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Ultimate strength of large scale reinforced concrete
thin shell structures
Hyuk Chun Noh
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA
Received 5 November 2004; received in revised form 1 April 2005; accepted 8 April 2005
Available online 23 June 2005
Abstract
This paper presents ultimate behavior of large scale reinforced concrete shell structures:
hyperbolic cooling tower shell and hyperbolic paraboloid (HP) saddle shell. Both geometrical and
material nonlinearities were considered in the analysis. To investigate the influence of concrete
tension cracks on the structural behavior, the smeared crack model having the capability of
representing double crack and crack rotation was used in the analysis. The biaxial stress state in
shells is represented by the improved work-hardening plasticity concrete model, where the ductility
increase phenomenon can be depicted. The load-displacement relationship, stress fields, occurrence
and propagation of cracks in concrete and steel yield patterns are presented. Due to the factors such
as modification in plasticity model, stiffness contribution in the doubly cracked elements, the model
predicts a more ductile behavior than the results reported in the current literature. The failure of
cooling tower shell seems to occur due to local yielding of meridional reinforcement in the windward
meridian. In the case of HP saddle shell, structural instability occurs due to severe tension cracks in
the shell part before yielding of reinforcement could occur.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Reinforced concrete shell structure; Nonlinearities; Ultimate strength; Cooling tower shell;
Hyperbolic paraboloid saddle shell
1. Introduction
Reinforced concrete has been extensively employed in the various fields of
constructions owing to the advantages in economic aspects and in the easiness in forming
Thin-Walled Structures 43 (2005) 1418–1443
www.elsevier.com/locate/tws
0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tws.2005.04.004
E-mail address: [email protected]
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1419
various shapes. Accordingly, the research on the concrete material and the behavior of
reinforced concrete (RC) structures have drawn great attentions for past several decades.
Especially, the RC shell structures are employed in many social branches owing to the
aesthetic beauty and the excellent performances. As a structure characterized by the
slenderness in their thickness, RC shell structures are designed to resist applied external
loads with internal membrane actions. Accordingly, the biaxial behavior of concrete is the
basic and crucial ingredients to be defined in an efficient and accurate way for the
numerical analysis of these structures. Some exemplary experiments related to this topic
are performed in the past [1–3], suggesting noticeable outcomes.
Numerical simulations of structural responses from the unstressed original state to the
ultimate state, until the failure of structure, enable us to have insights into a structural
behavior for its complete life-cycle. It is a challenging work, however, because such
simulations involve general nonlinear phenomena not only in its structural but also in its
material levels. This means that we need highly efficient numerical tools for analysis and
delicate material models as well to treat such problems. Especially when we deal with the
large scale reinforced concrete shell structures, numerical modeling not only in geometry
but also in material is one of the most important factors to be fulfilled with caution.
The examples of large scale thin shell structures most frequently constructed include
the cooling tower shell, which is a part of electricity generating nuclear power plants or
large-scale industrial factories, and the hyperbolic paraboloid (HP) saddle shells in the
public facilities. They definitely belong to the largest and thinnest concrete structures at
present. Even though the concern on the ultimate behavior of structures is originated from
academic interests, the concern on the ultimate behavior of cooling tower shells and HP
saddle shells is triggered by the failure events of these structures in the past. The cooling
tower shells have two events of collapse: the failure of the Ferrybridge towers in 1965 in
United Kingdom and the failure of the Ardeer tower in Scotland in 1973 [4–6]. The roof
shells, especially in the form of hyperbolic paraboloid, also collapsed in United State in
1975 [7], 15 years after the completion of construction.
It is noticed by Mang [8] that the full nonlinear analysis must be performed to forecast the
ultimate strength of the cooling tower shell because the load factor obtained by nonlinear
analysis is shown as considerably smaller than that obtained by the buckling analysis.
Actually, the linear buckling analysis gives ultimate load over five times that of the
nonlinear analysis. In the literature, many pursues have been made on the ultimate behavior
of cooling tower shells and show variety of results on the ultimate load and the cause of
failure [6,9–12]. Min [13] had used supercomputer for the nonlinear analysis of cooling
tower shells. In their extensive study, however, the effects of large displacement, tension
stiffening and the nonlinear stress strain relationship of concrete are ignored, thus leading to
higher ultimate loads. In case of HP saddle shells many works has been done as follows: Lin
and Scordelis [14], Mueller and Scordelis [15], Akbar and Gupta [16], Cervera et al. [17],
Shenglin and Cheung [18], Min [19]. The variation of ultimate load bearing capacity of this
structure, however, seems even more scattered than that of cooling tower shells.
In this study, the nonlinear behavior of reinforced concrete shells, taking into account
of the various nonlinear factors in material, geometry and physical domain, is examined.
The material nonlinearity in the concrete is taken into account by the adoption of the
plasticity theory with flow rule [20]. The stress state dependent constitutive matrix is
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431420
constructed by the work-hardening behavior of the applied plasticity model. To take into
account of the strength and strain increase in the biaxial stress state, a simple modification
is made to the plastic work-hardening model. The geometrical nonlinearity is included by
the large displacement formulation taking into account of the second order nonlinear strain
terms in the Green–Lagrange strain tensor. To represent the tension cracks in concrete, a
smeared crack model is employed. The contribution to structural stiffness from cracked
concrete is considered by tension stiffening model with strain softening branch. The steel
reinforcement is assumed to behave in one dimension and is modeled as bilinear material.
For the nonlinear control scheme, the arc-length control scheme is employed. The
reinforced concrete is modeled with the layered degenerated shell element [21], where use
is made of the substitute shear strain field to overcome the shear and membrane locking
phenomena. With the layering approach, the through thickness material plastification and
the crack penetration can be investigated.
2. Material modelling
2.1. Concrete in compression
The work-hardening plasticity is employed to model the non-linear behavior of the
concrete in compression. In this model, not only the initial yield surface but
also the definition of the subsequent yield (or loading) surfaces is defined [22,23]. In
other words, the yield surface is not fixed in the space of stress and the stresses are allowed to
go outside the yield surface. Here, the movement of the subsequent loading surfaces is
described by the hardening rule. In this study, the isotropic hardening rule is adopted: the
subsequent yield surfaces are expanded isotropically in accordance with the applied loads
during the plastic deformation.
2.1.1. Yield criterion
The Drucker–Prager yield criterion, which is a function of the two stress invariants, I1,
J2 is used.
f ðI1; J2Þ Z faI1 Cbð3J2Þg1=2 Z so (1)
where soZeffective stress. The initial yielding is assumed to occur at the stress state of
30% of the compressive strength, soZ0:3f 0c [22]. Following the experimental results of
Kupfer (1969) [1], where the ratio between uniaxial and biaxial yield stress is obtained to
be fcb Z1:16f 0c , the material parameters a and b in Eq. (1) are derived as aZ0.355so, bZ1.355. Accordingly, the yield function can be written as a function of stresses as follows:
f ðsÞ Z f1:355½ðs2x Cs2
y KsxsyÞC3ðt2xy Ct2
xz Ct2yzÞ�C0:355soðsx CsyÞg
1=2
Z so (2)
2.1.2. Hardening rule
The hardening rule defines the behavior of the subsequent loading surfaces in
accordance with the stress state and load history. In this study, the so-called Madrid
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1421
parabola is taken to describe this phenomenon.
s Z Eo3 K1
2
Eo
3o
32 (3)
Substituting the total strain 3Z3oC3p and using the relationship 3oZ2f 0c =Eo, the above
equation is rearranged as a function of plastic strain 3p, which defines the hardening
parameter for current stress state.
s ZKEo3p C2ffiffiffiffiffiffiffiffiffiffiffiffiffiEof 0c3p
q(4)
Since the hardening rule describes the movement of the yield surfaces, the stress in
concrete should be in the range of 0:3f 0c !s! f 0c . In the plasticity model adopted, however,
there is not any capability of representing the ductility increase effects when the concrete is
in the biaxial stress states [24]. To overcome this defect and to improve the behavior of the
model, the hardening rule is investigated and modified with following relationship.
f Z 5:2Rs C2:0; 3c Z ff 0cEc
(5)
where, Rs(Zs2/s1) is the principal stress ratio in the local stress plane, and f defines the
strain at the compressive strength of concrete. As a special case, when Rs is equal to zero,
the model behaves in the same way as in the uniaxial compression case.
2.1.3. Flow rule
Applying the normality condition of 3pijZdlðvg=vsijÞ [20,22,23], an ‘associated flow
rule’ is employed where the plastic potential g is substituted by the yield function f. In this
case, the stress–strain relationship is determined to be
Dep Z D KDaaT D
H CaT Da(6)
where aZflow vector defined by the stress gradient of the yield function; and DZconstitutive matrix in elastic range. The second term in Eq. (6) denotes the contribution
from the plastic flow and shows the effect of degradation of the material during the plastic
loading. The hardening parameter H is determined as
H Zvs
v3p
(7)
2.1.4. Crushing condition
Since the crushing of the concrete is the strain governing phenomenon, it is presumed to
be traced by the strain dual of the yield function, with the replacement of so by the ultimate
strain 3u, over which all the material properties are assumed as lost.
2.2. Concrete in biaxial stress state
Fig. 1 compares the influence of the biaxial stress states on the strength of the concrete
in compression–compression, compression–tension, and tension–tension regions between
–0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
–0.2
0.2
0.4
0.6
0.8
1.0
1.2
Initial Yield Surface
Subsequent Loading
Surface Region
Surface of maximum
compressive strength
Kupfer's Experiment
f'c /σ2
f'c /σ1
Fig. 1. Biaxial strength envelops.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431422
numerical model and Kupfer’s experiment [1]. The line denoted with hollow circle is
evaluated through one-to-one calculation of each stress state using the code constructed
applying the plasiticity model outlined in the previous section. The initial yield line as well
as the failure line are shown and compared with that of the Kupfer’s experimental results.
As shown in this figure, the biaxial strength of 1.16 time the uniaxial strength, as
mentioned in Section 2.1.1, is well depicted along the line of s2/s1Z1.0.
In the region of tension–tension stress state, due to the tension cut-off postulation, the
decrease in the tensile strength, observed in the experiment, is not represented.
2.3. Concrete in tension and shear
2.3.1. Cracks in concrete
The cracking of concrete is recognized as the most significant source of nonlinearity in
concrete. Generally, the behavior of concrete in tension is assumed to be governed by the
tensile strength of concrete f 0t . Stresses exceeding the tensile strength invoke the cracks in
concrete which render the anisotropy to the concrete. After the occurrence of cracks, the
elasticity modulus and the Poisson’s ratio is set to be zero in the crack normal direction and
the cracked shear moduli are applied. With this assumption, the constitutive matrix for
concrete cracked in 1 direction becomes
s Z diag½ 0 E Gcr12 Gcr
13 5=6G �3: (8)
For the concrete cracked in both directions, the stress–strain relationship becomes
s Z diag½ 0 0 Gcr12:Min Gcr
13 Gcr23 �3 (9)
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1423
Since the crack width is revealed to be the most governing factor for shear transfer
mechanism [25,26], the following formulae are adopted for the cracked shear moduli with
the shear retention factor b [27]. In case of cracked in 1 direction
Gcr12 Z bG 1 K
31
0:004
� �or Gcr
12 Z 0 if 31R0:004 (10)
Gcr13 Z Gcr
12
G23 Z5
6G
For concrete cracked in both direction
Gcr13 Z bG 1 K
31
0:004
� �or Gcr
13 Z 0 if 31R0:004 (11)
Gcr23 Z bG 1 K
32
0:004
� �or Gcr
23 Z 0 if 32R0:004
Gcr12:Min Z 0:5MinðGcr
13;Gcr23Þ
In this study, the value of b is assumed as 0.25, and the maximum strain in the crack
normal direction to maintain the shear transfer capability is assumed to be 0.004, as seen in
Eqs. (10) and (11).
2.3.2. Rotating crack model and constitutive relation for cracked concrete
After the first formation of cracks in concrete by the tension stress which exceeds the
tensile strength f 0t , the crack direction can change as the load proceeds. And the final state
of the crack is governed by the limit state of the reinforced concrete element, and this
phenomenon is demonstrated by some experiments [28]. In this study, therefore, the so-
called ‘rotating crack model’ is employed.
For concrete element cracked in one direction, the constitutive matrix is well defined,
and with the aid of geometric matrix Gcr [28], the incremental relationship is given as
follows:
Ds �x Z Dcrc1D3 �x Z ½Dc1 CGcr�D3 �x (12)
where Dc1 denotes the constitutive matrix for concrete cracked in one direction, which
represents the contributions from shear resistance in cracked section such as aggregate
interlocking, dowel action, friction along the crack surface, and so forth.
In case of concrete cracked in two-directions, the constitutive matrix for the concrete
element with cracks in two directions Dc2 can be derived as having only the contribution
from the shear resistance as follows:
Dc2 Z bG
sin22q Ksin22q Kcos 2q sin 2q
sin22q Kcos 2q sin 2q
symm: cos22q
2664
3775 (13)
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431424
where b and q denote shear retention factor and crack direction, respectively. The
geometric matrix Gcr, representing the effects of crack rotation, can be derived to be the
same as for the case of the concrete element with cracks only in one-direction. Mahmoud
and Gupta [29], in their study, assumed that the elements with cracks in two directions lose
all the stiffness, and the contribution to the global stiffness is ignored. However, due to
shear resistance in the cracked sections, some stiffness is retained. The constitutive matrix
given in Eq. (13) represents this contribution. Therefore, including the effect of crack
rotation with the geometric matrix, the constitutive matrix can be given as follows.
Dcrc2 Z ½Dc2 CGcr� (14)
2.3.3. Tension stiffening
One of the well known features of reinforced concrete structures is the tension
stiffening effect which explains the tensile stiffness contribution from the cracked concrete
due to bond between concrete and reinforcement. This phenomenon is included in the
finite element programs in various suggested ways [14,30–33]. Among the several
sophisticated models of tension stiffening, in this study, the tension cut-off strategy of the
tension stiffening is applied. The schematic view of this model is given in Fig. 2. The
ultimate strain 3m controls the degree of tension stiffening and is given as constant n times
the tension crack strain 3t.
If the maximum strain ever reached in the loading process is denoted as 31, the stresses
due to the increase and decrease of the load can be determined as follows.
s1 Z af 0t 1 K31
3m
� �; 3t!31!3m (15)
si Z s1
3i
31
; 3i!31
(n= TS)
s
f t
a f t
s1sisj
et ei e1 ej eme
a = 0.7
e m = ne t
Ec
EiEj
Fig. 2. Concrete stiffness after crack.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1425
The parameter ‘n’ will be denoted as ‘TS’ in the following sections to designate the
degree of tension stiffening.
2.4. Reinforcement
Even though the reinforcing steels in the layered concrete shell element are modeled as
plane type layers, the actual in situ embedding of the reinforcing steels is one dimensional.
Therefore, it is not necessary to introduce a complex multidimensional constitutive
relationship for steel [22], and will be modeled with one-dimensional bilinear
approximation. In the elastic range, steel behaves elastically with the elastic modulus of
Es, and after exceeding the yielding point the plastic modulus Esp is assumed. Generally,
the yielding strain 3y is assumed as around 0.002. In case of unloading, the path follows the
elastic slope Es.
3. Degenerated shell element with assumed shear strain field
In this study, a four node degenerated shell element [21,34] is adopted for the analysis
of reinforced concrete shell structures. The performance of degenerated shell [35]
deteriorated as the shell thickness is reduced, resulting in locking phenomena. To solve the
locking problems, various techniques such as the non-conforming modes [36,37], reduced
integration [38], and substitute shear strain fields [21,34,38,39] are suggested. In the
adopted shell element, the assumed shear strain field is applied to remove the locking
problems.
The geometric coordinates in the shell element can be determined by the following
expression, where the nodal coordinates at the mid-surface and the coordinate differences
between the top and bottom of each corner of the shell element are used.
x Z
x
y
z
8<:
9=; Z
X4
iZ1
Niðx;hÞ
xi
yi
zi
8<:
9=;
mid
Cz
2
X4
iZ1
Niðx;hÞ
Dxi
Dyi
Dzi
8><>:
9>=>;;
Dai Z ai;top Kai;bottom
(16)
The displacement vector u at any point in the shell is defined by the three translational
degrees of freedom, u, v, w, and two rotational degrees of freedom a and b along the two
vectors Vk2;V
kl which are normal to the nodal vector Vk
3.
u Z
u
v
w
8<:
9=; Z
X4
iZ1
Niðx;hÞ
ui
vi
wi
8<:
9=;
mid
CX4
iZ1
Niðx;hÞzhi
2
Vi1x KVi
2x
Vi1y KVi
2y
Vi1z KVi
2z
2664
3775 ai
bi
( )
(17)
It should be noted that the two rotational degrees of freedom are given in the nodal
coordinate system and dependent on the two nodal vectors Vk2;V
kl .
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431426
The stresses and strains are defined in the local coordinate system �x, and are given as
follows:
�3 Z h 3 �x 3 �x g �x �y g �x�z g �y�z iT (18)
�s Z hs �x s �x t �x �y t �x�z t �y�z iT Z �Dð �3 K �3oÞC �so (19)
where the stress and strain tensor with subscript ‘o’ denotes the initial ones.
Applying the stationary condition on the total potential energy consisting of
contributions from membrane, bending, shear, and external loads, the element stiffness
is divided into three components in accordance with the respective contribution to the total
potential energy as
Kmij Z
ðV
BTmiDmBmjdV ; Kbij Z
ðV
BTbiDbBbjdV ; Ksij Z
ðV
BTsiDsBsjdV (20)
3.1. Layering approach for reinforced concrete
In this study, an implicit layering scheme, introduced by Lin and Scordelis [14], is
employed to model the reinforced concrete structures. In the layered RC shell finite
element, the steel reinforcements are modeled as steel layers, which are located
intermediately between concrete layers. The location of each layer, designating the steel
and concrete, is given by the natural coordinate z which varies in the range [K1.0, C1.0],
and is determined at the middle point of each layer: a mid-point rule integration scheme.
The amount of reinforcement is represented by the layer thickness equivalent to the actual
sectional area of reinforcement. In accordance with the mid-point rule, the evaluation of
element stiffness consists of thru-thickness integration in z direction and area integration,
viz.,
k Z
ðA
ðC1
K1BTDBjJðx;h; zÞjdz dA (21)
In practice, the thru-thickness integration is converted into a summation of
contributions from each layer, i.e.,
k Z
ðC1
K1
ðC1
K1
Xn
iZ1
BTi DiBijJðx; h; ziÞj
2Dhi
hdx dh (22)
where jJ(x,h,zi)jZdeterminant of Jacobian matrix at each layer; DhiZthickness of i-th
layer; and nZtotal number of layers.
The layering scheme is a very useful technique in investigating the thru-thickness
gradual plastification and crack development in the concrete. The number of layers is
reported to be sufficient from 6 to 10 for each finite element [13,23]. In this study, 10
concrete layers plus the 4 steel layers, 2 inner and outer layers respectively, are used
exclusively.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1427
4. Formulation for geometrical non-linearity
4.1. Stress–strain matrix
The Green–Lagrange strain tensor consists of the linear infinitesimal and nonlinear
large displacement components as
3 Z 3L C3NL (23)
where the detailed expression of 3L and 3NL is obvious from the expanded form of the
Green–Lagrange strain tensor. With vector d composed of the derivatives of
displacements with respect to the local coordinate system �x as
d Z hE �x E �y E�z iT (24)
where
E �x Z h �u0 �x �v0 �x �w0x iT ; E �y Z h �u0 �y �v0 �y �w0 �y i
T ; E�z Z h �u0�z �v0�z �w0�z iT (25)
the linear and nonlinear components of the strain tensor can be conveniently written as
3L Z
1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 1 0 1 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 1 0 1 0
266666664
377777775
d Z Hd (26)
3NL Z1
2
ET�x 0 0
0 ET�y 0
ET�y ET
�x 0
ET�z 0 ET
�x
0 ET�z ET
�y
2666666664
3777777775
d Z1
2Ad (27)
Therefore, the strain is expressed in the matrix form as
3 Z H C1
2A
� �d (28)
The expression of the displacement field, as given in Eq. (17), can be rewritten as
dependent on the nodal displacement vector dk as
u ZX4
kZ1
�Nkðx; h; 2Þdk (29)
where dk Z h uk vk wk ak bk iT . Therefore, the vector d in Eq. (24) becomes dZGa
and G contains the shape functions as elements, and a consists of all nodal variables.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431428
Taking the variation of the Eq. (28) to get the strain-displacement matrix, we get
d3 Z dðHdÞC1
2dðAdÞ Z ðH CAÞG da (30)
In deriving the above expression, the relation dAdZAdd [40] is applied. As seen in Eq.
(30), the strain-displacement matrix is obtained to be
�B Z BL CBNL Z ðH CAÞG (31)
4.2. Expression for tangential stiffness
The residual force J in the typical finite elements is given by
J Z F K
ðV
�BTs dV (32)
Taking the variation of the assumed residual force, and noting that dsZD d3ZD �B da
and dJZ0, the tangential stiffness KT is derived as
KTda Z
ðV
dBTNLs dV C
ðV
BTLD �B dV da C
ðV
BTNLD �B dV da Z Ksda C �K da (33)
whereÐ
V dBTNLs dV ZKsda is used. Accordingly, the tangential stiffness is found to be
KT ZKsC �K. Furthermore, the matrix �K can be rearranged as �KZKLCKLD if the
relation of �BZBLCBNL is used. Therefore one can note that the tangential stiffness
matrix KT consists of the linear contribution KL, the stress dependent geometric stiffness
Ks plus the term representing the effect of the large displacement KLD, as
KT Z KL CKs CKLD (34)
4.3. Geometric stiffness Ks
The geometric stiffness matrix accounts for the effects of membrane forces. The
geometric stiffness is influenced only by the element geometry, the displacement field, and
the state of stress, and is free from elastic properties of the material. With the variation of
strain-displacement matrix BNL in Eq. (31), the geometric stiffness is derived as
Ksda Z
ðV
dBTNLs dV Z
ðV
GT dAT s dV Z
ðV
GT SG dV da (35)
where, the relationship of dATsZGSda is used [40]. The matrix S in Eq. (35) is defined as
S Z
sxI3 txyI3 txzI3
syI3 tyzI3
symm: 0
264
375 (36)
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1429
5. Numerical verifications
For the verification of the applied theories and the coded programs for the nonlinear
analysis of reinforced concrete structures, some example problems, tested in the past, are
investigated: McNeice slab [42], beam type plate [43] and Duddeck slabs [44]. Here, the
results on McNeice slab are given in detail. This two-way slab is tested by McNeice [42]
and analyzed by some researchers [31,45]. The material and geometrical properties of the
example are as follows: EcZ28,610 MPa, EsZ200,000 MPa, f 0c Z37:9 MPa, ftZ3.2 MPa,
fyZ345 MPa, slab depthZ44.5 mm, reinforcement is located 33.3 mm from top surface
with steel ratio of 0.0085. To trace the load-displacement path, the arc-length method [41],
which takes an indirect way of load increment/decrement control by using the arc-length
constraint, is employed.
A point load of 1000 N is applied at the center of the slab. Recognizing the symmetry of
structure and applied load, a quarter is modeled with 36 shell elements as shown in Fig. 3.
The displacement is found at the point ‘A’ in Fig. 3.
The result of present analysis which is denoted by the middle solid line is compared
with experimental result (heavy solid line) and the result of other research (dotted line)
[45]. As seen in Fig. 4, initial path follows that of [45], however, the overall path
shows good agreement with the experiment. The ultimate load is also close to that of
experiment. The tension stiffening effect is included with TSZ10 and the slope aZ0.8
as given in Fig. 2. In fact, the convergence is revealed very hard around the ultimate
load of the experiment showing the closeness to the experimental result. The failure is
investigated as occurred by the concrete crushing just after the yielding of
reinforcements near the loading point. It has to be mentioned here that the results of
this study on the beam type plate [43] and Duddeck slabs [44] are also in good
agreement with the experimental results.
The cracks in each concrete layer and reinforcement yield at the failure load are
illustrated in Fig. 5. The stresses in the concrete at the Gauss point 1 of element 36 is
shown in Fig. 6. At the failure, it is investigated that the concrete stress reaches about 66%
: Coner Supports
A
914
mm
914 mm
LC
CLLC
A
CL
Fig. 3. Dimension and mesh of McNeice slab.
0 5 10 15 20 25 300
1
2
3
4
Displacement(mm)
Load
Fac
tor : Experiment
: Present: Crisfield
Fig. 4. Load-displacement curve of McNeice slab.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431430
of the compressive strength in the top layer. The yielding of reinforcement occurs first
around the load factor 2.4 in the vicinity of the loading point.
6. Evaluation of ultimate strength
6.1. Cooling tower shell
The ultimate behavior of Port Gibson cooling tower at Mississippi, subject to the design
wind pressure that is modeled as a quasi-static pressure load in accordance with the quasi-
steady aerodynamic theory [6,10,13,29], is investigated.
(a) (b)
Fig. 5. Crack and yield pattern of McNeice slab at PZ3212 N (Load factorZ3.212) (on the deformed shape
magnified 10 times). (a) Cracks in each concrete layer. (b) Reinforment yielding.
–1 1 2 3 4 5
–20
–25
–15
–10
–5
5
Strain (x103)
Str
ess
(Mpa
)
Fig. 6. Stress–strain path in concrete.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1431
6.1.1. Geometry and materials
The Post Gibson tower, which has total height of 150.5 m and radii at top, throat, and
bottom are 38.6, 36.3, and 59.7 m, follows the meridional equation as
a~z2 CbR~z CcR2 Cd ~z CeR C f Z 0 (37)
The meridional equation takes hyperbola above the throat and ellipse below the throat.
The constants a to f in Eq. (37) can be found in Ref. [6]. The material constants for
concrete are: elastic Modulus EcZ28268.0 MPa, compressive strength f 0c Z27:6 MPa,
tensile strength ftZ3.0 MPa, Poisson’s ratio nZ0.2 and specific weight gcZ24.25 kN/m3,
and those for steel are: elastic modulus EsZ200600.0 MPa, yield stress fyZ413.7 MPa.
The reinforcement ratios are given in Table 1.
In Table 1, rm, rc denotes the reinforcement ratio in meridional and circumferential
direction, and zsm, zsc means the normalized distance from the mid-surface to the steel
reinforcement. As seen in Table 1, the shell thickness assumes relatively small values as
compared with that of the total height. The ratio of minimum thickness to total height of
the shell is as small as 0.135%.
6.1.2. Loads on cooling tower
Self weight of the reinforced concrete is assumed as 2.47 ton/m3. In the analysis, self-
weight is applied at first, and then the design wind load is applied with increments. The
wind load in the static linear and nonlinear analyses is taken as a quasi static pressure with
gusty wind effect [46]. The mathematical expression of the wind pressure is given as a
function of circumferential degree q from the luff of the shell and the vertical coordinate z
from the base as
pðq; zÞ Z qoGðqÞMðzÞ (38)
where qoZdynamic head given as rav210=2, the air density raZ0.126 kg/m3, and v10Z
design wind velocity. In this study, the design wind velocity is taken to be 40.2 m/s
Table 1
Reinforcement ratio along the meridional and circumferential directions
Elevation (m) Thickness
(mm)
rm zsm rc zsc
150.5 1017
143.52 203 0.00649 0.8689 0.00649 0.8654
112.74 203 0.00407 0.6059 0.00407 0.6039
97.35 203 0.00636 0.6059 0.00407 0.6039
81.96 203 0.00918 0.6529 0.00407 0.6036
56.76 240 0.00808 0.6529 0.00560 0.6501
31.56 287 0.00668 0.7133 0.00463 0.7110
6.36 342 0.00580 0.7508 0.00402 0.7488
0.0 763 0.00585 0.8552 0.00601 0.8522
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431432
(90 mph), having return period of 100 years. In the analysis by Mang [8] and Hara [10],
the value of design wind load of 42.6 m/s was used. The two functions in Eq. (38) are as
follows:
GðqÞ ZX12
nZ0
AncosðnqÞ; MðzÞ Zz
10
� �2=7
ðz in meterÞ (39)
The Fourier coefficients An of equation for G(q) can be found in Ref. [6,47].
The distribution of wind-induced pressure load on the outer surface of the cooling
tower is illustrated in Fig. 7(a), where the internal suction effect is not taken into
account.
Fig. 7(b) depicts the 3D distribution of the wind pressure on the cooling tower shell.
The elevation at the center of element of each circle of elements are 12.32, 53.30, 93.67,
128.48 and 156.15 m (1, 9, 17, 24, 30th row of elements in the 27!30 mesh). For wind
load in German guide, see VGB-BTR [48].
6.1.3. Nonlinear behavior of cooling tower shell
The port Gibson tower is modeled with 810 (27!30) elements and 868 (28!31) nodes
with 14 layers, 10 for concrete and 4 for reinforcement. Taking into account of the
symmetries in the structure and applied wind load, a half model is employed. For the nodes
in the plane of symmetry, the symmetric boundary condition is applied as required. The
base columns are omitted and assumed as hinge support [6,11,13,56]. In calculating the
wind pressure load, however, the height of the base columns is taken into account. In
applying arc-length control, 0.5% tolerance in both the displacement and residual forces is
taken to determine the state of convergence.
In Fig. 8, the axial forces developed in the shell due only to the gravity load is
presented. As expected, almost all part of the shell is in the compressive state of stress,
showing that the gravity load contributes the counter resistance to the tensile action
induced by wind loads.
Fig. 9 shows the load-displacement paths at the nodes along the windward meridian
at the specified location. The last data points of each load-displacement path are
CircumferenceHeight
Zero pressure line
Pressure
push
pull
1.064
(a) (b)
–0.400–1.350
0o
180°
Fig. 7. Wind pressure distributions.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1433
the unconverged results after 50 iterations at the last increment. Due to the arc-length
control, there appears the effect of energy release due to the formation of cracks or yield of
reinforcement. The ultimate load factor has reached up to 2.34 with throat (zZ130.7 m)
displacement of 58.1 cm and then the steel in the windward meridian is yielded.
Axial forces (N/m)
Z-c
oord
inat
e (m
)
–1.000.000 –800.000 –600.000 –400.000 –200.000 0 200.0000
15
30
45
60
75
90
105
120
135
150
165
Fc (Circumference)Fm (Meridian)
Fig. 8. Axial forces along windward meridian.
Displacement (m)
Load
fact
or
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
z=60.8z=81.0z=101.4z=130.7z=159.6
Fig. 9. Load-displacement path.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431434
The yielded steel reinforcements propagate in the circumferential direction, and then no
convergence is achieved after throat displacement of 64.5 cm, i.e., the failure occurs.
Fig. 10 shows the effect of degree of tension stiffening on the load displacement path
and the ultimate load capacity of the cooling tower shell. The ultimate load factor is
obtained as 2.06 when TSZ10, which is about 88% of the case when TSZ40 is assumed.
The load-displacement path with lower tension stiffening shows relatively severe
discontinuity than that of TSZ40, which indicates the abrupt changes of stresses due to
low tension stiffening effect.
The comparison of the load-displacement history with other research results is given
in Fig. 11. The discrepancy of the histories comes from various factors such as
Displacement (m)
Load
fact
or
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.5
1
1.5
2
2.5
Tension Stiffening 40Tension Stiffening 10
Fig. 10. Effect of Tension stiffening.
Displacement (mm)
Load
Fac
tor
0 200 400 600 8000
0.5
1
1.5
2
2.5
Mahmoud & Gupta (1995)Milford & Schnobrich (1984)Present (TS=40)
Fig. 11. Comparison with other results.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1435
the material constants, design wind load, modeling of structure, applied crack model,
and the small deviation of nodal point coordinate on which the displacement history
is found, and so on. Though the present analysis gives more ductile behavior than
the others, the global trend and the ultimate strength agree well each other. Due to the
inclusion of the height of base column of 30 ft, the crack load in this study is
more or less lower than the other results. The ultimate load is obtained as 2.34
(a) (b) (c)
Fig. 12. Crack pattern just before the failure load (deformation magnified 25 times). (a) Cracks on inner surface,
(b) cracks outer surface, (c) reinforcement yield at failure.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431436
(2.06 when TSZ10, see Fig. 10) which is greater than 2.07 of Milford [6] and 2.16 of
Mahmoud [11].
6.1.4. Cracks and steel yield
The first cracks are formed at the elevation of 40% of the total height of cooling
tower at the load factor of 1.4. The cracks are formed through all the layers: through-
thickness cracks. The crack patterns just before the failure load are given in Fig. 12(a)
and (b). The cracks on the inner surface are shown in Fig. 12(a), where the doubly
cracked elements appear along the meridional line at the windward meridian. In case of
the outer surface, Fig. 12(b), the doubly cracked elements appear along the line of the
maximum curvature near and below the throat. The cracks are formed mainly in the
horizontal direction which is caused by the cantilever effect of the wind pressure.
The second cracks in the vertical direction are occurred due to bulge-type bending
deformation. It is investigated that almost all the horizontal cracks are the through-
thickness ones.
The first yield of steel is occurred at the elevation of 61 m from the ground in the
windward meridian. Due to the yielding of steel, discontinuity in deformation slope
appears. The subsequent yield of steel is formed along the circumferential direction
leading to the failure of the structure, Fig. 12(c). As contrary to the results in Min [13]
and Mahmoud [11], where the yield of steel is observed in almost all part of the shell
from the bottom to the top and reached 308 from the windward meridian in
the circumferential direction, only localized yielding of reinforcement is observed in
this study.
6.1.5. Stress distributions
Fig. 13 shows the distribution of stresses in an array of elements in circumferential and
meridional directions. The array of elements in meridional direction is located at the
stagnation and the circumferential array of elements is located at the elevation of 81 m.
The stresses, after the commencement of non-linear behavior, are denoted with dotted
lines without symbols. With these figures, the gradual redistribution of stresses can be
investigated along the increase of applied wind pressure. It is noted that the tensile stresses
show some irregularity due to occurrence of cracks. It is worth of note that some
meridional (Fig. 13(b)) and circumferential (Fig. 13(c)) compressive stresses exceed the
initial yield stress of 0:3f 0c at the load factor 2.190.
6.2. Hyperbolic parabolid saddle shell
This shell is analyzed by many researchers including Min and Gupta [13], Lin and
Scordelis [14], Mueller and Scordelis [15], Akbar and Gupta [16], Cervera et al. [17],
Shenglin and Y.K. Cheung [18], and Min [19]. However, except for Mueller and Scordelis
[15], all the other results are discarded due to the inappropriate material modeling and the
poor results due to too coarse meshes. In this study, mesh with 1024 elements is used in
full model.
Circumferential angle
Str
ess
(Mpa
)
0 20 40 60 80 100 120 140 160 180–7.5
–5
–2.5
0
2.5
5(a) (b)
(c) (d)
LF = 0.575LF = 0.959LF = 1.343LF = 1.800LF = 2.190
Circumferential angle
Str
ess
(Mpa
)
0 20 40 60 80 100 120 140 160 180–10
–7.5
–5
–2.5
0
2.5
LF = 0.575LF = 0.959LF = 1.343LF = 1.800LF = 2.190
Stress (Mpa)
Ele
vatio
n (m
)
–9 –8 –7 –6 –5 –4 –3 –2 –1 00
40
80
120
160LF = 0.575LF = 0.959LF = 1.343LF = 1.800LF = 2.190
Stress (Mpa)
Ele
vatio
n (m
)
–1 0 1 2 3 40
40
80
120
160LF = 0.575LF = 0.959LF = 1.343LF = 1.800LF = 2.190
Fig. 13. Stress distributions at each loading steps: (a) sc and (b) sm in the array of elements in the circumferential
direction; (c) sc and (d) sm in the array of elements in the meridional direction.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1437
6.2.1. Geometry and material
Fig. 14 shows the geometry of the HP saddle shell which can be constructed by arrays
of rotating straight lines. The two corner points A is fixed at the base and the other two
corner points B are free. The four edges are stiffened by the square section edge beam with
side length of 20 inches. The viewpoint of configuration in Fig. 14 is (2,3,0). The shell has
4 in (10.16 cm) thick. The material properties for concrete: EcZ25663.2 MPa, nZ0.15,
f 0c Z25:856 MPa, ftZ2.578 MPa, and for steel: EsZ200,000 MPa, fyZ413.7 MPa.
The steel reinforcement ratio for shell part is AsZ0.2 in2/ftZ0.0004233 m2/m and for
edge beam AsZ5.1 in2/20 inZ0.006477 m2/m. The specific weight of concrete is WcZ23.562 kN/m3. The applied load on the HP saddle shell is dead weight plus the upper
distributed load of 20 psf(Z957.6 Pa).
6.2.2. Nonlinear behavior of HP saddle shell
Fig. 15 compares the edge beam displacement due to self-weight of reinforced
concrete. The displacements are in good agreement with the results of Min [13] even
though a coarse mesh is employed in this study. Table 2 compares the displacement due to
self-weight and the ultimate strength of the HP saddle shell with those of other results
Y
X
1.67ft
1.67ft
80.0ft
16ft
16ft
80ft
(a) (b)
A
B
B
A
4.88m
4.88m
24.38m
0.509m
0.509m
Fig. 14. Geometry and configuration of HP saddle shell. (a) Geometry, (b) configuration.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431438
available in the literature. As given in the table, the results of present study are in good
agreement with those of others.
In Fig. 16, the load-displacement history at the free tip point is compared with other
research results. The initial displacement is evaluated by the pre-analysis applying the self
weight of the structure. As can be noted in Fig. 16, the tension stiffening affects not only
the ultimate strength but also the ductility of the structure as well. The higher the degree of
tension stiffening, the more ductile the structure, and vice versa. Even though some works
X-coordinate
Vert
ical
dis
plac
emen
t (cm
)
–15 –12 –9 –6 –3 0 3 6 9 12 15–5
–4
–3
–2
–1
0
1
Min (64x64) [13]Present (32x32)
Fig. 15. Edge beam displacement.
Table 2
Comparison of ultimate load for HP shell
Elements Dead load displa-
cement (cm)
TS parameter Ultimate live load
(kPa)
Muller–Scordelis
(1977)
Triangle 105 4.4 3.5 7.8
Min (1997) 4-node 1121 4.8 3–20 2.73–9.62
Present 4-node 576 4.43 40 7.34
4-node 1024 4.84 40 7.36
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1439
delivered the load-displacement relationship of almost straight line up to the abrupt brittle
failure [15,19], in this study, a curved load-displacement relationship is obtained.
6.2.3. Crack formations and stresses
Since the crack is formed in the upper and lower surfaces during the loading process as
seen in Fig. 17, and the occurrence of cracks is generally assumed to cause reduction in
stiffness, the curved equilibrium path seems to be more acceptable. In the analysis by Min
[19], it is reported that no cracks are developed in the bottom surface excepting the region
near the edge beam boundaries. However, in this study, even in the bottom surface of the
shell, extensive tension cracks are developed. As expected, the beam-type edges are stiffer
to the great extent than the shell itself. Besides, the distributed loads on the top surface of
the HP shell push down the shell, making the edge-beams to exert tensile forces to the shell
in the diagonal direction in both the upper and lower surfaces. Therefore, it will be
more acceptable to have cracks in the lower surface of the shell. That is, due to the
initial geometry and the action of the rigid edge beams, the expanding action becomes as
severe as the bending action. Actually, it is investigated that the deflection at the free
corner points in the in-plane direction exceeds a half that of the vertical deflection.
Tip displacement(m)
App
lied
load
(kP
a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7–2
0
2
4
6
8
10
12
1
2
3
4
5
6
1 Min w/o TS [13]2 Min w/ TS(3,5,10,15,20) [19]3 Y.K.Cheung [18]4 Present(TS=40 400 ele)5 Present(TS=20 400 ele)6 Present(TS=40 1024 ele)
Fig. 16. Load-displacement history.
(a)
(b) (c)
Fig. 17. Deformed shape and crack patterns. (a) Deformed shape (TSZ40), (b) bottom surface cracks, (c) top
surface cracks.
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–14431440
The deformed shape (magnified 20 times) and the crack patterns just before the ultimate
load are given in Fig. 17.
7. Summary and conclusions
In this study, nonlinear behavior from unstressed virgin state to the ultimate state is
given for large scale reinforced concrete thin shell structures such as cooling tower and HP
saddle shells. In the analysis, not only the material nonlinearities in concrete (such as
work-hardening plasticity, cyclic behavior due to unloading and reloading, tension cracks
and crack rotation, tension stiffening and shear transfer) and steel reinforcement but also
the geometrical nonlinearities based on Green–Lagrange strain tensor are taken into
account.
The ultimate load bearing capacity of cooling tower shell is obtained as 2.34 times that
of design wind pressure, which corresponds to the wind velocity of 40.2 m/s (90 mph).
The initiation of the nonlinear behavior is triggered by the formation of tension cracks in
the windward meridian at the middle height of the cooling tower shell. As the applied load
increases, cracks in concrete spread along meridional and circumferential directions. At
the ultimate load, the yield of steel reinforcement is occurred in the windward meridian
and abrupt increase in the along-wind displacement appears invoking the energy release
phenomenon in the load-displacement path. Once the steel yields, the yield zone
propagates along the circumferential direction, reducing tremendously the load bearing
H.C. Noh / Thin-Walled Structures 43 (2005) 1418–1443 1441
capacity of the cooling tower shell, and the structure fails. That is, for the cooling tower
shell under consideration, the failure is investigated to be caused by the local yielding of
meridional reinforcement in the windward meridian.
In case of HP saddle shell, due to the extraordinary features in geometry and sensitivity
to the modeling of material and geometry, large variations in ultimate load are delivered
depending on researchers. In this study, as contrary to the brittle behavior with almost
straight line of load-displacement history of some research results, a ductile behavior with
curved history is investigated. The ultimate load is evaluated to be around 7.4 kPa, which
corresponds to load factor of 7.7. In the deformation process, the expanding action of edge
beams on the shell part is examined, resulting in severe tension cracks in the shell. The
failure of HP saddle shell is investigated to be caused by the structural instability due to
severe tension cracks in the shell. The yielding of steel is not observed in any of
reinforcements.
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