Testing the Asymptomatic Behavior

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 54:421452 (DOI: 10.1002/nme.436)

Testing the asymptotic behaviour of shell elementsPart I:

the classical benchmark tests

D. Briassoulis;

Department of Agricultural Engineering; Agricultural University of Athens; Iera Odos 75; Athens 11855; Greece

SUMMARY

The asymptotic behaviour is considered to be one of the most demanding levels of benchmark testingfor shell elements. In the present paper, the asymptotic behaviour of classical benchmark problemsis analytically and numerically investigated. The aim is to examine the possibility of using the clas-sical benchmark tests for testing the asymptotic behaviour of shell elements. Appropriate analyticalapproaches are introduced to investigate the asymptotic behaviour of the classical benchmark problems.The reformulated four-node shell element (RFNS) is employed in the numerical analyses. It is shownthat the classical benchmark tests, in addition to testing the reliability and robustness of shell elements,also represent strong challenging tests for the asymptotic behaviour of shell elements. In the course ofthe numerical investigation of the asymptotic behaviour of the classical benchmark problems, the reli-ability and eciency of the RFNS element already established by means of the classical congurationof the benchmark tests is re-conrmed in all cases of the corresponding asymptotic test congurations.Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS: nite shell elements; asymptotic behaviour; benchmark tests

1. INTRODUCTION

The performance of a large variety of shell nite elements and the development of newelements represent an open research eld, a rather attractive one, for many years now. Shearand membrane locking problems, as well as zero energy modes phenomena have been themost common problems for the classical C0 shell elements. Despite the eorts made so far toeliminate these problems and the remarkably successful, in a few cases, schemes proposed,the discussion on the performance, robustness and reliability of the corresponding elementsstill goes on [1; 2].

The present paper has been motivated by the work of Chapelle and Bathe [3], where some

fundamental considerations regarding the reliability of the nite element analysis of shellstructures are presented. The rst attempt in this course was made through the introduction ofan analytical test for the evaluation of plate bending elements. However, the corresponding

Correspondence to: D. Briassoulis, Agricultural Engineering Department, Agricultural University of Athens,Iera Odos 75, 11855 Athens, Greece

E-mail: [email protected]

Received 19 April 2000Copyright ? 2002 John Wiley & Sons, Ltd. Revised 7 September 2001


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422 D. BRIASSOULIS

infsup condition for ReissnerMindlin plate elements [4] was found to be very dicult toevaluate analytically [3]. Therefore, a numerical test methodology was developed to evaluatethe infsup condition numerically. The infsup test was furthermore elaborated in recentworks with special emphasis placed on shell elements [5; 6].

Plate elements which pass the infsup test are considered [5] to provide optimal discretiza-tions, that is the convergence rate is the maximum possible for the interpolations used and itis independent of the plate thickness used. ReissnerMindlin theory-based elements are veryeective when they satisfy two crucial mathematical conditions: the ellipticity and the relevantinfsup conditions (this has already been shown with MITC elements) [5]. Concerning shellelements, the ellipticity condition is satised if the formulation of the element ensures thatit contains the six physical rigid body modes and no spurious zero energy modes indepen-dently of the application or discretization [5]. This is an easy test to perform. Satisfactionof the infsup condition, on the other hand, means that the convergence properties of thenite element spaces ensure that the optimal rate of convergence is obtained for sucientlysmooth solutions. However, an analytical approach to identify whether the infsup conditionis satised is very dicult to develop (or many limitations have to be introduced) [5], be-cause the geometry and the boundary conditions are crucial ingredients in such an approachand the meshes used should also be considered. Thus, numerical tests may be used instead.Such a numerical infsup test methodology suitable for general shell elements, which is alsoconsidered to be a very useful tool in the search for improved shell elements is presented inReference [5].

Nevertheless, one should bear in mind also the limitations of the infsup test results. Thus,a shell element which satises the infsup condition is not guaranteed to perform well inmembrane-dominated problems [5]. Moreover, the evaluation of the discrete infsup conditionis geometry- and mesh-dependent: an element which is shown to pass the test under a specicconguration may not pass the test for a dierent conguration in terms of dierent geometryand=or mesh sequences [5]. Hence, since the evaluation of the discrete infsup condition

is problem-dependant, an appropriate problem should be selected for this purpose. Such agood problem is considered to be the hyperbolic paraboloid shell [7] (this problem will beconsidered in a future work).

Numerical evaluation is evidently the best available tool for testing the asymptotic behaviourof shells. The basic idea proposed in Reference [3] is that any test for shell elements becomesmore drastic if the thickness is allowed to go to very small (e.g. a set of drastic tests isobtained by allowing the thickness of the classical benchmark test congurations to decrease;t 0). To this end, some of the classical benchmark tests for shell elements were analysed[3] in view of their asymptotic behaviour. In the same work, a new set of limit tests, cleanerwith respect to identifying the asymptotic behaviour of shell elements, was proposed [3].

The present work is divided into two parts, dealing with the asymptotic behaviour of theclassical benchmark problems (Part I) and the newly introduced limit tests (Part II), respec-

tively. In the present paper (Part I), the behaviour of classical benchmark problems withrespect to their asymptotic behaviour and thus, their performance in testing the asymptoticbehaviour of shell elements is both, analytically and numerically investigated. Appropriateanalytical approaches are developed to investigate the asymptotic behaviour of the classical

benchmark tests. In some cases, simple physical models are introduced that oer a simplemeans of explaining the asymptotic behaviour of the problem under investigation. In ad-dition to the analytical solutions, extensive numerical analyses are employed to study the

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng2002; 54:421452


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TESTING THE ASYMPTOTIC BEHAVIOUR OF SHELL ELEMENTSPART I 423

asymptotic behaviour of these classical problems. The performance of the nite element mod-els is monitored against the physical convergence characteristics of the solutions obtained aswell as against the analytical solutions developed in the present work. The degree to whicheach limit test investigated represents a challenging test, with the development of strong

boundary eects over very narrow boundary layers, as the shell thickness is reduced, possi-bly also associated with the development of competing deformation modes in those zones, ispointed out in the two papers along with the specic diculties experienced.

The main purpose in the present work is to oer insight into the mechanisms developedby the shells of the corresponding benchmark problems as the shell thickness is reduced.It is expected that the investigation of these mechanisms would allow for an evaluation ofthe critical parameters in employing the particular classical problems also as benchmark testsfor testing the asymptotic behaviour of shell elements. In that way, physical convergencecharacteristics may be described in terms of the behaviour of a shell element in modellingextremely thin local boundary layers (e.g. emphasis is placed on the relationship betweenconvergence characteristics and behaviour within the boundary zones in connection with themeshing schemes used; of course, physical convergence characteristics may be considered inaddition to the classical appropriate convergence characteristics as described elsewhere in theliterature [6]).

The reformulated four-node shell (RFNS) element is employed in the present work forthe numerical investigation of the asymptotic behaviour of the classical benchmark problems.The RFNS element, a promising reliable and robust shell element, was introduced recently,applicable to both at and warped shell element congurations [8]. This shell element doesnot experience any locking phenomena or zero energy modes, performing equally well in atas well as in warped shell element congurations. The basic element formulation is basedon physical considerations and does not include heuristic numerical procedures aimed atavoiding locking or possible development of mechanisms. The performance of this elementhas been found to be favourably comparable to that of the few well-established shell elements.

The analysis of the RFNS element performance in this work has a parallel, supportivecomparative character, next to the analytical investigation of the asymptotic behaviour ofthe limit tests. Along this line, the asymptotic behaviour of the RFNS element is comparedagainst the corresponding analytical solutions in a way that: (a) the asymptotic behaviour ofthe element is investigated with respect to both, the classical (Part I) and the newly proposedlimit tests (Part II); (b) the converged solutions obtained numerically verify, indirectly, thecorresponding analytical solutions and allow for the investigation of the physical behaviourand modelling diculties of each limit test through their convergence characteristics.

2. THE RFNS ELEMENT UNDER THE CLASSICAL BENCHMARK TESTS

The formulation of the reformulated C0 shell element is presented in detail in Reference [8].The performance of the RFNS element as it has been veried against a wide range of themost well-known severe benchmark tests for shell elements [8] may be described briey asfollows:

The RFNS element was shown to pass successfully the thin plate bending patch test, aswell as the plane stress patch test [8]. The classical benchmark tests examined include the two



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424 D. BRIASSOULIS

cases of the pinched cylindrical shell with free ends, representing a relatively thick and thinshell. Also, the pinched cylinder supported by rigid diaphragms at the two ends, a problemrepresenting a severe benchmark test for shell elements was employed. Another classical testis the ScordelisLo roof problem. This problem is known as representing a critical test for

the ability of a shell element to represent complex states of membrane strain. The pinchedhemispherical shell, designed with a 18 hole at the top allowing an easier modelling withat shell elements, represents a problem which tests the ability of a shell element to representinextensional bending modes. The main question in this test concerns the ability of the shellelement to model rigid body modes (rotations) without developing spurious strain energy.In a second modelling version, the full hemisphere is modelled by a mesh which requiresthe use of highly distorted, warped, elements. In this form, the problem is used to test the

behaviour of shell elements with respect to warping, in addition to testing their behaviour inmodelling inextensional bending modes. The twisted beam subject to in-plane and out-of-planetip concentrated loads, represents one of the most severe benchmark tests for testing the eectof warping on the behaviour of shell elements. Two cases were employed along with this

problem, one corresponding to a thick twisted shellbeam and another one corresponding toa thin twisted shellbeam.

The results obtained in all problems mentioned above indicated a fast convergence for theRFNS element models to the corresponding theoretical values [8] (note that these problemswere tested in their classical conguration and not in the form of limit asymptotic problems).

3. ANALYSIS OF THE ASYMPTOTIC BEHAVIOUR OF THE CLASSICALBENCHMARK PROBLEMS

The asymptotic behaviour of the problems under investigation is traced numerically followingphysical convergence characteristics of the nite element solution as the shell thickness is

reduced. In addition, analytical asymptotic solutions are derived for the problems investigatedin order to allow monitoring and comparisons of the numerical analysis results.

3.1. The pinched cylinder with free ends

A circular cylindrical shell with free ends, pinched at the middle by two opposite forces Pact-ing along a diameter represents, in its asymptotic behaviour, a clear case where inextensionaldeformation prevails [9].

The analysis of the shell, assuming that the three strain components in the middle surfacevanish, yields an analytical series solution for the deformations. The radial deformation isgiven by [9]

w =2Pr3

DL

n=2;4;6;:::

1

(n2 1)2 cos n (1)

where D is the exural rigidity of the shell, r is the radius and L is the shell length.The nite element model employed in the numerical analysis is shown in Figure 1. Two

uniform meshes were used in this case, a 1515 and a more rened 17 30 mesh, modellinga symmetric quadrant of half the cylinder (note: the scheme 17 30 stands for a number of17 nodes in the meridional direction times and a number of 30 nodes in the circumferential



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X

Free edge

Line of symmetry

L

Line of symmetry

r

t

P=1

P=1Z, wz

Y

w

u

v

Figure 1. Finite element model of one-eighth of the classical benchmark test of the pinched cylindrical

shell with free ends (E= 10:5 106, = 0:3125, r= 4:953, L = 10:35, t= 0:094).

direction; the use of a 25 35 mesh, rened along the free boundary, simply conrmed theresults obtained by the 17 30 mesh in terms of displacements). The deformed shell obtained

by the nite element analysis for the classical problem (t= 0:094) [10], reveals a ring-type

load-carrying mechanism (Figure 2; just the same picture was obtained for the limit thinshell case with t= 0:000094). That is, the load is mainly carried by means of the ring-type

bending resistance of the shellarch structure. This may be readily conrmed by calculatingthe deformation of a ring pinched by two opposite forces. The maximum ring deection underthe load may be obtained analytically (herein calculated by using energy methods) as follows:

w =Pkr

8AG +

Pr

8AE +

Pr3

2EI

4

2

(2)

where A is the cross-sectional area of a ring of width equal to the shell length L, G is theshear rigidity, I is the moment of inertia of the cross-section of the ring-shell (including thePoisson ratio eect) and k is the shear correction factor. The results obtained by the two

equations are almost identical (for t= 0:000094, the ring solution diers from the convergedseries solution by 0.096 per cent). In the limit of tbeing very small, the asymptote may besimply approximated by the last term of the analytical solution of Equation (2).

The analytical [9] and the nite element solutions for the radial deection along thecrown line ( = 0), as depicted in Figure 2 for a very thin shell (thickness t= 0:000094;t=r= 1:90 106), coincide. Note that in this very thin shell case there is no apparent bound-ary layer development in the displacements eld along the free ends and it is because of



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426 D. BRIASSOULIS

-2.50E+02

-2.00E+02

-1.50E+02

-1.00E+02

-5.00E+01

0.00E+00

0.0 0.2 0.4 0.6 0.8 1.0

x/r

0.094/17x30

0.000094/17x30

0.000094/15x15

0.094/15x15

-2.02E+02

-2.00E+02

-1.98E+02

-1.96E+02

-1.94E+02

-1.92E+02

-1.90E+02

-1.88E+02

0.00 0.10 0.20 0.30 0.40 0.50

y/L

wz/(r/Et3)

wz/(r/Et3)

0.000094/15x15

0.094/15x15

anal

0.094/17x30

0.000094/17x30

ring

(a)

(b)

Figure 2. Circumferential distribution: (a) of vertical displacements wz at the middle (y =L=2) andmeridional distribution; and (b) along the crown line ( = 0) of the pinched cylindrical shell with free

ends (normalized with respect to r=Et3).

this behaviour that the problem represents a rather simple test for testing the asymptotic be-haviour of shells. In the case of the relatively thicker shell (t= 0:094; i.e. the classical bench-mark test), the radial deection varies slightly along the crown line, but it still remains veryclose to the asymptotic one. The maximum displacement in this case, obtained by the RFNSelement using the 1730 mesh is 0.1136, in agreement with the exact solution of 0.1139[8]. The slight variation of the displacements in the thick shell case may be attributed to the

increased contribution of the membrane strain energy component to the total strain energy asthe shell thickness increases. This is due to: (a) The fact that a relatively small portion of theconcentrated load is taken-up by membrane-exural action [11] in the meridional directionof the relatively thick shell, while most of the load is still carried by ring-bending action.(b) The need that the boundary conditions (membrane stress resultants due to (a) along withthe associated bending moments and eective transverse shear stress resultants distribution)are satised at the free ends. The corresponding weak membrane-exural action is extended



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-1.00E-01

-8.00E-02

-6.00E-02

-4.00E-02

-2.00E-02

0.00E+00

2.00E-02

0.00 0.01 0.02 0.03 0.04

y/L

yz

/(r/Et2) 0.000094/25x35

0.00094/25x35r

0.000094/25x35r

0.0094/25x35

-2.00E+02

0.00E+00

2.00E+02

4.00E+02

6.00E+02

8.00E+02

1.00E+03

0.00 0.01 0.02 0.03 0.04

y/L

/

(r/Et2)

0.000094/25x35

0.000094 an

0.00094 an

0.00094/25x35r

0.0094 an

0.0094/25x35

(a)

(b)

Figure 3. Meridional distribution of rotations : (a) and transverse shear strain yz; (b) along the

crown line of the pinched cylindrical shell with free ends (mesh 25 35; normalized with respect tor=Et2).

over the whole length of the relatively thick cylinder. However, this action is limited overa small portion of the shell in the circumferential direction, representing a relatively narrowzone in the vicinity of the load (refer to (a) above; Figure 2(a)). Note that the use of a rened25 35 mesh revealed the presence of a weak non-uniform displacement pattern, analogousto that of the thick shell, also in the displacement eld of the thin shell cases. This coupledboundary layermembrane load-carrying mechanism eect is detectable only at a very localscale. As the shell thickness is reduced, the boundary layer eect becomes much weaker and itis limited to a very narrow boundary zone, until at t=r= 1:90106 it eventually disappears.

In the thin shell cases, signicant boundary layer eects may be detected in the rotationsand the shear strain elds. The investigation of the rotations eld with the rened 25 35(Figure 3(a)) mesh reveals a highly localized boundary layer developing along the free bound-ary in the case of the very thin shell with t=r= 1:90106 (the rotations vector is tangentto the cylinder, perpendicular to the meridional line). The local disturbance of the rotationseld stems from the requirement that the meridional bending moments My= M due to thePoisson ratio eect on the ring-type bending moments M are zero along the free boundary.



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428 D. BRIASSOULIS

This may be investigated analytically by calculating the bending moments My (based on thering analysis, or on Equation (1)), and then by applying opposite bending moments alongthe free boundary (note that the analytical calculation of the bending moments was conrmedwith the nite element results). These end moments are responsible, in turn, for the devel-

opment of local disturbances in the rotations and transverse shear strain elds (Figure 3;boundary zone). In the case of thin shells, the local disturbances are absorbed by the beamon elastic foundation action of the cylinder very quickly and they fade away within a veryshort distance from the boundary [11]. The local disturbances represent a

t=r scale eect, as

shown by the analytical solution of the rotations in Figure 3(a) (the analytical solution wasobtained by employing the beam on elastic foundation theory; note: length scale is the eec-tive length, measured from the boundary, over which the disturbances retain a signicant

presence; this length in the present case is in the order of

t=r). In addition to the

t=rscale eect, this particular boundary layer also contains a length scale eect proportional tot=r. Such eects require extremely rened meshes along the boundary to capture. Especiallyfor the very thin shell cases much more rened meshes should be used in the circumferentialdirection. A similar, however quite stable, behaviour within an extremely thin boundary layerwas observed with the nite element solution of the transverse shear strain elds yz. A s aresult of the inadequacy of the mesh schemes used to capture accurately the extremely thin

boundary layer of the rotations eld in the very thin shell cases, the transverse shear straindistribution fails to converge to a zero value at the free boundary (cases of t=r= 1:90 106

and 1:90 105; Figure 3(b)). Apart from the accuracy of the simulation of the boundarylayer eects, in energy terms, in the case of thin circular cylindrical shells with free ends, the

boundary layer eects represent a pure beam bending localized action [11]. Thus, they justadd up to the prevailing ring-type bending action so that a pure bending mode of deformationdominates the shell behaviour. Thus, if convergence is measured in terms of some type ofenergy characteristics, one may be misled as to whether the boundary conditions are modelledappropriately.

The asymptotic characteristics of the problem may be described in terms of the relativeenergy shares for bending, shear and membrane energy as shown in Figure 4. The nite ele-ment results for the energy components (using the 1730 mesh) conrm that the transverseshear strain energy is very low even for the t= 0:094 case and dissipates rapidly and asymp-totically to zero. This conrms that no shear locking develops with the RFNS element in thiscase. The bending strain energy dominates the total strain energy by 98.35 per cent in the caseof the relatively thick shell and this domination becomes absolute very quickly as the shell

becomes thinner. The membrane energy has a small contribution of 1.23 per cent in the thickshell and fades away asymptotically as t decreases. As a result, the pure bending-dominated

behaviour is numerically and analytically conrmed and well modelled by the RFNS element.Thus, the problem of the pinched cylinder with free ends, apart from being a classical

benchmark problem for shell elements may also be considered as a simple test for testing the

asymptotic behaviour of shell elements under bending domination. It may not be consideredto be a drastic test though, except for the boundary layer eects.

3.2. The pinched cylinder with diaphragm-supported ends

The pinched cylinder supported by rigid diaphragms at the two ends (Figure 5), representsone of the most severe classical benchmark tests for shells. The nite element analysis of



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0

20

40

60

80

100

120

1E-061E-050.00010.0010.010.11

t/r

Energy(%) Bending

Membrane

Shear

Figure 4. Asymptotic behaviour of the pinched cylindrical shell with free ends in terms of the bending,

shear and membrane energy shares (mesh 17 30).

X

Diaphragm

Line of symmetry

L

Line of symmetry

r

P=1

P=1Z, wz

Y

w

u

v

Figure 5. Finite element model of one-eighth of the pinched cylindrical shell with end diaphragms(E= 3:0 107, = 0:3, r= 100, L = 200, t= 1:0).

the classical problem reveals a deformed shell which implies the activation of a complexshell-mode load-carrying mechanism [8; 10] (in contrast to the arch-mode mechanism of the

previous problem).



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430 D. BRIASSOULIS

In order to be able to investigate the asymptotic behaviour of this problem, an analyticalsolution was obtained based on the exact relations of Flugges bending theory of a circularshell [12]. In particular, the normal load component pr (i.e. the load in the radial direction)is expressed in the following double Fourier series form (tangential p and meridional pyload components are zero):

pr=

m=0

n=1

prmncos(m)sin

y

r

(3)

where = nr=L. The normal load harmonic term prmn is calculated so that the series yieldstwo anti-diametrical concentrated loads at = 0; y =L=2 and = ; y =L=2 (i.e. Equation(3) represents a singular periodic function).

The deformations (and in an analogous way the rest of the symmetric and antisymmetricstrain and stress components) are also given in terms of double Fourier series of the followingform:

u =

m=0

n=0

umnsin(m)siny

r

v =

m=1

n=1

vmncos(m)cos

y

r

w =

m=0

n=1

wmncos(m)sin

y

r

(4)

where each set of the superimposed harmonic terms umn, vmn and wmn is calculated for thecorresponding term of the series of Equation (3) describing the load, as a solution to a set ofthree linear equations. These equations are derived from the corresponding three dierentialequations for u; v and w (i.e. the displacements at the shell middle surface) of Flugges

bending theory of a circular shell by substituting the corresponding expressions of Equation

(4) [12].The convergence characteristics of the derived analytical solution are shown in Figure 6 forthe classical problem (t= 1) and for the limit case of t= 0:0001. It is shown that convergencefor the series describing the deformations of Equation (4) is achieved for a total numberof terms equal to m = n = 600 and 4000, respectively. The maximum displacements underthe point load (i.e. the most critical quantity for achieving convergence) for the classical

problem (t= 1) and for the limit case of t= 0:0001 are found to be 0:548105 and 5440:0,respectively (reported in the literature value [8] for t= 1 is 0:5475 105; converged valuefor t= 0:0001 at n = m = 6000 is 5446.0).

By comparing the numerical against the analytical solution, the vertical component of thedisplacements wz (Figure 5) is used to describe the corresponding deformation elds. Thedisplacement distributions obtained by the RFNS element for the classical case of t=1, as

shown in Figure 7, coincide with the analytical results, re-conrming the excellent behaviourof the element.

The investigation of the asymptotic behaviour of the RFNS element, when reducing theshell thickness did not reveal any special problems, other than the need to use much betterrened meshes in the vicinity of the load. The nite element analysis in the limit case oft= 0:0001 (t=r= 106) resulted in a deformed shell structure which, in general terms of theload-carrying mechanisms, appears to be similar to the corresponding deformed shell of the



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-6000

-5500

-5000

-4500

-4000

-3500

-3000

-2500

-2000

0 1000 2000 3000 4000 5000 6000

n

wmax

t=0.0001

-5.484E-06

-5.480E-06

-5.476E-06

-5.472E-06

-5.468E-06

-5.464E-06

-5.460E-06

0 500 1000 1500 2000 2500

n

wmax

t=1

(a)

(b)

Figure 6. Convergence of Fourier series solution for the deformations of pinchedcylindrical shell with end diaphragms (wmax is the max displacement under load;

n is the number of harmonics in each direction).

classical problem (t= 1). The main dierence observed may be described by the fact thatthe ring bending mode exhibits a pronounced attenuation in the circumferential direction,away from the load. This may be attributed to the increased equivalent elastic foundationaction of the shell against the ring-bending mode of deformation (the elastic foundationaction is due to the membrane type of resistance of the meridional bres supported by the

diaphragm at the ends). This behaviour is analogous (in general, reversed, terms of course) tothe phenomenon of the rapid attenuation of the edge eects of the meridional beam-bendingmode of deformation of cylindrical shells due to the relatively increased equivalent elasticfoundation eect of the ring-shell, when the shell thickness is reduced [13].

The vertical displacements wz obtained in the limit case of t= 0:0001 (t=r= 106), as it

is shown in Figure 8, are quite similar to the corresponding analytical results. Improvingthe mesh renement in the zone near the concentrated load yields results even closer to



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432 D. BRIASSOULIS

-2.0E+00

-1.5E+00

-1.0E+00

-5.0E-01

0.0E+00

5.0E-01

1.0E+00

0.0 0.2 0.4 0.6 0.8 1.0

x/r

wz/(r/Et2)

1-anal-600

1-17x30

-1.8E+00

-1.6E+00

-1.4E+00

-1.2E+00

-1.0E+00

-8.0E-01

-6.0E-01

-4.0E-01

-2.0E-01

0.0E+00

0.0 0.1 0.2 0.3 0.4 0.5

y/L

wz/(r/Et2)

anal-1-600

1-17x30

(a)

(b)

Figure 7. Circumferential distribution: (a) of vertical displacements wz at the middle (y =L=2) andmeridional distribution; and (b) along the crown line ( = 0) of vertical displacements wz of thepinched cylindrical shell with end diaphragms (t= 1) obtained by the RFNS element (mesh 17 30) and

analytically (Fourier series solution; n = m = 600), (normalized with respect to r=Et2).

the analytical solution. Even in the case of very thin shells, t= 0:00001 (t=r= 107), theRFNS element exhibits a stable behaviour simulating the general shell behaviour very wellas described by the analytical solution (Figure 9). The most demanding criterion, that iscalculating the maximum displacement under the load, was used to monitor the asymptotic

behaviour of the nite element solution. The same criterion was used to evaluate the eectof mesh renement on the convergence characteristics of the solution. Attention should be

paid to the fact that, in very thin shell applications, the convergence characteristics withrespect to the highly localized region where the displacement obtains its maximum value(i.e. the meridional narrow zone under the load; Figure 9(b) and 9(c)) do not necessarilyrepresent the overall convergence behaviour of the nite element model (refer to Figure 9(a)and (b)). In terms of the asymptotic behaviour of the nite element under investigation the



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-2.0E+01

-1.5E+01

-1.0E+01

-5.0E+00

0.0E+00

5.0E+00

1.0E+01

0.0 0.2 0.4 0.6 0.8 1.0

x/r

wz/(r/Et2)

1-anal-600

0.0001-17x30

0.0001-17x30r

0.0001-anal-4000

0.0001-24x23

0.0001-35x25

-2.0E+01

-1.5E+01

-1.0E+01

-5.0E+00

0.0E+00

5.0E+00

1.0E+01

0.00 0.05 0.10 0.15 0.20

x/r

wz/(r/Et2)

0.0001-17x30

0.0001-17x30r

0.0001-anal-4000

0.0001-24x23

0.0001-35x25

-1.8E+01

-1.6E+01

-1.4E+01

-1.2E+01

-1.0E+01

-8.0E+00

-6.0E+00

-4.0E+00

-2.0E+00

0.0E+00

0.0 0.1 0.2 0.3 0.4 0.5

y/L

wz

/(r/Et2)

anal-1-600

0.0001-17x30

0.0001-17x30r

0.0001-24x23

0.0001-35x25

anal-0.0001-400

(a)

(b)

(c)

Figure 8. Circumferential distribution: (a, b) of vertical displacements wz at the middle (y =L=2)and meridional distribution; and (c) along the crown line ( = 0) of the pinched cylindrical shellwith end diaphragms (t= 1; 0:0001) obtained by the RFNS element (17 30r: more rened near theload than 17 30) and analytically (Fourier series solution; n = m =600 for t=1 and n = m = 4000

for t= 0:0001), (normalized with respect to r=Et2).



14/32

434 D. BRIASSOULIS

-3.5E+01

-3.0E+01

-2.5E+01

-2.0E+01

-1.5E+01

-1.0E+01

-5.0E+000.0E+00

5.0E+00

1.0E+01

1.5E+01

0.0 0.2 0.4 0.6 0.8 1.0

x/r

wz/(r/Et2) 0.00001-anal-4000

0.00001-17x30r

0.00001-24x23

0.00001-35x25

-3.5E+01

-3.0E+01

-2.5E+01

-2.0E+01

-1.5E+01

-1.0E+01

-5.0E+00

0.0E+00

5.0E+00

1.0E+01

1.5E+01

0.00 0.05 0.10 0.15 0.20

x/r

wz/(r/Et2) 0.00001-anal-4000

0.00001-17x30r

0.00001-24x23

0.00001-35x25

-3.5E+01

-3.0E+01

-2.5E+01

-2.0E+01

-1.5E+01

-1.0E+01

-5.0E+00

0.0E+000.0 0.1 0.2 0.3 0.4 0.5

y/L

wz/(r/Et2)

anal-0.0001-4000

0.00001-17x30r

0.00001-24x23

0.00001-35x25

anal-0.00001-4000

(a)

(b)

(c)

Figure 9. Circumferential distribution: (a, b) of vertical displacements wz at the middle (y =L=2) andmeridional distribution; and (c) along the crown line ( = 0) of the pinched cylindrical shell withend diaphragms (t= 0:00001) obtained by the RFNS element and analytically (Fourier series solution;

n = m = 4000), (normalized with respect to r=Et2).



15/32


-3.5E+01

-3.0E+01

-2.5E+01

-2.0E+01

-1.5E+01

-1.0E+01

-5.0E+00

0.0E+00

1.E-071.E-061.E-051.E-041.E-031.E-02

t/r

wz-max/(r/Et2

)

15x15

24x23

analytical

35x25

Figure 10. Asymptotic behaviour of the numerical solution of the pinched cylindrical shell with end

diaphragms in terms of the maximum vertical displacements wzand mesh renement.

convergence characteristics concerning modelling of a highly localized region (or a boundaryzone) should be distinguished from the overall convergence behaviour of the model. Thisis because the local convergence characteristics may simply describe the compliance of themesh conguration used with the particular demanding meshing requirements imposed by theextremely thin loadboundary zone in very thin shells (t=r= 107). The results shown inFigure 10 leave no doubt about the performance of the RFNS element and the stability ofthe solution as the convergence of the maximum displacement under the concentrated loadfollows steadily the local mesh renement within the narrow zone near the load. Away fromthis zone, convergence has already been achieved with much coarser meshes.

Another important measure of the asymptotic performance of the element is the rotations

eld. The numerical results obtained in the two very thin shell cases oft= 0:0001 (t=r= 106)and t= 0:00001 (t=r= 107) by the RFNS element are presented in Figure 11, respectively. Itis apparent that the highly demanding dramatic change of the rotations eld in the vicinity ofthe load is simulated in a rather stable way. The distributions of the rotations near the load arefound to be in excellent agreement with the corresponding analytical solutions. Note that theanalytical solutions for the rotations eld were obtained by using a higher number of harmonics(m = n = 6000). This is due to the fact that the convergence requirements for the rotationsare more demanding than the corresponding requirements for the displacements (still a highernumber of harmonics would be required to achieve complete convergence for the rotations).

Thus, it is shown that this problem represents a mixed-deformation mode problem. Boththe membrane and the bending load-carrying mechanisms remain active even for extremelythin shells (at least very thin, as practically, i.e. in engineering applications terms, is needed

to test safely the asymptotic behaviour of shell elements). The fact that both load-carryingmechanisms remain active when the shell thickness is reduced was investigated analyticallyas follows. A dimensionless quantity k dening the ratio of the exural rigidity K to theextensional rigidity D:

k= K

Dr2=

t2

12r2 (5)



16/32

436 D. BRIASSOULIS

-6.0E+00

-5.0E+00

-4.0E+00

-3.0E+00

-2.0E+00

-1.0E+00

0.0E+00

1.0E+00

0.0 0.1 0.2 0.3 0.4 0.5

y/L

/(r/Et2)

0.0001-35x25

anal-0.0001-6000

-1.4E+01

-1.2E+01

-1.0E+01

-8.0E+00

-6.0E+00

-4.0E+00

-2.0E+00

0.0E+00

2.0E+00

0.0 0.1 0.2 0.3 0.4 0.5

y/L

/(r/Et2)

0.00001-35x25

anal-0.00001-6000

(a)

(b)

Figure 11. Meridional distribution of rotations along the crown line of the pinched cylindrical shellwith end diaphragms ( = 0; t= 0:0001 and 0.00001) obtained by the RFNS element (mesh 35 25)

and analytically (Fourier series solution; n = m = 4000 and 6000), (normalized with respect to r=Et2).

was set equal to zero (k= 0) and to an articially high value (k 107), in turn. The resultsobtained from the corresponding analytical solutions for the case of t= 0:0001, shown inFigure 12 (normalized appropriately for illustrative purposes), suggest that

fork= 0, the bending stiness is articially removed and the membrane load-carryingmechanism dominates the shell behaviour. It is clear in Figure 13 that this mechanism

does not involve any ring action in the circumferential direction. Instead, the load iscarried along the meridional direction through the membrane action of the bres

passing under the load in combination with the highly localized elastic out-of-planedeformation of the diaphragm (like a cable supported by springs; the spring-likeresistance to the localized out-of-plane deformation of the rigid diaphragm is not

provided by the diaphragm itself, since no physical diaphragm exists). The elasticend-resistance is provided indirectly by the membrane action of the meridional bres



17/32


-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

0.0 0.2 0.4 0.6 0.8 1.0

x/r

normalisedwz

m-asymptote

b-asymptote

Figure 12. Circumferential distribution of vertical displacements wz at the middle of the pinched cylin-drical shell with end diaphragms (y =L=2; t= 0:0001) obtained analytically for the cases ofk= 0 (mem-

brane-asymptote) and k 107 (bending-asymptote; Fourier series solution; n = m = 1600), (normalizedwith respect to 107 r=Et2 and 1010 r=Et2 for the membrane and bending asymptotes, respectively).

Figure 13. Deformed pinched cylindrical shell with end diaphragms (t= 0:0001; scaling factor

for displacements: 3 109) obtained analytically for the cases of k= 0 (membrane-asymptote;

Fourier series solution; n = m = 1600).

adjacent to those carrying the load, in combination with the coupling of the enddeformation of the bres to simulate a diaphragmatic action (Figure 13).

fork107, the ring bending action dominates the shell behaviour. Here, a rather softequivalent elastic foundation action is oered by the corresponding weak membrane



18/32

438 D. BRIASSOULIS

0

10

20

30

40

50

60

70

1.E-071.E-061.E-051.E-041.E-031.E-02

t/r

energy(%

)

Bend-17x30

Membr-17x30

Bend-17x30r

Membr-17x30r

Membr-35x25

Bend-35x25

Bend-Anal

Memb-Anal

Shear-Anal

Shear-35x25

Bend-15x15

Membr-15x15

Figure 14. Convergence characteristics of membrane, shear and bending strain energy shares of the

pinched cylindrical shell with end diaphragms as the shell thickness is reduced.

resistance of the meridional bres, as compared to the articially increased bendingstiness. This is obvious by the absence of a pronounced attenuation of the dominantring-type bending deformation.

It is apparent from comparing this illustrative example (of a very thin pinched cylindersupported by rigid diaphragms; t=r= 106) against the corresponding exact analytical andnumerical results, that a mixed deformation mode dominates the limit problem as the shellthickness is reduced. That is, the asymptotic behaviour of the shell is the result of the com-

bination of two main modes, each activated along one of the two principal directions of

the cylinder: the membrane-exural mode is dominant along the meridional direction and thering-bending mode of deformation is dominant in the circumferential direction. This behaviourmay be further examined in terms of the strain energy components.

The asymptotic characteristics of this problem in terms of the relative energy shares forbending, shear and membrane strain energy are shown in Figure 14 (the shear strain energyis negligible and it is not shown here for the clarity of the gure). Figure 14 suggests anon-uniform convergence rate to a mixed situation where both, membrane and bending strainenergies have a signicant contribution. By using a coarse mesh (15 15), a trend appearstowards a membrane domination. However, mesh renement (already shown to give resultsfor the displacements steadily converging to those of the analytical solution, Figure 10) re-sults in a relatively increased contribution of the bending strain energy share and in a moreuniform convergence rate. The slight tendency towards an increase in the membrane en-

ergy share, which is still shown to be present with intermediate mesh congurations (e.g.17 30 mesh), becomes systematically much weaker with better-rened meshes, until themembrane and bending shares are stabilized, converging to the corresponding values calcu-lated analytically. Note that a still better renement (than the one used herein) is neededfor achieving a complete convergence in the very thin limit shell cases of t=r= 106 and107 (refer to Figure 10). Nevertheless, the membrane and bending energy shares obtainedwith the 35 25 mesh are quite similar to the corresponding values obtained analytically,



19/32


Z, wz

Y

X

Line of symmetry

Lt

r

Free edge

Diaphragm

40

wv

u

-

Figure 15. Finite element model of one quadrant of the ScordelisLo cylindrical roof problem.

(E= 3:0 106, = 0:0, r= 300, L = 600, t= 3:0, q = 0:625 unit force=unit area).

of 62.46 and 37.54 per cent, respectively. Furthermore, as shown in Figure 14, the shearstrain energy (1.97 per cent numerically, 2.08 per cent analytically) dissipates rapidly withthe decrease of the shell thickness, re-conrming the locking free behaviour of the RFNSelement.

Thus, it has been shown numerically and analytically that the asymptotic behaviour of theproblem of the pinched cylinder supported by rigid diaphragms is described by a rather stablecontribution of both membrane and bending strain energies. As far as benchmark testing of

the asymptotic behaviour of shell elements is concerned, the problem should be consideredas a challenging well-dened mixed mode problem. This asymptotic behaviour test may also

be considered as a strong test for error estimators and mesh renement techniques.

3.3. The ScordelisLo roof problem

Another strong benchmark test for shells is the classical ScordelisLo roof problem. Thenite element model used is shown in Figure 15. The deformation of the shell in the caseof the classical problem (t= 3), obtained by the nite element analysis [8; 10] exhibits amixed membranebending mode of deformation [14]. The bending load-carrying mechanismis activated in transferring the load away from the free edges. The membrane load-carryingmechanism is mainly responsible for transferring the load to the end diaphragms (details on

the behaviour of the RFNS element in this case may be found in Reference [8]). There is noclear boundary zone apparent in the solution of the classical thick shell case.

The analytical solution of the classical problem is usually given in the literature based on theshallow shell theory [15]. However, in order to be able to investigate the asymptotic behaviourof the shell as t is reduced and compare this behaviour against the corresponding behaviourof the RFNS element, an analytical solution was derived in the present work based on theexact Flugges shell theory [12]. To this end, the uniform shell dead load pd is expressed in



20/32

440 D. BRIASSOULIS

terms of a Fourier series, as follows:

p =4

pd

n=1;3;5;:::

1

nsin

ny

L (6)

Accordingly, the membrane theory was used rst to derive the basic membrane solution for thedisplacements um ; vm, wm of the shell in the form of Fourier series expressions in y (wherethe index m stands for membrane). This solution results, of course, in the development ofnon-zero membrane hoop and shear stress resultants, Nm and N

my, respectively, along the free

edges of the shell, which are also expressed in the form of Fourier series. Subsequently, theexact Flugges bending shell theory was employed to eliminate the non-zero membrane edgestress resultants and also, to satisfy free end conditions in terms of bending stress resultants.The general bending theory was applied to the particular case of a symmetric stress system[12] (i.e. a shell having boundaries at =0=40

and a stress system symmetric withrespect to = 0). The solution was obtained by imposing along the free edge boundariesthe following line loads:

Nb=Nm

=4

pdrcos 0

n=1;3;5;:::

1

nsin

ny

L

Nby =Nm

y=8

pdsin 0

n=1;3;5;:::

L

n2cos

ny

L

M = 0

S = Q+@My

@y = 0

(7)

where b stands for bending. The solution of the three dierential equations describing themid-surface bending deformation of the shell (with no simplications introduced) is obtained

by expressing the nth harmonics ubn ; v

bn , w

bn of the displacements in the following exponential

form [12]:

ubn =Aem

vbn=Bem

wbn= Cem

(8)

and arriving at three equivalent linear equations for the constants A; B and C. The numberm corresponds to the eight complex roots mi of the characteristic equation of the system(condition of zero determinant for the nine coecient matrix of the three linear equations;the complex roots were calculated by using appropriate IMSL routines). The exact bending

solution obtained by satisfying the boundary conditions of Equation (7) was superimposed onthe membrane solution in order to get the nal analytical solution to the general ScordelisLoroof problem (i.e. a solution to the roof problem with the thickness t being a variable).

In studying the asymptotic behaviour of the shell problem analytically, the convergence ofthe Fourier series solution was investigated rst. The convergence of the analytical solutionthat was found to be excellent with a total number ofn = 100 was adequate enough even forvery thin shells with t= 0:00003 (t=r= 107).



21/32


-4.00E+06

-3.50E+06

-3.00E+06

-2.50E+06

-2.00E+06

-1.50E+06

-1.00E+06-5.00E+05

0.00E+00

5.00E+05

1.00E+06

0 10 20 30 40

theta

wz/(r/Et2)

3

0.3

0.03

0.003

0.0003

0.00003

Figure 16. Asymptotic behaviour of the circumferential distribution of the vertical displacements wz

at the middle of the roof shell (y =L=2) obtained analytically.

The rst observation made with the analytical solution was that, as the shell thicknessis reduced, a strong boundary zone develops along the free edge (Figure 16). The solutionobtained exhibits a stable behaviour with no singularities or any problem of this kind near thefree edge as the shell thickness is reduced from t=3 down to t= 0:00003 (t=r= 107). The

boundary zone eects that were developed are clear, they become stronger and develop over anarrower zone along the free boundary as the shell thickness is reduced. The deformations eldis expressed by means of the vertical component of the displacements for comparison purposesagainst the numerical solution. The vertical component of the displacements is denoted by wzhereafter (Figure 15).

The numerical analysis of the asymptotic behaviour of the ScordelisLo roof problem byusing the RFNS element revealed no problem of numerical instabilities=oscillations due tosingularities along the free edge. The only diculty encountered, an expected one, was theneed to use very well-rened meshes along the boundary zone as the shell becomes thinner. Inthe limit case oft= 0:0003 (t=r= 106), the nite element analysis results describe a deformedshell with a stable highly localized and very much pronounced edge zone deformation [10].The bending load-carrying mechanism is apparently limited to a very strong action developedalong the free edge zone. This strong action is attenuated dramatically away from the free

boundary. Thus, even though very much restricted in terms of the inuenced shell area, thebending load-carrying mechanism has a strong presence with a signicant contribution to theoverall shell behaviour in this limit thin shell case. Away from the free edge zone, it isthe membrane load-carrying mechanism that takes care of transferring the load to the end

diaphragms.The numerical solution is shown to be in very close agreement to the analytical solution

in terms of the overall behaviour of the shell. The results obtained for the classical case oft=r= 102 by the RFNS element are shown in Figure 17 along with the analytical solution.The excellent agreement obtained here was expected [8]. As the shell thickness is reduced,the simulation of the shell behaviour becomes more and more demanding though. Thus, inthe limit case of t= 0:0003 (t=r= 106), the exact simulation of the strong boundary zone



22/32

442 D. BRIASSOULIS

10 15 20 25 30 35 40

theta

3-17x30

anal-3

3-24x23

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

1.0E+06

0.0 0.1 0.2 0.3 0.4 0.5

0

y/L

wz/(r/Et2)

-4.0E+00

-3.5E+00

-3.0E+00

-2.5E+00

-2.0E+00

-1.5E+00

-1.0E+00-5.0E+01

0.0E+00

5.0E- 01

1.0E+00

wz/(r/Et2)

crown-3-17x30

edge-3-17x30

anal-crown-3

anal-edge-3

anal-crown-0.0003

anal-edge-0.0003

5

(a)

(b)

Figure 17. Circumferential distribution: (a) of vertical displacements wz at the middle (y =L=2) andmeridional distribution; and (b) along the crown line and the free edge ( = 0, 40) of the roof shell

(t= 3) obtained by the RFNS element and analytically (normalized with respect to r=Et2).

eects depends very much on the nite element mesh renement (Figure 18). However, thisneed for improvement with mesh renement strictly concerns an extremely thin outer zone ofthe boundary layer eects as the rest of the boundary layer eects and the overall behaviourof the shell are simulated satisfactorily even without resorting to very well-rened meshes.Moreover, it is clear that no singularities-related problems are present with the nite elementsolution (at least by using the RFNS element) as it is the case also with the exact analytical

solution. Even in the case of t= 0:00003 (t=r= 107

), the nite element solution followsthe analytically predicted development of the really strong boundary zone eects very closely(Figure 19). One should notice that, even in this case, apart from the extremely thin boundarylayer, modelling of the overall behaviour of the shell is excellent. Adequate modelling of the

boundary layer eects in this extreme very thin shell case depends heavily on local meshrenement especially along the free edge. The fact that the mesh renement, especially alongthe boundary, improves the simulation of the extreme outer zone of the boundary layer eects



23/32


-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

35 40

Theta

wz/(r/Et2)

0.0003-17x30

0.0003 anal

0.0003-17x30r

0.0003-24x23

0.0003-30x19

0.0003-39x22

0.0003-35x25

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06-5.0E+05

0.0E+00

5.0E+05

1.0E+06

0 5 10 15 20 25 30 35 40

Theta

wz/(r/Et2)

0.0003-17x30

3 anal

0.0003 anal

0.0003-17x30r

0.0003-24x23

0.0003-30x19

0.0003-39x22

0.0003-35x25

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

0.0 0.1 0.2 0.3 0.4 0.5

y/L

wz/(r/Et2)

crown-0.0003-17x30

edge-0.0003-17x30

anal-crown-0.0003

anal-edge-0.0003

crown-0.0003-17x30r

edge-0.0003-17x30r

edge-0.0003-30x19

crown-0.0003-30x19

crown-0.0003-39x22

edge-0.0003-39x22

crown-0.0003-35x25

edge-0.0003-35x25

(a)

(b)

(c)

Figure 18. Circumferential distribution: (a, b) of vertical displacements wzat the middle (y =L=2) andmeridional distribution; and (c) along the crown line and the free edge ( = 0, 40) of the roof shell

(t= 0:0003) obtained by the RFNS element and analytically (normalized with respect to r=Et2).



24/32

444 D. BRIASSOULIS

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

0 5 10 15 20 25 30 35 40

Theta

wz/(r/Et2)

0.00003-17x30

0.00003-17x30r0.00003 anal

0.00003-24x23

0.00003-30x19

0.00003-35x25

0.00003-39x22

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

35 40

Theta

wz/(r/Et2)

0.00003-17x30

0.00003-17x30r

0.00003 anal

0.00003-24x23

0.00003-30x19

0.00003-35x25

0.00003-39x22

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

0.0 0.1 0.2 0.3 0.4 0.5

y/L

wz/(r/Et2)

crown-0.00003 -17x30

edge-0.00003 -17x30

edge-0.00003 -17x30r

crown-0.00003 -17x30r

anal-crown-0.00003

anal-edge-0.00003

edge-0.00003 -30x19

crown-0.00003 -30x19

edge-0.00003 -35x25

crown-0.00003 -35x25

crown-0.00003 -39x22

22edge-0.00003 -39x

(a)

(b)

(c)

Figure 19. Circumferential distribution: (a, b) of vertical displacements wzat the middle (y =L=2) andmeridional distribution; and (c) along the crown line and the free edge ( = 0, 40) of the roof shell

(t= 0:00003) obtained by the RFNS element and analytically (normalized with respect to r=Et2).



25/32


-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

5.0E+05

1.0E+06

1.E-071.E-061.E-051.E-041.E-031.E-021.E-01

t/r

wzmax/(r/Et2)

wc-17x30

wb-17x30

wc-24x23

wb-24x23

wc-anal

wb-anal

wc-30x19

wb-30x19

wc-35x25

wb-35x25

wc-39x22

wb-39x22

Figure 20. Asymptotic behaviour of the numerical solution of the ScordelisLo cylindrical roof-shell

in terms of the maximum vertical displacementswzand mesh renement (normalized with respect tor=Et2).

in a rather stable and converging way is shown graphically in Figure 20. It is shown therefore,both analytically and numerically, that the asymptotic behaviour of the classical ScordelisLoroof problem is stable and it represents, along with the pinched cylinder with end diaphragms,a pair of very good tests for testing the asymptotic behaviour of shell elements.

The rotations eld developing along the centre line in the thin shell cases, also exhibitsa dramatic disturbance within the boundary zone as shown in Figure 21 (the rotations vector is directed along the meridional line). The stable behaviour of the nite element solutionand the excellent convergence characteristics obtained with the RFNS element are conrmedanalytically (by using a higher number of harmonics, n = 200, for the calculation of rotations).It should be noticed that, as in the case of the displacements eld, no singularities-related

problems of any kind are detected in the rotations eld calculated by either the numerical orthe analytical approach.The analysis of the asymptotic behaviour of the ScordelisLo roof problem in terms of

the relative energy shares for bending, shear and membrane energy is depicted graphically inFigure 22. Coarse meshes appear to yield a non-uniform convergence towards a membrane-dominated limit problem. However, local mesh renement results in a more stable and uniformtrend with the bending and membrane relative energy contributions, that converge and sta-

bilize to the corresponding rates of 37.3 and 62.7 per cent, respectively, which rates werecalculated analytically. This behaviour is probably due to the fact that with local mesh rene-ment a better simulation of the bending-dominated extreme boundary zone eects is obtained(Figure 20). The transverse shear strain energy is very small in the case of the classical

problem (0.022 per cent numerically, 0.0083 per cent analytically) and it fades away rapidly

with the decrease of the shell thickness, suggesting that the nite element solution is free ofany locking mechanisms.

As a nal analytical investigation of the asymptotic behaviour of this problem, the bendingstiness of the shell of the classical roof problem was allowed to be articially and signi-cantly weakened by setting the k factor of Equation (5) equal to k 106. Note that inthis way, only the relative contribution of the bending stiness is weakened. The membraneaction (membrane deformations and stress resultants) remains the same as that calculated for



26/32

446 D. BRIASSOULIS

-1.0E+05

-5.0E+04

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

4.0E+05

0 10 20 30 40

theta

/(r/Et2)

0.0003-39x22

0.0003-analytical

-2.0E+05

-1.0E+05

0.0E+00

1.0E+05

2.0E+05

3.0E+05

4.0E+05

5.0E+05

6.0E+05

7.0E+05

0 10 20 30 40

theta

/(r/Et2)

0.00003-39x22

0.00003-analytical

(a)

(b)

Figure 21. Circumferential distribution of rotations along the centre line of the ScordelisLocylindrical roof-shell (y =L=2) obtained by the RFNS element: (a) t= 0:0003; and (b) t= 0:00003

(Fourier series solution; n = 200), (normalized with respect to r=Et2).

the corresponding classical conguration. The analysis of this articially modied, in termsof bending stiness, conguration resulted in strong singularities in the asymptotic analyticalsolution. In particular, the solution obtained for the classical thick shell case of t= 3 wasdominated by a very strong bending boundary zone eect (Figure 23). This is due to thefact that the bending load-carrying mechanism developed by the articially weakened bendingstiness was made up at the expense of the development of a very much exaggerated bending

mode (note that no solution exists fork= 0). As the shell thickness was reduced to t= 0:0003,strong singularities developed, not allowing for any analytical solution to be obtained. Thissuggests that an articially made membrane-dominated ScordelisLo roof problem wouldinfact be an ill-posed problem (see also Reference [3]). However, the numerical and theanalytical results imply that the stable asymptotic shell behaviour of the ScordelisLo roof

problem strongly depends on the simultaneous activation of both, the bending and the mem-brane load-carrying mechanisms, no matter how thin the shell is. It is this combined action that



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0

10

20

30

40

50

60

70

80

90

100

1.E-071.E-061.E-051.E-041.E-031.E-02

t/r

Energy(%

)

Bend-14x14

Membr-14x14

Bend-39x22

Membr-39x 22Bend-35x25

Membr-35x25

Bend-Anal

Membr-Anal

Shear-Anal

Shear-35x25

Figure 22. Convergence characteristics of membrane, shear and bending strain energy shares of theScordelisLo cylindrical roof-shell as the shell thickness is reduced.

-4.0E+09

-3.5E+09

-3.0E+09

-2.5E+09

-2.0E+09

-1.5E+09

-1.0E+09

-5.0E+08

0.0E+00

5.0E+08

1.0E+09

0 5 10 15 20 25 30 35 40

Theta

wz/(r/Et2)

3

0.03

Figure 23. The eect of the articially weakened bending stiness on the behaviour of the ScordelisLoroof, in terms of the circumferential distribution of vertical displacements wz at the middle of the

shell (y =L=2) (normalized with respect to r=Et2).

oers stability to the shell even in extremely thin shell applications. Thus, it is shown that theScordelisLo problem, as far as its asymptotic behaviour is concerned represents another mixedmode (in energy terms) challenging problem for testing the asymptotic behaviour of shell el-ements. Adaptive mesh renement techniques may also be evaluated under this strong test.

3.4. The pinched hemispherical shell

The pinched hemispherical shell represents a non-inhibited pure bending problem widely usedto test the ability of a shell element to represent inextensional bending modes. The analyticalapproach of Reference [3] conrms that this classical benchmark test problem represents awell-posed non-inhibited pure bending problem.



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448 D. BRIASSOULIS

Y

Z

X

Line of symmetry

Free edge

Line of symmetry

Free edge

r

P=-1

P=1

18

Figure 24. Finite element model of one quadrant of the pinched hemispherical shell with a hole

(E= 6:825 107, = 0:3, r= 10:0, t= 0:04).

The nite element model used in the present analysis, with a 15 15 mesh, is shown inFigure 24. The use of a more rened 25 25 mesh conrmed the results obtained by the15 15 mesh. The deformed shell obtained for the classical case of t= 0:04 describes a clear

bending mode of deformation. The behaviour of the RFNS element in modelling this classicalbenchmark problem is excellent, as shown in detail in Reference [8].

The asymptotic behaviour of the problem is very stable and easily traceable by the niteelement analysis. In particular, as the shell thickness is reduced down to 0.00004 ( t=r= 4107)the deformation pattern remains almost unchanged and conrms the stable asymptotic

behaviour of the bending load-carrying mechanism dominating this case. The asymptotic be-haviour of the problem is depicted graphically in Figure 25. Shown in this gure is thevariation of the maximum displacement calculated by the RFNS nite element model under

one of the horizontal concentrated loads as a function of the non-dimensional thickness. Itis apparent that the maximum displacement stays very close to the corresponding analyticalsolution of the classical problem [8] (t= 0:094; wmax= 0:093). As the thickness is reduced, themaximum displacement is stabilized to a normalized (with respect to r=Et3) value of approx-imately 40. In the limit shell case of t=r= 4 107, a slight deviation from the convergedvalue is observed (probably due to some minor machine-related numerical problems in this

particularly very thin shell application).



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0

5

10

15

20

2530

35

40

45

1.E-071.E-051.E-031.E-01

t/r

w-max/(r/Et3)

15x15

analytical, t=0.04

Figure 25. Asymptotic behaviour of the numerical solution of the pinched hemispherical shell in terms

of the maximum displacements w (normalized with respect to r=Et3).

0

20

40

60

80

100

120

1.E-071.E-061.E-051.E-041.E-031.E-02

t/r

Energy(%)

Membrane-15x15

Bending-15x15

Membrane-25x25

Bending-25x25

Shear-25x25

Figure 26. Convergence characteristics of membrane, shear and bending strain energy shares of thepinched hemispherical shell as the shell thickness is reduced.

In energy terms, the convergence characteristics shown in Figure 26 also suggest a stableasymptotic behaviour with the bending energy, completely dominant even for a relativelythick hemispherical shell (99.3 per cent). The shear strain energy share is insignicant anddissipates rapidly with the decrease of the shell thickness. The slight deviation in the maximumdisplacement in the very thin shell case of t=r= 4 107 is also reected in the energy

components.

4. CONCLUSIONS

The asymptotic behaviour is considered to be one of the most demanding levels of bench-mark testing for shell elements. Following some recent discussions on the investigation of



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450 D. BRIASSOULIS

the reliability of shell elements, it was considered interesting to investigate analytically andnumerically the asymptotic behaviour of selected classical benchmark problems. Accordingly,in the present paper, the asymptotic behaviour of classical benchmark tests is analytically andnumerically investigated. The aim is to oer an insight into the mechanisms developed by

the shells of the corresponding benchmark problems as the shell thickness is reduced. It isexpected that an understanding of these mechanisms would allow for a better evaluation ofthe critical parameters pertaining to the use of the particular classical benchmark tests fortesting the asymptotic behaviour of shell elements.

The reformulated four-node shell element (RFNS) is employed in the numerical analyses.The asymptotic behaviour of the problems under investigation is traced numerically following

physical convergence characteristics of the nite element solution as the shell thickness is re-duced. Appropriate analytical approaches are developed to investigate the asymptotic behaviourof the classical benchmark tests. In addition, the analytical asymptotic solutions derived areused for monitoring and comparing the numerical analysis results.

Some of the benchmark limit tests considered lead asymptotically to the development ofvery strong boundary zone eects. Such boundary eects, which vary very much depending onthe shell thickness, are not easily modelled if the mesh of the nite element models used is notwell designed (e.g. as suggested by the corresponding analytical solutions) and of course, ifthe element is not capable of capturing adequately such dramatic changes within a narrow zoneand of modelling the particular deformation modes developing without experiencing problems(e.g. locking problems). In the case of strong boundary eects and competing deformationmodes, one may talk about a challenging limit test. If a shell element model exhibits goodconvergence characteristics to the corresponding analytical solution, then the element (alongwith the model=meshing used) is considered to be adequate with respect to modelling thespecic problem (e.g. a membrane-dominating problem, or a bending-dominated problem, ora mixed-dominated problem). If the element exhibits a stable converging asymptotic behaviourfor all kinds of deformation modes and boundary layers (in the limit of t 0), then the shell

element is considered to be adequate in terms of its asymptotic behaviour in general. Thisis shown to be the case with the RFNS element. In fact, the reliability and eciency of theRFNS element already established by means of the classical benchmark tests is re-conrmedin all cases of the asymptotic tests examined.

Concerning the asymptotic behaviour of the classical benchmark tests, it is shown that

The problem of the pinched cylinder with free ends, apart from being a classical benchmarkproblem for shell elements may also be considered as a simple test for testing the asymptoticbehaviour of shell elements under bending domination. It may not be considered to be a drastictest though, except for the boundary layer eects. In particular, a well-rened mesh should beused to reveal the presence of a weak non-uniform displacement pattern in the displacementeld of thin shell cases, detectable only at a very local scale. As the shell thickness is reduced,

this boundary layer eect becomes much weaker and it is limited to a very narrow boundaryzone, until it eventually disappears. However, in the thin shell cases, signicant boundarylayer eects may be detected in the rotations and the shear strain elds. Well-rened meshesare needed, especially in the circumferential direction, to reveal the highly localized boundarylayer developing along the free boundary in the case of very thin shells. As the boundary layereects represent a pure beam bending localized action, they just add up to the prevailingring-type bending action so that a pure bending mode of deformation dominates the shell



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behaviour. If convergence is measured only in terms of energy characteristics, one may notbe able to conclude as to whether the boundary conditions are modelled appropriately.

The asymptotic behaviour of the problem of the pinched cylinder supported by rigid dia-phragms is described by a rather stable contribution of both membrane and bending strain

energies. Thus, as far as benchmark testing of the asymptotic behaviour of shell elementsis concerned, the problem should be considered as a well-dened mixed mode problem. In

particular, in this benchmark problem, the asymptotic behaviour of the shell stems from thecombination of two main modes: the membrane-exural mode dominant along the meridionaldirection and the ring-bending mode of deformation dominant in the circumferential direction.This asymptotic behaviour test may also be considered as a strong test for evaluating errorestimators and mesh renement techniques. The basic characteristics of the asymptotic be-haviour in this case may be summarized as follows: The convergence characteristics withrespect to the highly localized region under the load in very thin shell cases do not nec-essarily represent the overall convergence behaviour of the nite element model. The localconvergence characteristics may simply describe the compliance of the mesh congurationused with the particular demanding meshing requirements imposed by the extremely thinmeridional zone under the load. Away from this zone, convergence may have been achievedwith much coarser meshes.

The ScordelisLo problem, as far as its asymptotic behaviour is concerned, also representsa mixed mode challenging problem (in energy terms) for testing the asymptotic behaviourof shell elements. In particular, the asymptotic behaviour of the ScordelisLo roof problemstrongly depends on the simultaneous activation of both, the bending and the membrane load-carrying mechanisms, each contributing in a stable way in terms of energy shares, no matterhow thin the shell is. Adaptive mesh renement techniques may also be evaluated underthis strong test. The basic characteristics of the asymptotic behaviour in this case may besummarized as follows: The exact simulation of the strong boundary zone eects developedin very thin shell applications does depend very much on the nite element mesh renement.

However, this need for improvement with mesh renement strictly concerns an extremelythin outer zone of the boundary layer eects as the rest of the boundary layer eects and theoverall behaviour of the shell are simulated satisfactorily even without resorting to very well-rened meshes. With mesh renement a better simulation of the bending-dominated extreme

boundary zone eects is obtained so that the bending and membrane strain energy sharesnally converge to the corresponding stable contributions predicted analytically.

The pinched hemispherical shell represents a clear non-inhibited pure bending problem. Inenergy terms, the convergence characteristics suggest a stable asymptotic behaviour with the

bending energy that was completely dominant even for a relatively thick hemispherical shells.In conclusion, it is shown that the classical benchmark tests also represent a set of strong

challenging and reliable tests for investigating the asymptotic behaviour of shell elements.

REFERENCES

1. Zienkiewicz OC, Taylor RT. The Finite Element Method, 4th (edn), vol. I. McGraw-Hill: London, 1989.2. Kerber T. Review of shell and plate nite elements in light of numerical locking and its remedies. Recherche

Aerospatiale 1990; 3:4576.3. Chapelle D, Bathe KJ. Fundamental considerations for the nite element analysis of shell structures. Computers

and Structures 1998; 66(1):1936.4. Chapelle D, Bathe KJ. The infsup test. Computers and Structures 1993; 47(4=5):537545.



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5. Bathe KJ, Iosilevich A, Chapelle D. An infsup test for shell nite elements. Computers and Structures 2000;75:439456.

6. Bathe KJ, Iosilevich A, Chapelle D. An evaluation of the MITC shell elements. Computers and Structures2000; 75:130.

7. Bathe KJ. The infsup condition and its evaluation for mixed nite element methods.Computers and Structures

2001; 79:243252.8. Briassoulis D. The four-node C shell element reformulated. International Journal for Numerical Methods in

Engineering 1996; 39:24172455.9. Tomoshenko S, Woinowsky-Krieger S. Theory of Plates and Shells. McGraw-Hill: New York, 1959.

10. Briassoulis D. On the asymptotic behaviour of shells. In preparation.11. Briassoulis D. Synthetic-structured nite element analysis of shells. Computer Methods in Applied Mechanics

and Engineering 1995; 121:411448.12. Flugge W. Stresses in Shells, 2nd (edn). Springer: Berlin, 1973.13. Briassoulis D. An integrated physical model for cylindrical shells. Journal of Structural Engineering, ASCE

1992; 118(8):21682185.14. Briassoulis D. Analysis of the load-carrying mechanisms of shells using the reformulated four-node C shell:

the ScordelisLo barel vault problem. Computers and Structures 1997; 64(14):253273.15. Scordelis AC, Lo KS. Computer analysis of cylindrical shells. Journal of American Concrete Institute 1964;

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