Reinforcement minimization of cylindrical shells

Int J Non-linear Mechamts. Vol 15. pp. 505 516 ©Pergamon Press Lid. 1980 Printed in Greal Britain

0020-7462,'80/120505 12 $02.00/0

REINFORCEMENT MINIMIZATION CYLINDRICAL SHELLS

OF

R. E. MELCHERS Department of Civil Engineering, Monash University, Clayton, Victoria, Australia, 3168

(Received 31 July 1979) Abstract--In the search for the absolute minimum amount of reinforcement to be provided in a structure to support predefined loading, most attention has been given to problems for which the relationship between unit cost of material and stress relationships is simple--usually linear. Such an assumption is convenient and reasonably realistic when reinforcement percentages are low. However, for higher reinforcement percentages, and when, as in the case ofcylindrical shells, there occur axial as well as radial loads, a more refined analysis procedure is desirable. This paper considers the optimal (absolute minimum reinforcement) strength design of closed cylindrical shells subject to uniform pressure and having rigid ends. Two cases are considered: the shell wall rigidly connected to the ends, and the shell wall hinged at the ends. For convenience, only internal pressure loading is considered in detail, although, using the theory given, results for external pressure cases can readily be obtained. It is assumed that buckling is not a critical factor in the design and that serviceability criteria can be met independently.

I. I N T R O D U C T I O N

The absolute minimization of reinforcement in reinforced (concrete) shells designed to ultimate load or plastic design criteria has received relatively little attention, presumably due to the relatively low cost of reinforcement material compared to the costs of installation, etc. Nevertheless, the problem is of interest where expensive or scarce reinforcement material must be employed, or where installation costs are proportional to the amount of reinforcement used. The problem is also of theoretical interest in providing a lower limit to other procedures for obtaining optimal plastic design solutions. Early work in applying plastic design or analysis principles and optimization to reinforced shells is reviewed by Olszak and Sawczuk [1]. Mroz [2] considered optimal design of reinforced domes using a specific cost relationship akin to that employed herein. Melchers and Rozvany [3], using a simplified linear specific cost function, considered the optimal design of reinforced concrete circular cylindrical water tanks with fixed and hinged bases. It was found that the character of the optimal design did not depend on the nature of the applied (radial) loading. Non-linear cost functions have found little application in optimization work so far. Mroz [2] considered a non-linear cost function for bending moment and axial force interaction; other authors (Prager and Shield [4], Rozvany [5]) have considered piecewise-linear cost functions, but applications have been scarce. In the present paper, circular cylindrical shells with either rigid or hinged endsand under uniform internal (or external pressure) will be considered. When such shells are independent of other structures, a unique relation between pressure and longitudinal stress exists--in general, however, this need not be the case (e.g., when subject to exterior loading as well as internal pressure). It is further assumed that the shell wall thickness is such that strength, not buckling, controls the design. No account is taken of serviceability criteria, such as maximum desirable crack widths, deformation or deflection. 2. O P T I M A L I T Y

CRITERION

For plastic structures, the optimality condition of Prager and Shield [4] is as follows. If the local state of stress at any point e in a structure is defined by generalized stress Qi(e) [i = 1. . . . . n] corresponding to a local state of strain qi(e) then the condition for minimum cost of the structure is a kinematic condition expressed by q(e) = G[Q(e)] 505

(1)

506

R.E. MELCHERS

where q is a kinematically admissible strain field corresponding to the statically admissible generalized stress field Q, and where the cost gradient vector G is defined as G(~)=(grad ~b)i= cSQ-~'63Q2 't~Q3""t] 0

'

(2)

The specific cost ~b (e.g., cost per unit area of plate) is related to the generalized stress field through the specific cost function

~b = ~b[Q(~)]

(3)

The total cost • of the structure is the integral over the structure of 4). If 4) is discontinuous at some generalized stress state Qn, the differentiation in (2) is strictly invalid, and the convex combination of gradient vectors G~ of adjacent regions (in each of which 4~(Q) is differentiable) is used: G~

= ~, ~,G,(a~)I ~ = ~ .

(2a)

i

where for all i, 2 i > 0 and Z i 2 i = 1. The optimal design for a given structure may thus be obtained by determining the cost function(s) ~b as functions of the generalized stresses, differentiating according to (2) or (2a) and interpreting the results as a set of kinematic conditions to be satisfied by the deformation of an "associated structure" with identical loading and boundary conditions. If the resulting optimal design is statically determinate, the deformation of the actual structure corresponds to that of the associated structure. 3. C O S T

FUNCTION

FOR

AXISYMMETRIC

SHELLS

Referring to Fig. l(a) for orientation, the stress resultants of interest for an axisymmetric shell are N~, M~ and No. It is assumed that small deflection theory is valid, so that Mo = O. The derivation of the cost function for Nx, M~ (see Fig. 2 with No= 0) is summarized below. It will be assumed that the reinforcement is present only in the extreme fibres of the shell wall thickness, the influence of any cover to the reinforcement being ignored [see Fig. l(b)]. For the idealized reinforced concrete shell wall section, strains and stress resultants shown in Fig. l(c), five different expressions can be developed depending on the stress state in the reinforcement layers AI and A2, and whether or not Aa or A2 or both, are present or not. In each case it is assumed that the concrete (matrix) material has an ideal-rigid plastic

~ L

Rigld End

-O

~L~

ac M

A1 Fig. 1. (a) Shell orientation. (b) Shell wall and reinforcement. Ic) Strain and stress state.

Reinforcement minimization of cylindrical shells

507

/'°'

/

f

.//

Fig. 2. m ~ - n. -no cost function.

material response with compressive strength tr¢ and zero tensile strength, and that the reinforcement is also ideal rigid plastic with tensile and compressive yield stresses a o. Introducing the reduced notation for unit width B = 1 ; mx

2M~ =-GcH2,

N~ n~=---, a,H

S=

Ata o a,H

and

(4)

S'= Aza° a,H

where all stress resultants are taken about the centre-line of each cross-section, and n~, S and S" are positive if tensile, it can be shown that the cost functions for m~-n~ (i.e. no---O in Fig. 2) are as set out in Table 1. With the introduction of the radial stress resultant N o, the expression for total specific cost becomes: =

=

+

Iso l

(5)

where the expressions for S~r are given in Table 1, and where Sot represents the reinforcement required to sustain the hoop force No. Using the reduced notation no= No/(afl) and So = Aoao/(afl) with no, So positive if tensile, it is evident that the following cases exist: -- 1 <~n o = S o + S~ - 1 0 ~ no ~< - 1 no = So + Sb

high compressive N o low compressive No

(6)

tensile No.

Since M o = 0 by assumption, it follows that with No taken about the section centre line, S'o= So. Hence, for the special case of internal pressure (No >10), the specific cost in the hoop Table 1. Cost functions for m~-n~ Cost contour section (see Fig, 2) ab bc cd de ef

v =0 0
S

S'

Eqn. No.

Cost fimction (m~-n~)

>0 >0 >/0 0 <0

>0 =0 ~<0 <0 <0

(4a) (4b) (4c) (4d) (4e)

sxr=n x S~r=(l+n~)--(l+nx--m~) °.5 S~r=mx--0.25 Sxr=--ilx--(--n~--mx) °'5 S~r=-n~-I

Range of validity 0~
508

R.E. MH.(m~r~s

direction is given by (7)

SOT = 2So = no.

The surfaces of constant cost in the m~, n~ no space (Fig. 2) are developed simply from (5) by assigning a given distribution of S T to each of S~T and SOT. Thus SOT= 0 produces the traces of the constant cost surface on the m ~ - n~ plane since no = 0 if SOT = 0. Similarly, for ~ By drawing no= ½, with S T - ~~, leaves only the contour for SOT = 0 to be plotted at no = 5. traces for other values of n0 the whole constant cost surface (Fig. 2) can be developed. An alternative representation is to plot cost contours as projected on to the m~-n0 plane, for given values of n~ (see Fig. 3). This representation is useful in establishing the stress regimes which may exist in an optimal solution. It follows that for no compressive, no reinforcement will be required until No exceeds the compressive strength of the concrete (matrix) material, a¢H. Hence in Fig. 2, the contours at n0=0 also apply for -l~
The strains from which the associated structure must be developed are obtained by applying (2) to the cost contours of Figs. 2 and 3. The cost surface may be considered to be a potential surface and then the strains are represented by the outward normals. For the typical locations A, B, C, D shown on Fig. 3, the stress regimes and the appropriate strains for the optimal solutions are as given in Table 2 for n~> 0. The results of Table 2 also apply for n~< - 1 ife~ is replaced by - ~ [-due to symmetry]. Similarly there is symmetry in m~. For -½~
mz

2 ,.

l"

i

I~ h

r~

•

I/2

-

n I

•

0

n i

" - I/2

rn

s.="2"

.

°n

+43-

I

~F__~t_~, m. °, }

nO

Fig. 3. m.~-no cost contours for various values ofn,. Table 2. Strains for given stress rcgimcs Location

Strcss state

Scc Fig. 3. Strain

A B C D

m~>~n~+~

C' D"

m~=-n~--n.~ no>~O O<~k~<~kt. So=% L m,<-n:,-n, n,~>0 k~=0 Co=%L

no>>-O O~m.,>n~no>~0 O<~k~<~kt. ln~=0 Ilo~>O O<~k~<~kL n~>m~>~0 n 0 ~ > 0 k~=0

O~<%~<~:0L ~:~=0 O<~o<~r,OL 0
O~,O~l~,OL O<~x~xL O<~CO~OL O
Reinforcement minimization of cylincrical shells 4. L O N G

CIRCULAR

CYLINDRICAL

SHELL

WITH

509 CLAMPED

ENDS

For the circular cylindrical shell shown in Fig. l(a), the axial force per unit length = N~ = pr/2, where p is uniform internal pressure. In what follows, this unique relationship

between Nx (and n~) and p will not be employed, to allow for the possibility of external axial loading. Hence n~ and p will be treated as independent parameters. The deformation of the shell under p and n~ will be as shown in Fig. 4(a) with the rigid ends connected to the shell wall so as to ensure clamping action. Restricting attention now to the internal pressure case only, it follows that only stress regimes A, B, C and D of Fig. 3 and Table 2 need be considered in order to determine the deformation of the associated structure.

D

£e'E '.~ I k, o0 I

t B~-.-C

,.oLd

I: !

r",,l'"

__

%1 "1

y

_

I~ltt

i ~c /I

(0)

¢o)

Fig. 4. Associatcd structure deformation field and gcneralised stress state for clamped base boundary condition long shell.

5. S T R E S S

REGIME

A

It is evident from Fig. 3 that for a state of stress m~ >/n~ + ¼and n0 4=0 (i.e. along gh) both k~ and e0 are defined, fixed values. Considering the nature of the shell deformation, it is readily apparent that these strain values are incompatible (cf. Fig. 4). The only way for stress regime A to exist is at g, when n0=0, and 0~<%~
6. S T R E S S

REGIME

B

A similar consideration to that for stress regime A indicates that stress regime B cannot exist unless it intersects the m~ axis at point i (i.e., he=0). This cannot occur for m~> 1¼(see Fig. 3), i.e., for ¼~i rn~>/nx, no=O and O<<.~o <~eOL.

7. S T R E S S

REGIME

C

Since mx = nx for this regime it can only exist along a line or at a point. Nevertheless, by considering an infinitesimal change in location along the shell, it is evident, using the same argument as for stress regime A that for both k~ and e0 to be defined values shell deformation is not possible in general. If, say e0 is to be allowed to vary, then ne---0 is necessary. Thus stress regime C is limited to S r -__!~, point I in Fig. 3.

510

R . E . MELCHERS

8. S T R E S S

REGIME

D

From Fig. 3, stress regime D is seen to have kx=0. This implies shell deformation limited to uniform expansion e0= e or to conical type deformation. In either case mx <<-n,,is required. 9.

DEFORMATION

FIELD

The deformation field for the associated structure, and the corresponding generalized stress state are given in Figs. 4(a) and (b). The appropriate stress regimes are indicated. The key parameters describing the deformation field and the stress-state are the boundary between the hoop force and the bending moment zones b, and the point of inflection for the bending moment zone a. These distances were also computed earlier for a linearized cost function (Melchers and Rozvany [3]). In order to evaluate a and b, the expressions for k~ and e0 for each region need to be evaluated using (2) and (5) together with appropriate substitutions. Consider a prescribed value of n~ >/0. If stress region A exists, it will be governed by (4c) [Table 1] for S r = S r ( m x , n~) and (7) for St=St(no). Hence, by (5) c = S r = n o + m~ - 0.25

(8)

No 2M~ c . . .(rcH . + ~ - / ~ -0.25

(8a)

or

or

c(a~H)=N°-~

2Mx H

a~H 4

(8b)

?,c 2 acH ~ M ~ - H

(8c)

k~= 1.

(8d)

Applying the optimality condition (2), there results: a~H _Oc_= 1 t3No

and

so that, to within the common constant a~H2/2, H %=~-,

These strains describe stress state A. For stress state B, equation (7) still applies for no, but (4b) is now applicable for ST = S,r(mx, n~)--see Fig. 2. Thus c = S r = no + (1 + n~)--(1 + nx --m~) 1/2 (9) from which it follows that (o~H 2) 0c 2 % = ~N-o = H/2

(9a)

and (acHE) kx=½(1 + nx-mx)-1/1 2

(9b)

At the junction of stress state A and B, i.e., at m ~ = n x + 3, equation (9b) reduces to [(acH2)/2]k~ = 1, which agrees with (8c). Stress states C and D can only exist with k~ = 0 as shown above. Application of (2) to the specific cost for these two stress states merely shows that %= ex; however ex is of no interest to the present problem. For convenience, in that which follows the constant term (oH2/2) will be ignored. 10.

OPTIMAL

DESIGN

The above information may be used to derive values for the parameters a and b which describe the optimal design. Referring to Fig. 4, two expressions may be developed in terms of a and b; one expressing slope continuity of the deformed shell surface, the other deflection continuity.

Reinforcement minimization of cylindrical shells

511

F o r slope continuity, ASu + AS, + AS~ + ASh + A S / - - ASe - ASg - AS, = 0

(i0)

where AS~ represents the slope change over section i. Due to symmetry of the bending m o m e n t diagram between x = a and x = b, it follows that u = f t = h and from the condition k~=0, in u, e a n d f A S u = A S I = A S e = O , so that 2ASt + AS~ - ASg + v = 0.

(10a)

T h e slope change ASt is obtained by integrating the curvature equation (9b) across deformation region t. F r o m equilibrium, m(x) = -- P [ x 2 - ( a + b ) x + a b ] .

(11)

Changing the variable x to p so as to integrate in the direction of increasing slope [see Fig. (4a)] there is obtained x = b - u - p , and from the boundary conditions: at and at

x=b-u

mx=nx

x=b-u-t

m~-n~+g __

3

it follows that: u ( = f ) = (b - a)/2 - [~b - a ) 2 - 2nffp] 1/2

(12)

t( = h) = [ ~ b - a) 2 - 2nJp] x/2 _ [~b - a) - 2(nx + 3)p],/2.

(13)

and

T h e slope change AS, can now be obtained by integrating (9b) from p = 0 to p = t using (11-13), with the initial condition that at p = 0 , the slope is zero. Eventually: l(2P )- ,,2 _ [~(b - a ) 2 - 2(nx + ¼)/p],/2-] AS,=(2p) -'/2 In 1_(2/p),/2 _ [ ~ ( b _ a ) 2 _ 2 n j p ] , / 2 _j.

(14)

T h e slope change ASg can be obtained in a similar manner, but with the initial condition that the slope at the v, g b o u n d a r y = o, it can be shown that e= -(b-a)/2+q

(15)

g = - - t l + t2

(16)

and A S o + ~= (2p)- 1/2 [arsin ( - t 1/t3) - arsin ( - t2/t3) ] + (a + b)/2 - t 2

(17)

where t l = E~(b - a ) 2 +

2nffp]'/2

(18a)

t2 = [~(b - a) 2 + 2(n~ + ¼)/p] 1/2

(18b)

t3 = [~(b - a ) 2 +2(n~ + 1)/p] '/2.

(18c)

Finally, AS~=S=b-a-h-f-u-t =2[(b-a)/2 -u-t] _ -

(19)

3 1/2 2 [ ~I ( b - a) 2 - 2(nx + g)p] .

(19b)

Deflection continuity can be expressed as: ADo + v + ASg + v e + f + h + ~

- ~

=0

(20)

where ADi is the change in deflection over region i relative to the undeformed shell. Using (12), (13) and (19b) the bracketted term in (2) can be shown to reduce to t 1. By integrating ASg+, it follows that: ADg + v = (2p)- I/2t~ [arsin (t,/ta) --arsin (t2/t3) ] + ½[{b + a)/2 - t,} 2 _ (t 2 _ t l)2] + (2p)- 1. (21)

R.E. MI!I.('III ~s

512

Substituting into (2) produces eventually: a " b - (2nx +

½)/p- rH =

0.

(22)

Equation (22) may be rearranged to solve for b in terms of a and substjtuted into (10a). The resulting equation is not explicit and must be solved numerically. Alternatively, (10a) and (22) can be solved simultaneously using an appropriate solution routine. If the parameters p, nx and rH are made dimensionless as a ratio of the shell radius r, then typical results for a/r and b/r, such as given in Figs. 5 to 8, are obtained. For (10a) and (22), the results are indicated as CASE 1, and these follow from the assumptions about applicable regions in the optimal solution shown in Fig. 4(a). These regions remain valid whilst p is large. For smaller values of p, region s becomes zero, and the condition mx = n~ + ¼ in Fig. 4(b) is no longer applicable. Hence (13) for t = h is no longer valid. As a result, (14) for the slope change over region t becomes: AS, = (2p)- a/2 In ( 2 / p ~ ~ ~

]

~~,/z_]

(23)

o8

b~ 07

H/R • 0 01

\\ 06

05

04;

i

03

o

,°L

o~ r

03 o~ o 5

°o

/

01 /

0 [ 10

L 30

[ 20

i 40

| 50

I

I 60

I 80

70

I 90

I 100

J 110

I t20

JP 130

Fig. 5. Optimal values a and b for H/'r =0.01. 09

08

"o

H/R=

0"05

% ,~ 0 07

06

05

O4

03

O3

0

0

% o

O" I t0

210

I 30

410

I 50

| 60

I 70

I 80

Fig. 6. Optimal values a and b for H/r=O.Ob.

I 90

I 100

I 1t0

I P 120

Rcirtlk~rccrncnt minimization of cylindrical shells

t0

t l

513

C~se 2

% - o,

09 0~8

b/r o

o

,

0 0.410

"

05

cose

~ = ~ - - ~ - . . _ . ~ ~ ~ _ ~ ~

2

b,~..o co~e~

05 0025

0

~

-

% 02~

0

IO

20

30

40

50

60

70

80

90

100

*.ZO

130

Fig. 7. Optin'tal "values a and b for H/,'=O.I.

~3

I 2

I

-./~ II

i

O8

:~-seI ~

¢ose 2 ~,0

0 5/0 ? O 025

o--* o/r 03

10

20

30

40

50

60

70

~0

90

I00

~10

120

p

Fig. 8. Optimal values a and b for H/r=0.2.

while ASs = 0.

(24)

Substituting (23), (24) and (17) into (10a) produces the modified requirement for slope continuity. The equation for deflection continuity [equation (22)] remains unchanged Solution of these two equations produces the results marked CASE 2 in Figs. 5-8. Finally, for still lower values of p, region v [Fig. 4(a)] will reach zero. As a result (16) for g will no longer be valid. Eventually the equations for CASE 3 can be solved numerically and plotted (see Figs. 5-8). The boundaries between the three cases can also be obtained. Thus s = 0 gives: b - a = 212(n~ +

¼)/p] 1/2

{25)

which can only be solved for a and b by trial and error and with the knowledge of the results of CASE 1 (or 2). Similarly, for v = 0, b = 2(n, + ¼)/(a • p) (26)

514

R.E. MELCHERS

which also requires a trial and error solution. Both region boundaries have been plotted in Figs. 5-8 where appropriate. 11. S H O R T

CIRCULAR

CYLINDRICAL

SHELL

WITH

CLAMPED

ENDS

For a shell of length L, the above results cease to be valid once b > L/2. At b = L/2 the no region (see Fig. 4) is reduced to a line at x = L/2 and consists of a concentrated reaction force resisting a total load of (b-a)p per unit length of shell circumference. When L/2
CYLINDRICAL

SHELL

WITH

HINGED

ENDS

For a long shell, when the clamped boundary condition at the rigid shell ends is replaced by a perfect hinge, regions e, g and v (i.e., a) in Fig. 4 must be zero, while the other regions need to be larger to accommodate the deflection rill2. Because of the hinged condition, slope continuity of the deflected shell shape is no longer enforceable and only deflection continuity is required. Referring to Fig. 4(a), with a =0, deflection continuity may be represented as (2 +h+f ) ASu+,+s+h+j--rH 2

(27)

But AS,,=ASr=O. AS,=AS~,. ASs=S and (s/2+h+f)=b/2, so that using (14) for AS, and (19b) for ASs, there is obtained

(AS, + t4)b -

rH

~

= 0

(27a)

¼)/p]'/2

(28a)

where t , = [ ~ b - a) 2 - 2(n~ +

Typical results obtained from solving (27a) numerically are shown in Fig. 9 for H/r = 0.01. Similar results can also be obtained for other values of H/r. When p is small, region s in Fig. 4(a), with a =0, will disappear. In the same way as for the clamped shell case the expression for AS, must now be replaced by (23), while t4 =0; thus:

b

rH

2AS; '~ - -~- = 0.

(29)

Solution of this equation is also shown in Fig. 9. When b in (27a) or (29) exceeds L/2, then exactly analogous to the situation for the clamped shell, hoop load carrying action is no longer applicable in the optimal design and direct spanning between the rigid ends is the optimal load carrying design. Since support at the ends is hinged, the stress state for longitudinal load carrying action is statically determinate and not subject to further optimization. 13.

DISCUSSION

The results sketched in Figs. 5-9 asymptote the results obtained using simplified linear theory (Melchers and Rozvany [3]) for nx=0 both near p ~ 0 and for p~oc,. For the clamped case, it was found earlier that b/r=2a/r=2[~H/r)] ~/2 while for the hinged case

Reinforcement minimization of cylindrical shells

515

OB

H/R , 0 0 1

07

O6

05

04

o@

05

02

_9 11

0 1

I

0 C,

20

40

60

60

10(3 300 5 0 0

7 0 0 900

4000 I J

2000

800C

6000

~C~C': :

Fig. 9. Optimal values b for hinged shell.

b/r= [H/r] t/z. It is readily verified that these results are also approached in the present solutions when p--* ~ , since the curvature ofthe m~ regions will then be Ik.I = 1 with s-,(b -a) and v ~ a for the clamped case, and with s--*b for the hinged case. At very low loads, however, regions s and v do not exist and indeed the moments I,".l will be only just greater than nx. For low values ofm~, it is evident from Tables I and 2 that for stress regime B, which will apply in regions t, h and g, the cost function is given by (9). The appropriate curvature k~ is given by (gb). It is evident that for very low moments ([mJ >>.n~), k~- , i" ~ This means that in contrast to the very high load case p--, oo, which has k~--*l wherever re:d=0, the very low load case has k~_,,t~. In particular, when G = 0 , k~--,½ extends over the whole region m~4:0. If the results of Melchers and Rozvany [3] are reworked with k~=-~, it is found that b/r = 2a/r = 2[H/r] ~l: for the damped case, and b/r = [2H/r] ~/z for the hinged case. Again it is readily verified that these values are attained in the present numerical results for n~=0 when p--,0. When G > 0 , it is seen that for p - * ~ , the values ofb/r and air still asymptote the earlier results from linearized theory, but more slowly so, with n~ increasing. More noteworthy, however, is the effect of non-zero n x at low values ofp. It is evident that b/r and a/r become unbounded as p--,0. The reason for this is evident when the deformation of the (associated) shell, Fig. 4(a), is considered with only regions t, h and g having finite curvature (k~½) but with regions u,fand e (k~ -~0) being very long due to the low value of Irn~l. Very large values of b and a are then required to attain the deflection rH/2. The present results show that the optimal plastic design for apparently simple problems can be markedly affected by the presence of even a very small axial load. The numerical results given in Figs. 5-9 were obtained by a standard subroutine to solve for the roots of a non-linear equation. However, due to the limited validity of the equation representing each case, some difficulty was experienced at times with selecting an appropriate trial solution. This problem was found to be particularly acute for low values ofp and high G values when it became very easy to transgress the various limits on the solution. The present results for b/r and air do not represent the final stage of the optimal design. They merely specify the size(s) of the appropriate regions. With that information the design proceeds by determining the bending moment diagram for the mx region and then, for a given shell thickness,/4, sizing the reinforcement to support mx and n~ acting in combina-

516

R.E. MELCHERS

tion. Similarly in the no region, the hoop reinforcement must be sized to support no, while longitudinal reinforcement is sized to support nx. Although no hoop reinforcement is theoretically required in an mx region, in practice it would be prudent to provide a minimal amount. 14. C O N C L U S I O N

It has been shown that the optimal design of a reinforced circular cylindrical shell having rigid ends, is dependent on the pressure distribution and the boundary conditions assumed at the rigid ends. The influence of longitudinal load, whether due to internal pressure, or externally applied or both, is very marked, particularly at low pressures. For the latter situation, hoop action is unlikely to be present in the optimal design. The results correlate well with earlier results based on a simplified linear theory when the longitudinal stresses are zero. REFERENCES 1. W. Olszak and A. Sawczuk, Inelastic Behaviour in Shells, P. Noordhoff, The Netherlands (1967). 2. Z. Mroz, Optimum Design of Reinforced Shells of Revolution, in: Nomlas.~ieal Shell Problems, lEd) W. Otszak and A. Sawczuk North Holland Publishing Co. (1964). 3. R.E. Melchers and G. I. N. Rozvany, Optimum Design of Reinforced Concrete Tanks, J. Engineerin9 Mechanics Division, A.S.C.E. 96, No. EM6, pp. 1093-1106, Dec. (1970). 4. W. Prager and R. T. Shield, A General Theory of Optimal Plastic Design, J. Applied Medlanics, A.S.M.E 34, 184-186 (19671. 5. G. I. N. Rozvany, Optimal Plastic Design with Discontinuous Cost Function, d. Applied Mechanics A.S.M.E. 41,309-310 (1974).

Zusammenfassung Auf der Suche nach der absolut geringsten Bewehrung fur eln Tragwerk mit gegebener Belastung wurde bisher die meiste Aufmerksamkeit Problemen geschenkt, fur die das Verhaltnis zwischen Materialkosten und aufnehmbaren Spannungen einfach - gewohnlich linear - ist. Diese Annahme ist nutzlich und bei geringen Bewehrungsgraden realistisch. Bei hoheren Bewehrungsgraden und bei Auftreten axialer und radialer Belastung wie im Fall yon Zylinderschalen ist jedoch ein genaueres Berechnungsver~hren erforderlich. Oieser Aufsatz bcfa~t sich mit dem hinsichtlich der aufnehmbaren Belastung bei absolut minimaler Bewehrung optimierten Entwurf geschlossener Zylinderschalen, die unverschieblich gelagert und gleichma iger Druckbelastung ausgesetzt sind. Es werden die zwei Falle der eingespannten und gelenkigen Lagerung der Zylinderschalenenden betrachtet. Zweckma~igerweise wird nut der Fall des Innendrucks detailliert behandelt; jedoch konnen mit den gegebenen theoretischen Abeleitungen leicht die Ergebnisse fur AuBendruck ermittelt werden.