# Buckling of Imperfect Thin Cylindrical Shell under Lateral Pressure

Authors: R. Lo Frano; G. Forasassi
Journal: Science and Technology of Nuclear Installations
 (2008)

---

## Abstract

The strength of thin shells, under external pressure, is highly
dependent by the nature of imperfection. This paper investigates
buckling behaviour of imperfect thin cylindrical shells with
analytical, numerical, and experimental methods in conditions for
which, at present, a complete theoretical analysis was not found
in literature. In general, collapse is initiated by yielding, but
interaction with geometrical instabilities is meaningful, in that
imperfections reduce the load bearing capacity by an amount of
engineering significance also when thickness is considerable. The
aim of this study was to conduct experiments that are
representative of buckling, in the context of NPP applications as,
for instance, the IRIS (international reactor innovative and
secure) and LWR steam generator (SG) tubes. At Pisa University, a
research activity is being carried out on the buckling of thin
walled metal specimen, with a test equipment (and the necessary
data acquisition facility) as well as numerical models were set up
by means FEM code. The experiments were conducted on A-316 test
specimens, tubes with and without longitudinal welding. The
numerical and experimental results comparison highlighted the
influence of different types of imperfections on the buckling
loads with a good agreement between the finite-element predictions
and the experimental data.


---

## Body

## 1. Introduction

The stability
of circular cylindrical shells under uniform lateral pressure has been widely
investigated. The behavior of cylindrical shells under external pressure is
very sensitive to geometric imperfections. There have been many theoretical
studies investigating the strength of cylinders with specific imperfection
forms, and it is well established that axisymmetric imperfections cause the
greatest reductions in strength
[1]. Many
researchers have studied buckling of circular cylindrical shell under external
pressure and more accurate solutions of the present problem were obtained for
short cylindrical shells and for anisotropic shells, respectively. However,  actual calculations seem to be confined mainly to some special ranges of the
shells geometries, boundary, and loading conditions [2].Moreover, the
literature concerning unequal wall transition joints in thin shell is limited. When thin shells were
subjected to external pressure, the collapse was initiated by yielding, which
was often the dominant factor but the interaction with the instability is
meaningful. In fact, the presence of imperfections reduces the load bearing
capacity by an amount of engineering significance; so the classical elastic
solution, like [3],
appears to be not adequate.The major factors that affected the collapse pressure of pipes were the
diameter-to-thickness ratioD/t, the Young's modulus and yield stress of
the material in the circumferential direction, and initial imperfections in the
form of ovality and wall thickness variations [4, 5], as for eccentricity and
presence of welding joint.The
current paper examined the buckling issue of a thin circular cylindrical shell
in the dimension range of possible interest for the nuclear steam generator
(SG) of type foresee in the IRIS Reactor. IRIS is currently under way of
preliminary design; it was an integral pressurized water reactor. Its reactor pressure vessel houses not only the nuclear core, but also all the major reactor primary coolant system components such as pumps, steam generators, pressurizer, control rod drive mechanisms, and neutron reflector.The
IRIS integral vessel is larger than a traditional PWR pressure vessel, but the
size of the IRIS containment results to be smaller than that of corresponding
loop reactors, resulting in a significant reduction in the overall size of the
reactor plant [6]. The IRIS SGs were once through,
with the primary fluid outside the tubes.In
general, buckling analysis is used to predict failure of long pipelines,
subjected to external over pressure.A
selected configuration was used to perform the different analysis significant
for the specified field of interests (numerical as well as experimental).The present study might also serve as a base
for other loading models, which include different boundary and geometrical
conditions. Buckling phenomenon occurs when most of the strain energy, which is
stored as membrane energy, can be converted to the bending energy requiring
large deflections.Nonlinear
buckling pressures can be evaluated using a nonlinear stress analysis by
observing the first change in the slope (i.e., stiffness of the structure) in
the load-deflection curve
[7, 8]. In the nonlinear buckling analysis; however, an initial imperfection,
either in terms of the geometry or load, is necessary to trigger the buckling
phenomenon. The load carrying capacity of shell subjected to an external
pressure could be characterized by the yield load. The first yield load was not considered as a failure
pressure, because it was no equivalent to an immediate loss of stability.


## 2. Theoretical Analysis

Elastic
structures can buckle under excessive loads. In order to determine whether a
given structure can become unstable or not, a first step is to apply a linear
stability analysis. A linear stability analysis using the normal-mode approach
always results in an eigenvalue problem, the solution of which gives a
bifurcation condition.The
classical analytical approach is based on the classic elastic solution like
Donnel or Flugge theories. Among these, Donnel’s theory represents a general
approach, while that of Flugge is considered to be more refined than the linear
Donnell-type stability one. From the Donnell stability equations in uncoupled
form [9], adopting an analytical procedure of minimization, one can obtain the
critical pressure load in the simplified form (1)p¯=(1+n2¯)2n2¯+1n2¯(1+n2¯)212π4Z2,Z=L2(Rh)(1−ν2)0.5, where ν is Poisson’s ratio; hR and L are thickness, the radius and length of the
shell; Z are the Batdorf parameter and n is the number of circumferential
waves.The
above formula of (1)
is function of the single geometric parameter Z, known as Batford parameter.Results of this procedure for large
values ofZ or L (long cylinder) are inaccurate because the associated membrane
stiffness component includes a nonnegligible error.Flugge’s
stability equation in coupled form the numerical solution is based on the exact
solution of three differential Flugge’s equations [10]. They describe the
buckling of cylindrical shell under the most general homogeneous membrane
stress action. The numerical calculation leads to a solution, which describes a
buckling mode with n half waves along the length of cylinder and 2 m half waves
around its circumference. In perfect agreement with the asymptotic value for
long shell, the simplified critical load is expressed as follows: (2)q1=(1−ν2)λ2+kλ4(λ2+2(2−ν))λ2−1,q1=paD, where k=h2/12a2 is a dimensionless parameters; h and a are thickness and radius of the shell; D
is the bending stiffness, p is the lateral pressure, while λ=nπa/1 is a modal
parameter, function of the radius (a), the circumferential wave number (n) and
length of shell.Of course also in this approach, if
the cylinder is very long, its central part is little influenced by the
stiffening effect of the bulkheads.Among the
analysed theories before, that of Cheng and Ho represents a general one, based
on a very accurate Flugge’s theory, which is considered to be more refined than
the linear Donnell-type stability theory. Their analysis is rather lengthy;
considerable numerical manipulation is required to obtain the critical load.
This approach allows to obtain the simplified minimum value of buckling load
for moderate-length cylinders [11]: (3)Pcr=2πEt236(1−ν2)3/4RLtR, where ν is Poisson’s ratio; t,L, and R are thickness, length, and radius of the shell;
E is Young’s modulus. It may be observed that for a certain range of L/R, the
calculations from the approximate formula of (3) are very close with the
general buckling condition.The resulting buckling pressures,
determined for a thin shell of AISI 316, according to the above mentioned
theories, are represented in Table1, for a wide range of geometries
(characterized by the diameter and thickness) and for the same length equal to
828 mm, and plotted in Figure 1. The material pressure-length curve is modelled
by a multilinear curve to show the influence of geometrical parameters on
studied theories. Results seem to indicate that
linear theories are inadequate to predict real behavior in the presence of the
imperfections.Table 1
Analytical values of pressure load.
DtFlugge theory (Pa)Donnel theory (Pa)Cheng Ho theory (Pa)151.22.59E+73.15E+81.70E+7161.22.764E+72.47E+81.545E+7171.53.67E+74.03E+82.46E+7191.23.28E+71.48E+81.19E+7201.54.32E+72.23E+81.93E+72025.76E+75.87E+81.05E+7301.25.18E+73.76E+71.11E+7Figure 1
Linear analyses pressure results comparison.


## 3. Numerical Analysis

The
presence of imperfections, such as the ovality, the eccentricity, or the
variation of the thickness, due to presence of longitudinal welding, makes
elastic analysis inadequate to the purpose of determining the critical load, so
a nonlinear analysis was required. In this paper, nonlinear analysis was
performed using a numerical analysis, in which either load or displacement was
used as a control parameter.This paper
would address in the first case, the effect of imperfections, such as ovality
or eccentricity, on buckling load, and in the second one the effect of a welded
joint presence, which involves unequal wall thickness transition, on the
strength of structure. Some different elastic-plastic buckling analyses were
carried out by means of the finite element program MSC.MARC
to evaluate the failure behavior of the transition joint and the effects of
geometry. The assumption of perfect plasticity permitted a better assessment of
the effects of circumferential instability. The tube was a long cylindrical
shell of lengthL, uniform thickness t, diameter D; the two ends were assumed
as fixed hedges, under simultaneous action of external lateral pressure.In such cases,
the models were realized with solid 3D finite elements.
The
cylindrical surface was modeled by use of 20-node brick elements, because shell
elements were not adequate to compute the contribution of radial compression. The adopted mesh involved 60 elements over the
circumferential section, and 5 elements on the thickness; moreover, the mesh in
the case of transition welded joint was more refined in the welded area. In the numerical analyses,
a nonlinear approach was adopted.In fact, three different imperfections
were considered: the eccentricity(e), the ovality (W), which is known to have
the most detrimental effect on tube collapse, and the longitudinal welding. The
imperfection parameters are defined as follows: (4)W=Dmax−DminD,e=tmax−tmintmin, where Dmax and Dmin are
the maximum and minimum tube’s diameter, while D is the nominal diameter. In
the same matter, tmax and tmin are the maximum and
minimum tube’s thickness.The welding
joint is characterized by a local thickness variation: a ridge occurred along
the longitudinal welding [12]. The width variation of welded joint covers a
range between 1.5 and about 4 millimetres on the inner and outer diameters. A
view of the welded joint, in one section tube, is plotted in Figure 2.Figure 2
View of the welded joint in the
section tube.Moreover, the
analyses of section’s tube by means of scanning electron microscope have
highlighted the presence of a skeleton metallographic microstructure, in the
weld joint quite different from the material alone shown in Figure3.Figure 3
Details of transition area between welded joints.Probably, this
alteration induced by welding process might increase the local strength of
tube. Therefore, this phenomenon was treated, in the numerical analysis,
assuming the presence of these two areas with different stiffness.Material properties, also,
have been experimentally verified by means of tensile and Vickers tests. The
tensile tests results, that were carried out to verify the material properties
certification and that it may observe in the stress-strain curves (Figures4), have confirmed the certified material properties, in the case of
seamless A 316 tubes, while the recorded yielding stress values are greater
than that certified (250 MPa) for welded specimen tube A 316.Diagrams stress versus strain.
(a)(b)In
fact, tensile and Vickers tests have confirmed the alteration induced by the
welding process which must be treated, in numerical analysis, assuming the
presence of two areas with different stiffness, that is the welded area and the
other tube part (see also in the sketched Figure3).Weld-induced
residual stresses are usually discontinuous at the weld in a transition joint
and can be significant in pipeline [13].However,
appropriate simulation has indicated that the differences of the failure
pressure at plastic collapse were not noticeable with or without weld residual
stress, thus this local effect, in the present analysis, was not considered.The
imperfections considered were of 20% of tube thickness for the eccentricity;
2.5% of nominal diameter for the ovality, while a thickness reduction of 12.5%
(from the certified tolerance) was considered in the weld joint.In the hypotheses of cylindrical shell in a bendingless
state, before buckling, and of homogeneous external pressure, the critical load
for long thin shell was performed adopting the equation;(5)Dh∇8ω+Ekx2∂4ω∂y4+2Ekxky∂4ω∂x2∂y2+Eky2∂4ω∂x4−σx(0)∇4(∂2ω∂x2)−2σxy(0)∇4(∂2ω∂x∂y)−σy(0)∇4(∂2ω∂y2)=0, where σx(0),σxy(0),σy(0) are the initial membrane stresses; ω is the
buckling deflection in the z direction; E is Young’s modulus, ν is Poisson’s
ratio; h and L are the thickness and the length of the shell; D=Eh3/[12(1−ν2)] is the bending stiffness; kxky were the curvatures in x,y direction
[14].The boundary and load conditions were
the same (real) ones, adopted for the experimental buckling tests on the AISI
316 tube.Rigorous tolerance criteria for
equilibrium iterations were applied and the size of increment was automatically
determined according to the number of iterations in the previous increments.
Mesh sizing used has demonstrated that is suitable to
assure the convergence of results and to accurately predict collapse behavior.
The deepening
of the initial buckle was either a result of a local buckling process,
characterized by a local instability, and/or of progressive stress-deflection
dependence.However, the reaching of a certain
depth of the initial on going buckle can lead to the formation of another adjacent buckle if the material local resistance
is larger than the local deformation one.To quantify the effect of
the imperfection parameters and the influence of the geometry on the load
bearing behavior, shells of a wide range of diameters and thickness were
investigated.The influence
of mentioned imperfections on the buckling loads is indicated for a wide range
of geometries and assuming the same tube length equal to 828 mm in Table2. In
this table, there are overviews of the obtained buckling loads for perfect and
for imperfect shells.Table 2
Buckling loads
for perfect and imperfect shells.
D (mm)t (mm)Buckling load (Pa)Perfect
shellEccentricityOvalityWeld joint151.23.117E+73.053E+72.840E+73.210E+7161.23.370E+73.316E+73.277E+73.373E+7171.54.307E+74. 230E+74.102E+74.418E+7191.24.127E+74.074E+73.958E+74.215E+7201.55.015E+74.620E+74.249E+75.306E+72025.691E+75.248E+75.375E+76.507E+7301.26.182E+75.714E+75.227E+76.514E+7It is evident
that the capacity of bearing load decreases in presence of important
imperfections, such as the ovality and eccentricity, while the presence of a
longitudinal material alteration, due to the welding, seems to have the
opposite effect.The
eccentricity and the ovality have shown the most detrimental effect on tube
collapse, while the longitudinal welding seems to induce positive stiffening
effect due to the material and mechanic characteristics of the welded joint
areas.The deformed
shapes of the thin cylindrical shells, for the cases of ovality, eccentricity,
and welding joint imperfections, as for a selectedD/t ratio (D/t=13.3) are
shown in Figures 5, 6, and 7 as obtained in the first numerical analyses.Figure 5
Deformed shape for shell with ovality.Figure 6
Deformed shape for shell with eccentricity.Figure 7
Deformed shape for shell with longitudinal weld.The
displayed graphs and the numerical results point out that collapse load level
is strictly dependent from imperfection; so the level of the first instability
load initially falls with increasing imperfection amplitudes.The number of
waves around the circumference decreases as the length increases and takes the
minimum value of two only for very long tubes(L/R≥50).


## 4. Experimental Device

The
experiments were conducted on commercially available stainless steel AISI 316
tube specimens with nominal diameters of 20 mm and nominal thickness of 1.5 mm.
For a better explanation, type A and B, in Table3, indicates specimens with and without welding joint.Table 3
Experimental buckling pressure load.
Test n°Buckling pressure (Pa)Type A (welded tubes)Type BExperimentalExperimental14.48E+73.72E+724.74E+73.80E+734.81E+73.90E+744.74E+73.71E+754.95E+73.63E+764.57E+73.71E+7The tests
specimens were cut, from the own certified piece, to the corresponding length
of 828 mm. Firstly, the diameter(D) has
been measured at twelve equidistant points along the meridional length. This
type of measure has been repeated along two equally spaced points on the same
circumference, for type A as well as for those B.Measured
values of external diameter are in agreement with the nominal diameter for each
test specimen. In fact, the small variations registered on the test specimens
allowed to evaluate that the real mean ovality is about the 0.3%, while the
thickness mean variation measured around the transition joint is about the 10%.
A schematic diagram of the hydraulic pressure device and the specimen’s support
are represented in Figure8.Figure 8
Buckling test machine and specimen’s support.These supports
maintain the tube in the vertical position, so any rotations or preloaded
states on the specimens, during the stages of the assembling, are prevented.The test
specimen was confined between two rigid cylindrical sliding base supports,
manufactured in the laboratory of Pisa University  .Buckling tests
adopted the same boundary conditions, end restraints, geometric properties, and
load acting on the tube, used in all the numerical simulations. The required
load and the strain relationship are recorded for each test and used later to
check the numerical buckling results.Data
acquisition system was based on strain gauge instrumentation, which allows
recording the shell deformation.A
piezoelectric pressure transducer was used to measure the pressure in the test
chamber, filled with oil, in which the test specimen and its supports were
positioned.Previously,
the data acquisition procedure was tested to verify the accuracy and
reliability of the measures.Some recorded
relations of pressure-strain circumferential in the upper part of diagrams and
the axial in the bottom, for the AISI 316 specimen types A and B, are shown in
Figures9 and 10.Figure 9
Diagrams of strain versus pressure for A tests.Figure 10
Diagrams of strain versus pressure for B tests.In
the A tests, the effect of the thickness mismatch had produced an increasing as
for the failure pressure (over 50 MPa) as for the hoop stress in the tube.The
influence of welding is shown in the strain shapes of Figure10. In this graph,
the weld effect is further limited at local area near the weld. In the B
tests, the tubes were buckled to a final value of about 40 MPa pressure.In the above
graphs, buckling phenomenon corresponds to the nonlinear part observed, more
evident for B tests.Small
deviation from the nominal or perfect geometry results in significant loss of
strength of such specimen, where imperfections of just one wall thickness in
magnitude can reduce the buckling stress to only one third the classical value
for perfect cylinders (according to Koiter 1945).As pressure
approaches bifurcation load, infinitesimally small buckles start to form, which
subsequently cause buckling of structure. Circumferential compressive stresses
accelerate the development of these prebuckling deformations as is shown in
Figure11(a).Effect of circumferential stresses.
(a)(b)In longitudinally welded cylinder,
circumferential stresses reach yield level in tension, therefore, opposite to
the effects (circumferential stresses) due to the buckling phenomenon.The
stabilizing effect of the tension stresses near the weld is illustrated in
Figure11(b).With the
increase of external pressure loading, the tensile stresses decrease and the
compressive stresses farther away from the weld are further increased until the
point is reached where the structure buckles [15].


## 5. Comparison between Buckling Pressure Loads

All
FEM cases discussed above, for clamped end and external pressure loading, have
shown numerical results that indicate a similar behavior and deformation
mechanism. However, some significant differences are also observed for the B
specimens: the radial expansions of tube and fitting are small, so the rotation
and the bending stress near the weld are small.In Table3,
the main buckling tests results are indicated referring to the tube specimens
geometries (length and diameter to thickness ratio equal, resp., to 828 mm and D/t=13.3). The experimental data correspond to the pressure value at
which the apparent linearity is loss in the test diagram.The
recorded results have been compared with the numerical ones in order to
highlight an agreement or discrepancy if any.A quite good
agreement has been obtained between the predicted numerical and experimental
buckling pressure values (42 and 38 MPa for seamless A316) in the case of
seamless as well as welded tube specimens. It is worth to note that the
alteration induced by welding process may increase the local strength of tube
(its recorded yielding stress value has been highlighted to be greater than that certified (250 MPa) for austenitic welded steel specimens and might result in an increased
strength of tubes themselves.As
already mentioned, from the comparison with the experimental results it is
clear that numerical values are higher than the experimental ones, due to other
geometrical imperfections not considered. The reason of this discrepancy may be
probably due to the other imperfections, which could not be detected on the
tube specimens.The presence
of the assumed geometrical imperfections on the shell produces a reduction of about
15% on buckling pressure respect to that of perfect shell.Experimental values of collapse, for
specimens A, have shown that the geometry discontinuities at the weld
were insignificant in the plastic collapse analysis of transition joints.Schematic
picture of collapsed tube, visible in Figures12(a) and 12(b), shows
the buckling modes obtained, for A and B specimens.Schematic A and B tubes deformation after buckling.
(a)(b)In the latter
case, it is shown that buckling mode will always follow the lowest mode without
additional constraints.


## 6. Conclusion

Noted that the tubes of an integrated PWR steam generator operate at a
differential pressure of about 10 MPa, it can be shown that the tube can bear
fairly high-external pressure levels.The
present work has investigated the plastic collapse failure behavior and
considered the effect of thickness mismatch in the weld joint, of ovality and
eccentricity.Most
of the analytical approaches proved to be suitable to foresee the collapse
loads of the investigated thin tubes, in absence of geometrical imperfections
(ovality, eccentricity, and welding), even if with discrepancies larger than
the ones characteristic of the numerical approach.The
buckling strength was shown to fluctuate greatly depending on the particular
shape of each imperfection and on the interaction between neighboring ones.The
pressure-strain relationship used in the mentioned analyses and showed in the
above graphs is crucial to correctly predict the tubes collapse behavior.The preliminary results show that the
numerical approach, implemented with FEM codes, as MSC.MARC, seems able to
achieve a good evaluation of the critical load, as a good agreement was obtained between test results and numerical
predictions.Moreover, it has been shown by means of extensive
analyses, that this method can accurately predict the collapse behavior of thin
tubes, both in linear analyses and under the combined effects of material
nonlinearity and geometrical imperfections, respectively.Numerical analysis has also shown that the
thickness mismatch and the strength of welded joint are parameters that control
the buckling of welded thin shell.



---

*Source: 685805-2008-04-08.xml*