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The following scenario for the buckling process is assumed:

                (i) When a
geometrically perfect cylinder is subjected to axial compression, it maintains
its load-bearing capacity up to the classical elastic critical stress under
which axisymmetric buckling (primary buckling) takes place.

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                (ii) After the
primary buckling, the deformation mode maybe bifurcated toward the
non-axisymmetric deformation mode. The curves shown in Fig. 1 have the
potential to become bifurcation paths. If the amplitude of initial imperfection
is large, this type of non-axisymmetric buckling may occur, bypassing the
axisymmetric buckling.

                (iii) In
accordance with the progress of deformation, the load-bearing capacity
continues to decrease until reaching the minimal value, and then begins to
increase. However, another bifurcation path with a lesser number of waves
exists adjacently (on the right-hand side in Fig. 1), and since this requires
smaller loads than the current path does to produce the same axial shortening,
the current deformed mode transitions to the next path in a snap-through manner
(secondary buckling).

                (iv) Secondary
buckling occurs repeatedly, accompanied by successive reductions in the number
of circumferential waves at every path jumping.

To summarize, Fig. 1 suggests the possibility such that the deformed
pattern of a cylinder under axial compression progresses discontinuously. A
group led by Yamaki in relation to the elastic buckling cylindrical shells and
results obtained from their experiments broadly support the above-mentioned
scenario

Geometric
Nonlinearity

Geometric nonlinearities arise from the presence of large strain,
small strains but finite displacements and/or rotations, and loss of structural
stability. Large strains, over 5% may occur in rubber structures and metal
forming. Slender structures such as bars and thin plates may experience large
displacements and rotations with small strains. Initially stressed structures
with small strains and displacements may undergo a loss of stability by
buckling.

Material
Nonlinearity

Material nonlinearities arise from the presence of time-independent behaviour,
such as plasticity, time-dependent behaviour such as creep, and viscoelastic/viscoplastic
behaviour where both plasticity and creep effects occur simultaneously. They
may result in load sequence dependence and energy dissipation (irreversible
structural behaviour).

Contact

Nonlinearity due to contact conditions arises because the prescribed
displacements on the boundary depend on the deformation of the structure.
Furthermore, no-interpenetration conditions are enforced while the extent of
the contact area is unknown.

FEM
models on buckling analysis

The numerical simulations were carried out using the general finite
element program ABAQUS 6.4-PR11. For this study, steel cylindrical shells with
three different lengths (L = 100, 150, 250 mm) and two different diameters (D =
42, 50 mm) were analysed. Cracks of various lengths and several orientations (?
= 0?, 30?, 45?, 60?, 90?) were created in the specimens. Furthermore, the
thickness of shells was t = 2mm. Fig. 1 shows the geometry of a cracked
cylindrical shell. According to this figure, the parameter a specifies the
crack length, and the parameter M specifies the orientation of the crack
direction measured from the horizontal axis. The distance between the centre of
the crack and the lower edge of the shell is designated by L­0, as
shown in Fig. 1. The specimens were denoted as follows:
L250–D42–?45?–?0.4–C0.5. The numbers following D and L show the diameter and
length of the specimen, respectively. The parameter C = L0/L specifies the
crack position. The cylindrical shells used for this study were made of a mild
steel alloy. The mechanical properties of this steel alloy were determined
according to ASTME8 standard 30, using the INSTRON 8802 servo hydraulic
machine. The stress-strain curves and the stress–plastic strain curve are shown
in Fig. 2, and the respective values are given in Table 1. Based on the linear
portion of the stress–strain curve, the value of the elasticity module was computed to be E = 195GPa and
the value of the yield stress was computed to be ?y = 340MPa. Furthermore, the value of
Poisson’s ratio was assumed to be ? = 0.33.

Boundary
conditions

For applying
boundary conditions on the edges of the cylindrical shells, two rigid plates
were used

that were
attached to the ends of the cylinder. To analyse the buckling of a shell
subject to axial load, by analogy with what was done in the experiments, a 30mm displacement
was applied centrally to the centre of the upper plate, which resulted in a
distributed compressive load on both edges of the cylinder. Additionally, all
degrees of freedom in the lower plate and all degrees of freedom in the upper
plate, except in the direction of the longitudinal axis, were constrained.

Analytical
process

The eigenvalue
analysis overestimates the buckling load, because in this analysis, the plastic

properties of
material do not have any role in the analysis procedure. For buckling analysis,
an

eigenvalue
analysis should be done initially for all specimens, to find the mode
shapes and corresponding eigenvalues. Primary modes have smaller eigenvalues,
and buckling usually occurs in these mode shapes. For eigenvalue analysis, the “Buckle” step was used
in software. Two initial mode shapes and corresponding displacements of all
specimens were obtained. The effects of these mode shapes should be considered in
nonlinear buckling analysis (“Static Riks”
step). Otherwise, the software would
choose the buckling mode in an arbitrary manner, resulting in unrealistic
results in the nonlinear analysis. For the “Buckle” step, the
subspace solver method of the software was used. It is noteworthy that, owing
to the presence of contact constraints between rigid plates and the shell, the
Lanczos solver method cannot be used for these specimens 32. In Fig. 4, two
primary mode shapes are shown for the specimen L150–D42–?90?. After completion of the “Buckle” analysis, a
nonlinear analysis was performed to plot the load–displacement
curve. The maximum value in this curve is the buckling load. This step is
called “Static Riks”
and uses the arc length method for
post-buckling analysis. In this analysis, the nonlinearity of both material
properties and geometry is taken into consideration.

 

Experiments
with cylindrical shells in axial compression

From the discussion of the preceding
article it is seen that only in the case of very thin shells will buckling
occur within the elastic range in which the theoretical formulas can be
applied. Quite naturally, practically all the early experiments were made with
comparatively thick tubes which fail, if longitudinally compressed, owing to
yielding of the material and not to buckling. Later, in connection with the use
of thin shells in aircraft structures, experiments were made with very thin
cylindrical shells under axial pressure. To realize a central application of
the load, steel balls testing is used as shown. The edges of the shell are
welded to the end plates. Owing to this additional constraint the edges are
stiffened and buckling usually occurs at some distance from the ends.  

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