Ultimate strength assessment of naval and commercial ships

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

Ultimate strength assessment of naval and

commercial ships

Malcolm J. Smith

Technical Report

DRDC Atlantic TR 2008-059

August 2008

Copy No. _____

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

This page intentionally left blank.

Ultimate strength assessment of naval and commercial ships

Malcolm J. Smith

Defence R&D Canada – Atlantic Technical Report DRDC Atlantic TR 2008-059 August 2008

Principal Author

Malcolm J. Smith

Approved by

Neil G. Pegg

Head/ Warship Performance

Approved for release by

James L. Kennedy

Chair/ Document Review Panel

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2008

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2008

Original signed by Malcolm J. Smith

Original signed by Neil G. Pegg

Original signed by Ron Kuwahara for

Abstract

The ultimate strength of several commercial and naval ship designs is studied using DRDC’s ULTMAT code. In ULTMAT, the compressive stress-strain response of local structure is predicted in a simplified manner using one of two methods: interpolation from libraries of pre-calculated load-shortening curves, or using simplified analytical formulas for the nonlinear load-shortening behaviour. The latter formulas are based on structural rules published by the International Association of Classification Societies (IACS). Load-shortening curves predicted by ULTMAT are compared with nonlinear finite element analysis (FEA) calculations for seventeen longitudinally stiffened panels and twelve transversely stiffened panels. Hull girder ultimate strength results are presented for six benchmark ship cross sections and are compared with previously published results. A further set of results for the HALIFAX class midships cross section strength is presented and is shown to agree well with the results of previous studies. The comparisons demonstrate the overall accuracy of ULTMAT’s prediction of structural strength, but reveal some deficiencies in specific areas. Using the IACS rules generally gives results of accuracy acceptable for design purposes. Interpolating curves from two of ULTMAT’s existing libraries generally gives results that are within 10% for hull girder strength. The accuracy can be improved by substituting load-shortening curves for selected panels with curves calculated directly from nonlinear FEA. Recommendations are made for an improved library of load-shortening curves.

Résumé

La charge de rupture de plusieurs navires commerciaux et de guerre est étudiée à l’aide du code ULTMAT de RDDC. Dans ce logiciel, le comportement d’une structure locale soumise à des contraintes-déformations est prédit de manière simplifiée grâce à deux méthodes : une interpolation entre des bibliothèques de courbes de contrainte précalculées, ou l’utilisation de formules analytiques simplifiées qui se fondent sur des règles de l’International Association of Classification Societies (lACS) quant aux structures. Les courbes de contraintes que prédit le logiciel ULTMAT sont comparées à des calculs exécutés par une analyse par éléments finis non linéaires (FEA) pour dix-sept panneaux renforcés sur le plan longitudinal et pour douze panneaux renforcés sur le plan transversal. Les résultats de charge de rupture des poutres de coque sont présentés pour les coupes transversales de six types de navire représentatifs et sont comparés à des résultats publiés antérieurement. Une autre série de résultats est présentée pour la résistance transversale du milieu des navires de la classe HALIFAX; elle se révèle bien correspondre avec les résultats d’études antérieures. Les comparaisons démontrent.que la précision générale des prédictions d’ULTMAT est acceptable quant à la résistance des structures, mais elles révèlent des lacunes sur certains aspects. L’utilisation des règles de L’lACS donne généralement des résultats d’une précision acceptable du point de vue conceptuel. L’interpolation des courbes provenant de deux bibliothèques existantes d’ULTMAT donne généralement des résultats de résistances de poutres de coques qui présentent un écart d’au plus 10 %. La précision des prédictions peut être améliorée par la substitution des courbes de contrainte de panneaux choisis par des courbes calculées directement par une analyse par éléments finis non linéraires. Le présent document fait des recommandations pour une meilleure bibliothèque de courbes de contrainte.

DRDC Atlantic TR 2008-059 i


ii DRDC Atlantic TR 2008-059

Executive summary

Ultimate strength assessment of naval and commercial ships: Smith, Malcolm J; DRDC Atlantic TR 2008-059; Defence R&D Canada – Atlantic; August 2008.

Introduction or background: Hull girder ultimate strength is the single most important structural design parameter for most naval surface ships and large commercial vessels. DRDC’s ULTMAT code has been used for many years to predict the ultimate strength of Canadian warships. Recent software developments for Cooperative Research Ships (CRS) have now extended ULTMAT’s application to commercial designs. The present study investigates the accuracy of ULTMAT for a wide range of ship designs, including bulk carriers, oil tankers, container ships, and naval frigates.

Results: Ultimate strength is the maximum bending load that a ship structure can withstand prior to breaking. It depends on a number of factors including the strength of the structural material, the dimensions and layout of the structural elements, and the tendency for local structure to buckle under compressive loading. The latter is investigated in this report in some detail, with comparisons of ULTMAT’s simplified predictions and results obtained using more detailed calculations. Strength results for the hull girder as a whole are given for six benchmark ship designs for which published data is available, and also for the HALIFAX class frigate.

Significance: ULTMAT provides a means of evaluating ultimate strength that balances accuracy and computation efficiency. Ultimate strength of a ship cross section can be evaluated with ULTMAT in a matter of seconds, making it suitable for design level assessments as well as more detailed studies of in-service ships. This report has addressed the accuracy of ULTMAT by examining the results from benchmark cases. These show that ULTMAT’s accuracy is comparable to other ultimate strength methods. Some deficiencies have been identified, and recommendations are made to improve ULTMAT’s accuracy for commercial ship designs.

DRDC Atlantic TR 2008-059 iii

Sommaire

Ultimate strength assessment of naval and commercial ships: Malcolm J. Smith; DRDC Atlantic TR 2008-059; R & D pour la défense Canada – Atlantique; Août 2008.

Introduction: La charge de rupture des poutres de coque est le paramètre de conception de structure le plus important pour la plupart des navires de guerre et des gros navires commerciaux. Le code ULTMAT de RDDC est utilisé depuis de nombreuses années pour prédire la charge de rupture des navires de guerre du Canada. Des progrès récents dans le domaine des logiciels pour le compte de Cooperative Research Ships (CRS) permettent maintenant d’élargir l’utilisation du code ULTMAT aux navires commerciaux. La présente étude examine la précision d’ULTMAT pour divers types de navires, y compris les vraquiers, les navires pétroliers, les porte-conteneurs et les frégates militaires.

Résultats: La charge de rupture est la contrainte de flexion maximale que peut supporter une structure de navire avant de se rompre. Cette charge dépend d’un certain nombre de facteurs, notamment de la résistance du matériau de la structure; des dimensions et de la disposition des éléments de structure, ainsi que de la tendance d’une structure locale à se déformer sous une contrainte de compression. Ce dernier aspect est examiné en détails dans le présent rapport, avec des comparaisons entre les résultats simplifiés du code ULTMAT et les résultats obtenus au moyen de calculs plus poussés. Les résultats d’ensemble pour la résistance de poutres de coque sont donnés pour six types de navires dont il existe des données publiées, et aussi pour les frégates de la classe HALIFAX.

Portée : Le code ULTMAT fournit un outil d’évaluation de la charge de rupture qui représente un bon compromis entre la précision et l’efficacité de calcul. La charge de rupture de la coupe transversale d’un navire peut être évaluée en quelques secondes à l’aide d’ULTMAT, ce qui en fait un choix indiqué pour des évaluations au niveau conceptuel, de même que pour des études plus détaillées de navires en service. Le présent rapport a évalué la précision du code ULTMAT en examinant les résultats pour des cas de référence. Ces résultats montrent que la précision d’ULTMAT est comparable à celle d’autres méthodes de calcul de la charge de rupture. Certaines lacunes ont été constatées et des recommandations sont faites pour améliorer la précision d’ULTMAT pour les types de navires commerciaux.

iv DRDC Atlantic TR 2008-059

Table of contents

............................................................................................................................................ i Abstract............................................................................................................................................. i Résumé

........................................................................................................................ iii Executive summary........................................................................................................................................ iv Sommaire

............................................................................................................................. v Table of contents................................................................................................................................ vii List of figures

................................................................................................................................... ix List of tables......................................................................................................................... x Acknowledgements

............................................................................................................................... 1 1 Introduction1.1 ........................................................................................... 1 Progressive Failure method1.2 ...................................................................................................................... 3 ULTMAT1.3 ....................................................... 4 Purpose and organization of the present document

............................................................................. 5 2 Load-shortening response of local structure2.1 ..................................................................................... 5 Longitudinally stiffened panels

2.1.1 ...................................................... 5 ULTMAT load-shortening curve libraries2.1.2 ....................................................................................... 6 IACS structural rules2.1.3 ............................................................................................... 6 Baseline curves

2.1.3.1 ..................................................................... 7 Boundary conditions2.1.3.2 ..................................................................... 7 Shape imperfections2.1.3.3 ........................................................................... 8 Residual stresses

2.1.4 ......................................................... 10 Comparison of load-shortening curves2.2 ...................................................................................... 21 Transversely stiffened panels

2.2.1 ......................................................................................... 22 TSP library curves2.2.2 .................................. 23 IACS structural rule for transversely stiffened panels2.2.3 ............................................................................................. 24 Baseline curves2.2.4 ................................................................................... 24 Comparison of curves

2.3 ................................................................................................................. 25 Conclusions ........................................................................ 32 3 Hull girder strength evaluation of intact ships

3.1 ...................................................................................................... 32 Benchmark models3.2 ......................................................................................................................... 39 Results

3.2.1 ................................................................................................ 39 Bulk Carrier 13.2.2 ............................................................................................... 39 Container ship3.2.3 ....................................................................................... 39 Double Hull Tanker3.2.4 ........................................................................................ 43 Single Hull Tanker3.2.5 ............................................................................................. 44 1/3 scale frigate3.2.6 ................................................................................................ 44 Bulk Carrier 2

DRDC Atlantic TR 2008-059 v

3.3 ................................................................................................................. 44 Conclusions ........................................................................................ 46 4 Application to the HALIFAX class

4.1 ..................................................................................................... 46 Cross section model4.2 .............................................................. 46 Ultimate strength results with ULTMAT 2.24.3 .............................................................................. 49 Comparisons with previous results

........................................................................................ 55 5 Conclusions and recommendations5.1 ............................................................... 56 Requirements for an enhanced curve library5.2 .................................................. 58 Automated calculation of an enhanced curve library5.3 .................................................................................... 59 Implementation and validation

..................................................................................................................................... 60 References..................................................................... 63 List of symbols/abbreviations/acronyms/initialisms

5.3.1 .............................................................. 63 Abbreviations/acronyms/initialisms5.3.2 ......................................................................................................... 64 Symbols

.............................................................................................................................. 65 Distribution list

vi DRDC Atlantic TR 2008-059

List of figures

......................................................................................................... 3 Figure 1: Structural unit types

.................................... 8 Figure 2: Boundary conditions for a longitudinally stiffened panel model.

.............................................................. 9 Figure 3: Shape imperfections for longitudinal stiffeners.

.................................... 9 Figure 4: Residual stress distribution due to welding a stiffener to plating

......................................................................... 12 Figure 5: Load-shortening curves for panels A-B

......................................................................... 13 Figure 6: Load-shortening curves for panels C-D

.......................................................................... 14 Figure 7: Load-shortening curves for panels E-F

......................................................................... 15 Figure 8: Load-shortening curves for panels G-H

............................................................................ 16 Figure 9: Load-shortening curves for panel J-K

....................................................................... 17 Figure 10: Load-shortening curves for panels L-M

....................................................................... 18 Figure 11: Load-shortening curves for panels N-O

....................................................................... 19 Figure 12: Load-shortening curves for panels P-Q

............................................................................. 20 Figure 13: Load-shortening curves for panel R

........................................................................................... 20 Figure 14: Collapse shape for panel P

.......................................................................................... 22 Figure 15: Collapse shape for panel Q

.......................................................... 26 Figure 16: Load-shortening curves for panels T-1 and T-2.




........................................................ 30 Figure 20: Load-shortening curves for panels T-9 and T-10.

...................................................... 31 Figure 21: Load-shortening curves for panels T-11 and T-12.

..................................... 33 Figure 22: Bulk carrier cross section showing beam and plate elements.

.................................... 34 Figure 23: Bulk carrier (top) and container ship (bottom) cross sections.

............................... 35 Figure 24: Double hull (top) and single hull (bottom) tanker cross sections.

.............................. 36 Figure 25: 1/3 scale frigate (top) and Bulk Carrier 2 (bottom) cross sections.

Figure 26: Ultimate strength in hog and sag for Bulk Carrier 1 (top) and container ship (bottom)....................................................................................................................... 40

Figure 27: Ultimate strength in hog and sag for double hull (top) and single hull (bottom) tanker models. ............................................................................................................. 41

Figure 28: Ultimate strength in hog and sag for the 1/3 scale frigate (top) and the Bulk Carrier 2 (bottom).................................................................................................................... 42

DRDC Atlantic TR 2008-059 vii

............. 47 Figure 29: Midships cross section model of the HALIFAX class: structural units view.

........ 47 Figure 30: Midships cross section model of the HALIFAX class: effective lengths in mm.

.................. 48 Figure 31: Interaction curves for the midships cross section of the HALIFAX class.

Figure 32: Hog and sag strength results for the HALIFAX class midships cross section with nominal scantlings....................................................................................................... 50

................................................................. 51 Figure 33: Hull bottom structure at two cross sections

Figure 34: Hog and sag strength results for the HALIFAX class with corrosion and rolling allowance and average imperfections. ........................................................................ 52

Figure 35: Hog and sag strength results for the HALIFAX class with corrosion and rolling allowance and severe imperfections............................................................................ 53

...................................................................... 57 Figure 36: Distribution of plate slenderness values.

................................................................. 57 Figure 37: Distribution of column slenderness values.

........................................................................................... 58 Figure 38: Distribution of area ratios.

viii DRDC Atlantic TR 2008-059

List of tables

................................................................................................................. 10 Table 1: LSP properties

Table 2: Normalized ultimate strengths for 17 longitudinal stiffened panels................................ 11 ................................................................................................................. 25 Table 3: TSP properties

....................................................................................................... 32 Table 4: Six benchmark cases

...................................................................................... 37 Table 5: Bulk Carrier 1 model properties

...................................................................................... 37 Table 6: Container ship model properties

............................................................................... 38 Table 7: Double hull tanker model properties

................................................................................. 38 Table 8: Single hull tanker model properties

................................................................................... 38 Table 9: 1/3 scale frigate model properties

.................................................................................... 39 Table 10: Bulk Carrier 2 model properties

.................................................................................. 51 Table 11: Corrosion and rolling allowances

....................... 53 Table 12: Plastic moment reduction factors for corrosion and rolling allowances

DRDC Atlantic TR 2008-059 ix

Acknowledgements

The author would like to thank DRDC’s James Nickerson for developing many of the ship cross section models discussed in the report.

x DRDC Atlantic TR 2008-059

1 Introduction

The midships ultimate strength is the single most important structural design parameter for large, ocean-going commercial vessels and for most major warships. It is the maximum bending load that a ship’s primary structure, or hull girder, can withstand. Ultimate strength can change throughout a vessel’s service life as a result of structural modifications during refit and in-service damage, so it is necessary to evaluate ultimate strength not just at the design stage, but at various stages of service life.

Ultimate strength is of primary interest at midships where the predominantly vertical bending loads are largest. It is however a sensible practice to evaluate ultimate strength at various stations and compare with the expected working load levels. Forward and aft of the midships position, bending loads are generally accompanied by predominantly vertical shear forces. For ships with large deck openings, such as container vessels, torsional effects may be significant. The horizontal components of bending and shear strength, and the torsional strength may all be important at sections where damage has occurred. In general, the structural capacity of the hull girder can be considered to be a function of five components: the vertical and horizontal components of both the bending and shear strength, and the torsional strength.

The focus of the present study is the ultimate bending strength in the midships region of intact ships. The effects of damage require special consideration, and this is left for future studies.

Various means have been developed for evaluating the ultimate strength, the simplest being analytical formulas based on the analysis of idealized box girders [1]-[2], and the most complex being three dimensional nonlinear finite element analysis of the hull girder [3]. In between these two extremes are specialized methods that retain some of the simplifying assumptions of the analytical methods as well as the accuracy of nonlinear finite element methods. One such method is the Independent Structural Unit Method (ISUM) developed originally by Ueda and Rashed [4] and now used as the basis of the ALPS code [5]. Another is the Progressive Failure method originally developed by Smith [6] for the UK NS94 codes, and which has also been adopted in the US for the ULTSTR code [7], and in Canada for the ULTMAT code [8].

A review study carried out by an International Ship Structures Committee (ISSC) panel [9] recommended the Progressive Failure method, recognizing that it offers the best combination of simplicity and accuracy. The method has also been adopted by the International Association of Classification Societies (IACS) common structural rules for oil tankers and bulk carriers [10]-[11].

1.1 Progressive Failure method

The Progressive Failure method, like the ISUM method, assumes that the hull girder behaves essentially like a box girder beam, and that plane sections remain plane under bending. It also assumes that structural failure occurs as a result of yielding on the tension side and combined yielding and buckling of the interframe longitudinal structure on the compression side. Interframe collapse means that the transverse frames remain intact, which rules out the possibility of more extensive grillage failures. The assumption of interframe collapse is consistent with modern ship

DRDC Atlantic TR 2008-059 1

design practices, but it may not hold for ships that have suffered gross damage. Ultimate strength methods like Progressive Failure must therefore be used with considerable care when assessing damaged vessels.

Pure bending loads create a stress-strain field that is aligned primarily in the fore-and-aft direction. It can therefore reasonably be assumed that little interaction occurs among the various structural elements that make up the cross section. This enables nonlinear analysis of elasto-plastic collapse of the compression-side structure to be carried out on small, isolated pieces of structure under uniaxial loading conditions.

In the progressive failure method, cross sections are subdivided into three types of structural units: longitudinally stiffened panels (LSPs), transversely stiffened panels (TSPs), and hard corners (HCs) (Figure 1). Each LSP contains one longitudinal stiffener and a portion of the attached plating. TSPs are generally used for longitudinal bulkheads and large web frames, while HCs occur at the major structural junctures.

The bending load is applied in the form of an imposed curvature, which in turn induces a distribution of fore-and-aft strains throughout the section. The resisting stress in each unit is determined from its average stress-strain curve. From the distribution of stresses in the section, the current neutral axis position can be determined, relative to which the net bending moment can be evaluated. The bending moment can be iteratively updated by incrementally increasing the applied curvature until the maximum value in a given bending direction is determined.

The stress-strain behaviour of the structural units in tension is relatively easy to define if elastic-perfectly-plastic material behaviour is assumed (i.e., elastic up to yield, with zero stiffness thereafter). A bilinear average stress-strain curve can then be used for all structural units. For compression, the Progressive Failure method uses average stress-strain curves for LSP and TSP units that define the nonlinear load-shortening properties of the local structure, including the elastic, plastic, buckling and post-buckling behaviour. For hard corners, elastic-perfectly-plastic behaviour is assumed in both tension and compression.

Because local structure can undergo collapse at various load levels, the hull girder does not collapse all at once but rather in a progressive manner. As the hull bends, the bending moment resisted by the hull increases until a maximum value (the ultimate strength) is reached. At this point, some parts of the structure may have already collapsed locally, and in these units it is the post-collapse, or load-shedding, part of their structural response that is contributing to the bending moment. Other parts of the structure will not have reached collapse and in these units it is the pre-collapse range that is contributing to the bending moment. That is why it is important to include both the pre-and post-collapse response via load-shortening curves in ultimate strength analysis.

Three general approaches are available for creating load-shortening curves for structural units. One is to calculate the load-shortening response directly from design formulas, which is the approach recommended by the IACS structural rules, and is now available in ULTMAT version 2.2. Another is to derive load-shortening curves directly either from experiment or from nonlinear finite element analysis. This is the approach used, for example, in the UK code NS94B linked with the nonlinear FE code FABSTRAN [12]. The third way is to interpolate a curve from a pre-existing library of curves. This approach is now used in ULTMAT and the UK code NS94D [13].

2 DRDC Atlantic TR 2008-059

Longitudinally Stiffened Panel

Hard Corner

ab

a

b

Transversely Stiffened Panel

Figure 1: Structural unit types

1.2 ULTMAT

The original version of ULTMAT predicted only the hog and sag ultimate strength by a direct iteration method using formulas for plating effectiveness and column strength curves of local structure [14]. This was originally a standalone program and was later implemented in the EBMS software [15]. Subsequent versions incorporated a load-shortening curve methodology for the strength of local structure and were able to calculate the vertical moment-curvature response of ship sections. Two libraries of load-shortening curves were developed based on the structural geometry of the HALIFAX class frigate [16],[17]. Later developments, leading up to Version 2.1, provided ULTMAT with fully biaxial bending strength assessment capabilities including interaction curve calculation, and tight integration of the load-shortening curve libraries [18]. A detailed description of ULTMAT’s capabilities is provided in the user’s manual for Version 2.1 [19].

The input for ULTMAT consists of the ship cross sectional geometry, including the properties of all structural elements and materials. Originally, the structural layout had to be input from ship drawings by hand or via a digitizing tablet in DIGFEM format [21]. The ULTSAS code, a joint development between Canada and the UK, provided a graphical interface through which cross sectional models could be automatically generated from three dimensional models [20]. Ultimate strength analysis could then be carried out using either ULTMAT or the NS94 family of codes. Damage modelling capabilities were also provided.

More recently, the Ultimate Strength Application was developed at DRDC for the CRS STRUC and SHIPSURV working groups [22]. This program provided an enhanced graphical user interface which allows cross sections to be either automatically generated from 3D FE models, or directly from drawings using GUI modelling tools provided in the software. Ultimate strength analysis can be performed on one or more cross sections at a time. Analysis of both intact and damaged conditions can also be performed at once, where one or more damage cases can be considered simultaneously.


1.3 Purpose and organization of the present document

The present document provides a summary of recent developments that have led to the creation of Version 2.2 of ULTMAT, and which were carried out in parallel with the development of the Ultimate Strength Application. This includes the implementation of IACS rules and a library of load-shortening curves for transversely stiffened panels. These developments are described in Section 2.

A larger purpose of the present work is to validate both the load-shortening curve methodology through detailed comparisons of load-shortening curves determined by (a) the IACS rules, (b) direct calculation with nonlinear FEA, and (c) interpolation from libraries of load-shortening curves. Comparisons are carried out for a variety of longitudinal and transversely stiffened panels, with dimensions and properties based on commercial and naval ship structures. Detailed discussion of the results is given in Section 2.1 for LSPs and in Section 2.2 for TSPs.

The ULTMAT code was developed primarily with naval surface combatants in mind, although the underlying assumptions and theory for ULTMAT are equally applicable to a wide variety of commercial ships. A further purpose of this study is to assess the accuracy of ULTMAT for typical naval and commercial structures. Cross sectional models for six benchmark ship designs (five commercial and a 1/3 scale warship) were developed using the Ultimate Strength Application and analysed with ULTMAT. Section 3 compares of ULTMAT predictions of hog and sag ultimate strength with previously published results. In Section 4, the midships cross section of the HALIFAX class frigate is analysed in some detail, and results are compared against those from two previous studies.

Section 5 summarizes the conclusions of the study and sets out recommendations for future developments. Among these is a recommendation to develop a more comprehensive load-shortening curve library, spanning a wider range of dimensionless parameters than is represented in the existing libraries, and with separate sets of curves for based on stiffener size and type.


2 Load-shortening response of local structure

2.1 Longitudinally stiffened panels

In this section, various methods for determining load-shortening curves for LSPs are discussed. In Section 2.1.4, the methods are compared for some representative LSPs.

2.1.1 ULTMAT load-shortening curve libraries

A typical cross section of a ship has over one hundred LSP units, making nonlinear FEA calculation of all the load-shortening (LS) curves impractical. In ULTMAT 2.1, three libraries of pre-calculated LS curves were available from which curves for specific structural units can be derived by interpolation:

1. The End-to-End (E-E) library 2. The Centre-to-Centre (C-C) library 3. NS94D library (NS94)

The first two of these were generated using nonlinear finite element (FE) analysis of three-dimensional LSP models. “End-to-end” refers for FE models that span a single frame space, with simple supports at the ends, whereas “centre-to-centre” refers to FE models extending from one mid-frame position to the next, with simple supports at mid-span, and symmetry constraints at the ends. By virtue of the symmetry constraints, centre-to-centre models act effectively extend over an infinite number of frame spaces, whereas end-to-end models only act over a single span. The stiffened panels used for these libraries were all based on symmetric tee stiffener sections. The 3D FE models were meshed in sufficient detail that all of the relevant failure modes could be included in the library curves, including Euler column-buckling, lateral-torsional buckling, and local buckling modes. Plate and stiffener imperfections were included, as were weld-induced residual stresses.

The library curves are derived from single stiffened panel models, each with a unique combination of four non-dimensional parameters. These four parameters determine the effective size, shape, stiffness and imperfection level under the assumption that the same LS curve applies to any panel with the same combination of parameters. The four non-dimensional parameters are:

Etb y /)/( σβ =Plate slenderness: (1)

Era y /)/( σπλ = Column slenderness: (2)

fwf

fwfs wtttdbt

wtttdAA

+−+

+−=

)()(

/ Area ratio: (3)


Imperfection ratios: and tp2/ βδ yr σσ / (4)

where definitions of all the symbols can be found in the List of Symbols. The C-C and E-E curve libraries were calculated for all combinations of β = (1, 2, 3, 4), λ = (0.3, 0.6, 0.9, 1.2), As/A = (0.2, 0.275, 0.35), and three imperfection levels: slight, average and severe.

The NS94 library was developed for the UK MoD’s NS94C/D ultimate strength analysis codes [13], and contains simplified bilinear LS curves that were derived from a combination of design curves and curves calculated using the FABSTRAN computer code [24]. This library was provided to DRDC and was implemented in ULTMAT with the kind permission of QinetiQ Ltd, for use by members of CRS. Non-dimensional parameters represented in the NS94 library are β = (1, 2, 3, 4), λ = (0.2, 0.4, 0.6, 0.8, 1.0, 1.2), and As/A = 0.2.

For the C-C and E-E libraries, trilinear interpolation of the library curves is used to derive a curve for a single panel with a given set of parameters (β, λ, As/A). If any of these parameter fall outside of the range of values represented in the library, then extrapolation of the library curves is performed instead. For the NS94 library, bilinear interpolation is used, since only one value of As/A is represented in that library. Interpolation/extrapolation for imperfection level is not performed by ULTMAT, since the three imperfection levels are discrete quantities and intermediate values are not permitted.

Extrapolation can lead to significant errors if one or more dimensionless parameters are too far outside the ranges of the library curves. Although parameter “cutoff” values are employed in ULTMAT to eliminate the more egregious errors, extrapolation is still a significant source of uncertainty in ULTMAT predictions.

2.1.2 IACS structural rules

Design formulas for the load-shortening behaviour of stiffened panels have been developed through the work of numerous researchers [25]-[28], and these have formed the basis of the hull girder strength rules in both the Common Rules for Double-Hulled Oil Tankers [10], and the Common Rules for Bulk Carriers [11], published by the International Association of Classification Societies (IACS).

As the IACS rules for hull girder strength are lengthy, a full presentation is omitted here and readers are referred to the referenced documents. Of interest here is applying the rules within the ULTMAT framework to generate load-shortening curves for stiffened panels. The IACS formulas relevant for this were implemented in Version 2.2 of ULTMAT, and the user is given the choice of either the IACS rules or one of the curve libraries when performing an ultimate strength analysis.

2.1.3 Baseline curves

To provide a baseline set of data for comparison purposes, load-shortening curves are also calculated directly with nonlinear finite element analysis (NLFEA). The LSPanel program [29],


originally developed by Martec Limited, is used to efficiently generate single-stiffened panel 3D FE models of specified dimensions, material properties and imperfections. In the original version of LSPanel, only tee section stiffeners were modelled, but the code has since been extended to angle section and flat bar stiffeners, and to allow both lateral and normal shape imperfections to be applied to the stiffeners.

Nonlinear FE calculations were carried out using the orthogonal-trajectory method in the VAST finite element code [30]. Only axial loading of the stiffened panels is considered in this study. The effect of lateral pressure loads (e.g. from water head, cargo, or heavy equipment) on the load-shortening response is not considered here, and this is consistent with ULTMAT’s existing curve libraries. Similarly, the effect of in-plane shear on the load-shortening response is not considered at present. The strength of ships under of combined bending and shear has been examined using the ALPS code [31], [32]. It was shown that for intact ships, vertical shear force only significantly reduces bending strength when it is greater than about 70% of the ultimate shear strength.

The details of the FE modelling used to generate the baseline curves is described in Sections 2.1.3.1 - 2.1.3.3.

2.1.3.1 Boundary conditions

LSPanel has the ability to create models with boundary conditions consistent with an end-to-end or centre-to-centre configuration, although only end-to-end models are used in the present study. The LSP models are oriented with y as the axial direction, and extend between y = -a/2 to +a/2. The transverse direction is x and the normal direction is z.

Simply supported boundary conditions are applied at each end in which all nodes located in the y = -a/2 and y = +a/2 planes are linked through multipoint constraints, as shown in Figure 2. The end displacements and rotations are then controllable through master nodes located in the centroidal position at each end. Boundary conditions at the two master nodes are sufficient to create simple supports in the normal and transverse direction, while still allowing axial compression. Compressive loads are applied through the axial degree of freedom of the master node at y = -a/2.

The assumption in ULTMAT is that the interaction between neighbouring units within the same frame space is negligible because the dominant load path is in the fore-and-aft direction. However, some interaction does occur at the outer edges of the plates and it is necessary to apply rotational constraints along the two transverse edges (x = -b/2 and x = b/2). Translations in the transverse direction are left free, however, as this allows Poisson effect to occur, is would normally happen if the panel were a component in a ship structure undergoing bending.

2.1.3.2 Shape imperfections

The conventions used to define geometric and constitutive properties are consistent with those used in Ref [9]. “Average” imperfections are assumed for which the plating imperfection ratio (δp/β2 t) is equal to 0.1, where δp is the maximum out-of-plane initial deflection of the plating, and t denotes the plate thickness. This imperfection is applied to the FE models as a double half-sine deflection shape with positive δ on the stiffener side of the panel. The initial maximum out-of-p


straightness of the stiffener normal to the plate is defined by the deflection δsv =0.001a, and a half-sine distribution of the stiffener imperfection is assumed. This imperfection is important for initiating column buckling collapse failures.

Figure 2: Boundary conditions for a longitudinally stiffened panel model.

Angle stiffeners are also assumed to have a maximum lateral out-of-straightness of δsh =0.001a at the flange, and diminishing linearly towards the toe, as shown in Figure 3. This lateral imperfection may be important for initiating lateral-torsional buckling modes. Both the vertical and lateral stiffener imperfections are oriented as shown in Figure 3, which is consistent with actual deformation patterns in welded ships.

In the nonlinear FEA calculations, shape imperfections are supplemented by a small amount of the first (local) elastic buckling mode of the panel. This has the effect of introducing both shorter wavelength components in the plating imperfection and realistic local imperfection to the web and flange of the stiffener, both of which are important for seeding local collapse modes. To allow for this a linear buckling analysis of the panel is performed prior to the nonlinear FE calculations, and the critical buckling mode stored on file.

2.1.3.3 Residual stresses

In addition to shape imperfections, “average” imperfections also include residual stresses in the plating equal to σ /σr y = 0.15. This has previously been determined to be the typical level of residual stress for in-service ships [33]. Residual stress can be applied directly to the FE models using the LSPanel program assuming a simplified distribution of the stresses inside and outside the heat affect zone (HAZ) around a weld. Figure 4 shows the distribution of axial residual stresses across the plating, based on a HAZ with a width of 2η t, where width η is chosen to maintain overall equilibrium. If it is assumed that stiffeners are rolled, (i.e. not fabricated by


welding the web to the flange), then axial residual stresses in the stiffener will assume high tensile values close to the weld (at the toe) and low compressive values away from the toe.

δsv

δsh

Figure 3: Shape imperfections for longitudinal stiffeners.

σ y σ r

2η t

Tension

Compression

Stiffener

Plating

Compression Tension

d

d/4

σ y σ y/4

Figure 4: Residual stress distribution due to welding a stiffener to plating


2.1.4 Comparison of load-shortening curves

The load-shortening curves for seventeen longitudinally stiffened panels are now compared. Load shortening curves are derived by interpolation/extrapolation from the three existing libraries, calculation using the IACS rules, and direct calculation with NLFEA. The properties of the benchmark panels are listed in Table 1. Panels A through J are based on naval ship designs and are all tee stiffened panels. Panels K through R are based on commercial designs (bulk carriers, tankers and container ships), and include a variety of tee, angle and flat bar stiffener geometries. Parameter ranges for the benchmark panels are: 1.317 and 4.491 for β; 0.047 and 1.038 for λ; and 0.146 and 0.438 for As/A.

Figure 5 – Figure 13 give the load-shortening curves for all of the panels. The imperfection level is not defined for the IACS rules, but average imperfections have been assumed for the other methods. Table 2 summarizes the normalized ultimate strengths for the seventeen panels, as predicted by the five methods.

Table 1: LSP properties

AName a (mm)

b (mm)

t (mm)

Beam section d×w×tw×tf (mm)

σyp (MPa)

σyb (MPa)

β λ s/A

A 1000 350 9.0 200×140×6.4×8.8 T 350 350 1.589 0.159 0.438

B 2000 350 9.0 200×140×6.4×8.8 T 350 350 1.587 0.318 0.438

C 2000 550 9.0 128×102×6.1×8.4 T 350 350 2.495 0.538 0.243

D 1000 450 9.0 128×102×6.1×8.4 T 350 350 2.041 0.264 0.281

E 2000 550 5.0 76×25×4.4×6.4 T 350 310 4.491 1.038 0.146

F 1000 550 5.0 76.2×25.4×4.4×6.4 T 350 310 4.491 0.519 0.146

G 2000 500 7.0 127×68×4.0×5.2 T 350 350 2.916 0.587 0.194

H 1000 450 9.0 127×68×4.0×5.2 T 350 350 2.041 0.317 0.172

J 457.2 202.7 2 38.1×14×1.78×3.3 T 245 245 3.487 0.373 0.211

K 5100 925 23.5 847×180×15×25 A 235.2 313.6 1.317 0.185 0.436

L 5100 925 23.5 549×125×11.5×22 A 235.2 235.2 1.317 0.289 0.288

M 5100 1000 25 830×200×15×33 T 313.6 313.6 1.546 0.198 0.426

N 5220 800 24.5 390×27 FB 392 392 1.411 0.612 0.349

O 860 880 25.5 395×25.5 FB 350 350 1.409 0.097 0.310

P 5100 1000 25 480×32 FB 313.6 313.6 1.546 0.429 0.381

Q 830 830 20 300×90×13×17 A 313.6 313.6 1.604 0.106 0.239

R 830 830 20.5 575×150×12×25 T 313.6 313.6 1.565 0.047 0.378


Panels A through H are all based on the HALIFAX class structure, so it might be expected that curves interpolated from the C-C and E-E libraries should show good agreement with the NLFEA curves. Generally this is found to be true, although the interpolated curves significantly underestimate the post-collapse response in several cases (C,D,G,H), while overestimating the strength of Panel E somewhat. Curves interpolated from the NS94 library curves sometimes give good agreement with NLFEA (B,H), but in most cases they give poorer agreement than the other library curves, and tend to oversimplify the post-buckling response. The IACS rules also give excellent agreement with NLFEA in a few cases (B,C), underestimate several others (D, F, H) and overestimate the strength of Panel E.

Panel J is a stiffened panel from a 1/3 scale model of a frigate. Agreement between the NS94 library curve and NLFEA is excellent, while the IACS rules give a load-shortening response that is about 15% lower than the NLFEA curve. The C-C and E-E library curves predict the ultimate strength of the panel accurately, but fall to the level of the IACS curve in the post-buckling range.

Table 2: Normalized ultimate strengths for 17 longitudinal stiffened panels

Panel IACS E-E C-C NS94 NLFEA

A 0.778 0.850 0.903 0.812 0.875

B 0.778 0.796 0.832 0.777 0.860

C 0.535 0.583 0.597 0.612 0.568

D 0.620 0.734 0.751 0.703 0.737

E 0.296 0.301 0.267 0.347 0.239

F 0.296 0.348 0.361 0.406 0.390

G 0.459 0.484 0.484 0.543 0.512

H 0.563 0.738 0.697 0.704 0.706

J 0.428 0.486 0.504 0.494 0.503

K 0.901 0.907 0.936 0.826 0.815

L 0.795 0.849 0.844 0.822 0.957

M 0.723 0.838 0.886 0.786 0.816

N 0.868 0.756 0.760 0.757 0.886

O 0.919 0.895 0.941 0.985 0.977

P 0.899 0.772 0.785 0.777 0.939

Q 0.696 0.843 0.884 0.924 0.924

R 0.725 0.871 0.945 1.062 0.918


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C-C NS94

NLFEA

A

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IACS E-E

C-C NS94D

NLFEA

B

Figure 5: Load-shortening curves for panels A-B


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NLFEA

C

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IACS E-E

C-C NS94

NLFEA

D

Figure 6: Load-shortening curves for panels C-D


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NLFEA

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C-C NS94

NLFEA

F

Figure 7: Load-shortening curves for panels E-F


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IACS E-E

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NLFEA

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C-C NS94

NLFEA

H

Figure 8: Load-shortening curves for panels G-H


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NLFEA

J

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IACS E-E

C-C NS94

NLFEA

K

Figure 9: Load-shortening curves for panel J-K


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IACS E-E

C-C NS94

NLFEA

L

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C-C NS94

NLFEA

M

Figure 10: Load-shortening curves for panels L-M


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C-C NS94

NLFEA

N

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IACS E-E

C-C NS94

NLFEA

O

Figure 11: Load-shortening curves for panels N-O


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NLFEA

P

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C-C NS94

NLFEA

Q

Figure 12: Load-shortening curves for panels P-Q


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IACS E-E

C-C NS94

NLFEA

R

Figure 13: Load-shortening curves for panel R

Figure 14: Collapse shape for panel P


Panels K through R are generally of stockier build with normalized strengths in some cases approaching one. β and λ values tend to be near or below the low end, and As/A values tend to be near or above the top end of the library ranges. The IACS rules agree well with NLFEA for two flat-bar stiffened panels (N and P). They overestimate the strength of K, but otherwise give conservative results (in some cases very conservative, such as in Q). The curves interpolated from the C-C and E-E libraries give good agreement with NLFEA for Panels L, Q, and R, despite the very low λ values in the last two cases. Agreement in the three flat-bar panels is not as good as in the tee- and angle-stiffened cases. Apart from panel L, the NS94 library curves are evidently too simplified to give good agreement in these cases. In three of these cases the NS94 curves increase without limit, which results from having to extrapolate the library curves for panels with λ < 0.2.

An interesting case is Panel P, where the IACS rules perform significantly better than all of the interpolated curves. The As/A value for this panel is slightly above the upper range of the C-C and E-E libraries, while β and λ are well within the library ranges. The collapse shape predicted with NLFEA is shown in Figure 14, indicating that Euler-column buckling occurs with buckling toward the stiffener, which is consistent with the initial shape imperfection. Combined compression and bending of the flat bar produces only minor instability in the stiffener, with the major instability occurring in the plating.

The IACS rules consist of three formulas defining critical stresses for beam-column buckling, torsional buckling, and local web buckling modes of a stiffened panel. The minimum of these three stresses, evaluated over a range of non-dimensionalized strain values, defines the IACS curve. For panel Q, the formulas predict local web buckling as the failure mode, with a critical stress significantly lower than for the other two modes. For comparison, the collapse shape predicted with NLFEA is shown in Figure 15. Failure is the result of buckling of the plating, combined with yielding of the stiffener. Despite the web shape imperfections, local buckling of the web is absent, suggesting that the IACS rule for this mechanism may be too conservative. Note that the IACS rules account for plate buckling effects indirectly by using an effective plate width in the formulas.

2.2 Transversely stiffened panels

In most modern ship designs, transversely stiffened panels appear primarily in longitudinal bulkheads, in the side shell structure of bulk carriers, and sometimes in the webs of heavy girders in the hull bottom. Ice breakers may also make significant use of transversely stiffened panels in ice-strengthened areas. In an ultimate strength analysis, they appear in ship cross sections mainly as wide sections of plating with hard corners at each end. Despite their large physical dimensions, their load carrying capacity is usually modest because of the low buckling threshold for wide sections of plating. Load carrying capacity can be increased by reducing the spacing of the transverse stiffeners, which is effectively the same as increasing the aspect ratio of the plating sections. Assuming that the transverse stiffeners themselves do not undergo out-of-plane deflections under axial (fore-and-aft) compression, the LS curves for these panels are a function of the plate slenderness, β, the aspect ratio, α, and the plate imperfection ratio δp/β2t.


Figure 15: Collapse shape for panel Q

The previous version of ULTMAT (2.1) did not have a curve library for TSPs. To remedy this deficiency, ULTMAT Version 2.2 now includes (a) a library containing published curves for unstiffened plates, (b) the NS94 library of simplified curves for TSPs, and (c) the IACS rule for TSPs.

2.2.1 TSP library curves

A set of curves for unstiffened plates of various slenderness values and aspect ratios were published by Smith et al [33]. The aspect ratio α is defined as a/b, where a is the width of plating in the transverse direction, and b is the transverse beam spacing in the longitudinal direction. For unstiffened plating, b defaults to the interframe distance. Note that the definitions of a and b differ from what is used for LSPs. This is consistent with what is used for TSPs in the IACS rules and the nomenclature of Ref [33].

These curves were digitized and added to ULTMAT for use in conjunction with the C-C and E-E libraries for longitudinal stiffened panels. Five values of β (1.0, 1.5, 2.0, 3.0 and 4.0), four values of α (0, 1.0, 3.0 and ∞), and three levels of imperfections (slight, average and severe), are represented in the library for a total of sixty curves. Standard linear interpolation is used for intermediate values of β, but because of the wide range of α values, an unconventional interpolation scheme is needed. Dow [13] proposed a factor K for interpolation between curves for α = 1 and α = ∞ that was adopted for the NS94 library curves:

)1(693.0exp(1 −−−= αK (5)


This was found to predict Smith’s α = 3 curves poorly. After some experimentation, the following values for the K factor were found to give acceptable results:

)1(5.1 −−= αeK for 1 < α < 3

(6))3(3.0 −−= αeK for 3 < α < 5

α11.0−= eK for α > 5

In the range 0 < α < 1, where the plating is longer than it is wide, the α = 0 curves are used without interpolation.

2.2.2 IACS structural rule for transversely stiffened panels

The IACS rule for TSPs is based on a similar formula for the normalized strength of unstiffened plate [28]:

2

211111.01⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛ −+=

βαφ

ασσ

uFry

up , α > 1

(7)

uFry

up φσσ

= , α ≤ 1

is the Frankland formula for the strength of long plates: where φuFr

225.125.2ββ

φ −=uFr , β > 1 (8)

0.1=uFrφ , β ≤ 1

(7) defines the ultimate strength for given values of α, β, and the yield stress σEquation y. The load-shortening curve is obtained by substituting the strain-dependent expression:

Etb yεσ

εβ =)( (9)

in place of β in (7) and (8). The result is an expression for normalized stress as a function of strain. The load-shortening curve is determined by taking the minimum of the perfect elastic stress-strain response and the strain-dependent strength expression:


⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=yyy

Eσεσ

σε

σσ )(,min (10)

This formula was implemented in ULTMAT along with other IACS rules as part of the version 2.2 development.

2.2.3 Baseline curves

To provide a set of baseline curves for comparison with the library, NLFEA calculations were performed for various transverse stiffened panel models. The FE models, generated using the LSPanel program, consist of one unstiffened bay section of panel plating with length b and width a. Transverse stiffeners are not included. Boundaries at the fore and aft ends (y = -b/2 and y = +b/2) are simply supported. This is enforced through multipoint constraints at each end that tie the displacements nodes to a single master node at which the simply support condition is maintained. Axial motion (y direction) is restrained at the master node at y = +b/2, while compressive loads are applied through the master node at y = -b/2. The other two edges of plate (parallel to the loading direction) are restrained against normal displacement only. The stiffening induced by this restraint allows TSPs to carry load beyond the initial buckling point.

“Average” shape imperfections, for which the plate imperfection ratio is 0.1, are included assuming a double-sine distribution across the plate. Residual stresses are not included for TSPs. It might seem logical to adjust the definition of β (Equation (1)) for TSPs by substituting a for b, since the plate is now loaded over an edge of length a. But for large aspect-ratio plates, this will produce β values and shape imperfections that are unrealistically large. Dimension b is therefore used in the definition of β for both longitudinally and transversely stiffened panels.

2.2.4 Comparison of curves

Library and IACS-derived load-shortening curves are here compared with the baseline curves for eight transversely stiffened panels listed in Table 3. For these panels plate slenderness values range between 1.94 and 5, and aspect ratios vary between 0.75 and 8.6.

Figure 16-Figure 21 compare load-shortening curves for the twelve panels. The normalized load carrying capacity of TSPs is generally much less than that of the LSPs, as expected. Overall, the ULTMAT library curves are within about 20% of the NLFEA curve, but are nearly always on the high side. One drawback of the Smith’s curves is that in their published form they are only available for normalized strains (ε/εy) up to slightly more than 2, which could limit their use in hull girder strength predictions.

The NS94 library curves do not agree as well with the shape of the NLFEA results and often greatly over-simplify the post-buckling response. The IACS curves exhibit poor agreement with the NLFEA curve except for values of β close to one. They predict the peak strength to be too high for most values of β and have it occurring at a strain level that is too low.


The main problem with the IACS formula is that the peak values of normalized stress predicted by Equation (10) are nearly always larger than the ultimate strength predicted by (7), and always occur for normalized strains less than 1. This is a direct result of including ε in the formula through the substitution of Equation (9). At a minimum, Equation (10) needs to be modified to cap the normalized stress at the ultimate strength given by (7).

Table 3: TSP properties

a (mm)

b (mm)

t (mm)

Name α σyp (MPa)

β δp (mm)

T-1 1000 667 5 350 5 1.5 12.5

T-2 2500 667 5 350 5 3.748 12.5

T-3 500 667 5 350 5 0.75 8.3

T-4 1000 667 7 350 3.89 3 10.6

T-5 2500 667 7 350 3.89 3.748 10.6

T-6 1000 333 5 350 2.71 3 3.7

T-7 2000 333 7 350 1.94 6 2.6

T-8 7400 860 17.5 315 1.903 8.605 6.3

T-9 4500 1500 22 315 2.64 3 15.3

T-10 2400 800 22 315 1.41 3 4.4

T-11 1200 400 22 315 0.786 3 1.36

T-12 2400 400 22 315 0.786 6 1.36

2.3 Conclusions

The LS curves interpolated from the libraries for longitudinally stiffened panels generally agree with NLFEA results for the naval structural panels (A-J) and the commercial panels (K-R). Accuracy of the curves is not significantly affected by the presence of the low column slenderness values and high area ratios among the commercial panels. But the library curves do not perform well for flat-bar stiffened panels, and this appears to be a limitation of curve libraries developed using tee stiffened models. The NS94 curves generally appear to be too simplified, and poor results are obtained where curves have to be extrapolated rather than interpolated. The IACS rules for LSPs give a very good agreement in selected cases, but overall provide conservative estimates adequate for design level calculations. Their accuracy is about the same for naval and commercial structure.

For transversely stiffened panels, curves interpolated from the Ref. [33] curves give the best agreement with NLFEA, in spite of the limited strain range. The IACS rule for TSPs generally performs poorly, and it is recommended that this rule be modified to correct an obvious deficiency in the predicted peak strength. It is also recommended that a new curve library be


generated with NLFEA that is similar to Smith’s curves but with an extended range of normalized strain.

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Figure 16: Load-shortening curves for panels T-1 and T-2.


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T-12



3 Hull girder strength evaluation of intact ships

The accuracy of strength predictions of ship hull girders is now examined with the objective of validating ULTMAT’s ultimate strength procedures for both naval and commercial structures. This is accomplished through evaluations of six benchmark case studies for which detailed structural information and associated numerical or measured results have been reported.

An ISSC study on ultimate strength published in 2000 [9] presented detailed results for five benchmark cases, along with cross section drawings and tables of properties. Among the five are two previously reported examples (MV Energy Concentration [34], and the UK 1/3 scale frigate [35]) for which accurate strength values have been determined. Each of the five cases was evaluated for ultimate strength by a minimum of six investigators using a variety of methodologies. The sixth benchmark case examined here is a bulk carrier cross section from a study by Paik [31]. A summary of the six benchmark cases is given in Table 4. Published results for these benchmark cases were obtained by various researchers using the Progressive Failure method, ISUM, or a simplified analytical method.

Table 4: Six benchmark cases

Ship type Length (m)

Breadth (m)

Depth (m)

Source

Bulk Carrier 1 285 50 26.7 [9]

Container ship 230 32.2 21.5 [9]

Double hull tanker 315 58 30.3 [9]

Single hull tanker, Energy Concentration 313 48.2 25.2 [9], [34]

UK 1/3 scale frigate 18 4.2 2.8 [9], [35]

Bulk Carrier 2 264 45 23.2 [31]

3.1 Benchmark models

For this study, cross section models of the six benchmark cases were developed using Ultimate Strength Application software developed by DRDC for Cooperative Research Ships [22]. The modelling procedure is as follows. Properties of all beam sections and structural materials are first entered into the software’s database. Then using ship drawings, coordinate locations for the main structural junctions are entered as keypoints. The basic arrangement of plates and beams is then constructed around these keypoints using special “stiffened-panel” and other wizards provided in the software. Properties of individual beam and plate elements are then adjusted as required. A


typical cross section, such as the one shown in Figure 22, can be created this way in about two hours.

The Ultimate Strength Application software automatically groups the plate and beam elements into the three types of ULTMAT structure units: longitudinally stiffened panels, transversely stiffened panels and hard corners. The grouping algorithm works as follows:

• Hard corners are identified at locations where three or more plate elements are connected together or where two plates are connected at an angle of greater than 20 degrees. Hard corners are sized according to the following rule: if s is the width and t is the thickness of an attached plate element, the width of plating included in the hard corner is the minimum of s/2 and 20×t.

• At remaining locations, longitudinal stiffened panels are formed where a beam and one or two plate elements are present. A portion of each connecting plate is included in the LSP unit. If the LSP is adjacent to other LSPs, the plating is subdivided equally; if it is adjacent to a hard corner, the LSP acquires the part of the plate not belonging to the hard corner.

• TSP units are formed from plate elements with hard corners at each end and with a minimum width of 40×t. The TSP acquires the part of the plate not belonging to the two hard corners.

Plots of the six cross sections developed for this study are shown in Figure 23 - Figure 25, in which the blue structural elements are longitudinal stiffened panels, the green elements are transversely stiffened panels, and the grey elements are hard corners. Each cross section structure is comprised predominantly of longitudinally stiffened panels, and the number of transversely stiffened panels is quite small. Detailed information on the dimensions, scantlings, material properties, etc can be found in the reference documents indicated in Table 4.

Figure 22: Bulk carrier cross section showing beam and plate elements.


Figure 23: Bulk carrier (top) and container ship (bottom) cross sections.


Figure 24: Double hull (top) and single hull (bottom) tanker cross sections.


Figure 25: 1/3 scale frigate (top) and Bulk Carrier 2 (bottom) cross sections.


To validate the cross section models, basic properties of the cross sections were compared with published values. Table 5 through Table 10 report the section modulus I, the vertical neutral axis position yG, the vertical first yield moment My, and the vertical plastic moment Mp obtained with ULTMAT and as reported in Ref [9]. These properties depend on the cross section geometry, scantlings and material properties, but do not depend on the load-shortening behaviour of the local structure. Therefore, they provide a convenient means of checking the properties of the model prior to ultimate strength analysis. The ULTMAT results agree very well with those from the literature, indicating that the modelling has been done correctly. Comparative results were not available for the Bulk Carrier 2 model.

The first yield moment reported here refers to the vertical bending moment at which either tensile or compressive yield begins. Yield moment is often reported as two separate values, one for the deck flange and one for the bottom flange. ULTMAT only calculates one value for each bending direction, and this is equivalent to the minimum of the deck and bottom flange values.

Table 5: Bulk Carrier 1 model properties

Source I (m4) yG (m) My (GNm) Mp (GNm)

ULTMAT 682.1 10.75 15.87 20.3

Chen 694.87 11.2 15.82 20.87

Cho 693.44 11.06 15.64 19.9

Masaoka 689.8 11.03 15.53 19.86

Rigo (1) - 10.94 - 20.26

Rigo (2) 702.48 10.66 15.45 20.03

Soares 679.31 11.15 15.41 19.64

Yao 682.5 10.87 15.21 20.12

Table 6: Container ship model properties


ULTMAT 682.1 8.63 15.87 20.3

Chen 694.87 8.86 15.82 20.87

Cho 693.44 8.84 15.64 19.9

Masaoka 689.8 8.54 15.53 19.86

Rigo (1) - 8.13 - 20.26

Rigo (2) 702.48 8.10 15.45 20.03

Soares 679.31 8.51 15.41 19.64

Yao 682.5 8.63 15.21 20.12


Table 7: Double hull tanker model properties


ULTMAT 1341 12.88 22.23 32.41

Chen 1347.3 12.88 24.12 32.4

Cho 1340.1 12.92 24.04 31.74

Masaoka 1360 12.79 24.22 32.07

Rigo (1) - 12.83 - 30.99

Rigo (2) 1382.3 13.18 25.17 32.86

Soares 1355.8 13.03 24.48 31.77

Yao 1344.7 12.84 24.02 32.96

Table 8: Single hull tanker model properties


ULTMAT 821.7 12.20 18.76 21.68

Cho 819.7 12.15 19.7 21.76

Dow 743.8 11.85 17.47 19.85

Masaoka 828.3 12.01 19.69 21.96

Rigo(1) - 12.31 - 20.35

Rigo(2) 812.86 12.81 20.57 20.39

Yao 800.89 12.27 19.43 21.75

Table 9: 1/3 scale frigate model properties


ULTMAT 0.0590 1.44 10.05 12.64

Chen 0.0649 1.376 11.17 13.77

Cho 0.0595 1.44 10.72 12.81

Dow 0.0627 1.407 11.03 17.01

Masaoka 0.0638 1.398 11.08 13.72

Rigo(1) - 1.433 - 14.32

Rigo(2) 0.0676 1.42 12 14.39

Yao 0.0608 1.424 10.83 13.24


Table 10: Bulk Carrier 2 model properties

Source I (m4) zG (m) My (GNm) Mp (GNm)

ULTMAT 477 9.96 10.49 15.71

3.2 Results

Figure 26 - Figure 28 give the predicted ultimate strengths in hog and sag for the six benchmark cross sections. In each plot, ULTMAT results are provided for each of the four available load-shortening options. The results for each case are discussed in the following subsections.

3.2.1 Bulk Carrier 1

The agreement between ULTMAT and the other results is quite poor overall, although it is better in hog than in sag. The poor sag strength results (about 20% less than the average of the published results) can be attributed to the main deck structure being comprised of slender, flat-bar stiffened LSPs. The dimensions of these LSPs are very similar to those of Panel N of Section 2.1.4. For that panel, the IACS rules performed better than the library curves, which accounts for better sag strength results obtained with ULTMAT/IACS. To determine if further improvement was possible, an additional calculation was performed with the E-E library, but using the Panel N NLFEA curve for the main deck LSPs. The result is given in Figure 23 (ULTMAT/N-curve), indicating that the sag strength has increased from 12.1 to 13.7 GNm. Agreement with the other results is greatly improved.

3.2.2 Container ship

Excellent agreement with the other results is obtained with all of the ULTMAT analyses. The scatter in the ULTMAT results is negligibly small in hog, and is about 10% in sag. It is typical to find greater scatter in sag strength than in the hog strength. This is because the structure above the waterline is usually more slender and is therefore more susceptible to Euler-column collapse than the stockier hull bottom structure.

3.2.3 Double Hull Tanker

The ULTMAT results for this cross section show a high degree of scatter, with the highest and lowest strength predictions bracketing all of the other reported results. To understand the scatter in the sag strength predictions, it should be noted that the main deck of this vessel is comprised of structural units very similar to Panel Q. For this panel it was noted that the IACS rules significantly underestimated the local strength and load shortening response, and this would explain the low sag strength prediction of ULTMAT/IACS. The NS94 LS curve for Panel Q predicted the strength to increase monotonically to the end of the curve. This results in artificially high sag strength predictions with ULTMAT/NS94. The E-E and C-C libraries agreed reasonably


well with NLFEA for Panel Q, and higher confidence is therefore expected for the ULTMAT/E-E and ULTMAT/C-C predictions.

10

11

12

13

14

15

16

17

18

19

20

10 11 12 13 14 15 16 17

Mus (GNm)

Muh

(GNm

)

ChenChoMasaokaRigo(1)Rigo(2)SoaresYaoUltmat/IACSUltmat/E-EUltmat/C-CUltmat/NS94Ultmat/N-curve

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

4 5 6 7 8 9

Mus (GNm)

Muh

(GNm

)

ChenChoMasaokaRigo(1)Rigo(2)SoaresYaoUltmat/IACSUltmat/E-EUltmat/C-CUltmat/NS94

Figure 26: Ultimate strength in hog and sag for Bulk Carrier 1 (top) and container ship

(bottom).


20

22

24

26

28

30

32

34

15 20 25 30

Mus (GNm)

Muh

(GNm

)ChenChoMasaokaRigo(1)Rigo(2)SoaresYaoUltmat/IACSUltmat/E-EUltmat/C-CUltmat/NS94Ultmat/Q-R-curves

17

18

19

20

21

22

15 16 17 18 19 20

Mus (GNm)

Muh

(GN

m)

AstrupChenChoDowMasaokaRigo(1)Rigo(2)YaoUltmat/IACSUltmat/E-EUltmat/C-CUltmat/NS94Ultmat/M-P-curvesRutherford (exp)

Figure 27: Ultimate strength in hog and sag for double hull (top) and single hull (bottom) tanker models.


10

10.5

11

11.5

12

12.5

13

13.5

14

8 9 10 11 12

Mus (MNm)

Muh

(MNm

)ChenChoMasaokaRigo(1)Rigo(2)DowYaoUltmat/IACSUltmat/E-EUltmat/C-CUltmat/NS94Dow (exp)

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

8.0 9.0 10.0 11.0 12.0 13.0 14.0

Mus (GNm)

Muh

(GNm

)

Paik

Ultmat/IACSUltmat/E-E

Ultmat/C-C

Ultmat/NS94Ultmat/O-curve

Figure 28: Ultimate strength in hog and sag for the 1/3 scale frigate (top) and the Bulk Carrier 2 (bottom).


Considerable scatter also occurs in the hog results. The structure of the double bottom of this ship is comprised of tee-stiffened panels similar to those of Panel R above. Neither the IACS rules nor the NS94 library curve agreed well with NLFEA for this panel, resulting in hog strengths that are too low with ULTMAT/IACS and too high with ULTMAT/NS94. The E-E and C-C libraries corresponded better for Panel R, and those hog strength predictions can therefore be expected to have acceptable accuracy.

Further assessment was performed using the NLFEA curves for Panel Q in the main deck structure and for Panel R in the double bottom structure, and using the E-E library curves elsewhere. This result is given in the top half of Figure 27, and shows that neither the hog nor sag strengths are changed much from the original ULTMAT/E-E result. The latter can therefore be used with some confidence. Neglected from this analysis is the effect of normal pressure (water head) on the local structural response, which might be expected to reduce the hog strength somewhat.

3.2.4 Single Hull Tanker

The single hull tanker is in this case MV Energy Concentration, which broke in half in Rotterdam in 1981. A forensic investigation by Rutherford and Caldwell [34] established the state of loading of the vessel at the time of failure. This is the dotted line shown in the lower plot of Figure 27, indicating that the ship failed in hog. The plotted results have taken into account the estimated amount of corrosion in the vessel at the time of the accident by uniformly reducing the thickness of the plating and stiffener webs by 1 mm and the stiffener flanges by 2 mm from their nominal values.

The ULTMAT hog results are in general agreement with the other values, while the sag results show a tendency to be conservative, especially where the library curves are used. The sag results can be understood by noting that the deck structure is mainly comprised of flat-bar stiffened panels similar to the Panel P configuration of Section 2.1.4. For this panel, none of the library curves agreed well with NLFEA, although the IACS rules performed much better. Slight improvement in the sag strength prediction might therefore be expected by using the NLFEA curve for Panel P.

Looking at the hog strength results more closely, it is noted that the three predictions using the library curves are 5-10% higher than the estimated failure load of 17.94 GNm. The bottom structure of this model is comprised of tee-stiffened panels similar to Panel M. The library curves slightly overestimate the strength of this panel, and this may partly account for the modest overestimation of the hog strength in relation to the experimental value. Neglect of the water head effect may also partly account for this discrepancy. Note that ULTMAT/IACS underestimates the hog failure load by only 2.5%.

The hog and sag strengths were re-evaluated using the NLFEA Panel-P curve for the deck structural units, the NLFEA Panel-M curve for the bottom structural units, and the E-E library curves elsewhere. The curves for these panels shown in Section 2.1.4 were for the intact (uncorroded) structure. Since corrosion effects are to be included here, the NLFEA curves for these two panels were recalculated with the uniform thickness reductions mentioned above. Very little difference was observed between the intact and corroded normalized curves, although the load carrying capacity of the corroded panel is less, by virtue of its smaller cross sectional area.


The ULTMAT result using NLFEA curves is given in the bottom half of Figure 27 (ULTMAT/M-P-curves) and indicates a large increase in sag strength (from 15.65 to 18.22 GNm) and a more modest reduction in the predicted hog strength (from 19.22 to 18.25 GNm). Adjustment of the load shortening curves in this case is effective at eliminating the conservativeness in the sag strength result, and also brings the predicted hog strength to within 2% of the accepted value.

3.2.5 1/3 scale frigate

The results for the 1/3 scale frigate model are shown in the top half of Figure 28, and indicate that ULTMAT’s prediction of the sag strength are somewhat conservative in relation to the experimental value (vertical dotted line) and the other numerical results. ULTMAT’s hog strength predictions are clustered at the lower end of the range of predictions. The deck structure is comprised of structural units similar to Panel J. The load-shortening curve plots for that panel showed that the NS94 library gave excellent agreement with the directly calculated curve, and so the ULTMAT/NS94 sag strength prediction should be better than the other ULTMAT results. Indeed, the ULTMAT/NS94 result of 9.26 MNm is only about 4% below the experimentally measured value of 9.64 MNm.

3.2.6 Bulk Carrier 2

For Bulk Carrier 2, only one set of comparison results are available (Paik’s result based on the ISUM code ALPS). ULTMAT’s hog strength results are in reasonably good agreement with Paik’s result, and the total scatter in the hog calculations is less than 10%. The sag results however do not exhibit good agreement and the scatter is over 25%. Some effort to find better agreement among the sag results is therefore warranted.

In this bulk carrier model, the main deck structure is primarily formed from flat-bar-stiffened panels with dimensions and properties similar to Panel O. This is a relatively stocky panel, and Figure 11 indicates that the curves from the C-C library gave the best agreement with NLFEA calculations, while the IACS and NS94 curves agreed poorly. This is consistent with the ULTMAT results, which show a low sag prediction with IACS, a high prediction with NS94, and the results with the other two libraries somewhere in between.

The sag and hog strengths were re-evaluated with the directly calculated curves for Panel O substituted for the C-C library curves. This change increases the sag strength from 11.2 to 12.7 GNm, and brings the ULTMAT result into better agreement with Paik’s result of 12.46 GNm. The hog strength is of course unaffected by changes to the deck load-shortening curves.

3.3 Conclusions

The analysis results for the six benchmark cases indicate some variability in ULTMAT’s accuracy when interpolated library curves or IACS rules are used. In some cases, agreement is very good, while in others ULTMAT shows a tendency to be conservative. Overall, the IACS rules produce overly conservative strength predictions for half of the models. With the E-E and C-C libraries, errors on the order of 10% are typical for the commercial designs investigated here,


although in one example, the sag strength was underpredicted by about 20%. The NS94 library curves exhibited an accuracy similar to the C-C and E-E libraries in four of six models. Because of the over-simplification of the curves and the resulting extrapolation errors, very large overestimates of hog and sag strength were found in the other two cases.

ULTMAT’s strength predictions can be improved through selective use of directly calculated curves. Because of the large number of nearly identical panels in the deck and bottom structure of the benchmark cases (a typical situation for commercial ship designs), one or two NLFEA curves is often enough to define the LS characteristics of the entire deck and bottom structure. This approach is less applicable in ships where the structural layout is less uniform, or when the horizontal component of bending strength is important.

Overall, the existing LS curve libraries are inadequate for predicting the strength of all commercial structural design types that have been investigated here. The development of an improved curve library is discussed in Section 5.


4 Application to the HALIFAX class

New results obtained using ULTMAT Version 2.2 are presented for the ultimate hull girder strength of the HALIFAX class frigate. Hog and sag strength results are compared with similar results from two previous studies.

4.1 Cross section model

Figure 29 and Figure 30 give two views of the HALIFAX class midships cross section. The longitudinal position of this cross section is 61.75 m from the forward perpendicular, or 250 mm forward of Frame 31. The cross section model was generated using an automatic cross section capability in the Ultimate Strength Application. This was applied to a detailed three-dimensional finite element model of the HALIFAX class structure to extract all the longitudinal structure at the specified cross section position. Longitudinally discontinuous structure (superstructure, tanktops, etc) was then removed from the cross section using windowing and deletion tools in the software.

Figure 29 shows the cross section structure grouped into ULTMAT structural unit types: longitudinally stiffened panel (LSP) units are coloured cyan; transversely stiffened panels are green; and hard corners are grey. Most of the LSPs in the cross section have dimensions and properties similar to Panels A through H of Section 2.1.4. For example, the main deck units similar to Panel C, while Deck 2 units are similar to Panel E, and Deck 3 units are the same as Panel F. With the exception of Panel E, fairly good agreement was obtained between the IACS, interpolated library, and directly calculated curves. Therefore, reasonably good accuracy in the section ultimate strength results can be expected with ULTMAT.

Figure 30 shows the cross section colour coded according to the effective length, or frame spacing. Frame spacing is one metre from the keel up to deck 3, and two metres above that level.

4.2 Ultimate strength results with ULTMAT 2.2

Fully biaxial bending strength calculations can be performed with ULTMAT. Biaxial strength information is most commonly presented in the form of interaction curves, describing the relationship between the maximum vertical and horizontal bending strengths. Ultimate strength interaction curves calculated for the midships section are given in Figure 31. In these results, nominal material properties, plate thicknesses and beam dimensions have been assumed; rolling and corrosion allowances have not been applied to the scantlings. Since the cross section is symmetric about the centreline, only the portion of the curves for positive horizontal bending are shown and symmetry about the vertical axis can be assumed. Average imperfections have been assumed throughout the cross section.


Figure 29: Midships cross section model of the HALIFAX class: structural units view.

Figure 30: Midships cross section model of the HALIFAX class: effective lengths in mm.


The four ultimate strength curves (labelled US) were obtained using ULTMAT 2.2 with the IACS rules, and with the three load shortening curve libraries as indicated. In these plots, ultimate strength values have been normalized with respect to the vertical fully developed plastic bending strength of the cross section. The plastic strength is determined assuming elastic-perfectly-plastic material properties throughout the section, but neglects elasto-plastic collapse effects. The plastic strength therefore represents the maximum possible bending strength that can be achieved with a given material (provided strain hardening is neglected). If the material has the same properties in compression and tension, the vertical plastic strength will be the same for hogging and sagging.

The convention used in Figure 31 is that the positive vertical axis represents sagging bending moments, and the negative vertical axis represents hogging. The curves therefore indicate that the sag strength of the HALIFAX class is approximately half of the hog strength. This is of course accounted for by the half framing below deck 3, which greatly reduces buckling and load-shedding effects and hence increases strength under hog loading.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Mh/Mp

Mv/

Mp

IACS US

E-E US

C-C US

NS94 US

Plastic Limit

Yield Limit

Sag

ging

Hog

ging

Figure 31: Interaction curves for the midships cross section of the HALIFAX class.

The four strength interaction curves in Figure 31 exhibit good agreement with one another. As noted above, the local load-shortening behaviour is generally well-represented by the IACS rules and the library curves. The discrepancies in the load-shortening curves for Panel E suggest that the strength of Deck 2 may be somewhat over-estimated in these results. But the Deck 2 structure


actually has a very small load carrying capacity (about 20-30% of its yield strength), and therefore contributes only modestly to the overall sag strength.

Two other curves are superimposed on the strength curves. The plastic limit envelope defines the biaxial plastic strength limit of the hull girder. Ultimate strength interaction curves cannot exceed the plastic strength envelope. Also given is the yield limit envelope, which defines the combinations of horizontal and vertical bending moment at which initial yield occurs in the section. The yield limit calculation assumes an idealized structure without shape imperfections or residual stresses. Yield is therefore determined by the section geometry and elastic material properties, and by the moment arm to the structural element that is furthest from the neutral axis. Note that the yield envelope is diamond shaped, with sharp corners where it intersects the horizontal and vertical axes. This is the result of the first yield location abruptly shifting for small variations in the bending direction from the vertical and from the horizontal.

The sag yield moment is about 50% larger than the sag ultimate strength for this cross section. The yield moment therefore does not always give a conservative estimate of the ultimate strength, and this example illustrates the dangers of structural designs based on first yield, unless a large safety factor is used. The current Canadian design standard in fact calls for ultimate strength analysis to be performed using a code such as ULTMAT, ULTSTR or FABSTRAN [36].

4.3 Comparisons with previous results

The midships ultimate strength of the HALIFAX class was evaluated in two previous studies [37], [38]. Cross sectional models near Frame 31 were used in both, but the models were not identical to each other or to one shown in Figure 29. The cross section model for the 1995 study was created from drawings using a digitizing tablet and stored in DIGFEM format. For the 2003 study, the model was created by applying an automated cross sectioning tool [39] to a 3D structural FE model of the Halifax class. Some manual editing of this model was performed to improve accuracy and correct errors. Neither of these earlier models included the longitudinal bulkheads between decks 1 and 2, and 2 and 3.

For a fair comparison of ULTMAT 2.2 with the earlier results it was decided to remove the longitudinal bulkheads from the model in Figure 29. Calculations were first carried out with average imperfections and nominal plating thicknesses and beam scantlings. Figure 32 shows the ULTMAT 2.2 results compared with the relevant results from the 2003 study. In this plot, strength results have been normalized with respect to the vertical plastic bending strength of the original Figure 29 section. By comparing ULTMAT 2.2 results in Figure 31 and Figure 32, the removal the longitudinal bulkheads can be determined to reduce the hog and sag strength by roughly 7-10%. The longitudinal bulkheads are therefore significant contributors to both the hog and sag strength, and it is recommended that this structure be included in future strength assessments.

Results were obtained in the 2003 study using ULTMAT 2.1 and the UK code NS94B. Note that the data point marked NS94B/NS94 indicates that the NS94B program was used with the NS94 curve library. In comparison to the 2003 results, ULTMAT 2.2 predicts strength values that are lower by about 5-10%. Several factors contribute to the observed scatter. One is that the 2003 study was carried out for a cross section close to Frame 28, about 6 metres forward of true


midships. Figure 33 shows that the hull bottom girders are somewhat larger at Frame 28 that at Frame 31. ULTMAT 2.2 analysis of these two cross sections showed that hog strength was about 5% larger at Frame 28, while the sag strength was essentially the same, irrespective of analysis method.

A further source of scatter is the grouping algorithms used in the two studies. In the Figure 29 cross section, hard corner units have been sized according to the newly adopted rule of using the minimum of (1) half the width of plating and (2) 20 × plate thickness. This differs from what was done in the 2003 study where half the width of plating was always included in the hard corner. The net result is hard corner units now have on average slightly smaller cross sectional areas, which should result in lower sag strengths, while hog strengths would be largely unaffected. A further difference is that the seven heavy girders in the bottom of the section, which were formerly modelled as oversized LSP units, are now treated as collections of hard corners and TSPs. This is believed to represent more accurately the way this structure behaves in compression, and would likely result in lower hog strengths.

All of the above factors seem to be sufficient to explain the observed scatter between the 2003 and present results.

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.30 0.35 0.40 0.45 0.50

Msag/Mp

Mho

g/M

p

Ultmat 2.1/E-E - 2003

Ultmat 2.1/C-C - 2003

Ultmat 2.1/NS94 - 2003

NS94B/E-E - 2003

NS94B/C-C - 2003

NS94B/NS94 -2003

Ultmat 2.2/IACS - 2008

Ultmat 2.2/E-E - 2008

Ultmat 2.2/C-C - 2008

Ultmat 2.2/NS94 - 2008

Figure 32: Hog and sag strength results for the HALIFAX class midships cross section with nominal scantlings.


Frame 28

Frame 31 (midships)

Figure 33: Hull bottom structure at two cross sections

In both the 2003 and the 1995 studies, the midships ultimate strength was evaluated including the effect of rolling and corrosion. For these analyses, plate thicknesses and beam scantlings were reduced by amounts listed in Table 11. The hog and sag strengths of the previous studies are compared with the ULTMAT 2.2 results in Figure 34, where results are for average imperfections and have been normalized with respect to the same vertical plastic bending moment used in Figure 32. The two figures can therefore be compared directly to assess the effect of nominal versus reduced scantlings; whereby it is seen that hog strength is reduced by about 12% and the sag strength by about 15%.

Table 11: Corrosion and rolling allowances

Location Reduction (mm)

External shell plating below line* 1.3

All other plating 0.8

Stiffeners attached to shell plating below line* 0.8

Other stiffeners 0.5

*Line is defined to be one metre above the deep water line


0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.30 0.35 0.40 0.45 0.50

Msag/Mp

Mho

g/M

p

ULTMAT 1.0 - 1995

NS94B - 1995

Ultmat 2.1/E-E - 2003

Ultmat 2.1/C-C - 2003

Ultmat 2.1/NS94 - 2003

NS94B/E-E - 2003

NS94B/C-C - 2003

NS94B/NS94 -2003

Ultmat 2.2/IACS - 2008

Ultmat 2.2/E-E - 2008

Ultmat 2.2/C-C - 2008

Ultmat 2.2/NS94 - 2008

Figure 34: Hog and sag strength results for the HALIFAX class with corrosion and rolling allowance and average imperfections.

The level of agreement between the ULTMAT 2.2 results and the 2003 results is about the same as for nominal scantlings. However, the 1995 results are significantly different, especially in the sag values. The 1995 study also gave ultimate strength results assuming severe imperfections in the structure. These are compared with ULTMAT 2.2 results in Figure 35, where analyses using IACS rules and the NS94 library have been omitted because severe imperfections cannot be specified for these cases. In comparison to the predictions with the E-E and C-C libraries, the 1995 results are now only slightly higher in sag, while they are now significantly lower in hog.

Two main factors appear to be responsible for the differences between the 1995 results and those of the more recent studies. One stems from that fact that the three cross sections are not geometrically identical. As noted above, the 2003 study used a cross section at Frame 28 instead of at Frame 31. While the 1995 study reported the cross section position to be close to Frame 31, the geometric and elastic properties of that section have been found to be significantly different from the midships cross section used for the present study. Again this seems to stem from differences in the dimensions of the large girders in the hull bottom. Indeed, the sizes of these girders vary widely through the midships engine room area according to local strength and equipment mounting requirements, and this appears to be a significant source of uncertainty in the ultimate strength modelling procedure.

The effect of this can be seen in the results for vertical plastic bending strength. Plastic strength results were reported in the 1995 study, and these are compared with similar results predicted by ULTMAT 2.2 in Table 12. These values have been again normalized to the original vertical


plastic strength of the Figure 29 section assuming nominal scantlings. Thus a normalized plastic strength of 0.809 indicates that removing the bulkheads and applying rolling and corrosion allowances reduces the plastic strength by 19%. Note that the ULTMAT 2.2 plastic moment is within about 6% of the 1995 values. This difference can be attributed entirely to the differences in cross sectional geometries. It seems reasonable then that these geometric differences could cause variations in the hog strength on the order of 5%, and this is the likely cause for the 1995 hog strengths being lower than the more recent results.

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.30 0.32 0.34 0.36 0.38 0.40

Msag/Mp

Mho

g/M

p

ULTMAT 1.0 - 1995

NS94B - 1995

Ultmat 2.2/E-E - 2008

Ultmat 2.2/C-C - 2008

Figure 35: Hog and sag strength results for the HALIFAX class with corrosion and rolling

allowance and severe imperfections.

Table 12: Plastic moment reduction factors for corrosion and rolling allowances

Code Normalized plastic strength

0.763ULTMAT 1.0 - 1995

0.788NS94B - 1995

ULTMAT 2.2 - 2008 0.809

A further difference is that in the methodology of ULTMAT 1.0 assumed that all structure acts at its ultimate load carrying capacity at the instant when the ultimate bending moment is achieved. In reality, some structure will not yet have reached its maximum load, while elsewhere the structure will be undergoing load-shedding. By assuming all structure acts at maximum capacity,


the ultimate bending strength is almost certainly overestimated, and may be greatly overestimated in structure where load-shedding effects are significant. This fact largely accounts for the higher sag strength with ULTMAT 1.0, since load shedding is a significant factor in the collapse of the decks. It is much less important to the hog strength of the HALIFAX class since the hull bottom is typified by low column-slenderness structure that is not susceptible to buckling failure.

Another difference between ULTMAT 1.0 and later versions is that junctions of plating were formerly not considered as hard corners. Instead they were analysed in the same way as stiffened panels, undergoing buckling failure in the direction of the minimum second moment of area. This modelling difference appears to have contributed little to the observed discrepancies in the strength values. As a result of the ULTMAT 1.0 modelling assumptions, the plate junction “stiffened panels” were computed to have relatively high rigidities and therefore would have acted in a manner quite similar to hard corners.

Finally it must be noted that the ULTMAT 2.2 results for severe imperfections are nearly the same as the average-imperfection results. This is a significant concern, since it suggests that the effects of imperfection level may have been improperly accounted for in the C-C and E-E curve libraries. This provides another reason for seeking improvement to ULTMAT’s curve library. The ULTMAT 1.0 results show very significant strength reductions that result from employing a residual stress term in the analytical formula for plate effectiveness.


5 Conclusions and recommendations

The methods presented in this study for evaluating the ultimate strength of ships have been shown to give results comparable to previously published results for a selection of naval and commercial ship designs. Using ULTMAT in conjunction with the IACS common structural rules usually gives conservative results, with an accuracy (i.e., agreement with previous results) that is acceptable for design assessments. When using ULTMAT in conjunction with libraries of pre-calculated load-shortening curves, one would generally expect to get better accuracy than with the IACS rules. In fact, the overall accuracy was not found to be significantly better. The E-E and C-C libraries gave more consistent results than the NS94 library. However, the curve libraries as a whole were found to be inadequate in a number of the commercial hull types investigated, although they generally performed well for the naval hulls.

In most of the example hulls investigated, the scatter in all strength predictions taken together is typically over 20%, and sometimes more than 20% scatter is observed among the ULTMAT results alone. In comparison to other published results, ULTMAT results tend to be skewed towards the low end of the range of strength predictions. Scatter is always larger in sagging than in hogging; this is expected because Euler-column buckling and load-shedding are more prevalent in deck structure than in hull bottom structure.

ULTMAT results obtained with the curve libraries can generally be improved by replacing the interpolated curves with NLFEA-derived curves in selected structural units. It was found that when NLFEA curves were used for large numbers of identical stiffened panel units in the main deck and bottom structure, ULTMAT predictions could be brought to within a few percent of the expected values.

The load-shortening response of transversely stiffened panels (TSPs) has also been examined in some detail. As with longitudinally stiffened panels, three general approaches have been explored: the IACS formula, interpolation from libraries, and direct calculation with NLFEA. The library curves developed by Smith et al [33] and implemented in ULTMAT are found to agree best with NLFEA calculations, even though these curves are limited in the maximum normalized strain represented. On the other hand, the IACS rule for TSPs has been found inadequate, particularly in its prediction of panel ultimate strength. Re-formulation of the rule is recommended such that it better complies with its own theoretical assumptions.

In the benchmark studies presented in this study, TSPs make up only a small proportion of the total number of units in the cross section structure but are often physically larger than the more numerous longitudinal stiffened panels (LSPs). In the example of HALIFAX class frigates, it was shown that TSPs can contribute as much as 7-10% of the total hull girder strength.

The remainder of Section 5 provides a series of recommendations for improving the accuracy of ULTMAT.


5.1 Requirements for an enhanced curve library

Based on comparison of load-shortening curves for individual longitudinal stiffened panels (Section 2) and comparison of hull girder strength results (Sections 3 & 4), the following points can be made regarding the accuracy of ULTMAT:

• The load-shortening curve libraries in ULTMAT were developed for tee-stiffened longitudinally stiffened panels. Although the curves appear to be valid for most tee- and angle-stiffened panels, their validity for flat-bar stiffened panels appears questionable;

• The ranges of dimensionless parameters represented in the existing curve libraries are such that LS curves often have to be extrapolated for commonly occurring stiffened panels in commercial hull designs. Notably absent from the libraries are curves for column slenderness of less than 0.3, and for area ratios of less than 0.2 or greater than 0.35.

• The existing curve libraries were determined primarily by numerical calculation using the relatively small stiffeners found in naval vessels (frigates and destroyers). Although the dimensionless parameters (β, λ, and As/A) largely account for the differences in scale encountered with commercial structures, they do not account for the different proportions in the stiffener dimensions (web height / web thickness, flange width / flange thickness, and web height / flange width). Whether or not this is a significant factor needs to be determined.

• The IACS rules are generally not sufficient to make up for shortfalls in the curve libraries, and sometimes give overly conservative strength estimates.

Improvement of ULTMAT’s curve libraries should therefore take into account the following factors:

• The range of dimensionless parameters represented in the curves should be larger enough such that only minimal amounts of extrapolation are necessary;

• Separate sets of curves should be provided for tee/angle and flat-bar stiffened panels;

• The effect of stiffener size on the resulting load-shortening curves should be investigated;

• Curves should be provided for three imperfections levels (slight, medium and severe), and the definitions of these levels should be reviewed.

The ranges of dimensional parameters in the six benchmark ship designs studied in Section 3 and the HALIFAX midships cross section of Section 4 were calculated and the results summarized in Figure 36 through Figure 38. Based on these distributions, it is recommended that new curve libraries should be developed using, at least initially, combinations of the following parameters:

• β = 1, 2, 3, 4;

• λ = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0;

• As/A = 0.1, 0.2, 0.3, 0.4.


Although these do not completely span the range of values of observed values, they should give sufficient coverage. Note that panels with λ > 1.0 seldom occur since this implies elastic buckling. Full effectiveness of plating is implied by β < 1, and should therefore be sufficient to use the equivalent β = 1 curve. Very little effectiveness remains in plating for β > 4, and for these extreme cases it is sufficient to use the equivalent β = 4 curve.

0

5

10

15

20

25

30

35

40

45

50

0-1 1-1.5 1.5-2 2-3 3-4 4-5 5-6 6-10

β

% o

f uni

ts

Figure 36: Distribution of plate slenderness values.

0

5

10

15

20

25

0-.05 .05-.1 .1-.2 .2-.3 .3-.4 .4-.6 .6-.8 .8-1 1-1.2

λ

% o

f uni

ts

Figure 37: Distribution of column slenderness values.


0

5

10

15

20

25

0-.15 .15-.2 .2-.25 .25-.3 .3-.35 .35-.4 .4-.45 .45-.5 .5-.6 .6-1

A s /A

% o

f uni

ts

Figure 38: Distribution of area ratios.

The variation of load-shortening behaviour with the area ratio parameter has never been adequately treated in the literature. An enhanced curve library developed along the above guidelines should illuminate the relative importance of this parameter, and may therefore be of significant research interest as well as practical importance.

5.2 Automated calculation of an enhanced curve library

The enhanced curve library recommended in the previous section would require a very large number of curves to be calculated by nonlinear finite element analysis using three dimensional shell/plate models of longitudinally stiffened panels. Using the recommended dimensionless parameters, the enhanced database would contain a minimum of 576 curves, assuming separate curve sets for tee/angle and flat bar panels and three levels of imperfections. This number would increase if more than one stiffener size is used.

Generating such a large number of curves is only practical if the model generation, FE analysis and post-processing steps can all be fully automated. The following considerations will have to be addressed to achieve a high level of automation:

• The LSPanel program can currently automate model generation for longitudinal stiffened panels, although the input data for LSPanel, including specification of the element mesh density, has to be manually generated.

• Nonlinear finite element analysis will in the majority of cases run to successful completion using LSPanel-generated models with “default” values for the nonlinear solution parameters. However, numerical failures occur frequently in nonlinear FEA, in which case alternative solution parameters have to be selected and the analysis restarted.


• The post-processing step consists of extracting a non-dimensionalized load-shortening curve from the finite element results file, a step which can be fully automated without difficulty.

What will be needed is a single master curve generation program that controls the model generation, analysis and post-processing steps. The master program will perform the following tasks:

• Generate LSPanel input for all of the curves, including appropriate meshing parameters and default nonlinear solution parameters;

• Perform system calls to LSPanel and the VAST finite element code for each curve;

• Extract and normalize load-shortening curves from the finite element data;

• Perform diagnostic tests to determine if the nonlinear analysis has run to successful completion and that the curve is “correct”;

• Where bad curves are encountered, the program will have to systematically adjust the nonlinear solution parameters and restart the FE analysis until a valid curve results.

The above approach can be applied to curve generation for longitudinally and transversely stiffened panels alike. The requirements for a transversely stiffened panel curve library are much less demanding and do not present any significant obstacles.

It was noted earlier that longitudinally stiffened panels have been modelled using both end-to-end and centre-to-centre models in the existing curve libraries, and the results of this study show that the two approaches give similar results. Smith et al [40] have noted that the centre-to-centre modelling approach simulates multi-span behaviour of the panel in which ultimate collapse can occur either towards the plating or towards the stiffener, while the end-to-end modelling approach produces only single-span models in which the direction of collapse is preferentially determined by the choice of imperfections. Centre-to-centre models therefore can give more realistic collapse behaviour overall and are therefore the best choice for generating a new curve library. However, the experience of the present author is that nonlinear calculations encounter greater numerical difficulties and are more time consuming when centre-to-centre models are used.

5.3 Implementation and validation

Once the enhanced curve library has been developed, it is proposed that it be implemented in a new version of ULTMAT (2.3). The implementation task is expected to be straightforward. Validation can be carried out using the same stiffened panel and hull girder models used in the present study.

Comparison of the enhanced curve library with previously developed curve libraries should also be performed where commonalities exist.


References

[1] Caldwell, J. B. (1965), Ultimate Longitudinal Strength, Trans. RINA. 107, 411–230.

[2] Paik, J. K., and Mansour, A. E. (1995), A Simple Formulation for Predicting the Ultimate Strength of Ships, Journal of Marine Science and Technology, 1, 52–62.

[3] Chen, Y-K., Kutt, L. M., Piaszczyk, C. M., and Bieniek, M. P. (1983), Ultimate Strength of Ship Structures, SNAME Transactions, 91, 149-168.

[4] Ueda, Y. and Rashed, S. M. H. (1984), The Idealized Structural Unit Method and Its Application to Deep Girder Structures, Computers & Structures, 18 (2) 277–293.

[5] Paik, J. K. and Thayamballi, A. K. (2003), Ultimate Limit State Design of Steel-Plated Structures, Chichester, England: John Wiley & Sons.

[6] Smith, C. S. (1977), Influence of Local Compression Failure on Ultimate Longitudinal Strength of a Ships Hull, In Proceedings of the International Symposium on Practical Design in Shipbuilding, 73–79, Tokyo.

[7] Adamchak, J. C. (1984), An Approximate Method for Estimating the Collapse of a Ship’s Hull in Preliminary Design, In Ship Structure Symposium ’84, 37–61. New York: Society of Naval Architects and Marine Engineers.

[8] Smith, M. J, and Pegg. N. G. (2003), Automated Assessment of Ultimate Hull Girder Strength, Journal of Offshore Mechanics and Arctic Engineering, 125, 211-218.

[9] Ohtsubo H. and Sumi. Y., eds. (2000), Ultimate Hull Girder Strength, In Proceedings of the 14th International Ship and Offshore Structures Congress, Vol. 2, 321-391, Kidlington England: Elsevier Science Ltd.

[10] International Association of Classification Societies (2006), Common Structural Rules for Double Hull Oil Tankers, Appendix A: Hull Girder Ultimate Strength. http://www.iacs.org.uk/.

[11] International Association of Classification Societies (2006), Common Structural Rules for Bulk Carriers, Chapter 5: Hull Girder Strength. http://www.iacs.org.uk/.

[12] Smith, C. S. and Dow, R. S. (1986), Ultimate Strength of a Ship’s Hull under Biaxial Bending, (ARE TR 86204), Admiralty Research Establishment, Dunfermline, Scotland.

[13] Dow, R. S. (1997), NS94C/D Ultimate Strength Design Method for Ships, (DRA/SS/SSSV/CR97215), Defence Research Agency, Dunfermline, Scotland.

[14] Pegg, N. G. and Cox, P. (1985), ULTMAT - A Program for Predicting the Ultimate Strength of Ship Cross Sections, (Technical Communication 85/313), Defence Research Establishment Atlantic.


http://www.iacs.org.uk/

http://www.iacs.org.uk/

[15] Hu, T. S. Z. and Heath, D. C. (1995). EBMS – A computer program for equivalent beam modeling of ship structures, (DREA Technical Memorandum 95/208), Defence Research Establishment Atlantic.

[16] Wallace, J. C., Jiang, L., and Chernuka, M. (1997), Generation of a Normalized Effective Stress-Strain Curve Database of Stiffened Panels for the Halifax Class Frigate, (Technical Report TR 97-39), Martec Limited. Halifax, Canada.

[17] Jiang, L., and Chernuka, M. (1998), Continued Development of a Normalized Effective Stress-Strain Curve Database of Stiffened Panels for the Halifax Class Frigate, (Technical Report TR 98-08), Martec Limited. Halifax, Canada.

[18] Smith, M.J. (2001), Progressive Collapse Method for Ultimate Hull Girder Strength Analysis, (DRDC Atlantic TM 2001-145), Defence R&D Canada – Atlantic.

[19] Smith, M. J. (2006), ULTMAT 2.1 User’s Manual, (DRDC Atlantic TM 2006-049), Defence R&D Canada – Atlantic.

[20] Smith, M. J. and Pegg, N. G. (2000), Automated Assessment of Ultimate Longitudinal Strength of Ship Hulls, (Technical Memorandum TM 2000-143), Defence Research Establishment Atlantic.

[21] Grant, R. M. (1991), DIGFEM – Digitizer for Finite Element Modelling of Planar Structures, (DREA Contractor Report 91/433), Defence Research Establishment Atlantic.

[22] Nickerson, J, and Smith M. J. (2007), User’s Manual for the Ultimate Strength Application, (DRDC Atlantic TM 2007-291), Defence R&D Canada – Atlantic.

[23] Lloyd’s Register of Shipping (2005), “Guideline notes on the determination of the Extreme Strength Capability for the NSR ESA or RSA Notations.” Naval Ship Rules consultation document.

[24] Dow, R. S., and Smith, C. S. (1986), FABSTRAN: A Computer Program for Frame and Beam Static and Transient Response Analysis, (ARE TR 86205), Admiralty Research Establishment, Dunfermline, Scotland.

[25] Faulkner, D. (1975), A Review of Effective Plating for Use in the Analysis of Stiffened Plating in Bending and Compression, Journal of Ship Research, 15(1), 1–17.

[26] Rahman, M. K., and Chowdhury, M. (1996), Estimation of Ultimate Longitudinal Bending Moment of Ships and Box Girders, Journal of Ship Research, 40 (3) 244–257.

[27] Hughes, O. F. (1983), Ship Structural Design, New York: John Wiley & Sons, chap. 14.

[28] Guedes Soares, C., and Gordo, J. M., (1997), Design Methods for Stiffened Plating Under Predominantly Uniaxial Compression, Marine Structures, 10, 465-497.


[29] Jiang, L., and Chernuka, M. (2000), Incorporation of UK NS94 Ultimate Strength Analysis in MGDSA, Part I: Engineering Considerations, (DREA CR 2000-112), Martec Limited.

[30] Martec Limited (2006), VAST— Vibration and Strength Analysis Program (VAST), Version 8.8 User’s Manual.

[31] Paik, J. K. (1993), Hull Collapse of an Aging Bulk Carrier Under Combined Longitudinal Bending and Shearing Force, Transactions of RINA, 135, 217-228.

[32] Paik, J. K., Thayamballi, A. K., and Che, J. S. (1996), Ultimate Strength of Ship Hulls Under Combined Vertical Bending, Horizontal Bending, and Shearing Forces, SNAME Transactions, 104, 31-59.

[33] Smith, C. S., Davidson, P. C., Chapman, J. C., Dowling, P. J. (1988), Strength and Stiffness of Ships’ Plating Under In-Plane Compression and Tension, Transactions of RINA, 130, 277-296.

[34] Rutherford, S. E., and Caldwell, J. B. (1990), Ultimate Longitudinal Strength of Ships: a Case Study, SNAME Transactions, 90, 441-471.

[35] Dow, R. S. (1991), Testing and Analysis of a 1/3-Scale Welded Steel Frigate Model, In C. S. Smith and R. S. Dow (Eds.), Advances in Marine Structures—2, pp. 749–773, London: Elsevier.

[36] Dept of National Defence (1993), Standard for the Structural Design of Canadian Forces Surface Ships, D-03-002-008/SG-002.

[37] Hu, T. S. Z. (1995), Ultimate Strength of HALIFAX / IROQUOIS Classes – Phase I, (DREA TM 95/224), Defence Research Establishment Atlantic.

[38] Smith, M. J. (2003), HALIFAX Class Midship Ultimate Hull Girder Strength, (DRDC Atlantic TN 2003-030), Defence R&D Canada – Atlantic.

[39] Lichodzijewski, M. and Jiang, L. (2006), Incorporation of UK NS94 Ultimate Strength Analysis in MGDSA. (DRDC Atlantic CR 2003-113), Defence R&D Canada – Atlantic.

[40] Smith, C. S., Anderson, N., Chapman, J. C., Davidson, P. C., and Dowling, P. J. (1992), Strength of Stiffened Plating under Combined Compression and Lateral Pressure, Transactions of RINA, 134, 131–147.


List of symbols/abbreviations/acronyms/initialisms

5.3.1 Abbreviations/acronyms/initialisms A Angle stiffener ALPS Analysis of Large Plated Structures (structural analysis program) C-C Centre-to-centre CPF Canadian Patrol Frigate CRS Cooperative Research Ships DIGFEM Digitizer for Finite Element Modeling of Planar Structures DND Department of National Defence EBMS Equivalent Beam Modelling of Ships E-E End-to-end FABSTRAN Frame and beam static or transient response analysis nonlinear (UK FE code) FB Flat bar stiffener FE Finite element FEA Finite element analysis GUI Graphical user interface HAZ Heat affected zone HC Hard corner IACS International Association of Classification Societies ISSC International Ship Structures Committee ISUM Independent Structural Unit Method LS Load-shortening LSC Load-shortening curve LSP Longitudinally stiffened panel LSPanel Program for generating stiffened panel models for FEA MoD Ministry of Defence (UK) MV Merchant Vessel NLFEA Nonlinear finite element analysis NS94 Family of UK ultimate strength codes T Tee stiffener TSP Transversely stiffened panel ULTMAT Canadian ultimate strength code


ULTSTR US Navy ultimate strength code

5.3.2 Symbols a Panel length, or frame spacing As/A Area ratio (stiffener area / panel cross-section area) b Plate width d Stiffener depth I Section moment of area K Interpolation factor for aspect ratio M Plastic bending strength p

Mus, M Ultimate strength in sag, hog uh

M Yield strength y

r Radius of gyration t Plate thickness t Web thickness w

t Flange thickness f

w Stiffener width y Neutral axis position G

Aspect ratio of plating α Plate slenderness β Maximum plate imperfection δp

Maximum vertical, horizontal beam imperfection δbv, δbh

Strain ε Parameter used to define width of heat-affect zone η Column slenderness λ Pi π Residual stress σr

Yield stress σy

Yield stress of beam, plate σyb, σyp


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The ultimate strength of several commercial and naval ship designs is studied using DRDC’sULTMAT code. In ULTMAT, the compressive stress-strain response of local structure ispredicted in a simplified manner using one of two methods: interpolation from libraries of pre-calculated load-shortening curves, or using simplified analytical formulas for the nonlinear load-shortening behaviour. The latter formulas are based on structural rules published by theInternational Association of Classification Societies (IACS). Load-shortening curves predictedby ULTMAT are compared with nonlinear finite element analysis (FEA) calculations forseventeen longitudinally stiffened panels and twelve transversely stiffened panels. Hull girderultimate strength results are presented for six benchmark ship cross sections and are comparedwith previously published results. A further set of results for the HALIFAX class midships crosssection strength is presented and is shown to agree well with the results of previous studies. Thecomparisons demonstrate the overall accuracy of ULTMAT’s prediction of structural strength,but reveal some deficiencies in specific areas. Using the IACS rules generally gives results ofaccuracy acceptable for design purposes. Interpolating curves from two of ULTMAT’s existinglibraries generally gives results that are within 10% for hull girder strength. The accuracy can beimproved by substituting load-shortening curves for selected panels with curves calculateddirectly from nonlinear FEA. Recommendations are made for an improved library of load-shortening curves.

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.) Ultimate strength Ship structure Load shortening curves Limit state design Nonlinear finite element analysis Hull girder bending moment Stiffened panel ULTMAT


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Ultimate strength assessment of naval and commercial ships