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Defence R&D Canada – Atlantic
DEFENCE DÉFENSE&
Prediction of the Effects of Cold Bending on
Submarine Pressure Hull Collapse
Liam Gannon
Technical Memorandum
DRDC Atlantic TM 2010-065
April 2010
Copy No. _____
Defence Research andDevelopment Canada
Recherche et développementpour la défense Canada
This page intentionally left blank.
Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapse
Liam Gannon
Defence R&D Canada – Atlantic
Technical Memorandum
DRDC Atlantic TM 2010-065
April 2010
Principal Author
Liam Gannon
Defence Scientist
Approved by
Kevin McTaggart
Acting Head/Warship Performance
Approved for release by
Calvin Hyatt
DRP Chair
© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2010
© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale,
2010
Original signed by Liam Gannon
Original signed by Kevin McTaggart
Original signed by Ron Kuwahara for
DRDC Atlantic TM 2010-065 i
Abstract
Submarine pressure hull frames and shell plating are shaped by cold bending during fabrication.
Cold bending introduces significant residual stress in these components which can be detrimental
to the strength of the structure. This study evaluates different methods of incorporating cold
bending residual stresses in the analysis of pressure hulls considering different out of circularity
mode shapes. Several methods of pressure hull collapse analysis are compared considering
interframe and overall collapse modes. These include an empirical method, a finite difference
method and the finite element method.
Collapse pressures predicted using the methods prescribed in the UK MoD submarine structure
design standard, SSP 74, are found to be conservative when compared with results from finite
element analysis. Collapse pressures predicted using effective stress-strain curves to incorporate
the effects of cold bending residual stress in finite element models agree well with those predicted
by explicitly modelling the cold bending process. This indicates that the use of effective stress-
strain curves is an acceptable means of accounting for the influence of cold bending residual
stress on the collapse pressure of a submarine pressure hull.
Résumé
Le bordé extérieur et les couples de coque épaisse des sous-marins sont formés par pliage à froid
pendant la fabrication. Le pliage à froid introduit une contrainte résiduelle importante dans ces
pièces, ce qui peut diminuer la résistance de la structure. La présente étude évalue différentes
méthodes permettant d’incorporer des contraintes résiduelles de pliage à froid dans l’analyse des
coques épaisses en considérant différentes formes rondes déformées. Plusieurs méthodes
d’analyse d’écrasement de coque épaisse sont comparées en considérant l’écrasement global et
l’écrasement entre couples. Ces méthodes sont la méthode empirique, la méthode par différences
finies et la méthode par éléments finis.
Les pressions d’écrasement prédites à l’aide des méthodes prescrites dans la norme de conception
de structure de sous-marin du ministère de la défense du Royaume-Uni, SSP 74, sont considérées
conservatrices lorsqu’on les compare aux résultats de l’analyse par éléments finis. Les pressions
d’écrasement prédites à l’aide de diagrammes effort-déformation efficaces pour incorporer les
effets des contraintes résiduelles de pliage à froid dans des modèles par éléments finis
correspondent bien à celles prédites en modélisant explicitement le procédé de pliage à froid.
Cela indique que l’utilisation de diagrammes effort-déformation efficaces est un moyen
acceptable de tenir compte de l’influence de la contrainte résiduelle de pliage à froid sur la
pression d’écrasement d’une coque épaisse de sous-marin.
ii DRDC Atlantic TM 2010-065
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DRDC Atlantic TM 2010-065 iii
Executive summary
Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapse
Liam Gannon; DRDC Atlantic TM 2010-065; Defence R&D Canada – Atlantic;April 2010.
Background: An accurate assessment of the structural performance of submarine hulls must
consider fabrication-induced residual stress and distortion. These arise from forming and joining
operations such as cold bending and welding. There are several ways of incorporating these
imperfections in a pressure hull collapse analysis. One such method is the effective stress-strain
curve method, which is a relatively simple means of characterizing these imperfections in a
structural analysis.
Results: This memorandum evaluates several methods of pressure hull collapse analysis
considering different ways of accounting for cold bending residual stress. Collapse pressures
predicted using the various methods are compared and it is found that the effective stress-strain
curve method of accounting for residual stress is both accurate and efficient.
Significance: An accurate, simple way of accounting for fabrication-related imperfections in
submarine structural analysis will make future investigations of complicated structural behaviour
easier to conduct and save a significant amount of time in future structural analyses.
Future plans: The effective stress-strain curve method could potentially be used to characterize
residual stresses due to welding. A future study will examine the use of this method to account for
residual stresses from both cold bending and welding.
iv DRDC Atlantic TM 2010-065
Sommaire
Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapse
Liam Gannon; DRDC Atlantic TM 2010-065; R & D pour la défense Canada –Atlantique; avril 2010.
Introduction : Une évaluation précise du rendement de la structure de la coque des sous-marins
doit tenir compte des distorsions et des contraintes résiduelles produites par la fabrication. Ces
distorsions et ces contraintes sont produites par les opérations de formage et de liaison comme
le soudage et le pliage à froid. Il y existe plusieurs façons d’incorporer ces imperfections à
une analyse d’écrasement de coque épaisse. Une de ces méthodes est le diagramme
effort-déformation efficace, qui est un moyen relativement simple de caractériser ces
imperfections dans une analyse de structure.
Résultats : Le présent document évalue plusieurs méthodes d’analyse d’écrasement de coque
épaisse considérant différentes façons de tenir compte de la contrainte résiduelle de pliage à
froid. Les pressions d’écrasement prédites à l’aide des diverses méthodes sont comparées, et il
s’avère que le diagramme effort-déformation efficace qui permet de tenir compte de la contrainte
résiduelle est précis et efficace.
Portée : Une façon précise et simple de tenir compte des imperfections liées à la fabrication lors
d’une analyse de structure de sous-marin devrait rendre d’ultérieures études sur le comportement
complexe des structures plus faciles à réaliser et permettra d’économiser beaucoup de temps dans
la réalisation des analyses de structure à venir.
Recherches futures : Le diagramme effort-déformation efficace pourrait être utilisé pour
caractériser les contraintes résiduelles dues au soudage. Une étude à venir examinera l’utilisation
de cette méthode pour tenir compte des contraintes résiduelles découlant du soudage et du pliage
à froid.
DRDC Atlantic TM 2010-065 v
Table of contents
Abstract ... .......................................................................................................................................i
Résumé.... .......................................................................................................................................i
Executive summary ..................................................................................................................... iii
Sommaire .....................................................................................................................................iv
Table of contents ...........................................................................................................................v
List of figures ...............................................................................................................................vi
List of tables ................................................................................................................................vii
Acknowledgements ....................................................................................................................viii
1 Introduction .............................................................................................................................1
2 Geometry .................................................................................................................................3
3 Cold Bending...........................................................................................................................5
3.1 SSP 74 ..........................................................................................................................5
3.2 Finite Element Cold Bending Simulation ....................................................................6
3.3 Effective Stress-Strain Curves ......................................................................................7
4 Methods of Analysis ................................................................................................................8
4.1 SSP 74 – Design of Submarine Structures ....................................................................8
4.2 Finite Element Analysis..............................................................................................10
5 Results and Discussion ..........................................................................................................12
5.1 Cold Bending Residual Stress.....................................................................................12
5.2 Effective Stress-Strain Curves ....................................................................................13
5.3 Model A Collapse Analysis ........................................................................................14
5.4 Model B Collapse Analysis ........................................................................................16
6 Conclusions ...........................................................................................................................18
References....................................................................................................................................19
Annex A .. Displacement Contours...............................................................................................21
A.1 Contours .....................................................................................................................21
Annex B .. Model B Stress History...............................................................................................25
List of symbols/abbreviations/acronyms/initialisms ....................................................................29
Distribution list ............................................................................................................................31
vi DRDC Atlantic TM 2010-065
List of figures
Figure 1: Overall and interframe collapse modes (n=4) ..................................................................1
Figure 2: Model A scantlings .........................................................................................................3
Figure 3: Model B scantlings ..........................................................................................................4
Figure 4: Frame and plate after cold bending .................................................................................6
Figure 5: Ring stiffened cylinder finite element model ................................................................10
Figure 6: Ring stiffened cylinder with out of circularity (n =4 OOC mode) .................................11
Figure 7: Comparison of cold bending residual stress predicted by K79 and 3D FEA .................12
Figure 8: Effective stress-strain curves for model A (interframe model) ......................................13
Figure 9: Effective stress-strain curves for Model B (overall model) ...........................................13
Figure 10: Displacement vector sum contour plot for Model A at failure for n = 3 OOC .............15
DRDC Atlantic TM 2010-065 vii
List of tables
Table 1: Methods used to predict cold bending residual stresses for the methods of analysis
used in this study ..........................................................................................................8
Table 2: Collapse pressure predictions for Model A (interframe collapse) ...................................14
Table 3: Collapse pressure predictions for Model B (overall collapse except where noted) .........16
viii DRDC Atlantic TM 2010-065
Acknowledgements
The author would like to thank Mr. John MacKay for preparing and plotting the effective stress-
strain curve data.
DRDC Atlantic TM 2010-065 1
1 Introduction
Submarine pressure hulls typically consist of internally stiffened cylindrical and conical
compartments separated by watertight bulkheads. The primary load for which pressure hulls are
designed is due to external water pressure acting on the hull when the submarine is submerged.
Pressure hulls are designed considering two primary failure modes: interframe collapse and
overall collapse. Interframe buckling occurs with little deformation of the stiffening frames. It is
characterized by large deformations of the shell plating between frames for a small increase in
external pressure. Overall buckling involves significant deformation of the frames and occurs
over the entire length of a compartment between bulkheads. Interframe and overall collapse
modes of a ring stiffened cylinder are illustrated in Figure 1.
During fabrication, the shell and frames of a pressure hull are shaped by cold bending. This
process induces significant residual stress in the material that may cause a reduction in the load
carrying capacity of the pressure hull. Consequently, an accurate pressure hull collapse analysis
must account for the cold bending residual stresses. Several different methods are available both
for determining the cold bending residual stresses and for incorporating them in a collapse
analysis. Collapse pressures determined from empirical data [1] implicitly account for cold
bending residual stress, while analytical approaches [2] often use an iterative elasto-plastic
approach based on the assumptions of beam theory to explicitly calculate the residual stresses due
to cold bending before the collapse analysis is carried out. The latter typically considers residual
stresses only in the circumferential direction. Accuracy in prediction of cold bending residual
stresses may be improved by application of the finite element method (FEM) to simulate the cold
bending process, allowing the full three dimensional residual stress field induced by cold bending
to be determined. This method has been used by Graham [3] to account for the effects of cold
bending residual stress in the collapse analysis of a ring stiffened cylinder.
Figure 1: Overall and interframe collapse modes (n=4)
The UK MoD submarine design standard, SSP 74 [1] may be used to determine the design
collapse pressure for both interframe and overall failure modes. The standard specifies that the
interframe collapse pressure for a cylindrical stiffened shell be determined by first calculating the
Von Mises interframe buckling pressure [4] normalized with respect to the pressure at which the
2 DRDC Atlantic TM 2010-065
mean circumferential stress at mid-bay reaches yield. This value is then compared with empirical
data to determine the interframe collapse pressure.
Kendrick [5], [6], [7] developed the first acceptable theory for overall elastic buckling of a ring
stiffened cylinder using energy methods. Three versions of the theory were developed
considering different buckled shapes and stress distributions. Kendrick's Part I theory for overall
buckling of a finite length cylinder assumes a buckled shape with a half sine wave between
bulkheads with no undulations due to the presence of the frames and uses an approximate stress
distribution prior to buckling. The Part II theory considered a uniformly framed cylinder of
infinite length, but with a more refined buckling shape where interframe undulations were
allowed and the correct axisymmetric stress distribution. Kendrick's Part III theory reverts to the
approximate stress distribution, but applies to finite length cylinders. Using his Part III theory,
Kendrick [8] found that compartment length is much more important for small out-of-circularity
mode numbers, n than for large ones. For n > 3, a compartment length of 11 frame spaces is
effectively infinite; however for n = 2, 28 frame spaces still resulted in an overall buckling
pressure significantly higher than that of an infinitely long cylinder.
In SSP 74 [1], the overall collapse pressure is taken as either the pressure to cause yielding in the
frame flange or in the plating in way of the frame. This is determined using either an elastic
method of analysis or an elasto-plastic method. In the elastic method, the stress is calculated by
adding the hoop stress and the bending stresses induced by the magnification of the initial out of
circularity. The bending stresses are calculated using a magnification factor which relies on a
simplification of Kendrick's Part I theory, derived by Bryant [9]. The Bryant pressure is a
simplification of the Von Mises buckling pressure where the Von Mises buckling pressure for a
cylinder without ring frames and the Bresse pressure for one frame plus a frame space of plating
are added together. Collapse is assumed to occur when the sum of the hoop and bending stresses
reaches yield. For the elasto-plastic analysis, the overall collapse pressure is determined by
analysis of a single frame and one bay of attached plating using the method of Kendrick [10]. A
number of approximate correction factors are applied to the collapse pressure of the single frame
and plating model to determine the collapse pressure for a complete compartment. Among the
limitations of this method is that it only applies to axisymmetric geometry. The elasto-plastic
method allows the limit state to be determined and may be more conservative than the elastic
method since the latter does not account for residual stresses. The application of the finite element
method allows many of the limitations inherent in analytical procedures to be overcome. Finite
element analysis (FEA) allows the actual geometry of the pressure hull to be modelled including
penetrations, geometric imperfections and the three-dimensional residual stress field resulting
from fabrication processes such as cold bending. The disadvantage of a detailed finite element
analysis is that significant time and effort are required on the part of the engineer to generate the
model and to specify initial conditions.
The objective of this study is to evaluate various methods of calculating cold bending residual
stresses and of incorporating them in a pressure hull collapse analysis. The effective stress-strain
curve method of incorporating cold bending residual stresses in a collapse analysis is of particular
interest due to its relative simplicity. This approach involves adjusting the material properties of
the model to implicitly account for the effects of cold bending residual stress in a collapse
analysis. This method may be significantly faster to implement than explicitly defining cold
bending residual stresses at each integration point in a three-dimensional finite element model
and may result in significant time savings for future projects.
DRDC Atlantic TM 2010-065 3
2 Geometry
Two different geometries are considered, designated Model A and Model B. The cylinder
scantlings were chosen such that one will fail by interframe buckling (Model A) and the other
will fail by overall buckling (Model B). The two geometries are similar; however the frame web
depth and flange width is reduced for Model B. This reduces the frame stiffness in order to
precipitate overall collapse. The scantlings for Models A and B are shown in Figure 2 and Figure
3 respectively. The material used is high strength steel used in submarine construction such as
Q1N or HY 80 with a yield strength of 550 MPa and elastic modulus of 207 GPa. For the finite
element analyses wherein cold bending is explicitly simulated and where residual stresses
determined by the method of Mitchell [2] are used, the hardening modulus is taken as 3.2 GPa as
measured by Bayley [11]. Although strain hardening should have little influence on the results, it
is included to enhance numerical stability of finite element solutions. For the collapse pressure
calculation using the methods of SSP 74 and for determination of effective stress-strain curves,
strain hardening is neglected.
Figure 2: Model A scantlings
4 DRDC Atlantic TM 2010-065
Figure 3: Model B scantlings
DRDC Atlantic TM 2010-065 5
3 Cold Bending
3.1 SSP 74
The elasto-plastic collapse analysis described in SSP 74 [1] requires that the cold bending
residual stresses be estimated using an iterative elasto-plastic procedure developed by Mitchell
[2]. This procedure is used to determine the overbend moment and springback curvature
necessary to achieve the required final curvature. The procedure begins by dividing the cross-
section of a single ring frame and one half bay of plating on either side of the web into a number
of horizontal strips. The material of each fibre in the cross-section is assigned an elastic,
perfectly plastic stress-strain behaviour.
The initial estimation of the required overbend moment is related to the maximum elastic moment
and the plastic moment by an approximate function, f(s), representing the variation in moment
from the limiting elastic moment to the fully plastic moment. The overbend moment is given by
[1]:
(1)
where is the initial estimate of overbend moment; Mel is the limiting elastic bending moment
and Mpl is the plastic moment capacity of the section. Assuming that the springback curvature is
linearly related to the elastic curvature, the initial overbend moment is used to calculate the initial
overbend curvature given by:
(2)
where Cob is the overbend curvature, Cr is the final required curvature and Cel is the curvature at
the maximum elastic bending moment. Recognizing that the final required curvature is equal to
the overbend curvature minus the springback curvature, (2) can be arranged to express the
springback curvature, Cs as:
(3)
After springback occurs, the section should be left with zero bending moment; however due to the
approximation in (1), the net internal bending moment will not be in balance with the externally
applied moment after springback, which is zero. This inconsistency is resolved by iteratively
adjusting the springback curvature until the net internal bending moment is zero. A new estimate
of the overbend curvature is then obtained and the process is repeated until Cs and Cob converge.
6 DRDC Atlantic TM 2010-065
3.2 Finite Element Cold Bending Simulation
Cold bending residual stresses are also predicted by finite element analysis. Three dimensional
FEA enables calculation of both the longitudinal and circumferential cold bending stress whereas
the iterative method prescribed by [1] only allows the circumferential residual stress to be
calculated. In the finite element simulations of cold bending, the shell plating and stiffener are
bent separately. Separate models are created for the frame and the plating using the ANSYS
SHELL181 element type with reduced integration option. In order to obtain a sufficient through
thickness resolution of residual stress, 21 integration points are used through the thickness of
shell plating and frame flange elements and 5 integrations points through the thickness of the
frame web elements.
Three bay widths of plating were used in the shell plating model in order to reduce end effects
and to reduce anticlastic curvature which is quite pronounced if only one bay width of plate is
used. Reducing the anticlastic curvature is thought to better represent the constraint of a typical
cold bending processes where three rollers are used to progressively shape the part. Both the
plating and frame models have lengths equal to one quarter of the total circumference and are
initially straight in order to reduce analysis times. The overbend curvature is determined using an
iterative procedure where rotations are applied to the nodes of each cross-section along the
circumference of the part to generate the overbend curvature as shown in Figure 4. Rotations
were applied at each cross-section in order to avoid excessive local deformations at the end of the
plate when rotations are only applied at the end cross-section. A small extension of the
frame/plate is added to both ends of both models in the circumferential direction. Rotations are
also applied to these end sections, so that a nearly axisymmetric stress field is generated and end
effects in the circumferential direction are eliminated. Once the overbend curvature is attained,
the rotations are removed and the frame or plate springs back elastically. Due to numerical
instability associated with springback to a new equilibrium position, the cold bending simulation
is run as a “slow dynamic” analysis which uses the Newmark algorithm for solution, allowing the
numerical instability to be overcome.
Figure 4: Frame and plate after cold bending
DRDC Atlantic TM 2010-065 7
3.3 Effective Stress-Strain Curves
The effective stress-strain curve technique is used to implicitly account for the effects of cold
bending residual stress by modifying the material stress-strain relationship. Once the residual
stresses are estimated, a frame or plate cross-section is divided into a number of horizontal strips
and incremental strains are applied to the cross-section. For each strain increment, the
corresponding stress is calculated and added to the residual stress in each fibre of the cross
section such that:
� �iyiiiri E )(,)(max ���� ��� for �i < 0 (4)
where �i is the total stress in the ith fibre; (�r)i is the residual stress; Ei is the elastic modulus, (�y)i
is the yield stress and �i is the strain in the ith fibre. The average stress, , acting on the cross-
section is then given by:
�i
ii
A
A�� (5)
where Ai is the area of the ith fibre. A stress-strain curve for each structural component (shell
plating, frame web or flange or complete frame) is constructed by applying compressive axial
strains until the entire cross-section yields. Effective stress-strain curves "soften" the material
response to account for the early onset of yielding due to residual stress. Instead of using the
entire cross-section to produce a single effective stress-strain curve, several subsets of fibres can
be selected to produce effective stress-strain curves representing the nonlinear behaviour of the
web and flange separately. Any number of subsets of adjacent fibres can be used to incorporate
the effects of residual stresses in this way. Although effective stress-strain curves should differ
for tensile and compressive loading, most material models for numerical analysis assume the
same material behaviour in both tension and compression. In the case of pressure hulls, the
external pressure results in primarily compressive stress until buckling occurs; therefore the use
of a compressive effective stress-strain curve should adequately represent the material behaviour
accounting for residual stress.
8 DRDC Atlantic TM 2010-065
4 Methods of Analysis
The following describes the methods of analysis used to determine the collapse pressure of a
cylindrical stiffened shell. A summary of the methods of used in this study is provided in Table 1.
Table 1: Methods used to predict cold bending residual stresses for the methods of analysis used
in this study
Cold Bending
Residual Stress
Prediction
Method
Method of Analysis
SSP 74
Interframe
SSP 74
Overall
FEA
InterframeFEA Overall
Implicit/Empirical x - - -
Mitchell [2] - x x x
3D FEA - - x x
Effective Stress-
Strain Curves- - x x
None - - x x
4.1 SSP 74 – Design of Submarine Structures
Interframe Collapse
Evaluation of the interframe collapse pressure of a stiffened cylinder begins with calculation of
the Von Mises buckling pressure, Pm1, normalized with respect to the mean circumferential stress
mid-way between frames, Pc5. The interframe collapse pressure, Pc, is then determined from an
empirical curve of the normalized interframe collapse pressure, Pc/Pc5 versus the normalized Von
Mises pressure, Pm1/Pc5. The pressures Pm1 and Pc5, are given by:
DRDC Atlantic TM 2010-065 9
(6)
(7)
where E is the modulus of elasticity; h is the of thickness of the shell plating; �yp is the yield
strength of the shell plating; n is the out of circularity mode; a is the radius of the cylinder to the
mid-plane of the shell plating, L is the length of the cylinder; � is Poisson’s ratio and G and � are
parameters defined in Chapter 6 of reference [1]. It is important to note that the empirical data on
which the interframe collapse pressure curve is based is only valid for an out of circularity of
0.5% of the radius a, or less.
Overall Collapse
The overall collapse pressure is taken as the lower of the pressure to cause yielding in the frame
flange or the plating in way of the frame. SSP 74 [1] requires that the overall collapse pressure be
calculated using both an elastic method and an elasto-plastic method. This is because the elastic
approximation is not always conservative if residual stresses are present or if elastic buckling
occurs. In the elastic method, the stress is calculated by combining the direct stress from linear
elastic axisymmetric analysis of an infinite ring-stiffened cylinder and the bending stresses arising
from the initial out of circularity. The calculations are carried out for a single frame including an
effective breadth of attached plating, recognizing that shear lag reduces the effectiveness of the
plating in bending. A range of out of circularity modes is investigated in order to find the
minimum collapse pressure.
The elasto-plastic collapse analysis is carried out using an algorithm developed at DRDC Atlantic
called K79 [12]. The collapse analysis begins with estimation of the cold bending residual
stresses using the method of Mitchell [2]. Once the cold bending residual stresses have been
determined, the collapse pressure for a single frame with one half bay of plating on either side of
the web is determined using the method of Kendrick [13], where the finite difference method is
used to solve the differential equation describing the response of the ring frame to increasing
pressure until a valid solution no longer exists. Bending moments and stresses are corrected to
account for the actual stress distribution in a stiffened cylinder and for the finite length of the
cylinder as specified in reference [1]. As in the elastic analysis, the elasto-plastic analysis is only
10 DRDC Atlantic TM 2010-065
valid for a single mode shape and so must be repeated for several out of circularity modes until
the minimum collapse pressure is found.
4.2 Finite Element Analysis
Three dimensional finite element models, created using ANSYS finite element software, are used
to determine both the overall and interframe collapse pressures of a ring stiffened cylinder.
Details on the geometry of the models were given in Section 2. Each model contains
approximately 54,000 elements and 54,000 nodes. All elements are 4-node shells of type
SHELL181 with six degrees of freedom at each node and employ reduced integration to prevent
shear locking and reduce analysis time. Two hundred elements are used around the circumference
with 10 elements between each frame. There are 10 elements through the depth of the frame web
and 6 across the width of the stiffener. For consistency between the cold bending simulation and
the collapse analysis, 21 integration points are used through the thickness of the shell plating and
frame flange and 5 integration points through the thickness of the frame web in all collapse
analyses. Residual stresses due to cold bending are included by mapping the residual stresses
from the cold bending simulation described in Section 2.2 onto the complete model. Cold bending
stresses determined using K79 [12] are also applied to this model in order to compare collapse
pressures predicted using the two methods of residual stress calculation.
Figure 5: Ring stiffened cylinder finite element model
DRDC Atlantic TM 2010-065 11
A preliminary elastic analysis is used to apply geometric imperfections in the form of out of
circularity mode shapes where the maximum amplitude of out of circularity is 0.5% of the
cylinder radius. This is done by starting with an initially flat stiffened plate and applying
displacements to each node to produce the required cylinder shape including out of circularity in
the desired mode n. A n = 13 interframe OOC mode with an amplitude of 0.01 times the shell
plating thickness was used in addition to the overall OOC mode for all analyses to enable
interframe collapse. This resulted in a combined maximum OOC slightly larger than 0.5% of the
cylinder radius at mid-bay. The initial elastic analysis allows the nodes to be moved to the correct
locations by specifying displacements and eliminates the need for time consuming equilibrium
iterations where convergence difficulty might be encountered. Following the elastic analysis, the
model geometry is updated so that its new undeformed shape corresponds to the deformed shape
from the initial elastic analysis. Figure 6 shows a model with an exaggerated n = 4 out-of-
circularity created in this manner. For the nonlinear collapse analyses, displacements of all nodes
at one end of the cylinder are constrained. The cylinder is loaded by a uniform external pressure
in addition to nodal loads applied in the longitudinal direction at the axially unrestrained end to
represent the hydrostatic pressure acting on the end closures. With the exception of the models
where effective stress-strain curves are used, the material model incorporates bilinear kinematic
hardening with Von Mises plasticity and associated flow rule. All geometry properties, material
properties, and solution steps are implemented through the ANSYS parametric design language,
allowing the entire analysis to be run from scripted macros.
Figure 6: Ring stiffened cylinder with out of circularity (n =4 OOC mode)
12 DRDC Atlantic TM 2010-065
5 Results and Discussion
5.1 Cold Bending Residual Stress
Cold bending residual stresses are predicted using the methods described in Section 3. The flange
and web of the frame are bent together with the shell plating bent separately. Circumferential
residual stresses in Model B, predicted using the method of reference [2] are compared with the
residual stress distribution predicted by the FEM in Figure 7. Results from the two methods of
predicting cold bending residual stress agree well with each other. The small difference between
peak values is attributed to different integration point locations between the two models.
Figure 7: Comparison of cold bending residual stress predicted by K79 and 3D FEA
465
470
475
480
485
490
495
500
505
-600 -400 -200 0 200 400 600
Rad
ial L
ocati
on
(m
m)
Circumferential Residual Stress (MPa)
3D FEA
K79
Flange
Web
Plate
DRDC Atlantic TM 2010-065 13
5.2 Effective Stress-Strain Curves
Figures 8 and 9 show the effective stress-strain curves calculated using the method described in
Section 3.3.
Figure 8: Effective stress-strain curves for model A (interframe model)
Figure 9: Effective stress-strain curves for Model B (overall model)
The 'softening' effect of the cold bending residual stresses is observed for all components once the
applied strain reaches approximately one half of the yield strain. The modified material model
allows the web to reach stresses higher than yield due to the net tensile residual stress state that
can be seen in Figure 7.
0
100
200
300
400
500
600
700
800
0.000 0.001 0.002 0.003 0.004 0.005 0.006
Str
ess
(MP
a)
Strain (-)
Shell Plating
Frame
Flange
Web
Stress-Relieved
0
100
200
300
400
500
600
700
800
0.000 0.001 0.002 0.003 0.004 0.005 0.006
Str
ess
(M
Pa)
Strain (-)
Shell Plating
Frame
Flange
Web
Stress-Relieved
14 DRDC Atlantic TM 2010-065
5.3 Model A Collapse Analysis
Table 2 provides a summary of collapse pressures predicted using the various methods to analyse
Model A. Separate effective stress-strain curves were used for the frame flange and web for all
effective stress-strain curve analyses.
Table 2: Collapse pressure predictions for Model A (interframe collapse)
OOC
ModeAnalysis Method - Cold Bending RS Method Collapse Pressure (MPa)
2
SSP 74 - Implicit 4.33
FEM - Mitchell 5.41
FEM - FEM 5.35
FEM - ���������-� 5.18
FEM - None 5.54
3
SSP 74 - Implicit 4.33
FEM - Mitchell 4.95
FEM - FEM 4.97
FEM - ���������-� 4.81
FEM - None 5.21
4
SSP 74 - Implicit 4.33
FEM - Mitchell 4.95
FEM - FEM 4.98
FEM - ���������-� 4.81
FEM - None 5.20
Collapse pressures predicted by the method of SSP 74 [1] are somewhat conservative when
compared with those predicted by finite element analysis. SSP 74 predicts interframe failure at
approximately 80% to 90% of the collapse pressure determined by FEA. This discrepancy is
greater than the ±10% scatter of the empirical data about the mean curve, although it should be
DRDC Atlantic TM 2010-065 15
noted that the empirical data are derived from a variety of different sources and consider
internally, externally stiffened and unstiffened cylinders. Comparison of collapse pressures from
the cases with no residual stress and those with cold bending residual stress shows that cold
bending residual stresses reduce the interframe collapse pressure by 3-5%.
Effective stress-strain curves are an effective means of representing the effects of cold bending
residual stress in a finite element model without the need for time-consuming mapping of the
stress field from one model to the other. Interframe collapse pressures predicted using effective
stress-strain curves are within 1-3% of those predicted by explicitly specifying the cold bending
residual stress as an initial stress. Figure 10 shows contours of the displacement vector sum at the
interframe collapse pressure for the n = 3 out of circularity mode shape predicted by the finite
element model with residual stresses predicted by FEA. A comparison of displacement contours
from the effective stress-strain model and the explicit cold bending model considering the n = 3
OOC mode over regular increments of external pressure is shown in Annex A.
Figure 10: Displacement vector sum contour plot for Model A at failure for n = 3 OOC
16 DRDC Atlantic TM 2010-065
5.4 Model B Collapse Analysis
Table 3 provides a summary of collapse pressures predicted using the different methods of
analysis for Model B.
Table 3: Collapse pressure predictions for Model B (overall collapse except where noted)
OOC Mode Analysis Method - Cold Bending RS Method Collapse Pressure (MPa)
2
SSP 74 - Elasto-plastic 4.44*
FEM - Mitchell 5.45*
FEM - FEM 5.33*
FEM - ���������-� 5.14*/5.14*†
FEM - None 5.53*
3
SSP 74 - Elasto-plastic 3.84
FEM - Mitchell 4.19
FEM - FEM 4.33
FEM - ���������-� 4.16/4.29†
FEM - None 4.5
4
SSP 74 - Elasto-plastic 3.18
FEM - Mitchell 3.57
FEM - FEM 3.86
FEM - ���������-� 3.65/3.85†
FEM - None 3.99
* Failure occurred in interframe collapse mode
† Separate curves for web and flange / combined curve for web and flange
DRDC Atlantic TM 2010-065 17
For the n = 2 OOC mode, all analysis methods predict failure by interframe buckling and follow
similar trends to those discussed in Section 5.2. The increase in overall collapse pressure above
the interframe pressure for the n = 2 OOC mode is consistent with Kendrick's [8] observation that
a finite compartment length is more important for small n than for large n. The SSP 74 method of
analysis provides conservative predictions of overall collapse pressure, predicting collapse
pressures 80-90% of those determined by finite element analysis using finite element predictions
for cold bending residual stress. Cold bending residual stresses are found to decrease the overall
collapse pressure by an amount comparable to the reduction in interframe collapse pressure. Cold
bending residual stresses reduce overall collapse pressures by 3-4%.
The effective stress-strain curve method of accounting for cold bending residual stresses predicts
collapse pressures in good agreement with those predicted using residual stresses from cold
bending simulations. Displacement contours from Model B with a n = 3 OOC mode and using
one stress-strain curve for the complete frame are compared with those from the explicit cold
bending model in Annex A. The use of a single effective stress-strain curve for the complete
frame provides collapse pressures in better agreement with the explicit cold bending simulation
model than the case where separate stress-strain curves are used for the frame flange and web.
The overall collapse pressures predicted when using separate effective stress-strain curves for the
frame flange and web are 4-6% lower than those predicted by the explicit cold bending
simulation. When one effective stress-strain curve is used for the entire frame, the effective
stress-strain curve analysis predicts collapse pressures within 1% of those predicted when cold
bending is explicitly simulated.
Circumferential stresses from Model B are compared at several critical locations for the n = 3
OOC mode in Annex B. When separate effective stress-strain curves are used for the frame flange
and web, yielding of the flange at approximately 400 MPa leads to plastic redistribution of stress
to the web causing a rapid rise in web stress. This is accompanied by a reduction in the effective
moment of inertia of the frame leading to premature failure in the overall buckling mode. This
behaviour is not observed in the case where a single effective stress-strain curve is used for the
entire frame. In that case, the flange is able to maintain a higher circumferential stress (525 MPa)
and yielding does not occur until a higher external pressure is applied. Otherwise, differences
between the stresses at these locations for cold-bent and effective stress-strain models can be
attributed to the differences in the initial residual stress values that are non-zero for the explicitly
cold-bent models and zero for the effective stress-strain models. Based on these observations, it is
recommended that a single effective stress-strain curve is used for the entire frame as opposed to
separate curves for the flange and web.
The reduction in overall collapse pressure due to cold bending residual stress is significantly less
in the present study than reductions predicted by Creswell and Dow [14] who found that cold
bending residual stresses have the potential to lower the overall collapse pressure by as much as
20-30%. They used separate effective stress-strain curves in the frame flange and web to
incorporate cold bending residual stress effects. Using the same approach, this study found that
there is an 8% decrease in overall collapse pressure compared to the stress-free structure.
Creswell and Dow [14] demonstrated that the reduction in collapse pressure due to cold bending
residual stress also depends on cross-section dimensions and both overall and interframe out of
circularity. This may be the reason for the large discrepancy between their results and those of
this study.
18 DRDC Atlantic TM 2010-065
6 Conclusions
Both interframe and overall collapse pressures predicted using the methods of SSP 74 were found
to be fairly conservative. SSP 74 under-predicts the interframe collapse pressure by 13-20%
compared with finite element solutions where cold bending is explicitly simulated. Overall
collapse pressures predicted by K79 to implement the elasto-plastic collapse analysis method
prescribed in SSP 74 are conservative and fall 11-18% below finite element predictions where
cold bending is explicitly simulated.
Circumferential residual stresses predicted by K79 using Mitchell's method agree well with those
predicted by simulating the cold bending process in a 3D finite element analysis. Cold bending
residual stresses are found to reduce interframe collapse pressure by as much as 3% and overall
collapse pressure by as much as 4%. Interframe collapse pressures predicted using cold bending
residual stresses predicted by Mitchell's method are similar to those predicted by the finite
element simulation where cold bending is explicitly simulated. Overall collapse pressures
predicted using cold bending residual stresses calculated using Mitchell's method were 3-8%
lower than those predicted by explicitly simulating cold bending. Since Mitchell's method
predicts only circumferential residual stress and the finite element cold bending simulation
provides the complete biaxial residual stress distribution, this result indicates that neglecting the
longitudinal component of residual stress is a conservative assumption.
The use of effective stress-strain curves to incorporate the effects of cold bending in a pressure
hull collapse analysis is efficient and allows collapse pressures to be predicted accurately when
compared with analyses wherein cold bending is explicitly simulated. Models using a single
effective stress-strain curve for the complete frame predict overall collapse pressures in better
agreement with models where cold bending is simulated explicitly than models using separate
stress-strain curves for the frame flange and web. When separate curves are used, the early onset
of yielding in the frame flange causes plastic redistribution of stress and a reduction in the
effective moment of inertia of the cross-section leading to premature failure. Although using
separate curves for the frame flange and web is less accurate, collapse pressure predictions are
conservative and fall below those predicted by explicit cold bending simulation by 1-3% for
interframe collapse and 3-4% for overall collapse. It is possible that further refinement of the
effective stress-strain curve method by using several effective stress-strain curves for multiple
fibres through the thickness of the flange may result in collapse pressures in better agreement
with the predictions where cold bending is explicitly simulated than using a single effective-stress
strain curve for the flange.
DRDC Atlantic TM 2010-065 19
References
[1] Defence Procurement Agency (2001). SSP 74 Design of Submarine Structures. Sea
Systems Publication No. 74. Defence Procurement Agency, Sea Technology Group, United
Kingdom.
[2] Mitchell, G.C. (1986). Overbend Prediction for Cold-Bent Beams. Computers and
Structures, 24(2), 187-196.
[3] Graham, D. (2007). Predicting the Collapse of Externally Pressurised Ring-Stiffened
Cylinders using Finite Element Analysis. Marine Structures, 20, 202-217.
[4] Von Mises, R. (1929). Stodola Festschrift.
[5] Kendrick, SB. (1953). The Buckling Under External Pressure of Circular Cylindrical Shells
with Evenly Spaced, Equal Strength, Circular Ring Frames - Part I. Naval Construction
Research Establishment Report R211.
[6] Kendrick, SB. (1953). The Buckling Under External Pressure of Circular Cylindrical Shells
with Evenly Spaced, Equal Strength, Circular Ring Frames - Part II. Naval Construction
Research Establishment Report R243.
[7] Kendrick, SB. (1953). The Buckling Under External Pressure of Circular Cylindrical Shells
with Evenly Spaced, Equal Strength, Circular Ring Frames - Part III. Naval Construction
Research Establishment Report R244.
[8] Kendrick, SB. (1964). Structural Design of Submarine Pressure Vessels. Naval
Construction Research Establishment Report NCRE R483.
[9] Bryant, A.R. (1954). Hydrostatic Buckling Pressure of a Ring-Stiffened Tube. Naval
Construction Research Establishment Report R306.
[10] Kendrick, S.B. (1977). The Elasto-Plastic Collapse of Ring Stiffened Cylinders. Naval
Construction Research Establishment Report R653.
[11] Bayley, C. (2007). Stress-Strain Characterization of NQ1 Pressure Hull Material. DRDC
Atlantic TN 2007-328. Defence Research and Development Canada.
[12] Smith, M.J. and MacKay, J.R. (2004). Overall Elasto-Plastic Collapse of Submarine
Pressure Hull Compartments. (DRDC Atlantic TM 2004-243). Defence Research and
Development Canada.
[13] Kendrick, S. (1979). The Influence of Shape Imperfections and Residual Stresses on the
Collapse of Stiffened Cylinders. Conference on Significance of Deviations from Design
Shape, 25-35. the Institute of Mechanical Engineers.
20 DRDC Atlantic TM 2010-065
[14] Creswell, D.J. and Dow, R.S. (1986). The Application of Nonlinear Analysis to Ship and
Submarine Structures. Advances in Marine Structures, Eds. C.S. Smith and J.D. Clarke,
175-200.
DRDC Atlantic TM 2010-065 21
Annex A Displacement Contours
A.1 Contours
Figure A-1 through A-6 show contours of the displacement vector sum at regular pressure
increments for Model A with an n = 3 OOC mode. Results of the effective stress-strain curve
method and results from the explicit simulation of cold bending by FEA are compared.
Displacements differ at failure because the failure loads differ slightly and near failure,
displacements increase significantly for a small increase in applied load.
Figure A-1: Model A displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 1 MPa (mm)
Figure A-2: Model A displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 2 MPa (mm)
22 DRDC Atlantic TM 2010-065
Figure A-3: Model A displacement vector sum contours from cold bending simulation (left) and
effective stress-strain curves (right) at 3 MPa (mm)
Figure A-4: Model A displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 4 MPa (mm)
Figure A-5: Model A displacement vector sum contours from cold bending simulation (left) and
effective stress-strain curves (right) at 4.5 MPa (mm)
DRDC Atlantic TM 2010-065 23
Figure A-6: Model A displacement vector sum contours from cold bending simulation (left) and
effective stress-strain curves (right) at failure (mm)
Figures A-7 through A-11 show contours of the displacement vector sum at regular pressure
increments for Model B comparing the results of the effective stress-strain curve method and
results from the explicit simulation of cold bending by FEA.
Figure A-7: Model B displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 1 MPa (mm)
Figure A-8: Model B displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 2 MPa (mm)
24 DRDC Atlantic TM 2010-065
Figure A-9: Model B displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 3 MPa (mm)
Figure A-10: Model B displacement vector sum contours from cold bending simulation (left) and effective stress-strain curves (right) at 4 MPa (mm)
Figure A-11: Model B displacement vector sum contours from cold bending simulation (left) and
effective stress-strain curves (right) at failure (mm)
DRDC Atlantic TM 2010-065 25
Annex B Model B Stress History
Figures B-1 through B-4 show the stress at critical locations for increasing external pressure for
Model B considering an n = 3 OOC mode. The cases where separate effective stress-strain curves
for the flange and web and a single stress-strain curve for the entire frame are compared with the
results of the analysis where cold bending is explicitly simulated.
Figure B-1: Stress history at the flange mid-width for a frame at mid-length, top of cylinder
-600
-500
-400
-300
-200
-100
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Cir
cum
fere
nti
al
Str
ess
(MP
a)
External Pressure (MPa)
Cold Bending
Effective Stress-Strain
Frame Effective Stress-Strain
26 DRDC Atlantic TM 2010-065
Figure B-2: Stress history at the web mid-height for a frame at mid-length, top of cylinder
Figure B-3: Stress history for shell plating directly above a frame at mid-length, top of cylinder
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Cir
cum
fere
nti
al
Str
ess
(MP
a)
External Pressure (MPa)
Cold Bending
Effective Stress-Strain
Frame Effective Stress-Strain
-400
-300
-200
-100
0
100
200
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Cir
cum
fere
nti
al
Str
ess
(MP
a)
External Pressure (MPa)
Cold Bending
Effective Stress-Strain
Frame Effective Stress-Strain
DRDC Atlantic TM 2010-065 27
Figure B-4: Stress history for shell plating at mid-length of cylinder (mid-bay)
-700
-600
-500
-400
-300
-200
-100
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Cir
cum
fere
nti
al
Str
ess
(MP
a)
External Pressure (MPa)
Cold Bending
Effective Stress-Strain
Frame Effective Stress-Strain
28 DRDC Atlantic TM 2010-065
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DRDC Atlantic TM 2010-065 29
List of symbols/abbreviations/acronyms/initialisms
Symbols
a Radius of cylinder to mid-plane of shell plating
Ai Area of ith fibre
Cel Limiting elastic curvature
Cob Overbend curvature
Cr Required curvature
Cs Springback curvature
E Modulus of elasticity
Ei Elastic modulus of ith fibre
f(s) Function representing variation in moment from limiting elastic to plastic
moment
h Thickness of shell plating
L Length of cylinder
Mel Limiting elastic bending moment
Mob Overbend moment
Mpl Plastic bending moment
n Out of circularity mode shape
Pc Interframe collapse pressure
Pc5 Mean circumferential stress between frames
Pm1 Von Mises buckling pressure
� Strain
�i Strain in ith fibre
µ Poisson's ratio
� Stress
Average stress
�i Total stress in ith fibre
��r)i Residual stress in ith fibre
��y)i Yield stress of ith fibre
�yp Yield stress of shell plating
30 DRDC Atlantic TM 2010-065
Abbreviations, acronyms and initialisms
DND Department of National Defence
DRDC Defence Research and Development Canada
DRDKIM Director Research and Development Knowledge and Information
Management
FEA Finite element analysis
FEM Finite element method
MoD Ministry of Defence
OOC Out of circularity
R&D Research & Development
RS Residual stress
SSP Sea Systems Publication
DRDC Atlantic TM 2010-065 31
Distribution list
Document No.: DRDC Atlantic TM 2010-065
LIST PART 1 – Internal Distribution by Centre:
4 Author (2 hardcopies, 2 CDs)
3 DRDC Atlantic Library (1 hardcopies, 2 CDs)
1 Scientific Authority, Submarine Scientific Support SLA (Mr. John Porter)
1 Project Manager, Submarine Scientific Support SLA (LCdr Wade Temple)
9 TOTAL LIST PART I
LIST PART II: External Distribution within Canada by DRDKIM
1 NDHQ/DRDKIM 3
2 NDHQ/DMEPM(SM) 4
2 NDHQ/DMEPM(SM) 4-2
1 Library & Archives Canada (Attn: Military Archivist, Government Records Branch)
6 TOTAL LIST PART II
15 TOTAL COPIES (3 hardcopies, 12 CDs)
32 DRDC Atlantic TM 2010-065
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DOCUMENT CONTROL DATA(Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified)
1. ORIGINATOR (The name and address of the organization preparing the document.
Organizations for whom the document was prepared, e.g. Centre sponsoring a
contractor's report, or tasking agency, are entered in section 8.)
Defence R&D Canada – Atlantic9 Grove StreetP.O. Box 1012Dartmouth, Nova Scotia B2Y 3Z7
2. SECURITY CLASSIFICATION (Overall security classification of the document
including special warning terms if applicable.)
UNCLASSIFIED
3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U)
in parentheses after the title.)
Prediction of the Effects of Cold Bending on Submarine Pressure Hull Collapse
4. AUTHORS (last name, followed by initials – ranks, titles, etc. not to be used)
Gannon, L
5. DATE OF PUBLICATION(Month and year of publication of document.)
April 2010
6a. NO. OF PAGES(Total containing information,
including Annexes, Appendices,
etc.)
44
6b. NO. OF REFS(Total cited in document.)
14
7. DESCRIPTIVE NOTES (The category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report,
e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.)
Technical Memorandum
8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development – include address.)
Defence R&D Canada – Atlantic9 Grove StreetP.O. Box 1012Dartmouth, Nova Scotia B2Y 3Z7
9a. PROJECT OR GRANT NO. (If appropriate, the applicable research
and development project or grant number under which the document
was written. Please specify whether project or grant.)
11GX03
9b. CONTRACT NO. (If appropriate, the applicable number under
which the document was written.)
10a. ORIGINATOR'S DOCUMENT NUMBER (The official document
number by which the document is identified by the originating
activity. This number must be unique to this document.)
DRDC Atlantic TM 2010-065
10b. OTHER DOCUMENT NO(s). (Any other numbers which may be
assigned this document either by the originator or by the sponsor.)
11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.)
Unlimited
12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to the
Document Availability (11). However, where further distribution (beyond the audience specified in (11) is possible, a wider announcement
audience may be selected.))
13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable
that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification
of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include
here abstracts in both official languages unless the text is bilingual.)
Submarine pressure hull frames and shell plating are shaped by cold bending during fabrication.
Cold bending introduces significant residual stress in these components which can be
detrimental to the strength of the structure. This study evaluates different methods of
incorporating cold bending residual stresses in the analysis of pressure hulls considering
different out of circularity mode shapes. Several methods of pressure hull collapse analysis are
compared considering interframe and overall collapse modes. These include an empirical
method, a finite difference method and the finite element method.
Collapse pressures predicted using the methods prescribed in the UK MoD submarine structure
design standard, SSP 74, are found to be conservative when compared with results from finite
element analysis. Collapse pressures predicted using effective stress-strain curves to incorporate
the effects of cold bending residual stress in finite element models agree well with those
predicted by explicitly modelling the cold bending process. This indicates that the use of
effective stress-strain curves is an acceptable means of accounting for the influence of cold
bending residual stress on the collapse pressure of a submarine pressure hull.
14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be
helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model
designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a
published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select
indexing terms which are Unclassified, the classification of each should be indicated as with the title.)
submarine structure; cold bending; residual stress; pressure vessel; nonlinear finite element
analysis; ring stiffened cylinder
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