A PARTIAL SAFETY FACTOR FOR PRESSURE HULL COLLAPSE PREDICTION USING FINITE ELEMENT ANALYSIS
2. SUMMARY OF THE VALIDATION EXERCISE
2.1. TEST MODELS
The mode of collapse of a pressure hull tends towards two extremes: if the stiffeners are sufficiently strong, the collapse will occur in the short sections of cylindrical plating between frames i.e. an interframe collapse; if the stiffeners are inadequate, or the model is long, collapse can occur in an overall mode. Obviously there is also a middle ground where interaction between the modes is possible.
Deliberately short models, typically 10 to 12 frames, were used to isolate the interframe collapse mode in a structure with scantlings representative of a submarine pressure hull. This ensured that the overall collapse pressure of the model was higher than the interframe collapse pressure without artificially increasing the frame dimensions. The results of a large number of tests, carried out over several decades, have been condensed into an empirical curve, which is considered accurate to r10% provided the magnitude of the OOC is no greater than 0.5% of the radius. Because of this accuracy, design is based on interframe collapse and frames are subsequently sized to avoid overall collapse [3].
Conversely, deliberately long models were tested to isolate the overall modes of failure. As well as the extremes of ‘short’ and ‘long’ cylinders there exist many examples of models representative of real submarine compartments.
2.2. FE MODELLING
The shape of any real pressure vessel will inevitably include departures from perfect circularity and these grow steadily as external pressure is increased from zero.
In fact, many of the models tested were fabricated deliberately with a dominant overall OOC, normally n=2, 3 or occasionally 4. Welding the frames to the plating also causes an indentation of the plating (‘hungry horse’
effect) which influences the interframe collapse.
Recognizing the importance of shape imperfections, the true shapes of the test models were extensively measured. Because overall collapse behaviour is limited to lower modes, n=2 (ovalisation), 3, 4, or 5, OOC data were typically measured at 15q intervals at frames.
Interframe buckling can be driven by higher buckling modes, n=12 to 15, therefore OOC was measured at 5q intervals when recorded at midbay. Frame spacing and
Warship 2008: Naval Submarines 9, Glasgow, UK
©2008: The Royal Institution of Naval Architects misalignment data were usually recorded at 90q intervals,
when taken.
Models were generated using PATRAN [4]. The geometry was created by generating points in space at measured locations. Curve and surface fitting techniques were then used to complete the geometry. Previous work has shown that perturbations in the geometry generated by the interpolation involved in these processes are of similar magnitude to those measured in the physical models.
Analysis was carried out using ABAQUS [5], which provides flexibility in terms of elements, material models and non-linear solution strategies. Previous experience with stress and collapse analysis of pressure vessels and associated structure informed modelling decisions such as element selection and mesh density; second order thin shell elements were used with a density of six or eight elements between frames.
The pressure hull plating of a submarine is cold bent, or cold rolled, a process which results in permanent plastic deformation of some of the material, which in turn locks in a pattern of residual stresses in the plate. Obviously this will affect the subsequent response of the structure.
In many models the T stiffeners were fabricated, with only the flanges being rolled and therefore subject to similar residual stresses.
A simple analytic solution, based on beam theory, exists for residual stresses in bent rectangular beams and this has been applied to cold-bent plate. However, there are limitations, e.g. it doesn’t account for material hardening or Poisson effects in the plate and it was decided to calculate the residual stresses directly using the FE method. A model of a typical section of plate used to form the shell of a model was analysed for each case.
Two steps were carried out: plastic bending, or rolling, which was idealised in a single step by applying a rotation at one end of the plate, and primarily elastic springback. An iterative procedure was used to determine the required overbend radius. The results of one such analysis are shown in Figure 1.
Figure 1 Contour plot of the residual circumferential stress on the inner surface of a typical plate section after cold-bending.
The calculated stresses showed substantial variation close to the edges of the plate. Away from the edges it was found that, although the residual circumferential stresses were high, they were not overly sensitive to modelling details such as exact plate thickness or material hardening behaviour, and agreed closely with the simple analytic solution for the stress s-xx in the bending direction, as shown in Figure 2. However, it was also found that significant residual longitudinal stresses, s-yy, were developed. Unlike the circumferential stresses, these were unbalanced with the resultant moment tending to deform the plate to an anticlastic surface after springback. The variation of the final radius of the plate was small and was ignored on the basis that the unbalanced residual moment would be reacted in the adjoining plate when the structure was welded up.
-400 -300 -200 -100 0 100 200 300 400
-4 -2 0 2 4
z (mm)
residual stress (MPa)
sig_x analyt ic s-xx s-yy
Figure 2 Residual stresses through the thickness of cold rolled plate.
The residual stresses were then simply included as an initial equilibrating condition in the appropriate components of the structural model. By default ABAQUS defines shell behaviour at five section points through the thickness but this was raised to 21 to give adequate resolution of the residual stresses.
A number of models were selected for analysis based on the amount of data recorded during the fabrication and testing of the models. Typically, OOC measurements were recorded at many, if not all, frame locations and for some models, at mid bay locations too. Frame spacing and misalignment were also recorded in many cases.
Plate thicknesses were usually limited to minimum, maximum and average values for individual plates, and scantlings were normally recorded for individual frames.
Extensive records of material properties were also taken, often as tables of proof stresses, in which case the material was modelled as elastic/perfectly plastic, but in some cases detailed load-displacement curves were available and more complex material models used.
Warship 2008: Naval Submarines 9, Glasgow, UK
©2008: The Royal Institution of Naval Architects 2.3. RESULTS
Figure 3 shows the interframe collapse of a short model (top) with the predicted collapse (bottom) and Figure 4 shows the interframe collapse of a longer, more representative model. In both cases the qualitative agreement is excellent. The results of the analyses of a series of short models is summarised in Table 1. In some cases excellent agreement between the calculated and observed collapse pressures was achieved but in a few cases the calculated pressure was significantly overestimated. Inspection of the raw data for these models revealed large variations in thickness across individual plates and further analyses were undertaken with minimum values. These results are shown in parentheses and all results agree to within 6% of the observed collapse pressures.
Figure 3 Interframe collapse in a short cylinder.
Figure 4 Interframe collapse in a model representative of a submarine compartment
Model Mode PFE/Pexpt
11 buckle 1.14 (1.006)
13 buckle 1.002
15 yield 1.098 (1.059)
17 buckle 1.062 (1.019)
18 yield 1.074 (1.054)
19 yield 0.963 (0.920)
20 overall 0.999
Table 1 Failure mode and predicted collapse/experimental collapse pressures of short models.
Figure 5 shows an overall collapse mode and Table 2 summarises the results of the analyses of longer models.
Only two of the models considered actually failed in an overall mode. The first (the 36 frame model), and two other cases which are not reported here, were predicted to within +6% but the 40 frame model was initially overpredicted by 8.7%. Further investigation into the source data appeared to show that the Young’s modulus of the plating used in the central section, and for some of the frame tables, was significantly lower than the assumed 207 GPa. When rerun with the measured value of 177 GPa, the collapse pressure was predicted to 4.2%.
Warship 2008: Naval Submarines 9, Glasgow, UK
©2008: The Royal Institution of Naval Architects This issue of variable modulus is also touched upon in
the next section.
Figure 5 Overall collapse.
The conclusion of the validation exercise was that if sufficient detail of shape, scantlings, residual stress and material properties are included in a finite element model then the collapse pressure of a submarine pressure vessel can usually be predicted to within +6%, providing the mode of collapse is identified and, if interframe, the analysis is repeated with the minimum measured plate thickness.
The major sources of error are likely to be inadequate knowledge of the structure and effects omitted from the model. Examples of the former include insufficient measurements of plate thickness and values of yield stress without stress-strain curves. Examples of the latter include residual stresses due to welding and anisotropic non-linear material behaviour. It was suggested that non- linear and anisotropic behaviour could be a result of production processes, e.g. a preferred rolling direction during the production of flat plate. Analysis of the production process has been carried out to inform plant design [6] but the output of such analysis may be useful in defining the initial condition of the material in the
further development of the collapse analyses described here.
Model Mode PFE/Pexpt
25 frame IF 1.017
28 frame IF 1.015
36 frame OA n=3 1.051
40 frame OA n=2 1.087 (1.042)
Table 2 Failure mode and predicted collapse/experimental collapse pressures of long models.