JOURNAL OF APPLIED ENGINEERING SCIENCES
VOL. 1(14), issue 4_2011
ISSN 2247- 3769 ISSN-L 2247- 3769 (Print) / e-ISSN:2284-7197
PRACTICALLY STRUCTURAL ANALYSIS OF LARGE COOLING
TOWERS
KOPENETZ Ludovic Gheorghe, CĂTĂRIG Alexandru*,
Technical University of Cluj-Napoca, *e-mail: [email protected] (corresponding adress)
ABSTRACT
For the structural analysis of large cooling lowers e x i s t different computation techniques developed to obtain the shell stresses
with sufficient accuracy. Natural draft cooling towers are mostly designed as thin shells supported along the circumference by a
system of columns. Hyperbolic, natural-draught cooling towers to be erected in seismically active zones are routinely analysed for
earthquake excitation. Thus the earthquake safety of natural-draught cooling towers grows more important. The response of
cooling towers assuming rigid-base translation excitation, can be approximated by beam elements with shear distorsion and rotary
inertia.
Keywords: cooling towers, practical analysis, high rise structures
Received: September 2011
Accepted: September 2011
Revised: October 2011
Available online: November 2011
INTRODUCTION
The cooling towers are rather peculiar structures being usually detached high-rise structures
with the height determined by the recirculated water flow and the difference of the temperature thet
has to be extracted from the water. For currently used power plant capacities (<500 Mw) the
cooling tower shells is about 100 m height with a corresponding basis diameter of 80 m and the
wall thickness of about 16 cm. For large geometrical dimensions the wall thickness chosen for
structural reasons required buckling safety as well as a sufficientlly high natural frequency.
MATRIALS AND METHODS
Usually, the cooling towers are structures made up of rotational shells supported by inclinad
columns forming a skeletal structure shaped as A, V or X letters ar, recently, by vertical columns
placed at large distances (8 – 18 m) between them (Fig.1, a).
The cooling procedure (wet or dry cooling) to determine the geometrical dimensions. The
size of the cooling tower is a direct function of the quantity of heat (Fig. l, b).
b
a
Fig.1. The cooling tower:
a. distances between vertical columns; b. size of cooling tower
PRACTICALLY STRUCTURAL ANALYSIS OF LARGE COOLING TOWERS
KOPENETZ L. et all., pp. 39-44
Figures 2 and 3 to depict wind-buckling of cooling tower-model without and with upper edge
beam.
Fig.2. Wind-buckling of
cooling tower-model without
upper edge beam
Fig.3. Wind-buckling of
cooling tower-model with upper edge beam
The loading acting on a cooling tower is due to its own weight. wind. earthquake and
temperature variation. The literature provides a large amount of both, practicai techniques and
refined methods for their analysis under axi – symetric loading. The same literature is still pour in
methods for analysis of cooling towers under asymetrical loading and efficient methods for their dynamic
analysis. After several faimous accidents, as that from FERRYBRIDGE (1965), when three cooling
towers of 113 m height collapsed during a storm and that from IMPERIAL CHEMICAL
INDUSTRIES Ltd (1973) when the cooling tower of ARDEER NYLON. Plant fall down, the
attention of the investigations has been drawn again to this type of structures and their dynamic
behaviour. The dynamic analysis of cooling towers is a difficult task even if the modelling is as
simple as using fing shaped finite elements. The paper presents an afficient free vibration analysis
procedure using bar type finite elements with inner nodes.
1. Structural discretization
The shell of the cooling tower is discretized in izoparametric finite elements of bar type [1],
with 3 or 4 inner nodes (Fig. 4).
Fig.4. Discretization of cooling tower shell
The support of the shell is modelled as a flexible connection emphasizing flexibility along the
three directions (Fig.5). The analytical model of the discretized structure is based on Timoshenko
type beams (Fig.6), i. e. taking into account the shear deformations. Following the usual FEM
JOURNAL OF APPLIED ENGINEERING SCIENCES
VOL. 1(14), issue 4_2011
ISSN 2247- 3769 ISSN-L 2247- 3769 (Print) / e-ISSN:2284-7197
technique [2], [3], [4], [5], a dynamic condensation procedure eliminates the displacements of the
inner nodes; 3 or 3 and 4 (Fig.4).
Fig.5. Cooling tower structural model
Fig.6. Cooling tower Timoshenko beam model
The interpolation functions for bar type finite elements [6], [7], [8], with three inner nodes are
of the form:
1
Φ 1 (r ) = r (1 − r )
2
1
Φ 2 (r ) = r (1 + r )
(1)
2
Φ 3 (r ) = 1 − r 2
while in the case of finite elements with four inner nodes the interpolation functions reads:
1
1
Φ 1 (r ) = r (1 − r ) + (− 9 r 3 + r 2 + 9 r − 1)
2
16
1
1
Φ 2 (r ) = r (1 + r ) + (9 r 3 + r 2 − 9 r − 1)
2
16
(2)
1
3
2
3
2
Φ (r ) = 1 − r + (27 r + 7 r − 27 r − 7 )
16
1
Φ 4 (r ) =
(− 27 r 3 − 9r 2 + 27 r + 9 )
16
The potenŃial energy π due to both, bending and shear deformations is given by:
PRACTICALLY STRUCTURAL ANALYSIS OF LARGE COOLING TOWERS
KOPENETZ L. et all., pp. 39-44
2
2
1
1
 dβ 
 dy

π = ∫ Ei I i   dx + ∫ k i Gi Ai  − β  dx
20
d
x
2
d
x




0
H
H
(3)
where:
- I i is the moment of inerŃia of the cross section,
- Ai is the cross section area,
- Ei is the longitudinalYoung modulus,
- Gi is the transversal Young modulus,
- k i is the shape coefficient,
- β is the rotational component due to bending only.
The equilibrium condition δπ = 0 in the form of stationarity of lunction π with respecting x
becomes
2
 dβ   dβ 
 dy
  dy

∫0 Ei I i  dx  δ  dx dx + ∫0 k i Gi Ai  dx − β δ  dx − β dx = 0
H
H
(4)
Using the interpolation functions (1) for a three inner node finite element its displacements y
and β may be expresed in terms of y i and θ i as:
 y1 
θ 
 1
1
2
3
 y  Φ
0 Φ
0 Φ
0   y 2 
 =
 
1
2
0 Φ
0 Φ 3  θ 2 
β   0 Φ
 y3 
 
θ 3 
while the corresponding deformations device from the kinematics in the form:
i
3
 dy   ∂Φ y 
∑

 dx   1 ∂x i 
 dβ  =  3

i
  ∑ ∂Φ θ i 
 dx   1 ∂x 
where:
∂Φ i  ∂r   ∂Φ i 
=  

∂x
 ∂x   ∂r 
and:
 ∂r 
−1
 ∂x  = J
 
is the invers of the Jacobian matrix J of r in terms of x.
(5)
(6)
(7)
Using (4) in a standard procedure the element stiffness matrix may be obtained. Having the
fundamental matrices at the element level an analysis technique at the structural level can be
performed. In the following, a nonlinear analysis procedure presented in [9] is applied. The
JOURNAL OF APPLIED ENGINEERING SCIENCES
VOL. 1(14), issue 4_2011
ISSN 2247- 3769 ISSN-L 2247- 3769 (Print) / e-ISSN:2284-7197
nonlinear equilibrium equations are solved via Newton-Raphson iterative technique which enjoy
independence from the finite element type. The dynamic equilibrium equations are numerically
integrated via either Newmark or Wilson methods. A computer program SUM01B has been
developed to carry out the above mentioned computations. The program SUM01B using a unique
vector and delivering the requested data through common blocks. The amount of memory required
depends on the vector’s dimension [9].
2. Numerical example
The structure presented in figure 7 has been analized in [10].
Fig.7. Column supported cooling tower [10]
R ( z ) = 35 . 5092 1 + ((106 .9421 − x ) / 88 . 8925 )
2
E = 2 . 758 ⋅ 10 7 kN / m 2
h = 0 . 3048 m
∂ = 0 . 167
ρ = 2 . 4 tf / m 3
44 INCLINED
WITH
(19 )
o
COLUMNS
1 .3208 ⋅ 0 . 6906 m 2 CROSS
SECTION
Bar type finite elements with three nodes have beenused and the curvature has been dealt
with by taking there are coefficients k y = 1.5 ⋅ 10 6 tf / m and kθ = 1.99 ⋅ 1010tf / m . The free vibration
analysis lead to the conclusion thet the first eigen modes are of transversal vibration type.
Indeed, the frequency of the transversal vibrations is much smaller than those of either longitudinal
or torsional vibrations [12], [13], [14], [15]. For the above inenlioned structure, modeiled using
eigh’t finite elements (Fig.8) the obtained frequency of the transversal vibrations is 2.472 Hz versus
2.296 Hz given in [10]; a difference of about 7.7%.
PRACTICALLY STRUCTURAL ANALYSIS OF LARGE COOLING TOWERS
KOPENETZ L. et all., pp. 39-44
Fig.8. Finite elements discretization of cooling tower [10]
CONCLUSIONS
* The theoretical bachgrounds and numerical example used the Timoshenko type beams
model are presented.
* Modelling the cooling towers by using bar type finite elements having three or four inner
nodes offers an efficient vibration analysis which emphasizes simplicity and an acceptable accuracy
for the practical engineering needs, [16], [17].
* The presented free vibration anaiysis technique may, therefore, be used in a seismic
analysis using the enforced aseismic design needs, [18], [19], [20].
* The tall shells for natural draft cooling towers show the necessity for continuous scientific
research, [21], [22], [23].
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VOL. 1(14), issue 4_2011
ISSN 2247- 3769 ISSN-L 2247- 3769 (Print) / e-ISSN:2284-7197
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