Analytical evaluation of local effects in cylindrical shells testing and design
A simple analytical evaluation of apparently anomalous results from tests on circular cylindrical shells that have been observed in the past is provided on the basis of the classical Ritz approach and modified Donnell relationships. Із використанням класичного підходу Рітца та модифікованих рівнянь...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2009
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| Цитувати: | Analytical evaluation of local effects in cylindrical shells testing and design / F. Guarracino // Проблемы прочности. — 2009. — № 5. — С. 28-35. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860258766957576192 |
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| author | Guarracino, F. |
| author_facet | Guarracino, F. |
| citation_txt | Analytical evaluation of local effects in cylindrical shells testing and design / F. Guarracino // Проблемы прочности. — 2009. — № 5. — С. 28-35. — Бібліогр.: 8 назв. — англ. |
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| description | A simple analytical evaluation of apparently anomalous results from tests on circular cylindrical shells that have been observed in the past is provided on the basis of the classical Ritz approach and modified Donnell relationships.
Із використанням класичного підходу Рітца та модифікованих рівнянь Доннелла виконано просту аналітичну оцінку явно аномальних результатів, які отримані при випробуваннях круглих циліндричних оболонок.
С использованием классического подхода Ритца и модифицированных уравнений Доннелла выполнена простая аналитическая оценка явно аномальных результатов, полученных при испытаниях круглых цилиндрических оболочек.
|
| first_indexed | 2025-12-07T18:52:50Z |
| format | Article |
| fulltext |
UDC 539.4
Analytical Evaluation of Local Effects in Cylindrical Shells Testing
and Design
F. Guarracino
Federico II University, Naples, Italy
УДК 539.4
Аналитическая оценка локальных эффектов в цилиндрических
оболочках при их испытаниях и проектировании
Ф. Гуаррачино
Университет им. Федерико II, Неаполь, Италия
С использованием классического подхода Ритца и модифицированных уравнений Доннелла
выполнена простая аналитическая оценка явно аномальных результатов, полученных при
испытаниях круглых цилиндрических оболочек.
К л ю ч е в ы е с л о в а : испытания труб, расчет предельного состояния, локальные
эффекты, аналитическая оценка.
Introduction . Pipelines are used worldwide, onshore and offshore, and have
now becom e vital components o f the energy systems o f all econom ically developed
countries. Pipelines are designed to accommodate the effects o f a w ide range o f
loading conditions resulting from internal and external pressure, bending, etc.,
during installation and operation [1]. Pipeline design calculations have been
traditionally based on the lim it stress approach, but since 1996 the lim it state code
has been developed. The use o f the lim it state approach provides a more
comprehensive basis for the calculation o f the ultimate conditions for pipes
subjected simultaneously to pressure and bending loads. The ultimate state o f the
pipeline deformation or loading is calculated using a m odel that describes the
characteristic ultimate m om ent or strain related to the geom etry and material
properties o f the pipe. The design factors are calculated using statistical
descriptions o f the scatter o f test results compared to the mean values together with
the statistical descriptions o f the variables com posing the particular m odel, e.g.,
material strength, elastic m odulus, etc. In the above process, it is generally
assumed that the scatter o f test results from minor and usually random variations in
the variables used in the model. In the case o f a pipe, these variations would
generally relate to the differences in the geom etries o f the test pipes from their
corresponding nom inal values, say for pipe w all thickness, or out-of-roundness.
The aim o f the present paper is to provide a sim ple analytical evaluation o f
apparently anomalous results from tests on circular cylindrical shells that have
been observed in the past, and can be useful to highlight the potential influence that
© F. GUARRACINO, 2009
28 ISSN 0556-171X. Проблемы прочности, 2009, № 5
Analytical Evaluation o f Local Effects
these anomalies m ight have on the process o f providing design calculation
guidance using the lim it state method without making resort to com plex finite
elem ent analyses.
A Survey o f Test R esults. U sually testing a section o f a circular cylindrical
shell in pure bending loading envisages that the test specim en deforms according to
sim ple bending beam theory. Primarily this im plies that w hile the material remains
elastic the application o f pure bending mom ent w ill induce the maximum tensile
and com pressive strains that are identical in magnitude. A typical test rig for a
m edium diameter pipe, o f about 700 mm diameter, is shown in Fig. 1. The test rig
applies a four-point bending condition with the central section o f the test pipe
assumed to be subjected to bending action only, without (or at m ost very little)
shear or axial forces.
The form o f load-deformation plot from such a rig is shown in Fig. 2 for a
pipe with a D / t ratio o f about 40.
FOUR POINT TEST ARRANGEMENT
L
MNNMMNMNSMNNMBNRMMMMMMNMMMNNMMMMNRMMMHNMNMMMMMNNHMMNMNNNMNNNNMMNRNMMMNMNM̂
STRUCTURAL LABORATORY REACTION FLOOR
Fig. 1. Typical four-point bending test arrangement.
Applied
(MN-m)
moment
Dit = 40
End rotation (rad)
0 0 .0 5 0,1 0.15 0.2 0.25 0.3 0 .3 5 0.4
Fig. 2. End rotation vs. applied moment.
From the limit state point o f view , the two relevant conditions are the
maximum m om ent and corresponding strain, for load-controlled conditions o f
design, and the strain at w hich the reduction o f load-bearing capacity first occurs,
which relates to displacement-controlled design conditions. Following the attainment
o f that strain, as the loading is further applied, the pipe develops a very local form
o f buckling.
Since the pipe is assumed to be an extrem ely simple structural element, and
the sim ple beam theory holds true, it has been com m on practice to assume that the
ISSN 0556-171X. npoôëeMbi npounocmu, 2009, N 5 29
F. Guarracino
axial strains have identical values in tension and com pression and that the strains
can be calculated directly from the curvature or the vertical displacements o f the
central section o f the pipe. The ultimate strain values from tests in w hich the pipe
has been loaded to the point o f local buckling have usually been inferred from
measurements o f the deformations. Only recently have strain gauges been attached
to the test pipe to measure axial strains directly.
Tests [2] were carried out on 152-mm diameter pipe to determine the
minimum curvature to w hich the pipe could be deformed prior to local buckling
occurring. A n arrangement similar to that in Fig. 1 w as used, and strain gauges to
measure axial and circumferential strains were attached at intervals o f 100 mm
apart along the central test section. In the design o f the test rig, it w as assumed that
a central test section o f about 5D w ould suffice to ensure that end effects due to the
loading conditions w ould diminish to a negligible level along the major part o f that
section. Figure 3 shows results o f the axial strain values along the top and bottom
o f the pipe section for two levels o f the applied loading. It is evident the axial
strains are fairly uniform along the length o f the test section but there are
significant differences in the averaged values o f the com pressive strains compared
to the tensile strains.
Fig. 3. Results from a 152-mm diameter pipe bend test [2].
A t that time the evident anomaly between the measured strains w ith the
expected values vs. the sim ple bending theory was not follow ed up, and even after
checking that the strain gauges were correctly positioned and the instrumentation
was functioning properly the cause o f the anomaly was not further investigated.
Som e time later, proving tests were carried out on a section o f 609-m m
diameter pipes containing a thin liner made from a corrosion-resistant material [3].
The purpose o f the tests was to determine accurately the level o f strain to w hich the
pipe could be bent before the liner buckled locally. The test arrangement o f Fig. 1
had a loading arm 2 m long to create the m oment in the central section o f the test
pipe. The test section was arranged to be 3.5D . The load was applied to the test
pipe using straight bars and loose yokes around part o f the pipe circumference. A
30 ISSN 0556-171X. npo6n.eMH npounocmu, 2009, N 5
Analytical Evaluation o f Local Effects
number o f axial strain gauges were attached along the top and bottom centre lines
o f the pipe at intervals from the support points. The values o f strain were
monitored as the load values were progressively increased. Figure 4 shows plots o f
the values for the top and bottom gauges averaged along the test sections and
plotted against the corresponding value o f applied load.
a — “ --------------------------------------------------------------------------------------------------------------------
0 0 1 0.2 0.3 0.4 0.5 0.6 0.7 O.S 0.9 1 1.1 1.1
Applied load (M N)
Fig. 4. Averaged strain values plotted against corresponding values of applied loading [3]: Maximum
ratio of tensile to compressive averaged strains = 1.28 (D = 609.6 mm, t = 18.9 mm, D /t = 32, X65
material).
It is evident from Fig. 4 that there is a system atic difference between the
averaged strains along the top and the bottom o f the pipe. A t the maximum load
level, the averaged axial tensile strains were 1.28 times the corresponding averaged
com pressive strains.
In v iew o f the importance o f the test results in providing the allowable strain
levels for the lined pipe, an investigation w as made with regard to the underlying
cause o f the anomaly. This is described extensively in [4], with the aid o f several
finite elem ent m odels intended to replicate the conditions in bending tests, or in
pipelines that have variations in cross section and are subjected to bending with
special attention paid to the constraint arrangements.
The investigation has yielded that the cause lays in the effect o f the im posed
ovalization applied by the saddles at the load application points. This result implies
recommendation for the m odification o f the loading application in w hich the loads
were applied, not through local stiffening o f the pipe w all or saddles, but through
the neutral axis o f the pipe, as shown in Fig. 5.
The test pipe was fitted with strain gauges, as before, as w ell as w ith gauges to
measure the ovality o f the pipe. The values o f the axial strains measured by the
gauges along the test section o f the pipe were quite uniform. A s expected, w ith the
m odified loading and support arrangement, the averaged measured values o f
com pressive strains agreed very closely w ith the corresponding values o f the
tensile strains.
ISSN 0556-171X. npoôëeMbi npounocmu, 2009, N 5 31
F. Guarracino
Fig. 5. Test arrangement with modified support and load application points.
A n alytica l Treatm ent o f the Problem . In order to provide a sim ple tool to
evaluate the effect o f the applied loads and o f the supports, reference is made to the
classical Ritz approach [5] and to a m odified set o f the Donnell strain and
curvature variations [6]. A s a matter o f fact, the Ritz method has been extensively
used by structural engineers w ell through the middle o f the twentieth century until
it has progressively lost ground to its more versatile localized form, i.e., the finite
elem ent method. Nevertheless, m any formulas o f primary practical importance
have been derived by this method, w hich still form the basis o f our understanding
o f a large number o f m echanical problems [7]. A difficulty o f the Ritz method
certainly consists in the extensive calculations required, but the appearance o f
computer algebra system s (CAS), that are software programs w hich allow
manipulation o f mathematical expressions in sym bolic form, has now made
possible the treatment o f many problems abandoned in the past.
The advantage o f the proposed procedure lies in the extreme sim plicity o f its
final expression, w hich can give a m eaningful physical insight into the parameters
w hich govern the problem at hand and can also offer a first validation to
subsequent three-dimensional and computationally expensive analyses.
The Donnell equations [6] have been used w ith a considerable degree o f
success for the analysis o f elastic and plastic buckling o f thin-walled circular
cylinders. The basic assumptions at the basis o f the Donnell theory have proved to
be able to deal with several deformation m odes w ith a satisfactory degree o f
accuracy and for this reason they can be considered to be able to represent also the
cases in w hich loading is not symmetrical with respect to the axis o f the cylinder.
However, the Donnell equations are not w ell adapted to solution by Fourier series
since som e o f the high-order derivatives found in the formulation som etim es lead
to divergent trigonometric series. Even i f in the present Ritz approach reference is
naturally made to an energy expression and no differential equations are involved,
a m odified set o f strain and curvature changes are employed.
A circular cylindrical shell is taken into consideration. With reference to an
elem ent in the middle surface o f the shell, the coordinate axes are directed with the
32 ISSN 0556-171X. Проблемы прочности, 2009, N 5
Analytical Evaluation o f Local Effects
x-axis in the axial direction o f the cylinder, the y-axis in the circumferential
direction, and the z-axis in the radial direction, u, v , and w are the components in
the x , y , and z directions o f the displacement o f a generic point. Said p the
central angle, the strains are assumed to be
du
dx ’
1 dv
r d p
w
r xp
du dv
+ ,
r d p dx (1)
where r is the radius o f the middle surface o f the shell.
In calculating the expression o f the strain energy, the changes o f curvature o f
the middle surface o f the shell are also required. To this scope, the follow ing
approximate expressions are assumed
d 2 u
X x ~ d 2 d x 2
1
X p - ^2
I
W +
2d 2w
d p
1
X xp -
I a 2 -j d w dv
+
dpdx dx (2)
and in order to evaluate the deformation induced by two opposite forces, F , acting
along a vertical diameter at a certain section x = 0 , the components o f displacement
varying along the length o f the cylinder are taken in the form
“ 1
- C 1R a e ~ ax [An sin(n p ) + B n cos(n p )],
n-1 n
X
- C 2e - ax 2 A cos(n p ) - Bn sin(n p )],
n-1
X
- C 3e ax 2 n [An sin(n p ) + B n cos(n p )],
(3)
n-1
x
r
where C j , C 2 , C 3 , A n , and B n are constants that must be calculated for the case
o f loading at hand.
Essentially, this is the key differentiation o f the present approach with respect
to classical formulations w hich assume the change o f curvature in the direction o f
the generatrix to be equal to zero [7]. However, as anticipated, the present
assumption im plies a considerable computational effort at a sym bolic level to
define the total strain energy by integration over the surface o f the shell o f the
strain energy per unit area in terms o f Eq. (3). Therefore, the w hole operation has
been performed by means o f ad hoc routines written w ith the aid o f the sym bolic
system MATHEMATICA® [8 ].
In the same manner, the equations for calculating the constants C j , C 2 , C 3 ,
A n , and B n have been first obtained by im posing the total potential energy to be a
stationary value and then solved. Finally, the results have been expanded in power
series.
ISSN 0556-171X. npoôëeMbi npounocmu, 2009, N9 5 33
F. Guarracino
The end result can be summarized in the follow ing formula, w hich provides
the top and bottom mid-surface strain on account o f the deformation induced by
two opposite forces, F , acting along the vertical diameter at the mid-span (the
formula is calculated for a circular cylindrical shell w ith L > > D ),
£ = (1-
2 n 6 E t 4
1 - - +
24 (4)
where t is the thickness o f the shell and E and stand for the Young modulus and
P oisson’s ratio, respectively, and f is given by
„ 4/(1- v 2 )
f = r ^ - (5)
6 4 ^ -J t3 / r
It is noteworthy that the expression o f what can be considered as the natural
half-wavelength o f the problem results proportional to the term t 3 / r , whereas in
the case o f circular shells subject to axial symmetric loading , it is proportional to
4 r t .
According to Eq. (4), Fig. 6 shows the value o f the top and bottom strains
along the axis x o f the pipe o f Fig. 4 induced by two opposite forces o f magnitude
1.118 MN.
E
Fig. 6. Top and bottom strains induced by two opposite forces of magnitude 1.118 MN in the pipe of
Fig. 4 (D = 609.6 mm, t = 18.9 mm).
Thus, the proposed formula can be straightforwardly em ployed to evaluate the
order o f magnitude o f the difference in top and bottom strains with regard to a
tested sample o f the previous Section. In fact, for the pipe characterized by
D = 609.6 mm, t = 18.9 mm and X65 material [3], for an applied load o f 1.118 MN
the absolute value o f the top and bottom strains calculated according to the simple
34 ISSN 0556-171X. npo6n.eMH npounocmu, 2009, N9 5
Analytical Evaluation o f Local Effects
bending theory is 0.002. Equation (4) yields the additional strain at the mid-span
on account o f the local deformation provoked by the concentrated loads, that is
0 .000308402 .(since the supports and the applied loads are not opposite, but distant
2 m apart, the values from each pair o f forces have been halved). B y adding this
quantity to the tensile strain and subtracting it from the com pressive one, it follow s
that the ratio o f tensile to com pressive strains is equal to 1.365, with a difference
from the measured ratio o f about 7%.
C onclusions. The aim o f the present work has been to provide a simple
analytical formulation to evaluate the effects o f test arrangement on the level o f
apparent strain.
B y means o f the obtained result, the seem ingly anomalous values o f measured
axial strain in tests can be explained quite straightforwardly. The proposed
formulation also offers a physical insight into the m echanics o f the problem in the
fashion o f many classical results still w idely used in the engineering practice.
Р е з ю м е
Із використанням класичного п ідходу Рітца та модифікованих рівнянь Д он
нелла виконано просту аналітичну оцінку явно аномальних результатів, які
отримані при випробуваннях круглих циліндричних оболонок.
1. F. Guarracino and V. Mallardo, “A refined analytical analysis o f submerged
pipelines in seabed laying,” A ppl. O cean R es., 21, 2 8 1 -2 9 3 (1999).
2. C. P. Ellinas, A. C. Walker, G. N . Langfield, and M. J. Vines, “A development
in the reeling method for laying subsea pipeline,” in: Proc. 1st Petroleum
Technology Australian Conference, Perth (1985).
3. A. C. Walker, A . Holt, F. Guarracino, and D. W ilmot, “Test procedure for
pipe and pipeline material,” in: Offshore Pipeline T echnology Conference,
Amsterdam (2003).
4. F. Guarracino, A. C. Walker, and A. Giordano, “Effects o f boundary conditions
on testing o f pipes and finite elem ent m odelling,” Int. J. P ress . Vess. P ip in g ,
in press (2008).
5. F. Guarracino and A. C. Walker, E n erg y M e th o d s in S tru c tu ra l M ech an ics,
Telford, London (1999).
6. L. H. Donnell, S ta b ility o f T h in -W a lle d T ubes u n d er T orsion , N A C A Report
N o. 479 (1933).
7. S. P. Timoshenko and S. W oinowsky-Krieger, Theory o f P la te s a n d Shells,
M cGraw-Hill, N ew York (1959).
8. S. W olfram, The M a th em a tica B ook, W olfram M edia/Cambridge University
Press, Cambridge (1999).
R eceived 05. 01. 2009
ISSN 0556-171X. Проблеми прочности, 2009, № 5 35
|
| id | nasplib_isofts_kiev_ua-123456789-48442 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T18:52:50Z |
| publishDate | 2009 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Guarracino, F. 2013-08-19T15:40:20Z 2013-08-19T15:40:20Z 2009 Analytical evaluation of local effects in cylindrical shells testing and design / F. Guarracino // Проблемы прочности. — 2009. — № 5. — С. 28-35. — Бібліогр.: 8 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48442 539.4 A simple analytical evaluation of apparently anomalous results from tests on circular cylindrical shells that have been observed in the past is provided on the basis of the classical Ritz approach and modified Donnell relationships. Із використанням класичного підходу Рітца та модифікованих рівнянь Доннелла виконано просту аналітичну оцінку явно аномальних результатів, які отримані при випробуваннях круглих циліндричних оболонок. С использованием классического подхода Ритца и модифицированных уравнений Доннелла выполнена простая аналитическая оценка явно аномальных результатов, полученных при испытаниях круглых цилиндрических оболочек. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Analytical evaluation of local effects in cylindrical shells testing and design Аналитическая оценка локальных эффектов в цилиндрических оболочках при их испытаниях и проектировании Article published earlier |
| spellingShingle | Analytical evaluation of local effects in cylindrical shells testing and design Guarracino, F. Научно-технический раздел |
| title | Analytical evaluation of local effects in cylindrical shells testing and design |
| title_alt | Аналитическая оценка локальных эффектов в цилиндрических оболочках при их испытаниях и проектировании |
| title_full | Analytical evaluation of local effects in cylindrical shells testing and design |
| title_fullStr | Analytical evaluation of local effects in cylindrical shells testing and design |
| title_full_unstemmed | Analytical evaluation of local effects in cylindrical shells testing and design |
| title_short | Analytical evaluation of local effects in cylindrical shells testing and design |
| title_sort | analytical evaluation of local effects in cylindrical shells testing and design |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/48442 |
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