Ultimate strength of large scale reinforced concrete thin...

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

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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


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.


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)


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)


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.


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 .


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.


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.


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)


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.


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.


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


(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.


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.


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.


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.


(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.


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.


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


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.


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


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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