Engineering Structures,Vol.
ELSEVIER
PII: S0141-0296(97)00048-5
20, Nos 1-2, pp. 75-85, 1998
© 1997 Elsevier Science Ltd
All rights reserved. Printed in Great Britain
0141-0296/98 $ 1 9 . 0 0 + 0.00
Finite element analysis of column
supported hyperbolic cooling
towers using semi-loof shell and
beam elements
Karisiddappa
Department of Civil Engineering, Malnad College of Engineering, Hassan Karnataka ),
India
M. N. Viladkar, P. N. Godbole and Prem K r i s h n a
Department of Civil Engineering, University of Roorkee, Roorkee 247 667, India
(Received October 1995; revised version accepted February 1997)
In most of the early works related to the analysis of hyperbolic
cooling towers, under either dead or the wind loads, only the tower
shell was considered in the analysis and a continuous boundary
condition in the form of fixity of the base of the shell was considered. However, the tower shell is supported by columns in the
form of A-frames. In order to consider realistic boundary conditions, it is essential to consider the supporting columns in the
analysis along with the shell. The present study considers this
problem and an attempt has been made to represent the tower
shell by semi-loof shell elements and the supporting columns by
semi-loof beam elements. The column ends are assumed to be
fixed at their bases. The analysis has been carried out for only the
dead load. The results have been found to compare with those of
Gould 'Finite Element Analysis of Shells of Revolution', Pitman,
London, 1984, and lyer and Appa Rao 'Studies of stress concentration at shell-column junctions of hyperboloid cooling towers',
Comp. Struct., 1990, 34, 191-20. Hoop forces have been found to
have altered significantly in the lower portion of the shell near the
column-shell junction. Moreover, the proposed model gives a better physical representation of a column supported hyperbolic cooling tower. © 1997 Elsevier Science Ltd.
Keywords: hoop force/moment, meridional force/moment, displacements, rotations, semi-loof shell/beam elements
1.
1965, at Ardeer in Scotland in 1973, at Bouchain in Northern France in 1979 and more recently at Fiddler's Ferry in
U.K. in 1984, attracted the attention of many investigators.
With an increase in the height of cooling towers and
reduction in shell curvature, the flexural response to loads
with unsymmetrical distribution of wind pressures became
important for the safety of the structure. The wind forces
acting on a cooling tower are random in nature and will in
practice be unsymmetrical on account of the very nature of
the wind phenomenon and due to influence of other structures in the vicinity of the tower. Thus, a cooling tower
is loaded with unsymmetrical dynamic loads. The simplest
Introduction
The cooling towers built for industrial purposes are
amongst the largest shell structures constructed in the form
of hyperbolic shells of revolution supported by closely
spaced inclined columns. The foundation for the columns
is usually in the form of an annular raft or raft supported
with raker piles depending upon the nature of the foundation soil. The main loading for natural draught cooling
towers is produced by wind except in those cases where
earthquake forces can also be significant. The spectacular
failure of the cooling towers at Ferry Bridge, England in
75
Column supported hyperbolic cooling towers: Karisiddappa et al.
76
approach for analysis is to compute the mean (static) symmetrical wind load, suitably modified by a 'gust factor'.
However, realistic analysis involves a dynamic analysis to
determine the response of the structure under the loading
as it actually occurs. Thus, these gigantic R.C.C. shell structures need suitable numerical modelling of wind loads
based on field measurements or wind tunnel studies and
physical modelling of the tower shell and supporting columns using finite elements.
2.
Earlier work
Cooling towers are generally supported on an annular ring
of closely spaced columns and produce a series of discontinuous, concentrated reactions at the base of the tower. In
earlier works, the presence of these discrete columns was
neglected in the formulation of the boundary conditions.
Davies and Cheung I presented a finite element solution
for the determination of membrane stresses, considering a
segment of the tower in the form of a deep circular ring
beam supported on columns and loaded with a uniform tangential load. Gould and Lee 2 with the aid of the geometrical
approximation of the meridional curve, extended their earlier study to the case of cooling towers supported on columns. This treatment was based on the assumption that the
ring beam is restrained in the circumferential and the horizontal directions. Abu-Sitta 3 attempts to examine, in some
detail, the influence of the practical boundary conditions at
the base of the tower. Compatibility between the column
supported boundary and the shell is compared to an equivalent shell extension having stiffnesses similar to the column supported boundary. The equilibrium equations given
by Novozhilov 4 were used to describe the general state of
stress expressed in terms of displacements and referred to
as the orthogonal curvilinear coordinates. To solve displacement equilibrium equations, a modified finite difference technique was used. Unlike the previous studies
(Gould and Lee2.5), the appropriate stress resultants have
not been forced to vanish at the base region between the
adjacent columns. Also, the effect of the thickened base
was not included.
A synthesis of earlier works 3'6 with the necessary modifications to arrive at an improved model for the base region
of the shell, has been proposed by Sen and Gould 7. The
authors have presented a finite element solution for a column supported hyperboloidal shell taking into account the
thickened base and the discrete column system. The stress
analysis was performed using a stiffness formulation with
the stiffness matrices of the shell and the support system
having been derived separately. The shell was idealised as
an assemblage of elements, each in the form of a frustum
of the meridional curve and connected at the inter-nodal
circles. The stiffness matrix for the column supports was
calculated explicitly. A typical tower was analysed for
quasi static wind and dead loads.
Han and Tu* presented another model suitable for the
analysis of column-supported shells of revolution. The
accuracy of this model was examined by conducting the
self-weight analysis of a column supported cooling tower.
Chauvel and Costaz 9 and Chauvel and Bozetto ~° have discussed the evolution of a new type of support from the
classical supports. The behaviour of the structure has been
analysed using the results of the design calculations and
the field monitoring of settlements during construction. The
ring beam and the shell, strongly reinforced by a lintel at
its base, were separated from the supports by horizontally
bound elastomeric bearings, whose thickness could eventually be adjusted in the event of any differential settlement.
Providing an independent supporting system to the shell,
which is very sensitive to the effect of foundation characteristics, looks satisfactory from the point of view of the operation of the cooling towers, but the idea proves costly for
the construction of such towers.
An axisymmetric shell element has nodal circles and displacements along a nodal circle expressed in terms of a
Fourier series. This means that the generalised coordinates
of an axisymmetric element are the coefficients of the Fourier series, whereas those for a conventional line element
are physical displacement components at nodal points.
Therefore, these two types of elements are not compatible.
To avoid this difficulty, several investigators have proposed
equivalent axisymmetric elements to represent the system
of column supports 3.~.~2. This approach reflects the flexibility of columns only in an average sense. The properties
of the equivalent shell elements are obtained by smearing
those of the individual columns over their tributary areas.
This process, in effect, transforms the columns into an axisymmetric shell.
Another approach used, for the analysis of a column supported cooling tower, is to represent the column supports
with more realistic line elements and the tower shell using
triangular or quadrilateral shell elements, which have nodal
points just like the line elements and which are compatible
with the line elements.
Gran and Yang ~3 employed a doubly curved membrane
quadrilateral shell finite element with 24 degrees-of-freedom to obtain the static response of a fixed base cooling
tower under dead load only. This element has been
developed with the intention of its application to study the
response of the column supported cooling towers. Yang and
Kapania ~4 have used multi-elements for the purpose of
physical idealisation. The model includes: (i) a doubly
curved quadrilateral general shell element (48 degrees-offreedom); (ii) a doubly curved triangular membrane filler
element (21 and 39 degrees-of-freedom); (iii) a doubly
curved general membrane element (42 degrees-offreedom); and (iv) a column element (16 degrees-offreedom). This finite element model (Figure 1) was used
to study the seismic response of column supported cooling towers.
Recently, Iyer and Appa Rao j5 used a doubly curved thin
shell triangular element, originally developed by Thomas
and Gallagher ~5, to study the stress concentration at the
s h e l l - c o l u m n junctions of hyperbolic cooling tower under
dead load only (Figure2). A three-dimensional beam
element with 12 degrees-of-freedom (Przemieniecki~V) was
used to model the column supports.
Karisidappa et al. l~ carried out a simplified analysis of
the tower shell for which an axisymmetric constant curvature meridional element was formulated (Figure 3). Each
element consists of a truncated, doubly curved, axisymmetric element of constant meridional curvature, with two
nodal circles at its ends and a linear variation of wall thickness. The element has 4 degrees-of-freedom (u, v, w and
0) per nodal circle (Figure 3) which were expressed in the
form of Fourier Series expansion. A large number of finite
truncated shell elements, each with a constant meridional
negative curvature, were used to discretise the shell having
varying meridional curvature. About 7 - 9 harmonics were
Column supported hyperbolic cooling towers: Karisiddappa
77
et al.
strained version of the element has the following Configuration (Figu re 4a):
r
~
l
7=
(i) four corner nodes and four mid-side nodes possessing 3 degrees-of-freedom U i, Vi, W i along the global X, Y, Z directions, respectively;
(ii) eight Loof nodes, two on each side, located at the
Gaussian quadrature positions (+1/~/3, -1/~/3) and
having two rotational degrees-of-freedom (0~,:, ~,~),
along and perpendicular to the edge, respectively;
and
(iii) the central node possessing all the 5 degrees-of-freedom, i.e. three displacements and two rotations.
Membrane
element
Filler
element
i
Transition
element
General
element
Column
element
Figure 7 Finite element model of column supported cooling
tower TM
S = S y m m e t r i c edge condition
Thus, the element possesses 17 nodes and 45 degrees-offreedom. Whereas three displacement degrees-of-freedom
at the corner and mid-side nodes are sufficient for defining
the membrane action, the rotations at loof nodes are necessary to impart C ~continuity and account for bending action.
The central node is required to satisfy the quadratic variation of normal deflections in the Lagrangian family.
A relationship between the cartesian coordinates of any
point in the shell element (Figure 4a) and the corresponding curvilinear coordinates is:
8
8
8
i=1
i=1
i=l
x=EN'x',y=EN'y',z=EN'z'
(1)
where N ~ are the shape functions at various nodes which
are defined below.
I .
.
.
.
.
Shell
3.1.1. Shape functions for interpolation of displacements: The shape functions for defining the variation of
I
I
I
I
I
I
I
I
displacements at the corner and the mid-side nodes
(Figure 4a) are given by:
1
_1
N' = ~ (-1+~+~'0+~12-~'02-~'0)
1
N~= ~ (1-n-~+~n)
1
N 3 = ~ (-1+~-¢~+'0:+¢'0:-~'0)
[
Columns
1
N 4 = 2 ( l + ~ - , - n 2 - ¢ n 2)
I
1
Figure2
Finite element mesh for studies on stress concentration15: a axisymmetric shell element; b longitudinal section
through element
N~ = ~ (-l+~+¢'o+n2+¢n2+~ "0)
1
found to be necessary for the convergence of forces and
moments.
1
g ~= ~ (-l+~-~m-n~-~n:+~n)
3.
N~ = ~ (1-~-n~+~n :)
Proposed finite element modelling
1
(2a)
3.1. Shell portion (semi-loof shell element)
Semi-loof shell elemenP 9 is perhaps the most efficient
element for the solution of shells having arbitrary geometry
and it accounts for both membrane and bending actions. It
is an isoparametric non-confirming element. The uncon-
The shape function for defining the variation of displacements at the central node is given (Figure 4a) by:
= ( 1 - ~ ) ( 1-n ~)
(2b)
Column supported hyperbolic cooling towers: Karisiddappa et al.
78
W
Nodal circle 1
x,,\O 1
f
-~
..4~ W
\\
",
!
r
\\
x
I
,F,\ ,,
-
',1,,
kk
Nodal circle 2
% I
""
"x
Figure3 Finite element geometry: a initial nodal configuration; b final nodal configuration
3.1.2. Shape functions
for
interpolation
of
rotations." The shape functions for the interpolation of
rotations at the loof nodes are given (Figure 4a) by:
1 [ /'~
3
9 2
L' = 8[~ ~-3n - ~ + ~,,~3~n+ ~n - 3 ~,,:~n:
3~,~
-~- 3~2'1'~ --
r---
]
(3a)
3 ~ ; ~ ( r / - - \,3
~
3
' ;;(,-~-,n
)]
~3 2
L ~ = §I f 3 ~ \ , ' ,~
3 n + ~9 ~ + \,,'3~n
- ~n
- 3~n 2
~_
3 ~,,~
+ 3\,'3~r/+
( 3
3
l
if
L
l
5 (~-h3-~nbJ
9 2
3
L" = ~ \,,3~+3n _ ~s~- \,"3s%+ 4n
(3b)
Kirchoff's shear constraints are applied by stipulating the
transverse shear at Loof nodes to be zero so as to eliminate
rotations parallel to the sides at the Loof nodes. The variables at the central node are also eliminated and the constrained version has only 32 degrees-of-freedom per
element including 24 translational degrees-of-freedom at
the comer and the mid-side nodes and 8 rotations at the
Loof nodes (Figure 4b ).
3.2. Supporting Columns (Semi-Loo[beam element)
~
3
,9 2
L 5 = 8 -\/3~q-3-q - 4 ~ + \/3~rt + 4r~ + 3\,3~r/-
3 {g
The shape functions for the interpolation of rotations at
the central node is given (Figure 4a):
3
L 9 = 1 - 4 ( ~ + ,02)
]
1[
/~
9
,~3 2
813~-+,v_ r t + ~ ( - \ , 3 ~ r / - ~'O - 3~/2
1[,,~
I_
3 \..';,3
+ 38n + 2(#~n-~n~)]
- 3~33 -
1[
~_
9
6
32
U = 8 - 3 ~ ' v ! 3 r / + 4 ~ + v'3~rt - ~7 + 3~n 2
]
3\/3
-
. 1
2~ (~3,}~__~,}~3)
1[ ,~
3
9 2
,,L 2 = 8[_V3~._3r / _ ~(2 _ \./"5~r/+ ~7 + 3,,3~r/-
g 3 =
3~,5
+ 3\/,~e'O - 2-(sc3r/-~c~/~)J
"_ 3V3~n-
The shell-beam combinations are the most important and
difficult problems in practice. The tanks and towers are provided with curved ring beams. Stiffners are provided in thin
shells to avoid buckling. Cylindrical shells are supported
with edge beams. To investigate these types of problems
an efficient beam element is often needed and the SemiLoof beam element should provide the means. A Semi-Loof
beam element compatible to a Semi-Loof shell element has
been formulated t9 22
3.3. Element geometry
- 3e,
3\/3
+ ;
l
1[
~
9
3,
L 7 = 8 [-3sr'-\/3rt + 4 ~ - \,,'Sscv/- 4rt - + 3sc~l2
As mentioned earlier, the first step is the selection of the
free nodal parameters. The nodal configuration of an
isoparametric Semi-Loof beam element is shown in
Figure 5a. The element possesses two types of nodes,
namely:
Column supported hyperbolic cooling towers: Karisiddappa et al.
79
Conventional nodes
~
Vi ID Ui
z
X
Y
Rj
= x
Global axes
X, Y, Z - local axes
(a) Initial nodal configuration
(b) Final nodal configuration
Figure 4 Semi-loof shell element
(i) End and mid-side nodes at which three displacements (u i, vi, w i) are taken as nodal parameters along
the global axes x, y, z (nodes 1 - 3 in Figure 5a). In
addition, 0~x, gy, ~ are taken as nodal parameters
along the global axes only at nodes 1 and 3.
(ii) L o o f nodes located at ---1/~/3 with rotations (0~x, 0~y,
0~z) along the local axes (X, Y, Z) as nodal parameters
(nodes 4 and 5 in Figure5a). These initial 21
degrees-of-freedom of the element are arranged in
two vectors as given below:
{~e} = {ulvIwI
~ U2V2W 2 ~ U3V3w3}T
{0e} = {0101,0/, ~X~y~z ~X~y~Z0~x0~y0~z
}T
(5)
(6a)
=
oP =
ONi . yi
~ON' zi
i=1
OZ "
This vector can be normalised into unit vector, .,~ as:
.~ = { x } =
(8a)
-Ixl
where
Ixl = L\@/ + \ ~ /
+ \@/J
(8b)
For the generation of a vector Z, an attitude point,
defined as shown in Figure 5c.
P(xa, Ya, za) is
Z--(P-A).8=
l xl [::1
-y.
• Xv
(9)
-- Za
The unit vector 2 is obtained as:
(lO)
The unit vector, l~ is obtained as the vector product of )¢
and 2 i.e.:
(6b)
To define strains and rotations, it is necessary to create
unit vectors X, 1) and 2 at any point, P(x,y,z) on the
element axis. A vector X tangent to the axis is given by:
(7)
i=1 O-;
2 = z/Izl
where N i is obtained on basis of the following polynomial
(1, ~, ¢ )
x
i=I
[axlO~
]ay/o~
(4)
The nodes and the variables are defined in such a way
as to match the configuration of the Semi-Loof shell
element and that each beam element conforms to the neighbouring beam elements in both the slope and the deflection.
Of the initial 21 degrees-of-freedom, the local rotations Ox
and Oz along the axes ~" and 2 at nodes 4 and 5 are constrained and, therefore, the final degrees-of-freedom left out
are 17 (Figure 5b).
The coordinates at any point, P(x, y, z) on the element
axis are interpolated using the coordinates of the end and
midside nodes 1, 2 and 3 and shape functions N i as
given below:
3
3
3
x = E N'zi; Y = E Uiyi ; z = E N'zi
i=1
i=l
i=1
'
0x.X ~
?:{r}:
(:)
rv =2x,~
(11)
Column supported hyperbolic cooling towers: Karisiddappa
80
Z
~k
L°ad
y
/
axes \
///"f
V-'/
4
w,
04
y
5
~V2
~
X
0
4
"
v2 ~
-
5
Oz
~
Oy
~ ~,w3
~ ,vz 3
~
~
v
/ . ~
(J.jI , ~ 0 ;
e t al.
3
Oy
u~
~=-. !m"
~=+1
ul
~=-1
/
Y
(a) Initial nodal configuration
D,
X
Global axes
w2
w3
v2
03
0x5
2
o,
I~1
ul
" " 1= u3
0x
x
Ox
(b) Final nodal configuration
^
A
Z
(c) Unit
Figure5
vectors
Semi-loof beam element: a initial nodal configuration; b final nodal configuration; c unit vectors
3.4. Displacements and rotations
1
The displacements along the global axes of a generic
point P(x, y, z}
{d} = {u, v, w}T
(12)
are interpolated using the corresponding displacements of
nodes ( 1 - 3 ) and shape functions, N' defined by equation
(6b). These shape functions are (refer to Figure5a)
given by:
N' = ~ ( ~ - ~c)
N~=(1-~)
I
N 3 = ~ ( ~ + ~1
(13)
Therefore
{d} : {N} {~e}
(14a)
Column supported hyperbolic cooling towers: Karisiddappa et al.
where
OX-
[N] =
o
o
o
o
NI
0
i
0
N:
0
$
0
N~
0
NI
i
0
0
N ~"
~
0
0
81
T OL
T OL
O~
T OL
T OL
O~
T OL
T OL
O~
o
(14b)
3
The displacements (U, V, W) along the local axes
{P, X, K Z} are given by:
U = { X } - {d} = {X}T{d} = { X } T [ N ] { ~ }
(15a)
V = { Y } . { d } = {Y}T{d} : { Y } T [ N ] {be}
(15b)
W = {Z}. {d} = {z}T{d} = {z}T[N] {be}
(15c)
OX
The relevant terms needed for defining the element matrices are assembled in a single vector {g} in equation
(21) as:
{g} =
The derivatives of these displacements with respect to the
local coordinate X, are now obtained easily as:
OOr OOz OV
OU Ox, Or, Oz,
u, v, w, 0X'
oX' oX' OX
OW
- Oz, ~
OU
T ON
T ON
T ON
T ON
O~
Of;
}T
+ Ov
(21)
(16a)
These quantities can be expressed in terms of nodal parameters {p} as:
e
{g} = [S] {p}
T ON
T ON
O~
where [S] is a matrix containing the shape functions and
derivatives as given in equations (14a), (16a, b, c), (19a,
b, c) and in equations (20a, b, c).
The two local rotations, 0y and 0: at Loof nodes were
eliminated by imposing the shear constraints:
The rotations at a generic point, P(x, y, z},
{0} = {0~, 0~., 0z}T
(22)
(17)
3'~ = 0 and %.~ = 0 at nodes 4 and 5.
are obtained using the global rotations at nodes (1,4,5,3)
and shape functions U (j" = 1,4,5,3), i.e.
{0} = [ L ] { ~ }
(18a)
The shape function U are defined using the following polynomial basis:
{1, ~ ~, ~}
(18b)
These shape functions are (refer to Figure 5a) presented in
equation (18c). The matrix [L] is defined in a same way
as equation (14b).
1
L':4(-
This element constraining follows the same lines as that for
Semi-Loof shell element.
1 + sc + 3 ~ - 3 ~
3)
4. Analysis of column supported tower under
dead load
4.1. Problem details
The tower considered for the analysis is a column supported
hyperbolic cooling tower under dead loads, earlier analysed
by Gould 23. The behaviour obtained on the basis of the
proposed model has therefore beeen compared with the
results obtained by Gould 23 and Iyer and Appa Rao ~5. The
geometry of the structure is shown in Figure 6 and can be
described by a general second degree equation for its meridian as follows:
Az: + 2 Brz + Cr 2 + 2Dz + 2Er + F = 0
1
L 3 = ~ (-1 - ~:4- 3 ~ - 3~ 3)
3
i-
,r~
z," = ~ ( l - v3 ~ - #~ + \/3 ~3)
L5 = 4 (1 + ,.3 ~ - Zy - \/3 ~3)
(18c)
Two different equations have been used: (i) to describe
the meridian above the throat; and (ii) to describe the meridian below the throat, with continuity of slope and curvature at the throat level. The voefficients of these equations
are presented in Table 1.
The shell is supported by 36 pairs of columns. The col-
Table 1 Coefficients of equation (23F3
The rotations in local axes (0~, Ov, 0~) are obtained as:
Ox= { x } T { 0 } = { X } T [ L ] { 0e}
(19a)
0v = { y}T{ 0} = { Y}T[L]{ 0 ~}
(19b)
0z = { z F { 0} = { Z F [ L ] { 0~}
(19c)
The derivatives of these rotations with respect to the
local coordinate, X will be
(23)
Coeff.
Above throat level
Below throat level
A
B
C
D
E
F
-0.10667
0.2755 x 10 s
1.0
0.3108 x 10-3
-0.002619
-13224.397
0.16734
0.3087 x 10-5
1.0
-0.2687 x 10 3
-0.7118 x 10-3
-13224.83
Column supported hyperbolic cooling towers: Karisiddappa et al.
82
I
I
250'
4
Top
5'T
15'
30'
1.33
1.25
1.00
0.75
50'
- 0.75
50'
Throat
• 0.75
230'
50'
" 0.75
50'
- 0.80
50'
"0.85
50'
• 0.90
50'
• 0.95
1.00
2.50
- 4.00
Bottom
400'
q
36 column pairs
at l0 ° spacing
Figure 6 Cooling t o w e r g e o m e t r y 23
umns are circular in cross-section with a radius of 2.0 ft.
The shell is of variable thickness (Figure 6). The geometrical details of the cooling tower are presented in Table 2a.
The material properties of the shell and the columns used
in the analysis are given in Table 2b.
Table2a
Geometrical details of hyperbolic cooling t o w e r 23
SI Parameter Description
No.
Symbol
Parameteric
value
1. Height above throat level
2. Height below throat level
3. Radius at t o p
4. Radius at bottom
5. Radius at throat level
6. Circular spacing of column
pairs (A-frame)
7. Diameter of columns
8 No. of column pairs
9. Height of A-frames
10. Shell thickness at bottom
11. Shell thickness at t o p
12. Shell thickness at throat level
Zt
Zb
r,
rb
a
150 ft
400 ft
125 ft
200 ft
115 ft
10 °
tb
tt
t
4 ft
36
30 ft
4 ft
1.33 ft
0.75 ft
4.2.
Finite element discretisation
The finite e l e m e n t mesh used in the m o d e l l i n g is shown in
Figure 7. The s e m i l o o f shell elements have been used to
m o d e l the t o w e r shell and s e m i l o o f b e a m e l e m e n t s to m o d e l
the columns. D u e to the a x i s y m m e t r i c nature of loading as
well as the g e o m e t r y , only a small strip of the t o w e r crosssection with symmetrical boundary conditions at edges, has
been analysed using the finite e l e m e n t idealisation shown
in Figure 7a.
In the second part o f the study of this problem, a small
portion o f the thick shell at the bottom o f the c o o l i n g t o w e r
b e t w e e n the t o w e r shell and the c o l u m n has been replaced
by a ring b e a m m o d e l l e d with s e m i l o o f b e a m e l e m e n t
Table2b Material properties used in analysis of column supported cooling t o w e r
Property
Shell
Column
Young's modulus, E(Kips/ft ~)
Poisson's ration, /~
Weight density, p (Kips/ft 3)
0.576 x 106 0.72 × 10e
0.15
0.15
0.15
0.15
Column supported hyperbolic cooling towers: Karisiddappa e t al.
83
9
Symmetric
edge conditions
14
~
Symmetric
edge conditions
9
Z
.,3
97
(c) Fixed at the
shell bottom
Global axes
(a) Column supported
(b) Column supported
with ring beam
Figure 7 Finite e l e m e n t idealisation (for dead load analysis): a c o l u m n supported; b c o l u m n supported with ring beam; c fixed at
the shell b o t t o m
(Figure 7b). The fixity condition has been assumed at the
column bases. The analysis has also been carried out by
assuming fixed end conditions at the level of shell bottom
(Figure 7c) and increasing the thickness of the shell at the
bottom from 4 to 4.758 ft, to include the flexibility of the
supporting columns, instead of considering the columns as
such in the analysis. This idea of replacement of columns
by an equivalent uniform shell thickness is based on the
equality of extra area of cross section of the shell to the
total area of cross section of the diagonal columns. A comparative study of the behaviour of hyperbolic cooling tower
for the three cases has been made and the results discussed
in the following section.
clear from Figure 8 that the variation of membrane stress
resultants all along the height, at cb = 0 ° meridian, is in
good agreement with the solutions given by Gould 23 and
Iyer and A p p a Rao ~5, with the exception of some difference
in values near the base of the tower.
Table 3 shows the comparison of column forces and
moments with the earlier solutions presented by Gould 23
and Iyer and A p p a Rao ~5. The results show a good match
Table3 Column forces (Kips) and m o m e n t s (Kips/ft) due to
dead load
lyer and Appa
Rao TM
Gould 23
Present study
4.3. Discussion and comparison of results
The results obtained from this case study have been
presented in Figure 8 and Tables 3 and 4. Figure 8a shows
the variation of meridional forces, Ns along the height of
the tower plotted for 0 ° meridian. For the purpose of comparison, the profiles obtained by Iyer and A p p a Rao ~5 and
Gould 23 have also been plotted in the same figure.
Figure 8b shows a similar plot for hoop forces, N+. It is
Nxx -1245.59
Vxv
6.58
Vxz
8.70
Mxy
146.51 (top)
Mxz
Mxx
-1245.71
-1242.93
0.00
5.19
5.14
8.34
76.66 (top)
113.96 (top)
163.80 ( b o t t o m )
107.88 ( b o t t o m )
147.64 (bottom)
117.31
115.71
113.99
8.70
5.14
7.11
Column supported hyperbolic cooling towers: Karisiddappa et al.
84
Table 4
Variation of m e m b r a n e forces along the height at 4)° meridian measur ed f r o m t h r o a t level
H o o p forces
N+ (Kips/ft)
M e r i d i o n a l forces Ns (Kips/R)
Column supported
Column s u p p o r t e d
Height (ft)
Fixed
base
Equivalent
shell
with
ring
beam
without
ring
beam
Fixed
base
Equivalent
shell
with
ring
beam
without
ring
beam
148.94
141.83
123.66
78.87
-10.57
-60.57
-110.57
-160.57
-210.57
-260.57
310.57
-334.23
-365.77
1.93
-1.39
0.96
0.07
3.15
4.72
5.96
7.16
8.17
9.17
12.6
11.2
9.04
-1.93
-1.39
-0.96
0.07
3.15
4.72
5.09
7.16
8.17
9.17
12.7
10.57
8.81
-2.22
-1.39
-0.96
0.07
3.15
4.72
5.98
7.16
8.17
9.17
12.37
16.41
17.76
-2.22
-1.39
-0.96
0.07
3.15
4.72
5.96
7.16
8.17
9.17
12.34
16.60
17.13
0.75
1.37
3.45
8.69
19.12
24.43
29.19
33.78
38.16
42.48
46.90
57.93
62.56
0.75
1.37
3.45
8.69
19.12
24.43
28.92
33.78
38.16
42.49
47.07
57.64
62.75
0.20
1.37
3.45
8.69
19.12
24.43
29.19
33.78
38.16
42.49
47.07
57.56
62.51
0.20
1.37
3.45
8.69
19.12
24.43
29.19
33.78
38.16
42.48
46.90
57.44
62.71
+ tension, - compression.
200-
Top
Throat
""
~ l x
~
' ¢ ~
.~ _200i
-~
-4001
0
o Nagesh-Appa Rao (1990)
o Gould (1984)
× Present study
~ u m n
supported)
I
-20
I
"Pk"~ I Bottom
-40
-60 -80
Ns (KIPS/ft)
(a) Meridional force (Ns) along height at 0 = 0 ° meridian
_-- 2°°i__i ................
0
.~
~ .............
,.
~Okr
)
_T_h_~o_~
....
t~ Nagesh-Appa Rao (1990)
o Gould (1984)
× Present study
(column supported)
-200 --
-400
10
5_'_P.....
I
Bottom
-30 -40 -50
N¢ (KIPS/ft)
(b) Hoop force (N~) along height at 0 = 0 ° meridian
Figure8
0
~
-10
-20
M e m b r a n e stress resultants at 0 = 0 ° meridian: a meridional force, Ns along height at 0 - 0 ° meridian; b h o o p force
No along height at 0 =0 ° meridian
except for some values of column moments. However, the
difference in the moments is within the acceptable limits.
The reason for the difference in the moments may be
attributed to the approximation of the compatibility of
rotations used in the formulation at the s h e l l - c o l u m n junctions.
Based on different finite element models (Figure 7a-c)
values of the membrane stress resultants have been
presented along the height of tower in Table 4 for all the
three models at 4)= 0 ° meridian. Surprisingly, it has been
found that both the hoop and the meridional forces do not
vary significantly for the solutions obtained with and without the ring beam in between the shell and the column junctions, but the stress resultants, particularly the hoop tension
at the bottom, differ considerably when a continuous
boundary condition has been imposed, though the effect has
been found to be local. The difference in hoop stresses has
been felt only in a height of 60 ft from the top of columns,
which is about 1/9th of the total height of the tower. This,
therefore, emphasizes the need for a practical boundary
condition which will be in the form of ' A ' frames represented by the column elements with column bases fixed.
The analysis with equivalent shell for diagonal columns
gives the results which are identical to the fixed based cooling tower, except with a slight decrease in the hoop force
and an increase in the meridional force at the bottom of
the tower. However, the hoop force obtained at the base is
less than half the value obtained for the column supported
tower. This again substantiates the need for the consideration of a practical boundary in the analysis.
The idea of introducing a ring beam at the bottom of the
tower has been dropped for the larger size problems, as
there is not much difference in stress resultants obtained
and a saving of a few elements leads to saving a lot of
computational effort. The magnitudes of the hoop and the
meridional moments have been observed to be quite small
and, hence, have not been presented.
5.
Concluding remarks
The semiloof shell and the semiloof beam elements have
been used for analysing the complex structures. The Semi-
Column supported hyperbolic cooling towers: Karisiddappa et al.
Loof shell element has given very encouraging results in
situations where both membrane and bending actions dominate the structural response. With this proposed physical
modelling and with relatively coarser finite element mesh,
the results with engineering accuracy have been obtained.
The behaviour of hyperbolic cooling tower shell has been
studied for three different cases, namely: (i) fixity at the
base of the tower shell; (ii) replacing the column support
by equivalent shell thickness at the bottom of the tower
shell; and (iii) column supports as such. The behaviour
observed is summarised below:
4
5
6
7
8
9
10
( 1) The membrane forces away from the bottom portion of
the tower have not at all been affected by the different
boundary conditions considered.
(2) There is a considerable difference in the membrane
forces observed in the lower 1/9th portion of the tower
shell above the top of columns for the tower under
dead load. Particularly, for the variation of hoop forces,
the difference is quite significant.
(3) Replacement of flexible diagonal columns by equivalent uniform shell thickness at the bottom of tower may
be thought of only for the preliminary analysis. However, it does not represent the actual behaviour of the
structure.
(4) Results obtained from the present study with practical
boundary conditions match very well with those of
Gould 23 and Iyer and Appa Rao ~5.
(5) The proposed finite element model consisting of SemiLoof shell and beam elements gives a better physical
representation of the column supported tower shell.
11
12
13
14
15
16
17
18
19
20
References
I
2
3
Davies, J. D. and Cheung, Y. K. 'The analysis of cooling tower ring
beams', Proc. Inst. Civil Engrs, 1968, 3 9 : 5 6 7 - 5 7 9
Gould, P. L. and Lee, S. L. 'Hyperboloids of revolution supported
on columns', ASCE, J. Engng, Mech. Div., 1969, 95, 1083-1100
Abu-Sitta, S. H. 'Cooling towers supported on columns', ASCE J.
Struct. Div., 1970, 96, 2575-2588
21
22
23
85
Novozhilov, V. V. 'Thin Shell Theory', Translated by P. G. Lowe,
Noordhoff, The Netherlands, 1959
Gould, P. L. and Lee, S. L. 'Bending of hyperbolic cooling towers',
ASCE, J. Struct. Div., 1967, 93, 125-146
Gould, P. L. and Lee, S. L. 'Column-supported hyperboloids under
wind loading', Int. Assoc. for Bridge Struct. Engng, Zurich, Swizterland, 1971, 47-64
Sen, S. K. and Gould, P. L. 'Hyperboloidal shells on discrete supports', ASCE J. Struct. Div., 99, 595-603
Han, K. J. and Tu, W. W. 'A finite element model for column supported shells of revolution', Int. J. Numer. Meth. Engng, 1987, 24,
1951-1971
Chauvel, D. and Costaz, J. L. 'Studies of column supported towers',
Engng Struct., 1991, 13, 310-316
Chauvel, D. and Bozetto, P. 'Studies of column supported towers',
National Seminar on Cooling Towers, New Delhi, India, 1990, pp.
TS IV/17-TS IV/24
Basu, P. K. and Gould, P. L. 'Finite element discretization of open
type axi-symmetric elements', Int. J Numer. Meth. Engng, 1979, 14,
159-178
Chan, A. S. L. and Wolf, J. P. 'Cooling tower, supporting columns
and reinforcing rings in small and large displacement analyses', Cornput. Meth. Appl. Mech. Engng, 1978, 13, 1-26
Gran, C. S. and Yang, T. Y. 'Doubly curved membrane shell finite
element', ASCE, J. Engng. Mech. Div,, 1979, 105, 567-584
Yang, T. Y. and Kapania, R. K. 'Shell elements for cooling tower
analysis', ASCE J. Engng Mech. Div., 1983, 109, 347-349
Iyer, N. R. and Appa Rao, T. V. S. R. 'Studies on stress concentration
at shell-column junctions of hyperboloid cooling towers', Comput.
Struct., 1990, 34, 191-202
Thomas, G. R. and Gallagher, R. K. 'A triangular thin shell finite
element: linear analysis', NASA CR-2482, 1975
Przemieniecki, J. S. 'Theory of Matrix Structural Analysis', McGrawHill, New York, 1968
Karisiddappa, Prem Krishna, Godbole, P. N. and Viladkar, M. N.
'Analysis of hyperbolic cooling tower for wind loads', Int. J. Wind
Engng, Japan Assoc. for Wind Engng, 1992, (52), 126-135
Irons, B. M. and Ahmad, S. "Techniques of Finite Elements', Ellis
Horwood, Chichester, England, 1986
Martins, R. A. F. and Oliveira, C. A. M. 'Semi-loof shell, plate and
beam elements--new computer versions, Part I, element formulation', Engng Comput. 1988a, 5, 15-25
Martins, R. A. F. and Oliveira, C. A. M. 'Semi-loof shell, plate and
beam elements--new computer versions, Part II, element programming', Engng Comput. 1988b, 5, 26-38
Albuquerque, F. A. 'A beam element tor use with the semi-loof shell
element', M.Sc. thesis, University of Wales, Swansea, U.K., 1973
Gould, P. L. 'Finite Element Analysis of Shells of Revolution', Pitman, London, 1984