Buckling strength of axially compressed cylindrical shells still attracts sizeable
amount of research. One specific topic has been devoted to buckling of variable length
(n. as a premodifier) cylinders under axial compression. When two or more
cylindrical segments form a prime load bearing structure then the interaction between
two neighboring segments become critical when the load is axial compression.
Typical application exists in aerospace where the gap between segments is filled by
shimming. Once axial compression is applied to two segments where there is a
variable gap between them, then the uneven load results. Diminishing axial gaps
result in a variable length of hoop contact between two cylinders and in localized
plastic deformations. These local effects at the imperfect end of the cylinder can
propagate along the shell’s length. They in turn can trigger asymmetric bifurcation
buckling or collapse, and as such they can pose design limitations. Numerical results
have highlighted a complicated nature of the interaction between the plate and
imperfect cylinder. A big drop in buckling strength has been obtained for relatively
small amplitude of waviness in length. The largest sensitivity of buckling strength
was associated with small amplitudes of axial length. For example, for axial length
imperfection amounting to 25% of wall thickness the buckling strength was reduced
by 40%. It appears that the number of sinusoidal waves in the imperfection profile
plays a secondary role, i.e., its role in reducing the buckling strength is not a dominant
factor.
Experimental program aiming at reassessing NASA Space Vehicle Design
Criteria guide SP-8007 is reported. Metallic cylinders with the radius-wall-thickness
ratio, R/t, varying between 250 and 1500 were buckled under static loading.
Experimental models were from copper, aluminum or stainless steel. Their diameter
was 135mm. The whole program consisted from 150 carefully conducted buckling
tests. The role of geometrical imperfections was of particular interest and it was
closely monitored during tests. It was noted, for example, that internal pressure
reduced the influence of imperfections on buckling, resulting in higher buckling
strength. A stable post-buckling behavior of copper model (R/t＝1350) was obtained
for the case of simultaneous action of internal pressure and bending. But for other
cases, internal pressure triggered local yielding and this, in turn, accelerated
elastic–plastic buckling. Some models were retested under a different set of loads.
While these tests were within elastic domain, the concerns were raised about the retest
data. In view of this, two sets of test data are provided, i.e., “single test” and “retest”
buckling results. In a separate study, the influence of the thermal insulation layer onto
buckling performance of cantilevered cylindrical shell is reported. This is both an
experimental and numerical study. Vacuum induced buckling tests of small steel
cylinders are reported. Models were mass manufactured industrial containers for
storage of paint. Initial geometry of cylinder’s generators was carefully measured and
the loading amounted to hydrostatic external pressure.
The cylinder made of Mylar is a commercially available membrane material used
by the aerospace industry for its resilience to fracture and tearing despite a thickness
here of just 0.044 mm. The cylinder has a nominal radius of 30 mm after wrapping a
flat sheet around a rigid Aluminium tube. The tube remains in place to form an inner
mandrel, which differs in radius from the Mylar by 1%: this gap is fashioned precisely
by inserting a rod-like shim between the Mylar and the mandrel before joining the
edges of the sheet along a narrow, overlapping seam of width typically less than 2%
of the circumference. The shim is then removed so that the annular gap is free of
obstruction, and the mandrel is slightly shorter so that only the Mylar cylinder is
compressed by the end-plates. As expected, local buckles form first at random
positions on the cylinder, but each has a distinctive diamond-like shape depressed
across its middle with edge ridges that eventually become highly creased. All
diamonds are oriented with their axes of symmetry aligned to the axial and
circumferential directions, and all have roughly the same size.
At the manufacturing stage, the confined cylinder may be subjected to cold
expansion, which results in an increase of the yield strength in tension, but may have
a significant effect on the material response in compression: more nonlinear behavior
and lower compressive yield strength. Compressive load strain diagrams are given as
measured for 20 inch diameter seamless pipe and UOE pipe in the same diameter.
UOE pipe is made by forming a long steel plate with a strong press in a U shape, then
with a special cap in an O shape. Then the longitudinal weld is made. Finally cold
expansion of the pipe is applied to minimize geometrical imperfections and to achieve
the same diameter for all pipes. The UOE pipes showed a clear Bauschinger effect in
the compressive stress strain diagram.
The Bauschinger effect may result in a drop of the collapse pressure of 20%
compared to the same geometry nonexpanded pipe. Much research has been
performed on this issue. It would be interesting to investigate how Bauschinger effect
works out on the strength of confined cylinders using the tools developed by Vasilikis
and Karamanos.
The reduction of strength induced by the Bauschinger effect can be greatly
rectified by proper heat treatment. DeGeer et al. have performed several theoretical
research programs and experimental work on this issue. Such heat treatment is applied
during
polyethylene coating for corrosion protection. The effect of such treatment on the
mechanical behavior of the steel and on the collapse pressure is described in a
research program aimed for a pipeline in a very deep part of the Gulf of Mexico
(deeper than 2000m).
Engineering dynamical systems are often subject to excitations such as winds,
sea waves, and seismic motions which inherently possess evolutionary characteristics.
That is, not only the intensity of the excitation, but also its frequency content is a
time-varying function. Thus, a realistic representation of these excitations requires, in
most cases, resorting to concepts relating non-stationary stochastic processes. In this
regard, certain recent advances include wavelet based representations of
non-stationary stochastic processes which can be viewed as natural extensions in the
wavelet domain of earlier work related to non-stationary stochastic process
representation in the Fourier domain.
Further, assessing the risk of failure, or performing a reliability based analysis of
dynamical systems is closely related to the determination of the probability that the
response of the system stays below a prescribed threshold over a given time interval.
The determination of the above time-dependent probability, known as survival
probability, has been a persistent challenge in the field of stochastic dynamics. An
alternative definition of this problem, known as the first-passage problem, relates to
the evaluation of the probability that the system response crosses a predetermined
threshold for the first time over a given time interval.
Considering nonlinear systems under non-stationary stochastic excitations
increases significantly the difficulty of the first-passage problem, since even in the
simplest case of a linear oscillator under white noise excitation, an exact closed form
solution for the survival probability has not been feasible so far. Clearly, Monte Carlo
simulation (MCS) based approaches are among the most potent tools for general
uncertainty quantification of systems of engineering interest. In this regard, several
research efforts have focused on developing versatile MCS based techniques such as
importance sampling, subset simulation and line sampling for reliability assessment
applications. However, there are cases where the computational cost of these
techniques can be prohibitive, especially when large scale complex systems are
considered. Thus, there is a need for developing efficient approximate numerical
and/or analytical methodologies for addressing the first-passage problem. One of the
first frameworks developed, restricted to linear systems, is based on knowledge of
system response statistics, such as mean out-crossing rates, and usually assumes
Poisson distribution based approximations. Further, various techniques such as the
probability density evolution method and numerical path integral schemes have
focused on solving directly the original (nonlinear) stochastic differential equation
(SDE) of motion and determining first-passage statistics. Note, however, that the
computational cost related to these schemes can be demanding for systems with more
than a few degrees of freedom.
In the course of our study, we examined several parameters that affect buckling
behavior of confined thin-walled cylindrical shells. Nevertheless, a few additional
parameters are stated by Prof. Gresnigt, namely residual stresses, Bauschinger effect,
and heat treatment that may influence the mechanical properties. Residual stresses and
the Bauschinger effect are mainly related to the manufacturing process. For
large-diameter, thin-walled cylinders (D/t>100), the most popular manufacturing
method is spiral welding of a steel coil. Unfortunately, there exists little information
on the residual stresses of such pipes. In the course of a European research program,
the authors are currently working on the finite element simulation of spiral-welding
pipe-manufacturing process to determine residual stresses, including the effects of
decoiling. Furthermore, residual stress measurements are also performed for
comparison purposes. By completion of this research, information on the distribution
of residual stresses will be available and appropriate stress distribution can be
considered as the initial stress state for external pressure analysis under confined
conditions.
Heat treatment is another factor for external pressure capacity. There exist several
publications for the beneficial effects of heat treatment on external pressure buckling
of thick-walled offshore steel pipes. On the other hand, there is no information for
such effects on the response of thin-walled confined pipes, mainly due to pipe coating.
Therefore, more research is necessary on this subject. Prof. Gresnigt also refers
extensively to “mechanically lined pipe”, a novel pipe product, very promising for
hydrocarbon pipeline applications. When this double-wall pipe is bent, the
thick-walled outer cylinder deforms rather smoothly, whereas the thin-walled liner is
susceptible to wrinkling under lateral confinement conditions, provided by the outer
cylinder, as pointed out by several recent publications, experimental and
computational. For lined pipes used in offshore applications, their structural behavior
and buckling under external pressure is of particular interest, not given attention in the
literature. This problem is somewhat different than the one described in the present
paper; external pressure is not applied at the interface between the outer cylinder and
the liner, but it is applied at the outer surface of the double-wall lined pipe. This
means that both the outer and the liner pipe shrink and eventually buckle, most likely
in the form of an oval shape. However, it would be interesting to analyze externally
pressurized lined pipes with the numerical tools of the present paper and, in particular,
to examine the detachment and the deformation shape of the thin-walled liner.
The effects of gravity loading on the structural behavior of buried thin-walled
cylinders under external pressure conditions have been examined, referring mainly to
buried steel water pipelines, with values of D/t ratio greater than 100. Initial
ovalization of the buried pipe due to gravity will reduce its external pressure capacity.
Using the finite element models of the present study, it would be helpful to extend the
present work in determining a rational allowable value of initial ovalization due to
gravity, so that external pressure capacity is not significantly reduced. This would be
of significant interest to the water steel pipeline community.
Buckling of thin-walled shell-like components, frequently found in pressure
vessels, has been the subject to numerous studies over many decades. A substantial
wealth of accumulated knowledge exists in the form of books, conference proceedings,
reviews, and other published material. The authors list some existing shell stability
technical issues—many of which are not addressed in the NASA recommendations.
Initial geometric imperfections, nonlinear prebuckling deformations, boundary
conditions, load introduction effects, combined loads, and variation in material
properties are just a sample of topics still awaiting further investigations. The reliance
on arbitrarily chosen knock-down factors is also echoed. There have also been a
number of published reviews of research on buckling. A review of the buckling
resistance of thin and slender structures typically found in nuclear industry, back in
1984, categorized vessel related components, prone to buckling, as:(i)stiff (buckling
in plastic range),(ii) medium (elastic/plastic buckling),and (iii)soft (elastic
buckling).The criterion used here was the ratio, REY, of elastic bifurcation-to-first
yield load. For REY≥ 5 the component was deemed to be stiff. For REY≤ 0 .2,
structure was classified as soft. In this context, the results of 42 experimental buckling
tests have been compared with computed predictions of buckling. From the difference
between tests and numerical predictions of buckling together with the number of tests
carried out, it is seen that the range of errors varies from -30% to +50%.The
discrepancies have been attributed to: (i) specimen geometry, (ii) boundary conditions,
and (iii) material data. It was also noted that in some cases the buckling was difficult
to observe experimentally and it was subjective (internally pressurized domes; for
example).
Progress has been made in assessing the issues leading to big discrepancies
between test data and theory. Testing methods, for example, have improved and they
have been more focused. It is true to say that buckling experiments are now better
instrumented than in the past. Equally, more rigorous computational models have
been developed. Recent review papers address structural behavior of domed ends and
add to the accumulated know-how base. Design methodology is available in the form
of various codes, and in a specialized stability handbook. As mentioned earlier, one of
the unresolved issues within structures prone to buckling is the effect of initial shape
imperfections on the magnitude of buckling load. Despite efforts aiming at finding a
universal answer to the dilemma, it still remains a subject of active research. It is true
to say that the detrimental effect of initial shape imperfections has been quantified for
a range of structural components and loading conditions. However, it is, by-and-large,
still based on a component by component basis. An attempt of a unified approach to
the design of imperfection sensitive structural components can be found. However,
research into the derivation of less restrictive knock-down factors still continues. A
different way of reducing imperfection sensitivity of axially compressed cylinders is
explored. Numerical results suggest that cylinders filled with, and/or surrounded by a
compliant core, can be less imperfection sensitive.
We are interested in characterising the buckled geometry of Horton and Durham
in a modern context, for promoting shape-changing, or “morphing”, structures in new
technologies reliant upon a controllable surface “texture”, for example, haptic
electronic displays, building facades with tuneable thermodynamical properties, and
aerodynamical surfaces that can switch from being smooth to rough and back again.
In this sense, the behaviour must remain elastic well into the post-buckled regime, so
we deal with very thin shells whose radius-to-thickness is greater than 1000, above
the value of 800 used by Horton and Durham, and much higher than the typical upper
limit of 500 for practical shells. Correspondingly, our cylinders resemble membrane
tubes rather than shells and, because of such thinness, post-buckling is dominated by
inextensible behaviour where a well-known, associated mode-shape is the
“Yoshimura” pattern, with commensurate features observed here. We were originally
focussed on the torsional “wrinkling” of membranes wrapped around tubes without
knowledge of Horton and Durham’s work when we noticed a distinctive
doubly-periodic buckled pattern after compressing the membrane ends; so we decided
to change our focus and concentrate on the axial case alone. There are also parallels
with recent studies on the buckling of thin films connected to compliant substrates,
where distinctive surface patterns are wrought by in-plane compressive stresses: the
substrate elastically restrains the out-of-plane deformations in the same way that the
mandrel provides a “hard” constraint. But many results come from simulations
because experiments are notoriously diffcult to arrange whereas our patterns can be
quickly observed with inherent uniformity by wrapping a thin sheet of paper loosely
around a cardboard tube before compressing it by hand. Another related study
concerns the “stamping” of initially constrained elastic plates by confining them
further between rigid surfaces. This shows, rather elegantly, the transition from a
buckled state to a wrinkled state, depending on the level of confinement.
Foremost, we are concerned with how the overall buckled shape performs when
the initial annular gap between the shell and mandrel is varied. In particular,
experiments show that the circumferential wavelength viz. the number of
circumferential buckles, appears to be governed only by the annular gap size and the
radius of mandrel (or shell), and not its thickness nor axial length. This differs from
the observations of Hunt et al. where, recall, no mandrel is present, and the choice of
length must be contrived to fit with the expected buckle size. In our case, length does
not matter because the mode-shape is re-stabilised by the mandrel and because
inextensibility during post-buckling implies that the thickness does not influence
matters provided it is small enough. Furthermore, the number of circumferential
buckles does not depend on the load: this empirical observation allows us to decouple
our views of the buckle geometry along and around the cylinder and thence to treat
them separately. The corresponding wavelengths are reasonably straightforward to
deduce using the assumption of inextensibility because we may view them as the
deformed loci of shell in these directions, as if packaged according to geometrical
constraints. This approach is novel in its kinematical basis: that we are able to deduce
the loaded shape—albeit approximately—without solving for the nonlinear governing
shell equations of deformation.
For transport of corrosive hydrocarbons, lined pipe can be an attractive
cost-effective solution, especially in deep-water applications for flow lines. In these
applications, lined pipe consists of a carbon steel outer pipe that provides the
structural strength and a corrosion resistant alloy (CRA) liner, protecting the carbon
steel outer pipe from the transported corrosive product. The liner pipe is mechanically
fitted inside the outer pipe. Altogether, it offers a much more cost-effective solution
than a single thick-walled CRA pipeline.
An important failure mode of lined pipe is liner wrinkling when the lined pipe is
bent. Nevertheless, in the case of off-shore pipeline applications, lined pipes are
subjected to externalpressure and the thin-walled liner might buckle in a form similar
to the one described by Vasilikis and Karamanos. It would be interesting to investigate
this problem as well.
In addition, as a consequence of a small leakage, there will be a pressure built up
between the liner and the outer pipe. If the transported product is gas and if the
internal pressure in the pipeline suddenly decreases (e.g., due to shutdown of the line),
a collapse could occur as described in the paper of Vasilikis and Karamanos.
Another interesting application may refer to thin-walled buried pipelines. Due to
the surcharge of the soil and possible traffic loads, the pipe tends to flatten. If external
water pressure if present, it will also contribute to the flattening, particularly in
crossings with water-ways and canals.
In water transportation pipelines, the operating pressure is low and the diameter
mostly large, leading to thin-walled pipes. For the design, there are two main criteria:
strength and allowable deflection.
The strength of the pipeline under external pressure is described in the paper by
Vasilikis and Karamanos: the bending moments in the pipe wall and the danger of
snap-through of the flattened top of the cross section. Nevertheless, there, the
allowable deflection to avoid such a failure mode is still an open issue. In present
design standards, there is a limit on the ovalization because of the requirements
related to avoiding obstructions during the use of the pipeline but, in particular, also
because of the danger of snap-through. The mechanics of this failure mode is not well
understood, and the design guidance is just a limit to the flattening. This limit is rather
arbitrary. The safety against snap-through is unknown. The work of Vasilikis and
Karamanos could be extended to derive a much better motivated criterion. It is noted
that, because of the popularity of higher strength steel, there is a tendency to even
thinner-walled water transportation pipelines, e.g., pipes with a diameter-to-thickness
ratio that exceeds 200, very prone to snap-through buckling.