Elasto-Plastic Buckling Stability Under Hydrostatic Pressure
Authors: Benoit J.M. , Bellamy Gilles P.J. 
Published: October 2008

ABSTRACT
A spherical submarine hull element is analyzed by means of a nonlinear finite-difference program. The imperfection sensitivity of this stiffened shell is greatly dampened by the structural discontinuities introduced through the stiffened portholes.
The influence of shape imperfection or geometric manufacturing tolerance is studied by varying the imperfection amplitude and by changing its distribution (different shapes).
Several grades of steel are considered for the present hull structure as well.
The analytical results are presented on various graphs visualizing the buckling characteristics of the present structure. Other simplified analytical tools are reviewed with the purpose of serving as a cross-check. Finally, model test results are compared with the finite-difference results.

INTRODUCTION
The development of deeper offshore oil and gas fields required the use of deep-sea diving vessels. These work submarines are called upon to perform various tasks ranging from inspection to equipment installation on the sea floor. Reliability and safety are essential, in particular from a structural point of view. The main design load consisting of hydrostatic pressure warrants a detailed analysis.
The cylindrical and spherical shapes are most common in marine vessels. The optimum load-carrying capacity of these structural shell shapes allows substantial material saving. The relatively thin structures developed can be subject to stability or buckling failure while perfectly well designed from a material stress point of view. Hence, a stability check is absolutely mandatory for these vessels. Loss of stability can be local, leading to stress redistributions without major damage, or it can cause buckling of larger parts, leading to total collapse.
The failure mode can be very progressive or sudden, the deformation pattern giving ample warning or no warning at all.
Buckling can occur in the elastic range or in the plastic range of stresses.

BASIC CONCEPTS OF STABILITY
The principal sources for nonlinear structural behavior are nonlinearities in the material properties and the geometric nonlinearities caused by rotations of structural elements. Since the latter is the physical reason for static instability, we shall consider the effect of material nonlinearity at a later stage.
Buckling of a structure occurs either at a bifurcation point or at a limit point. A bifurcation point indicates a load level at and above which some new deformation mode is possible. Hence, a bifurcation analysis yields a rigorous solution of the problem only if the failure mode is orthogonal to the prebuckling deformation pattern. The existence of a bifurcation point indicates only that the equilibrium on the primary path loses its stability. The structural behavior at and beyond this point is governed by the conditions on the secondary path. This secondary equilibrium path can be stable (Fig. 1) or unstable (Fig. 2), symmetric or asymmetric. An unstable bifurcation point indicates an imperfection sensitive structure.
A limit point (Fig. 3) corresponds to the maximum of the load-displacement curve visualizing the primary equilibrium path. Under a load exceeding this maximum, there exists no equilibrium configuration in the immediate neighborhood. The structure collapses according to an amplified prebuckling deformation mode.