Finite element techniques for the analysis of cooling tower shells with geometric imperfections

Thin-Walled Structures I (1983) 239-263

Finite Element Techniques for the Analysis of Cooling Tower Shells with Geometric Imperfections S. S. J. M o y a n d S. M. N i k u Department of Civil Engineering, University of Southampton, Southampton SO9 5NH, UK

ABSTRACT Two finite element methods for analysing geometrically imperfect cooling tower shells are presented. In the first the geometry of the imperfection is modelled by the elements; in the second the imperfection is represented by an equivalent load on the shell. Axisymmetric and general shell elements have been considered. Results are given which show that the first approximation to the equivalent load is sufficiently accurate and that it is possible to represent local imperfections by axisymmetric imperfections which require less computation. It is also shown that axisymmetric elements should be used wherever possible, because of their greater efficiency, following the geometry of an axisymmetric imperfection but representing local imperfections by equivalent loads.

NOTATION F ( Z , O) i, j (superscripts)

m , , m0, m,A M , , M0, M,0 n~, ne, n,~

Typical function of Z and 0. Indicate Fourier series constants (for a particular value of Z) for the ith or ]th harmonics. U n b a r r e d constants, e.g. q~, refer to cosine terms; barred constants, e.g. q,, -i refer to sine terms. Changes in bending stress resultants. Bending stress resultants. Changes in m e m b r a n e stress resultants.

239 Thin-Walled Structures 0263-8231/83/$03'(~) O Applied Science Publishers Ltd, England. 1983. Printed in Great Britain

241)

N , , No, N,o NH p (superscripts)

[P] q. q.*

Is] Z AF.,

S. S. J. Moy and S. M. Niku

Membrane stress resultants. Number of harmonics in truncated Fourier series. Indicates perfect shell. Nodal forces. Total normal pressure. Normal pressure due to N,,, similarly qn~, q,~,~. Stiffness matrix, Vertical distance above shell base. Nodal displacements. Normal force due to N,, similarly AFo,, AFo,o.

Terms not shown here are defined in the text or figures.

1 INTRODUCTION There has been considerable interest in the behaviour of cooling tower shells since the collapse of three new towers at Ferrybridge power station, UK, in 1965. The problem there was insufficient resistance to wind uplift. Further problems were highlighted by the collapse of another tower at Ardeer, Scotland, even though the shell had been redesigned to provide significantly greater resistance to uplift. The shell of the Ardeer tower had been constructed with imperfections which w e r e s h o w n 2"3 to have contributed to the collapse. The geometry of modern towers is defined by a hyperboloid of revolution. The Ardeer tower shell had an overall height of 106.5 m and a base diameter of 78-8 m. The difficulty of setting out such a shell during construction has been known for a long time, and all cooling towers have imperfections. (This is immediately apparent when standing at the base of a tower and following the line of the shell.) The imperfections in the A r d e e r tower were particularly severe--a survey during construction showed the worst meridian of the tower to be out of line by nearly 600 m m (compared to a basic shell thickness of 150 mm). The realisation that imperfections could modify the behaviour of the tower shell resulted in research into the extent of that modification. Croll and co-workers have published a series of papers 4-7 giving the results of analyses which represented the imperfection by an equivalent load. M u n r o first suggested this approach in 1967 in an unpublished report and a theoretical justification for the case of an axisymmetric imperfection

Finite element techniques for cooling tower shell analysis

241

was given by Calladine. 8Croll and Kemp obtained their solutions by using the finite difference technique and considered mainly axisymmetric imperfections. The finite element method has been used extensively for analysing perfect cooling tower shells. There have been two basic approaches. The first, which was outlined by Zienkiewicz,9 was to use general shell elements, such as flat triangles, to represent the complete shell. The second was to use axisymmetric shell elements. The earliest of these elements had a straight meridian. Grafton and Strome '° described the use of this element with axisymmetric loading. Percy e t a l . , H among others, developed the method for general loading, representing the circumferential variation of load by a Fourier series. Great economy was achieved because of the orthogonality properties of Fourier series. The straight elements produced geometric discontinuities at each node when representing curved shells. Jones and Strome ]: introduced one of the first doubly curved elements, which gave radial and slope continuity at each node but still produced discontinuity of meridional curvature. These curved elements caused difficulties with rigid body motion and higherorder terms were added to element shape functions to overcome these problems. The approach of Mebane and Stricklin 13is typical. There is no readily available work on the application of finite elements to cooling tower shells with imperfections. This paper presents two techniques for this. In the first the actual geometry of the imperfection is modelled by the elements. A new curved axisymmetric element was developed for axisymmetric imperfections. Flat shell elements were used for local imperfections. In the second technique the imperfection, axisymmetric or local, was represented by an equivalent load on the perfect shell. Results are presented from both techniques, and comparisons of accuracy and efficiency, in terms of computer time and storage, are made.

2 FINITE ELEMENT ANALYSIS OF COOLING TOWER SHELLS It is necessary to describe briefly how finite elements are applied to cooling tower shells before considering i n detail the analysis of imperfections. A typical cooling tower shell is shown in Fig. l(a). The principal loadings are self-weight and wind. The magnitude of the wind load (as determined by wind tunnel tests) varies with height above the ground but

S. S. J. Moy and S. M. Niku

242

i

[J , , ' ~ ~ ~'

• I wind ~-direction

/~ ~,

/ meridian

/

® ~

~..._

~-- ~

\

~ --

"',.,T ~os ~ ,~

a)

b) Fig. 1. Details of wind loading.

the circumferential variation is assumed constant and as shown in Fig.

l(b). Using finite elements there are two possible approaches to the problem: 1. The shell can be represented by axisymmetric ring elements as shown in Fig. 2. At each node there are four displacements u, v, w , / 3 , and the corresponding forces. Each of these variables can be represented by a Fourier series to give circumferential variation. In general IA

IA

F(Z.O) = F°(Z)+ ~ Fi(A)cosjO+ ~ ~'J(Z)sinjO j=l

j=t

(1)

but because there is an axis of symmetry for the wind loading, the individual series for each variable contains only the cosine or the sine terms. From these series and the assumed geometry and shape

I

node 1 d~sp[acemen~ u

-,v

Hg. 2. Axisymmetric thin shell element.

Finite element techniques for cooling tower shell analysis

243

functions of the element the displacement approach will give the element stiffness matrix. In linear analysis the harmonics of the Fourier series decouple to give [0q[PJ] = [0q[Sq[~]

(2)

where [0~] is a diagonal matrix containing terms cos jO or sin jO. Thus

[lPq = [Sq[~]

(3)

Solution of eqn (3) for each harmonic in turn will give the Fourier series constants for each displacement. From these the Fourier series constants for the stress resultants are calculated. The complete shell is represented by two-noded elements and effectively the problem is reduced to one dimension. Relatively few elements are required and thus only a small number of simultaneous equations have to be solved, although there is a set of equations for each harmonic. (The so-called most adverse wind loading used in Britain is represented by an eight harmonic series.) This approach is extremely economical in terms of both computer time and storage. The axisymmetric shell requirement is potentially a major limitation when considering imperfections. 2. The alternative approach is to use general shell elements (flat triangles and quadrilaterals were used in the work presented) to represent the complete shell. Figure 3 shows a typical mesh. It is

symmefry-~_ \

/~

O=O

Fig. 3. Typicalmesh for flat triangularelements.

244

S. S. J. Moy and S. M. Niku

immediately apparent that the size of the tower requires many elements and degrees of freedom with consequent penalties on both computer time and storage. It is however possible to follow the geometry of a localised imperfection.

3 IMPERFECTION ANALYSIS

3.1 Geometricrepresentation The obvious approach is to follow the geometry of the imperfection with the elements.

Z2 L

0effect ', shett tine ~ - ' , ! Z~

L

Fig. 4. Detailsof axisymmetricimperfection. A typical axisymmetric imperfection could be modelled as in Fig. 4 using'doubly curved ring elements. The shape of the imperfection would be given in this case by

a[ =

2zr(Z- Zt) ] 1

-

-

z"zSS

(4)

and that of the shell by R ( Z ) = RP(Z) + r(Z)

where R p is the horizontal radius of the perfect shell.

(5)

Finite element techniques for cooling tower shell analysis

245

F i g u r e 4 also s h o w s v a r i o u s g e o m e t r i c p a r a m e t e r s w h i c h a r e r e q u i r e d to d e f i n e t h e e l e m e n t . T h e s e can b e d e r i v e d f r o m e q n (5). T h u s cot 6 -

dR dR p dr -- - -+ dZ dZ dZ

R,

dZ-~ d2R

where

dZ 2

1+

~-~

(6)

d2R p d2r - - ~ dZ: dZ 2

d R P / d Z a n d d: RP/dZ 2 a r e f o u n d f r o m the g e o m e t r y o f t h e p e r f e c t shell, a n d f r o m e q n (4) dr dZ

a~r -

d 2r dZ 2 -

(Z2 - Z,)

sin

21r(Z - Z 0 (Z2 - Z , )

2a 2zr 2rr(Z - Z l ) (Z2 - ZI) 2 cos (Z2 - Zl)

(7)

T h e a x i s y m m e t r i c i m p e r f e c t i o n w a s similar to the s t a t e o f p a r t o f the shell o f t h e A r d e e r t o w e r . H o w e v e r , t h e r e w e r e also localised ' p a t c h ' i m p e r f e c t i o n s as in Fig. 5, which c o u l d b e d e f i n e d b y a r = ~

E

1 - cos

(Z: - Zl)

1 - cos

(02 - 01)

1

Fig. 5. Details of patch imperfection.

(8)

246

S. S. J. Moy and S. M. Niku

The imperfection is no longer axisymmetric and must be modelled by general shell elements. The implementation of the geometric representation is discussed in Section 4.1. 3.2 Equivalent load method The imperfection is represented by equivalent additional loading applied to the perfect shell. This loading will be derived for a general patch imperfection, and then simplified for the axisymmetric imperfection. The applied loads (wind and self-weight) cause a system of stress resultants in the imperfect shell. Since the imperfect shell is inclined to the perfect shell, the membrane stress resultants of the imperfect shell have components tangential and normal to the perfect shell. The tangential components are membrane stress resultants in the perfect shell. The normal components can be considered as additional normal forces on the perfect shell and applying these forces to the perfect shell is equivalent to analysing the imperfect shell. Figure 6 shows the meridional membrane stress resultants in an element of the shell. Resolving normal to the perfect shell gives AFn, =

- N , sinot+ N , +

as,

8s, sin ct+

ass

8s,

8s~

aa 8s, )+aN~,aas,]as, = [ - N, ct + N, ( ct + as, as,

~ ~C-~ ~

N

c~s~ ~

~

~(pe~

unit

~-"~'~-Imper fe c: she{[ "-'"

~"~perfecf shell

0 (circun~fe~ Axes ~ (meridionat) Fig. 6. Meridional membrane stress resultants in the imperfect shell.

Finite element techniquesfor cooling tower shell analysis

247

assuming a is a small angle and ignoring higher-order terms.

Oa

aF.,=

N, Ts-£, as,

J

Since tan~, = a -

AF., =

Or 0s®

OZr ON, Or )Ss, Sso N, ~ + Os~ Os,

AF~ a2r ON, Or q"* - 8s, Ss-------~e- N , ~ + as~, as,

(9)

q . , is the normal pressure on the perfect shell due to the meridional m e m b r a n e stresses in the imperfect shell. A similar approach for the circumferential stress resultants gives

02 r ONe Or q.e = N, ~ -t as, Os,

(10)

T h e r e will also be a twisting deformation of the shell as shown in Fig. 7. Resolving normal to the perfect shell

A o. ;

(N.-,.-

\perfect shett

Fig. 7. Shear membrane stress resultants in the imperfect shell.

248

s. s. J. May and S. M. Niku

Following the same procedure as before this leads to a:r O"r a N ~ Or aNo, Or q.,o = N , ~ +N o , ~ + --- + as, aso asoas, c?s~ aso aSo as~

and since N~ = No, because of the equality of cross-shears in thin shells a:r

q.,o = 2 N e a ~ + as, aso

aN,o Or aNo~ Or -+ -Os~ Oso aSo as,

(11)

Adding eqns (9), (I0) and (11) gives the total normal pressure due to the imperfection a:r q. = N , ~

a~-r O: r + 2N,o aS, aso + No as-'-~--o

{ON, O N , o ) ar (aNo a N , o ) ar + !t as, + aso ' + It Oso + as, ~ aso

(12)

The last two terms in eqn (12) are very small compared to the other terms and can be ignored in the remainder of the analysis. The derivatives 02 r/Os:,, etc., can be found directly from eqn (4) or eqn (8) but an iterative approach is required to find the stress resultants. Initially the stress resultants in the perfect shell are found, ignoring the imperfection. These are then used to give the first approximation to q~ aZr aZr aZr q~. = NP, _---r + 2 N ~ + Ngo - ~. os~ as, Oso as'o

(13)

W h e n this load is applied to the shell it causes changes in the stress resultants, n~, etc., and rn~, etc., and corrected stress resultants N~

= N~ + n I

Mt

= M $ + ml,

N~

= N~ + n~

M~ = M~ + m~

Ng

= Ng + ng

Mg = M~ + mg

(14)

For a linear analysis the changes in the stress resultants can be used to find a second-order normal load a2

a2r + 2 n ~ a Z r

qld = n k 0 7 ,

Os, Oso

+nlo----r-r as~

(15)

Finite element techniques for coohng tower shell analysis

249

When this is applied to the perfect shell it gives further changes in the stress resultants n~, etc. This process can be continued until convergence is achieved. In the case of an axisymmetric imperfection 02r

O2r --

Os, Oso

- -

--

0

Os~

so that 02r q . = N , oS ~

(16)

The only significant changes in the membrane stress resultants caused by this pressure are in No, the circumferential component. Since n~ is insignificant only one cycle of iteration is required.

4 I M P L E M E N T A T I O N OF THE IMPERFECTION ANALYSIS

4.1 Geometric representation using axisymmetric ring elements A new ring element had to be derived to model the axisymmetric imperfection. The element geometry was developed from the Jones and Strome element. ~2 Only a brief description of the element is given; full details can be found in Reference 14. The geometry of the element is defined using the actual values of 4), R, and R (see Fig. 4) at each end of the element. Continuity, or discontinuity where required in an imperfection, of radius, slope and curvature can be maintained at each node. This is a significant improvement over previous elements. Rigid body motion can be a problem with ring elements, and various shape functions have been proposed. '2'3 The Mebane and Stricklin '3 functions were chosen for the new element. In these functions various internal degrees of freedom of the element are included. Since these internal degrees of freedom are zero at each end of the element the corresponding terms in the element stiffness matrix must be eliminated by static condensation.

25O

S. S. J. Moy and S. M. Niku

4.2 Geometric representation using general shell elements Only flat triangle and quadrilateral elements have been used in this work. The derivation of the element stiffness matrix is a standard operation?' Flat quadrilaterals can be used for the perfect shell or the shell with axisymmetric imperfection because the element nodes can be made to lie on the lines of principal curvature of the shell and be co-planar. For the patch imperfection the analysis is restricted to triangular elements.

4.3 Equivalent load representation using axisymmetric elements The applied load (wind and self-weight) is symmetric about the windward meridian (0 = 0°), consequently the membrane stress resultants in the perfect shell are given by Fourier series involving only cosine or sine terms /.4

N~ = ~ N~]cosjO j = 0 1.4

N~o = ~ N~sinjO j=l IA

Ng

=

N¢'cosj0

(17)

j = 0

The equivalent pressure representing the imperfection is given by the series

q. = ~ q'.cosiO+ ~ ?/~sini0 i=0

(18)

i=1

where the constants of each harmonic must be calculated. Considering first the axisymmetric imperfection, from eqn (16) the first approximation to q, is

q. = N~ 32r - o2r ~ NgJcosjO Using standard techniques of Fourier analysis

q'~= 1 yf~ [32r ~

j=oNgJco

jO

)

3Zr

dO = -cr-

(t9)

Finite element techniques for cooling tower shell analysis

251

since f_',, cos jO dO = 0 except where j = O. Similarly

'f'r

q" = - -

¢92r

~

.

~ Ng'cosjO

- ~s2Ng '

i= ltolA

= 0

i>IA

q', = -~

)

cosiOdO

(20)

~s~ j~=° N~,JCOslO cosiOdO

= 0 for all i

(21)

so that q° = ~

;=,,Ng~c°siO

(22)

This series is symmetric about the # = 0 ° meridian and has the same n u m b e r of harmonics as the original series for the applied load and m e m b r a n e stress resultant N , , because of the properties of the integrals with limits - r r and zr. W h e n considering a patch imperfection, the series for q. is m o r e complicated. F r o m eqn (13) 02r 02r N k + 02rNg q. = ~s2, N~. + 2 0 s , Oso "~sZo

(23)

Considering the first term on the right-hand side of eqn (23) a2r

=

0~r ~' N~Jcosj0

for01<0<02

= 0 for other values of 0

(24)

252

S. S. J. Moy and S. M. Niku

T h e expressions for the constants in the series for this first term are 1

~ 102 r la

(q.U')° - 2zr ;_,~

_

1

\

(~s~,j~=,)Ng'cosjO)dO ~OZr la

)dO

--~ . -~s2j~=oN~'cosjO cosiOdO (=~),= I~r

;f2 / 02 r IA

)

, (~s,i~=oN~,tcosjO siniOdO

(25)

r e m e m b e r i n g that r is also a function of 0. The calculation for each constant in eqn (25) involves 1,4 + I separate integrations because in general none of the integrals is equal to zero. There are similar expressions for the constants corresponding to the second and third terms o f eqn (23). It is necessary to include at least 40 harmonics (NH) in the series for q.. The resulting changes in stress resultants are given by

NH

NH

n~ = ~ n~cosjO+ ~ h•sinjO, etc. j=

1=0

(26)

I

which has the same n u m b e r (NH) of harmonics as the q. series. W h e n considering the series for the second-order pressure, a typical expression for the constant in each harmonic would be = -TJ"

_--v Los~

~ j = 0

n~cosjO+ ~ J =

I

h~sinj

cosiOdO

(27)

which involves many more integrations than the first-order calculation (compare the upper limit on the summation). Since the other constants must also be determined, the calculation of the second-order pressure is a m a j o r computation which requires a large amount of computer time. A simplification which halves the computation is to assume a pair of patches symmetrically placed about the 0 = 0° meridian. This makes the (qU,) terms in eqn (25) zero with similar savings in the constants corre-

Finite element techniques for cooling tower shell analysis

253

sponding to the second and third terms of eqn (23). This simplification is only valid when the applied load is symmetric about a diameter of the shell.

4.4 Equivalent load representation using general shell elements Since the membrane stress resultants are calculated directly for each element, it is possible to calculate the equivalent pressure simplymthe difficulty lies in the size of the problem. An isolated patch imperfection destroys all symmetry and a mesh for the complete tower must be used. In this work symmetry was maintained by using a pair of patches. 5 RESULTS The methods described above have been used to analyse a shell with similar geometry to the Ardeer tower. Figure 8 gives the principal dimensions and the location of the imperfection. The amplitude of the 2¢.1m

thickened cornice at top (O00mrn max.)

22.2m

basic shetl 150rnrn thick thickened ring beam at base (580ram max) imperfecfion ZOlle

\; 39.¢m

J

F~. 8. Cooling tower geometry.

I

J

J

/

-200

-!00 N#(kN/m)

I

_ _ j r ~--flat shell

- axIS~P~tm / /-~e

/

I

0

/Z

0 Ne (kN/m)

30

,\

j

d

1

?S

0

He(kNm/m)

1

0

?

....

/'le (kNmimi

f/jl shell

5[') a×ls/rh

,--~/

5O

.

75

d)

75

loo

Fig. 9. Stress resultants in the perfect s h e l l

-30

----

25

5C

50

25

75

75

"~ i axlsymmetnc

~oo =_o ~_

,oo E

b)

Finite element techniques for cooling tower shell analysis

255

imperfection (both axisymmetric and local) was chosen to be 300 mm. The base of the shell was considered to be fixed, but the effects of base boundary conditions are very localised and would be negligible at the level of the imperfection. It is only possible to include a representative sample of the very large quantity of results obtained.

5.1 Analysis of perfect shell Figures 9(a), (b), (c) and (d) show plots of the membrane stress resultants N,~ (meridional) and No (circumferential) and the bending stress resultants Ms and Mo at the 67.5 ° meridian, which is close to where the wind suction is greatest. The stress resultants are due to wind load only, with a free stream wind pressure of 1 kN m -2. The agreement between axisymmetric, fiat triangle and fiat quadrilateral elements is excellent for N~, No and Mo, but is not good for Ms, particularly near the base of the tower. However, Ms has a small magnitude compared to the other stress resultants. TABLE 1

Comparison of Element Efficiency--Perfect Shell Analysis

Element type Axisymmetric Flat triangle Flat quadrilateral

Number of elements 34 1156 578

Totaldegrees Computertime of freedom (ICL 2970 units) 140 3780 3780

220 1600 3200

It is worth comparing the computational effort to obtain these resultants. Table 1 shows number of elements, degrees of freedom and computer time for the three types of element. (Computer time is machine dependent and is only included for comparison.) The economy of the ring elements is obvious. The agreement between the three types of element was such that it was considered worth while to use them all in the imperfection analyses.

5.2 Axisymmetric imperfection An axisvmmetric imperfection, as defined by eqn (4), with an amplitude of 300 mm and from 47-9 to 60"7 m above the base of the shell was

elevafi0n(m) oJ

:z

± cr

m o

_ w

..~

re

~

m

-

-

z

G ',~

I~

elevahon(m)

o~ oJ x

$

I r

I

i

3 ro

l

~

~Z

i

os ca.

i

t~

=r t

v

e[evahon{m)

sre ~

r~

--

o

['l t f r

=

2

T. c

i

i

.g

,

\x.// i

i i

• _

i

,

i t

/ i

i j,

,J

eteva~'q0n(m/

Finite element techniques for cooling tower shell analysis

257

analysed in four ways. In the graphs given in Figs 10-14 the wind load stress resultants in the imperfect shell are plotted. Figures 10(a), (b), (c) and (d) show N,, N0, M, and Me, respectively. The results using axisymmetric elements, either following the geometry of the imperfection or using the equivalent load, are very close to each other. The results for the fiat triangle and flat quadrilateral elements are also close, but they do not coincide with the axisymmetric element results. The general trend is the same with every type of element. The changes in stress resultants caused by the imperfection are local to the

TABLE 2

Comparison of Analyses--AxisymmetricImperfection

Type of analysis Axisymmetric elements, geometric approach Axisymmetric elements, equivalent load Fiat triangle elements, geometric approach Flat triangle elements, first-order equivalent load Flat quadrilateral elements, geometric approach

Number of elements

Total degrees Computer time of freedom (ICL 2970 units)

34

140

220

34

140

320

1156

3780

1600

1156

3780

2300

578

3780

3200

imperfection area and follow the changes in curvature of the imperfection, as predicted by eqn (16). The difference in the magnitude of the stress resultants is due to the differences in the elements themselves. There are chaages in the circumferential membrane stress resultant (Ne) and the meridional bending stress resultant (M,) of up to 10-times the values for the perfect shell. Table 2 compares the efficiency of the various methods. It is obvious that the axisymmetric elements use much less computer time than the general shell elements. The poor efficiency of the quadrilateral elements is because they give a larger bandwidth than the triangles.

S. S. J. Moy and S. M. Niku

258

5.3 Local imperfection In order to compare the various methods for analysing local imperfections a standard local imperfection defined by eqn (8) was adopted. The amplitude was assumed to be 300 mm and the vertical extent was from 47"9 to 60"7 m above the base, as in the axisymmetric imperfection. Circumferentially the imperfection was assumed to be centred over the 72 ° meridian with a total angular extent of 54°° The analysis of a local imperfection requires significant computer time whichever method is adopted. Figures l l(a) and (b) show the changes in the membrane stress resultants N , and No at the 67-5 ° meridian caused by the first- and second-order equivalent loads with axisymmetric elements. The secondorder changes are much smaller than those caused by the first-order load. particularly in No. This trend was also shown when using flat triangle elements and at other meridians. It was decided that for a hyperbolic cooling tower shell it was sufficiently accurate to use only the first-order equivalent load in subsequent analyses. As was discussed in Section 4.3 the use of double, symmetric local

a) E

70

~ ii

7C

g

first-

r

60 L

i

f I

i

second-

second-o~der

. order

50' "I

j

sun,re

:

i

do~b[e

•

50

q ]

f

!

smgte

:.

double

•

I

!

1

i

. first

mJL

-50

./

J 0

n~(ktl!m)

b

50

-100

0

"00

r g ( k i'q/rn 1

Fig. 11. Comparison of first- and second-order changes in membrane stress resultants, shell with local imperfection.

Finite element techniques f o r cooling tower shell analysis

259

imperfections considerably reduces the computational effort, compared to a single imperfection. Figures 1 l(a) and (b) also show changes in N , and Ne when using the first-order equivalent loads for single and double imperfections. There is no practical difference between the results. This is due to the localised effect of the imperfections on the stress resultants, which is further illustrated by Figs 12(a) and (b) which show the variation a~

~mperfectionfrom t,5"to99"

,,,/ No

1

l

j \,

,7

ftat-

axisymmetriz~ ~,, ,," L- tr*angte

(kN/m)

el vahon 20

50

b)

80

110 meridian(0")

imperfectionfrom t,5"fo'99"

]O0

,'' axlsymmefric

""'

,

I I

',

flat ..... trianqte

i eIe~ahonL ~ 490mj 20 ~0

i

J 80

110 meriaian(O')

Fig. 12. M e m b r a n e stress r e s u l t a n t s in shell with local i m p e r f e c t i o n .

of N, around the circumference at two levels within the imperfection. The stress resultant changes fall away very rapidly towards the edges of the imperfection and are very small outside the imperfection. The imperfection was centred over the 72 ° meridian where the compressive stresses from the wind loading on the perfect shell were a maximum. Another point of interest would be the 0~ meridian where the wind produces maximum meridional tension. In both cases use can be made of symmetric imperfections. Other loadings or oth.er shell geometries could preclude the use of symmetry, but for cooling tower shells under dead and wind load the double imperfections can be adopted. Figures 13(a) and (b) show the effect on N, and No of extending the

260

S. S. J. Moy and S, M. Niku

angular extent of the local imperfection. For both N, and Ne the stresses tend towards those for the axisymmetric imperfection as the angular extent of the imperfection increases. However, the meridional stress resultant N , decreases whilst the circumferential stress resultant No increases as the angular extent increases. The magnitude of the changes in stress resultants compared to their total magnitude is far more significant for the circumferential component Ne. Thus a further saving in computer time could be made by considering all imperfections as axisymmetric The ~) _ 70

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0

100

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Fig, 13. Effect on changes in membrane stress resultants of extending the angular extent of the local imperfection.

tests on imperfection sensitivity have been made using smooth curves to define the imperfections. Such curves only introduce curvature discontinuities in the shell. It is unlikely that local imperfections which cause slope or even radius discontinuity in the circumferential direction could be simplified in this manner. Finally the results of local imperfection analyses using axisymmetric elements and the equivalent load were compared with those from analyses using fiat triangle and quadrilateral elements. At an early stage the quadrilateral elements were discarded. It was not possible to follow the geometry of the imperfection with them (see Section 4.2) and the computer time required for the equivalent load approach was prohibitively large. Typical results are shown in Figs 14(a), (b), and (c). The

261

Finite element techniquesfor cooling tower shell analysis a} _ 70

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Fig. 14. Comparison of membrane stress resultants, obtained by various methods, in a shell with local imperfection.

geometric and equivalent load methods gave almost identical results using flat triangle elements but the discrepancies between these and the results using axisymmetric elements are very similar to those found with an axisymmetric imperfection. Table 3 shows the computer time required TABLE 3 Comparison of Analyses--Local Imperfection

Type of analysis Axisymmetric elements, single imperfection, first-order equivalent load Axisymmetric elements, double imperfections, first-order equivalent load Axisymmetric elements, double imperfections, second-order equivalent load Flat triangle elements, geometric approach Flat triangle elements, first-order equivalent load

Number of elements

Totaldegrees Computertime of freedom (ICL 2970 units)

34

140

1200

34

140

700

34

140

120

1156

3780

1600

1156

3780

2300

262

S. S. J. Moy and S. M. Niku

for the various methods. The use of more complicated curved general shell elements could lead to more efficient analysis of local imperfections but it is unlikely that the savings in computer time would be significant. The trends shown are in excellent agreement, but which method is the most accurate? The obvious approach to this was to check convergence by refining the meshes but, as Table 3 shows, this was not possible because of the amount of computer time involved. It would seem prudent to use the largest changes in stress in a practical cooling tower problem. These are predicted by the axisymmetric elements.

6 CONCLUSIONS Geometric imperfections in cooling tower shells have been analysed by the finite element techniques described in Sections 2 to 4. A comparison of the results from these techniques yields the following conclusions: 1. The use of single and double symmetric, local imperfections gives the same results for changes in stress resultants in a cooling tower shell. 2. In the equivalent load method it is sufficiently accurate to use only the first-order equivalent load to determine the changes in stress resultants. 3. When the angular extent of a local imperfection is increased the changes in the stress resultants tend to the changes produced by an axisymmetric imperfection. The axisymmetric imperfection gives an upper bound for the changes in circumferential membrane stress resultant N,, which is most affected by imperfections. 4. Because of points 2 and 3, in many problems with cooling tower shells it would be sufficient to consider only an axisymmetric imperfection. 5. As Tables 1-3 clearly show, the axisymmetric elements require far less computer time than the general shell elements. Thus it is recommended that wherever possible the axisymmetric elements be used. For axisymmetric imperfections the geometric approach is best, but a local imperfection has to be analysed by the equivalent load method.

Finite element techniques for cooling tower shell analysis

263

REFERENCES 1. Central Electricity Generating Board, Report of the committee of inquiry into collapse of cooling towers at Ferrybridge, Monday 7 November 1965, 1966. 2. ICI Ltd., Petrochemicals Division, Report of the committee of inquiry into the collapse of the cooling tower at Ardeer nylon works, Ayrshire, on Thursday 27 September 1973, 1974. 3. Kemp, K. O. and Croll, J. G. A., The role of geometric imperfections in the collapse of a cooling tower, The Structural Engineer, 54 (January) (1976) 33-7. 4. Croll, J. G. A. and Kemp, K. 0., Design implications of geometric imperfections in cooling towers, IAAS World Congress on Space Structures, Montreal, 1976. 5. Croll, J. G. A., Kaleli, F. and Kemp, K. O., Meridionally imperfect cooling towers, Proc. American Society of Cdvil Engineers, 105 (EM5) (1979) 761-77. 6. Croil, J. G. A., Kaleki, F., Kemp, K. O. and Munro, J., A simplified approach to the analysis of geometrically imperfect cooling tower shells, Engineering Structures, I (January) (1979) 92-8. 7. Croll, J. G. A. and Kemp, K. O., Simplifying tolerance limits for meridional imperfections in cooling towers, Journal of the American Concrete Institute, 76 (January) (1979) 139--58. 8. Calladine, C. R., Structural consequences of small imperfections in elastic thin shells of revolution, Int. J. Solids and Structures, $ (1972) 679-97. 9. Zienkiewicz, O. C., The finite element method, London, McGraw-Hill, 1977. I0, Grafton, P. E. and Strome, D. R., Analysis of axisymmetrical shells by direct stiffness method, AIAA Journal, 1 (October) (1963) 2342-7. 1 I. Percy, J. H., Pian, T. H. H., Klein, S. and Navaratna, D. R., Application of matrix displacement method to linear elastic analysis of shells of revolution, AIAA Journal, 3 (November) (1965)2138--45. 12. Jones, R. E. and Strome, D. R., Direct stiffness method analysis of shells of revolution utilizing curved elements, AIAA Journal, 4 (September) (1966) 1519-25. 13. Mebane, P. M. and Stricklin, J. A., Implicit rigid body motion in curved finite elements, AIAA Journal, 9 (February) (1971) 344-5. 14. Niku, S. M., The effects of various boundary conditions and geometric imperfections on stress distributions of hyperbolic cooling tower shells, University of Southampton Report, March 1982.