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.

(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.