30.18. iass paper andres harte n 37

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JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: IASS

BUCKLING OF CONCRETE SHELLS: A SIMPLIFIEDNUMERICAL APPROACH

MATTHIAS ANDRES a and REINHARD HARTE b

a Dr.-Ing. Wayss & Freytag Ingenieurbau AG, Frankfurt am Main,b Prof. at Bergische Universitt, Pauluskirchstrasse 7, 42285 Wuppertal, Germany

Editors Note : Manuscript submitted 28 October 2005; revision received 14 June 2006; accepted for publication 4 September 2006. This paper is open for written discussion, which should be submitted to the IASS Secretariat no later than August2007.

SUMMARY

In 2004, the paper of Stefan Medwadowski on Buckling of Concrete Shells: An Overview [16] presented anexcellent overlook over the state-of-the-art of classical stability methods and its application to concrete shells.

The respective algorithms in finite-element-method have only been touched, but not discussed in detail. The present paper will give a review of former and new developments in the numerical simulation procedures of buckling phenomena. On one hand, the progress in the development of numerical tools to simulate nonlinearityin concrete structures is significant. On the other hand, there is still a vital need for engineering experience and imagination in order to interpret and value the results correctly. Thus the present paper will consecutively fit tothe Medwadowski paper [16]. Besides a more realistic consideration of material nonlinearity in reinforced concrete composite as well a more practical numerical approach will be given.

Keywords: Buckling, Stability, High-Performance Concrete, Shell Structure, Nonlinear, Material Law, Damage, Cooling Tower

1. INTRODUCTION

The paper [16] was concentrated on the analytical procedure to describe the buckling mechanism of concrete shell structures. The progress in describingand evaluating of buckling loads of concrete shellsstarted already in the Seventies. The analyticalmethods have the advantage to use few diagramsand more or less simple formulas to analyse theglobal (acc. Kollar) or local (acc. Mungan) bucklingload of a shell. The main disadvantage is the limitedapplicability. This only allows the prediction for some theoretical cases, but is not appropriate for complex shell structures with non-symmetric

boundary conditions and/or loadings.

The present paper describes an approach to findmore realistic buckling loads for various shellstructures. This approach is developed as analternative to a time-consuming, fully nonlinear analysis. It will show a procedure to evaluaterealistic buckling loads and corresponding mode

patterns with less effort, but still considering the

nonlinear concrete behaviour by assuming adequateconcrete damage. This might be helpful for thedesigning engineer to test the structure against

buckling failure during the design phase and toevaluate the optimal shape of the shell.

2. FINITE ELEMENT FORMULATION

For the discretization of a shell structure within thescope of a finite element model, an isoparametricfour-noded finite element (ASE4-element) [18]

formulated in Reissner-Mindlin theory has beentaken. The shell element is formulated as anassumed-strain element to avoid locking effects[22]. For the realistic description of the nonlinear material behavior of the material concrete it isnecessary to get information about the stress andstrain distribution over the height of a cross-section.Thus the layered shell element approach, as it wasused in [24] for a different element approach, has

been adopted in the following. The different layerscan describe both the concrete and thereinforcement steel layers so that the stiffness of the


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VOL. 47 (2006) No. 3 December n. 152

cross-section is considered realistically. To receivethe overall stiffness matrix of the structure, which isthe central purpose of each finite element analysis,the multi-level strategy [11] (figure 1) is used. Thisallows the description of the material behaviour onthe level of the material point and the improvementof the model for various material properties.

An extremely important issue in buckling analysisis the precise simulation of geometric imperfectionsof a shell structure. In general, these imperfectionslead to a drastic reduction of the overall bucklingload so that they have to be considered close toreality. For this purpose an artificial displacement

vector is added to the perfect geometry, andconsidering this vector all relevant geometric

properties will change and have to be defined new.At the beginning of an analysis it is not easy toidentify the most conservative pattern of thisadditional imperfection vector with respect tolowest buckling loads. In general an imperfection

pattern affine to the lowest eigenmode has beenused [9], but some actual publications have beenshown that this assumption might not necessarily bethe worst imperfection pattern in any case [7].

Nevertheless this approach has proved to be a goodstarting point so that it was applied for thefollowing investigations, too.

3. ELASTO-PLASTIC MATERIAL MODELFOR HIGH STRENGTH CONCRETE UNDER CONSIDERATION OF DAMAGE

The recent progress in the development of highstrength concrete was one of the main motivationsfor the presented research work. To describe thematerial properties of this kind of concrete manyexperiments have been performed in variousresearch centres in the near past. These experiments

serve as a source of each material description andlead to the specific regulations in international andnational code. In Germany the new code DIN 1045-1 for concrete structures has been installed in 2001[8]. In this code the use of nonlinear analysismethods has been regulated for the first time inGermany.

For numerical modelling of the nonlinear material properties within a finite-element-system, theapproach of [6] has been chosen in order to get a

most flexible formulation for the material law. Thisapproach transforms the mainly biaxial stress stateof a concrete shell structure into two separatedirections, which can be described with uniaxialmaterial models, which in general are the onlyinformation given in code regulations. Thus themodel assumes an incremental linear orthotropicstress state in the concrete. With the equivalent

Poissons ratio of 21 = the material law

results in Equation (1).This biaxial stress state then can be expressed bytwo equivalent stress-strain relations in principalstress directions. After a transformation the strainsresults in Equation (2).

Under consideration of the biaxial fracture criterionof [4] the stress-strain relations can be formulated.The ascending branch of this uniaxial relation can

be described by the use of [8]

structure finite element integration point material point

ijij, sTK IPpI

pT ,k p

a}

3a

2a

1a

3a 2a

1a

}qmn=abb

{

qmn=a bb

{ }

s

sa

1Q

2Q

3Q1a

2a

3a

V

global displacement vector local displacement

pv )()( }{ ga bb a

strain in middle-surface

ijij, g

strain in each layer

stresses in each layerstresses in middl e-surfacelocal stiffness matrix /local load vector

global stiffness matrix /global load vector

Figure 1. Multi-level structural analysis concept


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( ) cRi f

k k

+=

21

2

(3)

with 1cic = and cccm f E k 11.1 = .

(with: ic - actual equivalent strain in direction i,1c - strain at maximum compressive stress, cm E -

mean value of Young`s modulus, cc f 1 -calculated mean value of compressive strength for nonlinear analysis).

With respect to [8] the descending branch can bemodelled with a linear and a nonlinear part withinthe finite-element-analysis. This makes it possibleto modify the slope according to the fracture energy

cG and the equivalent length of the finite element

intn Al e

eq = (with e A - area of the finite element,intn - number of integration points) in every

material point [20].

The composite material reinforced concrete has thewell-known ability to carry higher loads than the

pure reinforcement layers in the cracked state. Theresidual load-carrying capacity of the concrete

between cracks is dependent on the bondcharacteristics between steel and concrete matrix.

To model this so-called tension-stiffening effect,the concept of iterative simulation according to [19]has been used in this analysis.

4. NONLINEAR ELASTIC AND INELASTICSTABILITY ANALYSIS

4.1 General

In case of conservative static loading the commonequation of motion can be reduced to the tangentialform:

iaT FPVK =+

. (4)

This equation is the base of any nonlinear finite-element-analysis. Here T K denotes the tangential

stiffness matrix, aP the given load vector, iF theinner force vector of the structure. The tangentialstiffness matrix can be decomposed into three parts,the elastic stiffness matrix eK , the initial stress

matrix K and the initial displacement matrix uK .Further decomposition with respect to linear (L)and nonlinear (NL) dependency on thedisplacements leads to

. NLu

NL Lu

Le

ueT

K K K K K

K K K K

++++=

++=

(5)

The numerical treatment of nonlinear finite-element-simulations and the tracing of the nonlinear structural response branches have been intensivelyinvestigated in the past [13, 21].

4.2 The definition of stability and criteria todescribe the loss of stability

In science the expression stability has variousdefinitions depending on the respective faculty of

science. In structural mechanics stability describesthe stable equilibrium state of a structure. This leadsautomatically to the definition of loss of stability,which means the change from a stable into anunstable state of equilibrium during the loadhistory. The neutral point between stable andunstable equilibrium states of a structure can becharacterized by its possible deformations. For stable equilibrium states a distinct displacement

pattern exists, and additional displacements directlycorrespond to rising loads on the structure. The

behaviour of the structure at the limit point of stable

( )

+

=

12

2

1

2121

221

211

2

12

2

1

241

00

0

0

11

d

d

d

E E E E

E E E

E E E

d

d

d

. (1)

=2

1

21

122

2

1

1

11

1

d

d

E E

E E d

d

u

u (2)


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VOL. 47 (2006) No. 3 December n. 152

equilibrium is completely different. Here theresponse of the structure due to the loading is notunique and at least various response paths are

possible.

Loss of stability at the limit point is characterized by two phenomena, snap-through behaviour or bifurcation. In both cases the nonlinear load-displacement branch before buckling is described asthe prebuckling state, the branch after buckling asthe postbuckling state. To find such multipleequilibrium states within a nonlinear analysis,different algorithms and criteria can be formulatedand checked. Equations (4,5) lead to theformulation of

0VK =+

alt T (6)

in the neutral equilibrium point. This formulationmeans that an increase of displacements is possiblewithout further load increase. Equation (6) isfulfilled, if the determinant of the tangentialstiffness matrix is zero. To evaluate the determinantof the stiffness matrix, the single values in thestiffness matrix in general have to be scaled in order not to exceed the range of computer-precision bymultiplications. The expressiveness of the

parameter is not high for the observation of reinforced concrete shell structures, because its

decrease down to the point of neutral equilibrium islow. Anyway, the occurrence of the neutral statecan be definitely identified by this parameter.

Another observation parameter is the so-calledstiffness parameter according to Bergan [2]. This

parameter, described in equation (7), combines theactual stiffness state with the actual displacementstate:

iTiT i

T T

i P S ++

++

=VK V

VK V 1112

1

. (7)

(with: i - incremental load step, TiK - tangential

stiffness matrix, i+V - incremental displacement

vector, i denotes step i).

In case of snap-through behaviour, the parameter will be zero, whereas in case of bifurcation

behaviour in the neutral equilibrium state the parameter will not be able to definitely identify theloss stability.

Another possible criterion is the observation of thediagonal values ii D of the tangential stiffnessmatrix. In the state of stable equilibrium alldiagonal values are positive, whereas in theunstable state at least one value is getting negative.

This is the reason why the authors of this paper have performed complete nonlinear analysesaccompanied by the observation of all parameterslisted in table 1.

State of equilibrium

Stiffness- parameter

determinant

T K Diagonal

values

stable 0 P S 0det >Td K 0> ii Dneutral(snap-through

point)0= P S 0det =Td K 0= ii D

neutral(bifurcation point)

0 P S 0det =Td K 0= ii D

unstable 0 P S 0det Td K 0


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VOL. 47 (2006) No. 3 December n. 152

The description of the damage of concrete due totension is much easier. Only two states of damagemay exist. If the layer is cracked the parameter is

1= D , whereas in the uncracked state it is 0= D .The overall damage of the cross-section then again

can be evaluated by integration over cross-sectionheight:

ht h

D crack ten= . (13)

The damage component of the reinforcement steelcan be defined with the help of the secant Youngsmodulus, where 1= D denotes the total yielding of the cross-section.

max,0

0

d

d steel

E E

E E D

= . (14)

These damage parameters make it possible toreduce the stiffness of each finite elementseparately, with respect to the actual state of loading and the corresponding stresses.

To identify, whether the damage has a reducingeffect on the stiffness or not, case differentiationswith respect to the different kinds of damage haveto be defined. For example table 2 shows the casedifferentiation for crack damage. As a criterion the

strains at the inner ( ( )2/h ) and the outer surface ( ( )2/h+ ) of an element have beentaken. These two values define the strain state, and

by linear transformation the stress state of theobserved element can be delivered. For example across section has been damaged by cracks andshould be stressed in tension again. Then bothtensile and bending stiffness will be reduced. If thecross-section will be stressed in compression, onlythe bending stiffness has to be reduced, as theconcrete cross-section can bear the compressive

load again by closing the cracks. From a theoreticalviewpoint this is not exact, because the bendingstiffness will as well rise in case of increasingcompression, but this stiffness increase is not verystable. In case of only small disturbances the cross-section would loss the bending stiffnessautomatically, which from a safety point cannot be

02

<

+ h 02

=

+ h

0

2>

+ h

02 <

h

B B B D D

ten ,0> B D

ten

B Dten

h

B D Dten ,00

B D Dten ,0<

B Dten

>0 B D ,

Table 2. Case differentiation for crack-damage ten D

s

e

0


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accepted anyway. Thus the increase of bendingstiffness with respect to the state of compressionwill neglected. More important is to clearly identifythe surface of the element on which the crack isinitiated.

With these assumptions and with the objective toestablish a simple method to calculate bucklingloads and modes considering damages, thefollowing iterative approach has been developed:

1. Determination of 1, =ied K with strain state zero.

2. Solving of 0

11, PVK = == iied .3. Determination of 1, =id K .4. Solving of eigenvalue problem

( ) 0

K K =+ === 1,11, id iied .5. Determination of 2, =ied K under consideration of displacement state 1=iV .

6. Solving of eigenvalue problem( ) 0K K =+ === 1,22, id iied .

7. Solving of equilibrium equation0

22, PVK = == iied .8. Determination of 3, =ied K and 2, =id K

considering the displacement state 2=iV .9. Solving of the eigenvalue problem

( ) 0K K =+ === 2,33, id iied .

10. Output of the eigenvalues 21 , == ii and 3=i .

The quality of the found buckling load factorsdepends on the quality of the given damage state.With the help of this approach it will become

possible to quantify the influence of damages on the buckling behaviour of a shell, while avoiding time-consuming fully nonlinear numerical analyses insome cases. To identify the appropriate damagestates and corresponding parameters the authorsrefer to the results of SFB 398 at the Ruhr University of Bochum [12].

Further damage effects on concrete structures can be initiated by the phenomena of creep andshrinkage, which have a significant influence on theoverall behavior of a structure. It will not be

considered in the present paper, but has beenactually described in the work of [3].

5. BUCKLING ANALYSES OF CONCRETESHELL STRUCTURES

5.1 Axially compressed cylindrical shell

The axially compressed shell panel is a well-known benchmark test for buckling analysis. Thedimensions and boundary conditions of the chosen

panel have been adapted from Wolmir [23]. The

radius of the panel is taken to m33.83= R , theshell thickness to m10.0=t and the concrete ischosen as a C100/115 with a reinforcement ratio

L

P

R

R

P

v22 1

displacement v 2 [cm]

l o a d f a c t o r

[ - ]

0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64

C 100/115, R = 83.33 m, t = 0.10 m, a s = 9.0% a c

nonlinear material prop.

nonlinear material prop.(imperfect, low amplitude )

linear material properties

nonlinear material prop.(imperfect, high amplitude )

nonlinear material prop. (30% relief)nonlinear material prop. (40% relief)

0.00

0.16

0.32

0.48

0.64

0.80

0.96

1.12

Figure 5. System and loading of an axial compressed shell panel (left) and load-displacement-curve from a modified

stability analysis (right)


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VOL. 47 (2006) No. 3 December n. 152

of %0.9= . The panel can be modelled by an 8x8finite element mesh, considering symmetric

boundary conditions with respect to both axes(Figure 5).

Figure 5 as well gives the load-displacement curvesfor different levels of nonlinear analysis methods, incomparison with results of a classical geometricallynonlinear analysis with elastic material behaviour.In all cases the load has been increasedincrementally close to the point of neutralequilibrium, characterized by snap-through

behaviour. After adding a perturbation vector, theresponse path in the postbuckling range can befound. The magnitude of this perturbationsignificantly influences the numerical convergenceof the solution.

In the geometrically nonlinear, but elastic analysisthe well-known response path with the postbucklingminimum on the level of 70% of the linear bucklingload is found.

The analysis with the nonlinear materialcharacteristics yields a postbuckling minimum,which drops down to a level of 50% compared withthat of the system with linear elastic material. Thismeans that the postbuckling minimum is found at30% level of the linear buckling load. This

minimum, which has to be interpreted as theultimate limit load of the system, can be detectedvia different analysis methods. The full analysis of the response path is very time consuming and has alow convergence rate cause of the high numericalinstability in the neighbourhood of the point of neutral equilibrium. This effort can be minimized

by an analytical decrease of load when approachingthis instability point. When decreasing the load, a

perturbation vector is added in a similar way as inthe full nonlinear analysis. Figure 5 shows also theanalysed response paths for different magnitudes of relief and definitely demonstrates that the

postbuckling minimum can be detected on the samelevel. This approach reduces the computation timeand leads to a safe identification of the lowestultimate load.

In addition, the nonlinear analysis was performedconsidering geometrical imperfections. As can beseen in Figure 5, the response paths are smooth tothe response path of the perfect system.

Further investigations have been made consideringdamage effects, as described in section 4.4. The left

part of figure 6 shows the results with differentkinds of presumed damage states, characterized bythe reduction of the compressive strength in thedifferent -directions. Best correspondence to the

physically nonlinear analysis can be found presuming a damage of 50% in both 1 - and 2 -direction.

Furthermore, the right part of figure 6 shows theresults of the modified linear buckling analysis (seesection 4.4). Here the damage state at the point of

the postbuckling minimum, calculated by the fullnonlinear analysis, is taken as an initial state. Evenin the first step of analysis a good approximation of the real postbuckling minimum is found. Thefollowing iteration step yields a value very close tothe postbuckling minimum of the full nonlinear analysis.

l o a d

f a c t o r

[ - ]

displacement v 2 [cm]0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64

50% damaged (comp.) in 1- and 2-dir.

50% damaged (comp.) in 1-dir.

undamaged, nonlinear material properties

C 100/115, R = 83.33 m, t = 0.10 m, a s = 9.0% a c

0.00

0.16

0.32

0.48

0.64

0.80

0.96

1.12

50% damaged (comp.) in 2

-dir.

displacement v 2 [cm]

l o a d

f a c t o r

[ - ]

0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64

= 0.336= 0.3581. step

1. iteration

0.00

0.16

0.32

0.48

0.64

0.80

0.96

1.12

nonlinear material properties

C 100/115, R = 83.33 m, t = 0.10 m, a s = 9.0% a c

Figure 6. Load-displacement-curve of an axial compressed shell panel with initial compression damage (left) and results of

modified linear stability analysis (right)


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Result: The resulting postbuckling minimum of ageometrical nonlinear analysis, which has often

been assumed as the lowest bound, will drop downsignificantly to 30 % when considering materialnonlinearities and initial damages. Thus a safetyfactor, i.e. = 5.0 in the old German concrete codeDIN 1045, ed. 1988, which has been taken in the

past for the design of thin concrete shells against buckling, can be confirmed by the previouscalculations. With this knowledge a reduction of this factor cannot be justified without further

parameter studies.

5.2 Cooling Tower Shell Grand Gulf

Cooling tower shells are typical thin reinforcedconcrete structures which might be exposed to

buckling phenomena. Thus, to apply the presentedtools and concepts, the cooling tower of the GrandGulf Nuclear Power Plant, Mississippi, USA builtin 1977 has been taken as example. Unusually, thiscooling tower did partly collapse during its erection

because of a tornado which hit the central crane.The shell was repaired and the cooling tower wasfinished, but nevertheless it has been in the focus of many researchers in the past. [14, 15, 17].

Figure 7 shows the discretization with 64x51ASE4-shell elements of half of the structure due tosymmetrical boundary conditions. As well the maindimensions, the distribution of the originalthickness and of the inner and outer reinforcementratio over height are given there.

First, all analyses were made with the materialspecifications of the originally used materials, takenfrom [14]. Later the calculations have been repeatedtaking the specifications of a high strength concreteC80/95. In this case the thickness was reduced by

about 4 cm to get comparable results with respect tothe critical load level. The reduction of thicknessleads to higher stresses due to wind action, whichresults in higher reinforcement ratios. These ratioshave been considered as well. The nonlinear analyses were made for the loadcombination ( )W G + , with G dead load, W wind load and - load factor.

The classical linear buckling analysis leads to a buckling factor of = 9.3 for normal concrete and = 9.6 for high-strength concrete. The corresponding

buckling modes do not differ significantly.

Figure 8 shows the load-displacement curves of both analyses considering linear and nonlinear material behaviour. The results with linear elasticmaterial specifications show the typical snap-through phenomenon as the relevant mode of failure. The corresponding displacement mode ischaracterized by a singular buckle in the region of the largest compressive stresses in the lower sectionof the tower (figure 9).

In case of considering material nonlinearity thestructure shows a completely different response.Again the shells failure is characterized by loss of stability, but now the tower shows a nonlinear load-displacement-relation in the pre-buckling phases

because of cracking of concrete in the windwardmeridian section. Snapping-through occurs at a loadlevel about 30% less that considering linear elasticmaterial behaviour. The correspondingdisplacement pattern shows a completely different

behaviour both in the pre- and in the postbucklingrange. The cracking of the shell in the windwardmeridian section and as well of parts of the upper ring beam lead to buckling in the flanks of thestructure sideward to the windward meridian.

150.54 m

143.22 m

82.03 m

63.02 m

43.95 m

25.02 m

6.02 m

0.00 m

1.02 m

0.20 m

0.23 m

0.26 m

0.76 m

0.34 m

0.20 m

0.30 m

0

15

30

45

60

75

90

105

120

135

150

0 5 10 15 20 25 30

circumferentialreinforcement

meridionalreinforcement

z

reinforcement (inner/outer)[cm/m]

h e i g

h t [ m

]

Fi ure 7. Discretization, ori inal thickness and iven rein orcement ratio on one side o the coolin tower Grand Gul


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VOL. 47 (2006) No. 3 December n. 152

In case of high strength concrete this behaviour does not differ significantly. But, because of smaller ductility, the post-buckling range is nowcharacterized by considerably less displacementstill total failure.

Additionally, the analysis was performed applyingthe approach presented in section 4.4. Under consideration of the damage state near to the failure

state a reduction of about 50% against the classical buckling factor is observed. Within this analysis thedamage induced displacements growth is notconsidered. The detailed results of this example aregiven in [1].

Result: The presented nonlinear algorithms aresuitable for practical application to large scale shellstructures. The consideration of materialnonlinearity yields buckling loads, which areapprox. 70 % lower in comparison to those of alinear elastic analysis. Because of a nearly linear

pre-buckling behaviour of a linear elastic analysisthe reduction as well is about 70 % lower theclassical load via eigenvalue analysis. Thus theretention of a high safety factor, like = 5.0 acc.DIN 1045, ed. 1988, is still justified. It as wellincludes further reductions of the buckling limit dueto imperfections, creep, shrinkage, etc. Thederivation of buckling loads via classical eigenvalueanalysis will only provide an acceptable approach

of the real critical load if the structural behaviour inthe pre-buckling state is nearly linear.

6. CONCLUSIONS

It has been demonstrated that the consideration of material nonlinearity may have a significantinfluence on the stability of reinforced concreteshells. Both in case of bifurcation (axiallycompressed shell panel) or snap-through (coolingtower), the relevant load limit can be down to 30 %of the results of a linear elastic analysis. This

justifies the retention of high safety margins when performing classical stability proofs.

The use of high-strength concrete does not changethe phenomenology of the stability behaviour. Theadvantage of higher strength and greater stiffnesscan be credited only within certain limits, as the realdesign of shell structures demands for minimumshell thickness, due to concrete cover in order toavoid corrosion of reinforcement. Thus bucklingfailure does not seem to be probable, the probabilityof buckling failure seems to be low. Nevertheless,the combination of high strength concrete withtextile, glass or carbon reinforcement [5] might leadto a minimized wall thickness of curved shells.Then the consideration of initial damages will getmore importance, and the presented algorithmsmight allow a realistic approach of the expectedstability phenomena.

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 -2.0 -2.2 -2.4

4.8

8.0

6.4

3.2

1.6

0.0

11.2

9.6=9.40 =8.94

=2.70=3.10

normal strength concrete / linear

normal strength concrete / nonlinear

high strength concrete / linear

high strength concrete / nonlinear

l o a d

f a c t o r [ - ]

horizontal displacement in [m] at = 0 and z = 89.63 m

(G+W)

Figure 8. Load-displacement-curve of different analysis of the cooing tower Grand Gulf


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ACKNOWLEDGEMENTS

The authors want to thank the German ResearchFoundation DFG for funding most of the research

work presented in this paper [1, 10].

REFERENCES

[1] Andres, M. Zum Stabilittsnachweis vonSchalentragwerken aus Hochleistungsbeton,Dissertation, Bergische UniversittWuppertal, Germany (2004).

[2] Bergan, P.G., Horrigmoe, G., Krakeland, B.and Soreide, T.H. Solution techniques for nonlinear finite element problems. Int. J.

Numer. Methods Eng. 12 (1978), 1677-1696.

[3] Bockhold, J. Modellbildung und numerischeAnalyse nichtlinearer Kriechprozesse in

Stahlbetonkonstruktionen unter Schdigungs-aspekten, Schriftenreihe des Instituts fr Konstruktiven Ingenieurbau, Heft 2005-6Ruhr-Universitt Bochum, Germany (2005).

[4] Curbach, M., Hampel, T., Speck, K. and Scheerer, S. Versuchstechnische Ermittlungund mathematische Beschreibung der mehraxialen Festigkeit von Hochleistungs-

beton bei zwei- und dreiaxialer Druckfestig-keit, Zusammenfassung des DFG-ProjektsCU 37/1-2 , TU Dresden, Germany (2000).

=2.84=3.10

=8.94 =7.63

prebuckling range postbuckling range

l i n e a r

( n o r m a l s t r e n g t

h c o n c r e

t e )

n o n

l i n e a r

( h i g h s t r e n g t

h c o n c r e t e )

| displacement | | displacement |

| displacement || displacement |

Figure 9. Displacement patterns in the prebuckling and postbuckling state of a linear and nonlinear analysis


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VOL. 47 (2006) No. 3 December n. 152

[5] Curbach, M., Wagner, A., Hegger, J. and Will, N. Textilbewehrter Beton - Eineinnovative Werkstoffkombination, VDI-

Ingenieurtag 2003, Fachkongress Innova-tives Planen und Bauen - Bautechnik fr dieGesellschaft von morgen VDI-GesellschaftBautechnik (2003), 51-70.

[6] Darwin, D. and Pecknold, D.A. Nonlinear biaxial stress-strain law for concrete, Journal of the Engineering Mechanics Division /ASCE 103(EM2) (1977), 229-241.

[7] Deml, M. Ein Finite-Element-Konzept zur Traglastanalyse imperfektionssensitiver Schalenstrukturen, Berichte aus demKonstruktiven Ingenieurbau, Nr.6/97, TUMnchen (1997).

[8] DIN 1045-1 Tragwerke aus Beton,Stahlbeton und Spannbeton Teil 1:

Bemessung und Konstruktion , Berlin: BeuthVerlag GmbH (2001).

[9] Eckstein, U., Harte, R., Krtzig, W.B. and Wittek, U. Solution strategies for linear andnonlinear instability phenomena for arbitrarily curved thin shell structures. Proc.7th Int. Conf. on Structural Mechanics in

Reactor Technology. Amsterdam: North-Holland (1983), 163-170.

[10] Harte, R. and Andres, M. Grundsatzunter-suchungen zum Stabilittsverhalten vonSchalenkonstruktionen aus Hochleistungs-

beton, Abschlussbericht des Projekts HA1031/3 (DFG) , Bergische UniversittWuppertal, Germany (2004).

[11] Krtzig, W.B. Multi-level modellingtechniques for elasto-plastic structuralresponses. In: Owen D.R.J., Onate, E.,

Hinton, E. (eds.): Computational plasticity , part 1. Barcelona: Int. Center for NumericalMethods in Engineering (1997), 457-468.

[12] Krtzig, W.B., Petryna, Y.S. and Stangenberg, F. Measures of structuraldamage for global failure analysis , Int. J. of Solids and Structures 37 (2000), 7393-7407.

[13] Krtzig, W.B., Wittek, U. and Basar, Y. Buckling of general shells. In: J.M.T.Thompson, G.W. Hunt (Eds.): Collapse ,

Cambridge: Cambridge University Press(1983), 377-394.

[14] Mahmoud, B.E.H. and Gupta, A.K. Inelasticlarge displacement behavior and buckling of hyperbolic cooling tower shells, NorthCarolina State University, USA (1993).

[15] Mang, H.A., Floegl, H., Trappel F. and Walter, H. Wind-loaded reinforced-concretecooling towers: Buckling or Ultimate Load,

Engineering Structures 5 (1983),163-180.

[16] Medwadowski, St. J. Buckling of concreteshells an overview, IASS-Journal Vol. 45(2004), n. 1, 51-63.

[17] Min, C.S. A study of inelastic behavior of reinforced concrete shells usingsupercomputers, PHD-Thesis, North CarolinaState University, USA (1992).

[18] Montag, U. Konzepte zur Effizienz-steigerung numerischer Simulationsalgo-rithmen fr elastoplastische Deformations-

prozesse, Techn.-Wiss. Mitteilung Nr. 97-6,Ruhr-Universitt Bochum, Germany (1997).

[19] Noh, S.-Y. Beitrag zur numerischen Analyse

der Schdigungsmechanismen von Naturzug-khltrmen, Dissertation, RWTH Aachen,Germany (2001).

[20] Plling, R. Eine praxisnahe, schdigungs-orientierte Materialbeschreibung von Stahl-

beton fr Strukturanalysen, Dissertation,Ruhr-Universitt Bochum, Germany (2000).

[21] Ramm, E. (ed.) Buckling of shells , Berlin:Springer-Verlag (1982).

[22] Simo, J.C., Hughes, T.J.R. On theVariational Formulation of Assumed StrainMethods, J. Appl. Mech., 53, 51-54 (1986).

[23] Wolmir, A.S. Biegsame Platten und Schalen .Berlin: VEB Verlag fr Bauwesen (1962).

[24] Zahlten, W. A contribution to the physicallyand geometrically nonlinear computer analysis of general reinforced concrete shells.Technical Report 90-2, Ruhr-UniversittBochum, Germany (1990).


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