion is not governing, and only affects the design in areas of large bending moment, such as the bottom of a reservoir wall with a fixed base, junctions between a domed roof and a ring beam, and branched junctions in elevated water towers.

Direct tension cracking in mature concrete The crack widths due to circumferential ring tension forces under service loads are deemed to be satisfactory if the stress in the hoop reinforcement does not exceed the permissible value off, given in Table L In addition, the maximum bar spacing is limited to 300 mm. This concludes the serviceability checks for cracking, since shear cracking is deemed to be satisfactory if the provisions for shear at the ultimate limit state are complied with. Ultimate limite state of flexure The ultimate limit state conditions are essentially checks on the section thickness and reinforcement provided by the serviceability conditions of cracking. In flexure, the ultimate moment of resistance is calculated using the equations in CP 110, and reproduced below. Based on the concrete section: M.=O.15 f~,bd 2 Based on reinforcement:

M.=O.87 fyAsz Flexural tension cracking in mature concrete BS 5337 suggests two methods for checking this case. The first involves a formula for the design surface crack width in terms of the average strain at the level at which cracking is considered (Appendix C, BS 5337). The second method, in which the crack widths are deemed to be satisfactory if the steel stress under service conditions does not exceed the appropriate value in Table 1 is more adaptable to programming and is incorporated in the DART program. Considering Fig. 2, the stress in the tension reinforcement is given by the product of a~ and the stress in the concrete: f~ = ct~.M/l,(d - x)

(12)

(13)

where the lever arm z is given by: z=(1 -- 1.1 ~A~/f~.bd)d

(14)

and f~u is the characteristic strength of the concrete. If the bending moment on the section due to ultimate loads exceeds either value from equations (12) or (13), the section is suitably modified in the program. A check is also made on the x/d ratio, which should not exceed 0.5.

Ultimate limite state of shear The shear stress v due to ultimate loads is checked using:

(9)

v= V/bd

(15)

where x is the depth to the neutral axis, d is the effective depth to the tension reinforcement, M is the bending moment, ~ is the modular ratio Es/Ec. The second moment of area I c of the transformed cracked section is given by:

l¢=bxa/3+e~p'bd(x-d')2 +c%pbd(d-x) 2

(10)

where p is the tension steel r a t i o = A J b d and p' is the compression steel ratio = A~bd. The depth to neutral axis x, for stresses within the elastic range, is given by,

o J Figure 2.

_x__[ C~eA s ~.

Transformed section

Adv. Eng. Software, 1981, Vol. 3, No. 1 37

Table 2.

Ultimate shear stress

Concrete grade (N/ram 2) 100 As/bd

0.25 0.50 1.00 2.00 3.00

25

30

0.35 0.50 0.65 0.85 0.90

0.35 0.55 0.70 0.90 0.95

where V is the shear force due to ultimate loads. The value of v is checked against the permissible shear stress vc given in Table 5 (CP 110), and reproduced in Table 2. Since it is inconvenient to provide shear reinforcement in the wall or dome of a reservoir, the shear stress is limited in the program to vc. In extreme cases, however, the optional facility for additional shear reinforcement is provided.

Local bond and anchorage bond This is checked according to clause 3.11.6 in CP 110, where the local bond stressfbs is given by:

fn~= V/Eu,d

(16)

where 2u s is the effective perimeter of the tension reinforcement. Required anchorage lengths are calculated from: Anchorage length = force in bar (anchorage bond stress)(Z bar perimeters)

(17)

whatever the initial state and initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision'. Consider a decision process consisting of N stages or activities each of which has a discrete set of state variables x~ associated with it. If the initial activity is the Nth activity, and successive activities are the ( N - 1 ) t h activity, and so on, the basic functional equation to determine the optimum xu that will minimize the set of cost functions .qi(x3, is given by: f,,(x)= min [9N(XN)+fN-t(X--XN)]

(18)

0<~x,v<~x

for N = 2, 3..... where the initial condition is determined by:

fl(x)=cll(X)

(19)

In this way equation (18) represents a dynamic allocation process in which resources are assigned to activities in a sequential manner. In the program, a slight variation of the functional equation is used, in that the cost is not evaluated as a series of separate costs for each stage, but rather as a single total cost for a particular combination of state variables. The basic principle is still valid, however, as the decisions are dependent on one another and each decision affects the system's response to subsequent decisions. The number of iterative operations required to reach an optimal solution is vastly reduced using the dynamic programming method since operations are additive rather than multiplicative. Examples analysed and designed using the DART program have been found to converge after typically fifteen to twenty iterations 9.

That concludes the reinforced concrete design to BS 5337. P R O G R A M CAPABILITIES PRESTRESSED CONCRETE DESIGN This part of the program is still under development, but at this stage the section for the design of a prestressed flat slab roof has been completed 7. When this facility is fully operational for general shells, it is intended to provide for: (a) circumferential prestressing, (b) meridional (vertical prestressing, (c) prestressing of ring beams, (d} prestressing of flat slabs with radially positioned columns or with an orthogonal arrangement of columns. O P T I M U M DESIGN A useful feature of the program, should this option be required, is that a range of values for the section thicknesses and the shape factor (i.e. diameter to height ratio for circular reservoirs) can be specified, and the program will automatically determine the optimum configuration based on various cost items input by the user. These include costs of concrete, reinforcement, formwork, waterproofing, construction joints, excavation, backfilling, underdrainage, and neoprene movement joints. An iterative search method, which is a modified version of the Dynamic Prbgramming Successive Approximation technique, is used. Dynamic programming is an iterative procedure for determining the optimum solution of a multistage decision process and is based on the Principle of OptimalityS: 'An optimal policy has the property that

38

Adv. Eng. Software, 1981, Vol. 3, No. 1

Analysis The analysis section is extremely versatile and features include: (i) Axisymmetric and non-axisymmetric loading: linear variation of distributed loading along an element; concentrated loading. (ii) Temperature loading, both axisymmetric and nonaxisymmetric which may vary quadratically through the shell thickness. (iii) Meridionally curved elements, with linear variation of shell thickness permitted. (iv) Branched shells and multiply-connected nodes. (v) Clamped, free, pinned-fixed and sliding boundary condilions, and prescribed nonzero displacements. (vi) Automatic generation of data for nodal co-ordinates, member incidences and shell thicknesses. (vii) Automatic generation of load data. (viii) Up to 29 Simpson stations per element for numerical integration, in deriving the stiffness matrix. (ix) Analysis of axisymmetric slabs or shells on elastic foundations. The Winkler hypothesis is used for the modelling of the soil behaviour, and the soil is assumed to consist of a number of concentric circular springs symmetrical about the vertical axis. (x) Maximum program limits: 150 elements, 151 nodes; 50 branches; 100 stations around the shell for approximating non-axisymmetric loading; 37 stations around the shell for output of stresses.

I'

9144

Example 1

m_C

II1 Bill

--

,c:,o,eo.o,. \P1=5977 kNim2

-

P2

~

-A

Figure 3. Sludge digester tank (xi) Printed output: 4 displacements at each node; 3 midsurface strains, 3 midsurface stresses and 8 stress resultants per element at mid-element locations; stresses and strains at up to 11 stations through the shell thickness (by extrapolation). The above printed output is given for each individual Fourier harmonic, and then summed to give the solution for the structure as a whole.

Design The design section of the program has the following features: (i) Reinforced or prestressed concrete design. (ii) Optional optimization: cost items taken into account include concrete, reinforcement, formwork, excavation, backfilling, underdrainage, construction joints, rubber movement joints, waterproofing, stone packing and ventilators. Constraints set by the user include required capacity and dimensional constraints (maximum diameter, height, etc). (iii) Design options set by the user: one or two layers of reinforcement at a section; choice of restraint in each of two directions, for restrained shrinkage and early thermal cracking; class A, B, or C exposure conditions on each face of each element; different material properties for each element (particularly useful for incorporating rubber joints, for example). (iv) Printed output: reinforcement and concrete stresses at the various stages in the design; table of reinforcement required for each element; anchorage lengths; prestressing losses, tendon eccentricities, effective prestress, and stresses at SLS and transfer; for cost optimization: a detailed breakdown of the costs and quantities for each iteration. (v) Dynamic core allocation to reduce the core storage requirements. (vl) A final bill of quantities. Although the paper refers specifically to liquid retaining structures, the program can be used for the design of general reinforced and prestressed circular shell structures.

The advantage of being able to model the soil behaviour under a floor slab is illustrated by this example, a sludge digester tank with a conical floor supported on an elastic foundation, shown in Fig. 3. The tank was originally analysed by Flfigge ~°, with the following assumptions: (i) the water pressure acting on the conical bottom is transferred directly through the wall into the ground, and does not enter the problem; (ii) the weight of the tank wall is assumed to be P = 8.028 kN/m acting as a point load around the base of the wall; (iii) the load P causes a normal soil reaction on the floor of P2 = 1.756 kN/m 2. The meridional bending haoment under these assumptions is shown in Fig. 4, and has been validated using an identical finite element idealization, using the DART program. The problem is now analysed as a slab on an elastic foundation, using 20 branched rings to support the floor. The loading is the same as that used by Flfigge except that the pressure load P2 is replaced by the water pressure load on the floor. The magnitude of the bending moments in the floor is a function of the floor settlement, and also of the spacing between the branched ring supports, i.e. the distance over which the floor is unsupported, which can correspond to a pocket of weak material beneath the floor. The expected settlement can be determined using the formula for a uniformly loaded circular elastic footing, given by Perloff11: Settlement

Sd= Cap.B(1 - v2)/Es

(20)

where Cd= 1.0 for settlement at the centre, p=uniform pressure load, B = diameter of footing, vs = Poisson's ratio for soil, E, = Young's modulus for soil. In this example, a vertical floor settlement of 17 mm is designed for, the branch thickness being varied until the desired settlement is achieved. The resulting meridional moment diagram, given by a computer plot, is shown in Fig. 5. The scale quoted can be ignored, since it refers to the original plot before photoreduction. The fluctation in the bending moment in the floor is due to the fact that the branches connect to every third node on the floor. The overall shape of the moment diagram is far more realistic than that given by Fliigge, however, since in practice one would expect a reversal of moment near the base of the wall due to the water load. CPU time (Univac 1100/81): 48 seconds.

Example 2 This example illustrates an analysis involving nonaxisymmetric loads, and optimized design. The circular

Future development envisaged Extension of the elastic analysis to a non-linear analysis. Plotting facilities for drawing the structure, showing all reinforcement details: at present the plotting capabilities include plots of the forces on a cross-section of the structure.

kNmlm

11.72~-

N U M E R I C A L EXAMPLES Three examples which illustrate some of the special features in the DART program, are documented.

Figure 4. Meridional bending moment M,

Adv. Eng. Software, 1981, Vol, 3, No. 1 39

loading conditions considered are: (a) hydrostatic pressure load due to water up to the level of the ring beam; (b) an imposed load of 1 kN/m 2 on the doomed roof; (c) self weight. The reservoir is optimized with respect to three state variables xi: x I is the ratio of reservoir diameter to wall height, x 2 is the wall thickness, x 3 is the floor thickness. The values of the state variables are given in Table 3, the floor thickness being constant at 150 mm. The desired reservoir capacity is 8665 m 3, and the dome thickness is maintained at 76 mm for all configurations.

Figure 5. Elastic foundation analysis - - meridional bending moment M s c~

e,,r,, v

__

/

I_L'Z<_To °o'~

t. e,: ~J. ...... /-.i I.__ 11 idealized/ actual' [ 1150 elevation (o) section i

--~-3.t,3 ~--~[ ~ kN/m2 I--I It,9.St, 20.60

-¢

Ioc) tn

Results The optimum structural design is found to be: diameter: 38.067 m; wall height: 7.613 m; wall thickness: 0.250 m; CPU time (Univac 1100/81): 10 min 51 sec (18 iterations). The optimum solution is clearly dependent on the cost rates chosen for materials and construction, and may vary somewhat for a reservoir constructed in another region. The optimum configuration here resulted in a cost saving of 15% over the most expensive configuration chosen from the state variables in Table 3 (diameter 47.961 m, height 4.796 m, wall thickness 0.40 m). Example 3 This example, an elevated water tower with a conical bottom, shown in Fig. 8, illustrates an analysis and design of a branched shell and the facility for plotting forces on the structure. The tower is analysed subject to self weight, and two imposed loads: (a) water pressure inside the tank; (b) 1 kN/m 2 imposed load on the flat roof of the tank. A computer plot of typical results is shown in Fig. 9. CPU time (Univac 1100/81): 39 seconds.

(b}

E=3/..l.86xl06

kNlrn 2

~,=0.15

Figure 6. Hillside reservoir. (a) Elevation and section; (b) idealized distribution of earth pressure reservoir with a doomed roof, subject to unsymmetric earth pressure loads, is shown in Fig. 6. Ahmad 12 analysed the reservoir using six Fourier series harmonics, treating the base of the wall as fixed. A comparative analysis using DART is carried out, approximating to the load functions with six harmonics at 60 theta positions around the shell, using 110 elements. Typical results are shown in Fig. 7. The same reservoir is then optimized with a view to obtaining the most economical design, based on realistic cost rates. For optimization the earth loading is simplified by backfilling to a uniform height around the wall. A non-symmetrical load analysis is thus not required, which vastly reduces the computer time. The major cost rates influencing the design are set at: concrete: R 35/m3; reinforcement: R 500/tonne; excavation: R 5/m3; backfill: R 4/m3; dome formwork: R (2H + 10)/m 2, where H is the height of the roof above the ground. R refers to Rands (approximately 50p). The formula for the roof formwork cost allows for variations in the cost of staging due to increased height above the ground. The two numerical constants in the formula are input by the user. The reservoir is backfilled to a uniform height, level with the ring beam. Additional

40

Adv. Eng. Software, 1981, Vol. 3, No. 1

--

t~*Ll~Tt

i 0

~

i

OART

T Figure 7.

Table 3. XI

Meridional bending moment M s at 0 = 180 °

State variables X2

X3

2.0

0.4

0.150

3.5

0.3

--

5.0

0.25

--

8.0

0.20

--

10.0

--

--

11/,

z~ 225

!25 --

~

/

,225

i

2990 ] 250

267/, m

_

E= 28x106 kNIm 2 v'= 0.2 Forces due to w~otor load inside l a n k and Imposed roof Iosd

main shell nodes: 1-119 branch nodes: t,4 -120-148-10/. 1

...~..

Figure 8.

im,el¢o L N H.rHr

Elevated water tower

xj.,_

CONCLUSIONS A c o m p u t e r p r o g r a m which efficiently produces the o p t i m u m design for circular concrete reservoirs and elevated water towers has been presented. Other structures such as sludge digester tanks, hyperbolic cooling towers, silos and chimneys can also be designed. The d y n a m i c p r o g r a m m i n g successive approximations technique has been successfully adapted to this problem, and c o m p u t e r time for an analysis involving a large n u m b e r of possible configurations is still relatively small. T h e a m o u n t of work involved in data preparation is reduced by automatic generation of nodal co-ordinates, m e m b e r incidences, shell thicknesses, material properties and distributed loading coefficients. The o p t i m u m reservoir configuration is highly dependent u p o n the cost rates for materials and construction, (and any variations in these values m a y result in an entirely different solution). The floor thickness will depend on foundation conditions, and the p r o g r a m can analyse the floor as a slab on an elastic foundation, using concentric ring elements with soil material properties to model the spring constants for a Winkler foundation. The user can decide on the spacing of the ring elements, which corresponds to the width of a pocket of weak material beneath the floor. This facility is particularly useful in the case where the wall is fixed to the footing, since any rotation of the footing will transmit a bending m o m e n t to the wall. REFERENCES 1

BritishStandards Institution, The structural use of concrete for retaining aqueous liquids, BS 5337, BSI, London, 1976

CIRcunr(M(HIIIII. IM0aP) fOII¢( I N l l

Figure 9.

2

3 4 5 6 7 8 9 10 11 12 13

I , ( I A : 0.O0 0 / G R ( ( S

i .o g n , t l . t ~ *TII

DART

Kit m*O ¢ ~ , O . a a u n

Circumferential (hoop)force No

Witmer,E. A., Pian, T. H. H., Mack, E. W. and Berg, B. A. An improved discrete - - element analysis and program for the linear-elastic static analysis of meridionally-curved, variable thickness, branched thin shells of revolution subjected to general external mechanical and thermal loads, MIT Research Report ASRL TR 146-4, 1968 Novozhilov, V. V. Theory of Thin Shells, P. Noordhoff, Groningen, 1959 BritishStandards Institution, The structural use of concrete, CP 110, Part 1, BSI, London, 1972 BritishStandards Institution, Design and construction of reinJbrced and prestressed concrete structuresfor the storage of water and other aqueous liquids, CP 2007, Part 2, BSI, London, 1970 Hughes,B. P. Design clauses in BS 5337 for controlling early thermal and shrinkage cracking, Struct. Eng. 1977, 55, (3), 125 Lloyd, A. R. and Doyle, W. S. Automatic analysis and design.of circular concrete reservoirs to BS 5337, Proc. Third Int. Conf. Finite Element Methods, Sydney, 1979, p. 129 Bellman,R. E. and Dreyfus, S. E. Applied Dynamic Programming, Princeton University Press, 1962 Lloyd, A. R. and Doyle, W. S. Automatic analysis and design of axisymmetric water retaining structures to BS 5337, Proc Int. Conf. Eng. Software, Southampton, September 1979, p. 252 FI/igge,W. Stresses in Shells, Springer, Berlin, 1973 Perloff, W. H. Foundation Engineering Handbook, (Ed. H. Winterkorn and H-Y. Fang) Van Nostrand, New York, 1975 Ahmad, S., Irons, B. and Zienkiewiez, O. Analysis of thick and thin shell structures by curved finite elements, Int. J. Num. Methods Eng. 1970, 2, 419 Anchor, R. D. Structural Design to BS 5337 (1976), Struct. Eng. 1977, 55, (3), 120

Ado. Eng. Software, 1981, Vol. 3, No. 1

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