2。Background: buckling of axially compressed cylinders

2。1。Theoretical background

The failure/buckling behavior of cylindrical shell is mainly characterized by the radius-to-thickness ratio of the shell。 Thin- shell cylinder usually buckles elastically, so failure by   buckling

controls the design criterion。 Whilst, thick cylinder fails in the elastic–plastic range, so failure is govern by collapse。 If a cylinder is subjected  to  uniform  axial  compression,  the  following    buckling

Table 1Set of material data obtained from uni-axial tensile tests on mild steel plate (E ¼ young's modulus, UTS¼ ultimate tensile strength)。 Note: The upper yield for

0。5 mm thickness specimen was taken from 0。2% proof stress。

modes  can  be  experienced:  (i)  axisymmetric  mode,  where the         

cylinder develops a corrugated appearance, with waves only in the axial direction, (ii) asymmetric mode, where the cylinder develops inward and outward displacement of the shell wall with several full waves around the circumference, and usually several waves up the height (note that the number of waves falls progressively as the shell becomes thicker), and (iii) symmetric mode, where the cylinder forms a single bulge around the circumference。

The classical buckling analysis of cylinder is based on the hy- pothesis of membrane pre-buckling state, i。e。, bending stresses are neglected, and of Donnell shallow shell theory。 From the above hypothesis, the classical elastic critical buckling load for  axially

compressed cylinder with axisymmetric mode is given by:

and at the other end, the cylinder is only allowed to move in  the

axial direction。 Fig。 1b shows the photograph of the experimental set up of the cylinder, taking a close look at the top edge after spring-back/unloading。 The cylinder is assumed to be made from

However, for relatively thick cylinder that fails within the plastic region, the reference buckling load, Fref, is taken as the load required to cause the cylinder to yield and it is designed according to Ref。 [15] as Eq。  (2):

mild steel with the material properties shown in Table 1。

The specimens were modeled using four-node three-dimen- sional doubly curved shell elements with six degree of freedom (S4R)。 The material is modeled as elastic perfectly-plastic。  Non-

fied  Riks

Fref   = πDtσyp

whereFcyl   is the cylinder classical elastic critical buckling   load

linear  static  analysis  was  carried  out  using  the   modi

method algorithm which is implemented in ABAQUS。

Fref   is the cylinder reference buckling load required to    cause

yieldE is the Young's Modulus of the   material

syp  is the yield stress of the material

ѵ is the Poisson's ratio of the  material

D  is the diameter of the  cylinder

t is the wall thickness of the cylinder

2。2。Modeling details

Consider a circular cylinder with diameter, D, radius, R and uniform wall thickness, t, having an axial length, L, as sketched in Fig。 1a。 It is assumed that the cylinder is subjected to axial com- pression。 The cylinder is assumed to be fully clamped at one end,

3。

Experimentation

3。1。Cylinder geometry and material properties

For this experiment, five mild steel cylinders with three dif- ferent thickness (t ¼ 0。5, 1。0, 2。0 mm) as shown in Fig。 2, were tested。 All cylinders were assumed to have nominal diameter, D ¼ 100 mm。 The axial lengths for all cylinders were kept at