Abstract
Pillars have various applications in ships and maritime structures. They can undergo bending, torsion and compression loads. Their limit state can be determined through finite element analysis, considering their geometric characteristics, material and applied stresses. However, finite element modelling can be less easy to handle than an analytical solution, especially when large numbers of calculations are needed.
In this study, we present the results from 1D analytical analysis considering an initial compression stress and two different moments at each end of the pillar — implying that a transverse force was applied at one end. This represents a typical ship pillar between two consecutive decks.
The physical problem statement consists of the resolution of a first order coupled differential system of four equations. Solving this problem numerically accounting for large displacements and elastic material behavior yields solutions very close to the finite element modelling. We finally developed an elastic-plastic analysis of the same problem. Analytical resolution consisted of the derivation of a moment-curvature law including elastic and perfectly plastic behaviors. To derive this law, compression and moment are applied on a cross section of the beam. In the elastic behavior, bending stress and curvature are related through Young’s modulus and inertia, whereas once yield stress is attained no additional stress can be carried in the yielded surface and a new moment-curvature relationship must be developed: In a given cross section, bending stress varies linearly around the neutral fiber (which position depends on the compression stress and curvature) until yield stress is eventually attained. Above and below the position where the yield stress is attained, bending stress remains constant and equal to the material yield stress.
The full analytical procedure is the following: let us first apply the compression stress in a uniform manner in the beam cross section. Then, curvature is increased step by step, and we let plasticity arise and penetrate from the inner part of the curved beam. Integrating the moment over the cross section at each curvature step allows to build the moment-curvature relationship. Analytical results were compared to finite elements analysis; and turned out to be similar with a discrepancy falling within a few percents for both full and hollow pipes.
Finally, we used the moment-curvature relation that was developed to derive the full elastic-perfectly plastic solution with a numerical model; and compared to FEA results.
