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Common Structural Rules for Bulk Carriers and Oil Tankers

Draft Technical Background for Rule Change Proposal 1 to 01 JAN 2020 version

Copyright in these Common Structural Rules is owned by each IACS Member as at 1st January 2014. Copyright © IACS 2014. The IACS members, their affiliates and subsidiaries and their respective officers, employees or agents (on behalf of whom this disclaimer is given) are, individually and collectively, referred to in this disclaimer as the "IACS Members". The IACS Members assume no responsibility and shall not be liable whether in contract or in tort (including negligence) or otherwise to any person for any liability, or any direct, indirect or consequential loss, damage or expense caused by or arising from the use and/or availability of the information expressly or impliedly given in this document, howsoever provided, including for any inaccuracy or omission in it. For the avoidance of any doubt, this document and the material contained in it are provided as information only and not as advice to be relied upon by any person. Any dispute concerning the provision of this document or the information contained in it is subject to the exclusive jurisdiction of the English courts and will be governed by English law.

COMMON STRUCTURAL RULES TECHNICAL BACKGROUND FOR RULE CHANGE PROPOSAL 1

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Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 1. Reason for the Rule Change According to GBS audit Findings in Observation No. IACS/2015/FR9-15/OB/02, which is “With regard to the inclusion of, or reference to, the IMO requirements in the Rules as per EC 9.3.6, the concept for rule formulation documented in the TB Document (2.1.2, p.990/1810 of CP2) is not consistently adopted during the formulation of rules. In some cases, the references are made as per the concept mentioned in the TB Document, while in certain other cases, similar references (or inclusion) are not made.”, the Corrective Action Plan is submitted to IMO and the Corrective Action is the following:

1) Review Rule and TB Rule Reference text and identify structural requirement content which is contained in the various IMO instruments.

2) Analyse whether the requirements, which are contained in the various IMO instruments, are necessary to be kept in CSR BC & OT. Some may be kept if found necessary while others may be removed.

3) A Rule Change Proposal and TB update will be carried out based on the conclusion of the analysis.

After identifying structural requirement content, which is contained in the various IMO instruments, it is analysed whether the requirements will be kept in CSR BC & OT, and Rule and/or TB reference amendment are proposed. The Rules and its TB will be amended according to following principles:

a) The requirements of the IMO Instruments are considered as a part of the Classification requirements when they are re-stated in the Rules or when the Rules explicitly require their applications even when using a reference. IMO references are provided either in Rules or in TB or in both.

b) The references to IMO Instruments are provided to draw attention of designer and Rules user. References to those IMO Instruments are given for information and are not part of the Classification requirements. The information status of the reference is given.

c) Where IMO Instruments address adequately the requirements from IACS viewpoint, the corresponding requirements are removed from the Rules and TB.

d) When it is considered that only part of an IMO requirement is relevant to CSR, the relevant portion of the text should be copied into the CSR and, where necessary, a restriction of its application scope should be provided; in any case the IMO requirement meaning should not be modified due to this partial extract.

e) CSR should not contain interpretations of IMO requirements and where - in developing a CSR Rule change proposal - it is considered that an interpretation is necessary, a UI against such IMO requirement should be developed to complement the CSR requirement.

2. Background These requirements for openings in watertight bulkheads are the same as some provisions in SOLAS, Ch II-1, Reg 13-1.1 (as amended). However, the provisions, which are to be complied with in SOLAS, are not introduced in CSR BC & OT completely. Therefore, the requirements are not complete duplications compared with SOLAS. It is decided that this requirement is to be removed. The rule text and relevant TB Reference are removed.


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3. Impact in Scantlings There is no impact in scantling

Pt 1 Ch 2 Sec 3 [4.1.1] 1. Reason for the Rule Change

See reason for Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 2. Background This requirement is based on Ch II-2, Reg. 4.2.2.3.1 of the SOLAS Convention (as amended). However, it is described more clearly in CSR BC & OT, that the fore peak and other compartments located forward of the collision bulkhead may not be arranged for the carriage of fuel oil or other flammable products. There is no need to keep this requirement as it is fully covered by SOLAS. The rule text and corresponding TB Reference are removed. 3. Impact in Scantlings There is no impact in scantling


See reason for Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 2. Background This requirement about fuel oil tanks references SOLAS Ch II-2, Reg. 4.2 and MARPOL, Annex I, Ch 3, Reg 12A. There is no need to keep this requirement as it is fully covered by SOLAS and MARPOL. The rule text and corresponding TB Reference are removed. 3. Impact in Scantlings There is no impact in scantling


See reason for Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 2. Background This requirement about stern tube references SOLAS Ch II-1 Reg.12.10. There is no need to keep this requirement as it is fully covered by SOLAS. The rule text and corresponding TB Reference are removed. 3. Impact in Scantlings There is no impact in scantling


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Pt 1, Ch 4, Sec 5, Symbols 1. Reason for the Rule Change The proposal is to correct the definition of fyB at Fore End (FE) to be used in calculation of external pressure. 2. Background The current pressure calculation at Fore End (FE) gives abnormal pressure jump due to “fyB = 0 and BX = 0”. This is not intended and correct. Therefore, fyB is to be “1.0” when BX = 0 at fore end to calculate correct external pressure. The pressure comparison between CSR BC & OT Jan 2020 and RCP is carried out and shown in Table 1 and Figure 1 below.

Table 1: Pressure comparison between CSR BC & OT Jan 2020 and RCP

CSR BC&OT Jan. 2020 Proposal

Water Line Case

x from AE FE - 5350mm FE - 250mm FE FE - 5350mm FE - 250mm FE

y (m) 1 0.1 0 1 0.1 0

z (m) Water Line Water Line Water Line Water Line Water Line Water Line

Bx (m) 2 0.2 0 2 0.2 0 Pex (kN/m2) - HSM1 155.517 157.514 218.748 155.517 157.514 157.603

fyB fyB ≠ 0 fyB ≠ 0 fyB = 0 fyB ≠ 0 fyB ≠ 0 fyB = 1

Below Water Line Case


y (m) 2 2 2 2 2 2

z (m) 3 3 3 3 3 3

Bx (m) 2 0.2 0 2 0.2 0 Pex (kN/m2) - HSM1 301.797 303.219 315.338 301.797 303.219 303.282

fyB fyB ≠ 0 fyB ≠ 0 fyB = 0 fyB ≠ 0 fyB ≠ 0 fyB = 1

Above Water Line Case


y (m) 5 5 5 5 5 5

z (m) 27 27 27 27 27 27

Bx (m) 2 0.2 0 2 0.2 0 Pex (kN/m2) - HSM1 105.241 107.238 168.472 105.241 107.238 107.327

fyB fyB ≠ 0 fyB ≠ 0 fyB = 0 fyB ≠ 0 fyB ≠ 0 fyB = 1 Bx= Moulded breadth at the water line. fyB=Ratio between Y-coordinate of the load point and Bx.

FE=Fore end of the rule length L is the perpendicular to the scantling draught waterline at the forward side of the stem.


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Figure 1: Pressure distribution between CSR BC & OT Jan 2020 and RCP

Position 1: FE – 5350 mm Position 2: FE – 250 mm Position 3: Fore End 3. Impact in Scantlings RCP is a correction of wrong pressure distribution so abnormal scantling requirement in way of fore end will disappear.


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Pt 1, Ch 4, Sec 6, [2.5.1] 1. Reason for the Rule Change This proposal is made to clarify that shear load due to dry bulk cargo should be considered not only for FE strength assessment but also for FE fatigue assessment. 2. Background In the current Rules, it is not clear how to consider the shear load due to dry bulk cargo when FE fatigue assessment is carried out. According to the Technical Background and technical point of view, it is considered rational to apply shear load due to dry bulk cargo for not only FE structural assessment but also FE fatigue assessment. Therefore, it is proposed to add “FE fatigue assessment” to clarify the application of this requirement. 3. Impact in Scantlings There is small impact (increase or decrease) on fatigue life and scantlings due to these changes depending on size of ship, cargo hold type, connection type as well as design as shown below. Please note that some of test ships (Ship-A, Ship-B and Ship-C) are designed in accordance with pre-CSR Rules so fatigue life can be lower than 25 years.

Table 1: Test ships and details for lower hopper knuckle connections

Ship ID Size Cargo Hold Type

Bilge hopper – inner bottom Connection Type

Ship-A Handy max Loaded Radiused Ship-B Handy max Empty Radiused Ship-C Panamax Loaded Welded Ship-D Panamax Empty Welded Ship-E Cape Empty Welded

Ship-F Cape Empty Radiused

Figure 1: Hot spots for lower hopper knuckle


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Figure 2: Fatigue life comparison for lower hopper knuckle connections

Table 2: Test ships for lower stool connections

Ship ID Size Cargo Hold Type

Ship-A Handy max Loaded

Ship-B Handy max Empty

Ship-C Panamax Loaded

Ship-D Panamax Empty

Ship-E Cape Empty

Ship-F Cape Empty

Figure 3: Hot spots for lower stool


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Figure 4: Fatigue life comparison for lower stool connections

Pt 1, Ch 4, Sec 8, [5.1.1] & [5.2.1] 1. Reason for the Rule Change For simplified fatigue stress analysis, the Rule change is to clarify loading height for tanks for fuel oil/other oil/fresh water when these are located in the cargo hold region. 2. Background Current rule requires half filling of the tanks in loading height, but the definitions of centre of gravity and reference point are not clarified, hence, different implementation of each class’s software has been identified. In addition, this is not in line with FE fatigue procedure (full-filling). For consistent approach to direct FE fatigue procedure, the current rule for the simplified fatigue stress analysis is amended to full-filling of the tanks. And it is also considered to be more conservative and rational. 3. Impact in Scantlings In order to analyse the proposal, a Capesize bulk carrier is taken as an example and comparison study was carried out. Fatigue life of longitudinal stiffeners in way of the bottom plate of top side wing tank was found to be more than 25 years. As can be seen in Figure 1 below, the approach with 50% filling of fuel oil tank is found non conservative since the stress from dynamic internal pressure is not considered for stiffeners located above 50% filling level of tank. It means only hull girder stress is considered for between A (Top of tank) and B (0.5 x height of tank) as shown in Figure 1 below. The reference point is assumed at the half height of the tank for dynamic internal pressure for fatigue strength calculations of stiffeners for lower part of the tank (below 50% filling, between B and C). RCP with full-filling can be considered conservative but more realistic for lower part of tank (Between B and C) comparing with the possible actual conditions with 100% filling at departure and 0% to 20% filling at arrival condition (See orange curve in Figure 1 below).


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Figure 1: Fatigue life for longitudinal stiffeners in way of the bottom plate of top side wing tank


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Pt 1, Ch 8, Sec4, [5.1.1] 1. Reason for the Rule Change The Rule Change is to clarify the application of buckling requirements for direct strength analysis.

2. Background Local plate buckling for cross ties is missing in Ch 8, Sec 4, [5.1.1] since cross ties are modelled using shell element and subject to local plate buckling with method UP-B in accordance with table 1 and Figure 2 in Sec.4. Only overall column buckling is required in the current rule text. 3. Impact in Scantlings There is no impact on scantlings since Rule Change is to clarify the inconsistency in the current rule text and table/figure. Pt 1, Ch 8, Sec5, Global Elastic Buckling 1. Reason for the Rule Change For some ship stiffeners, such as side frames and inner bottom stiffened panels (mainly subject to transverse loads) of single side bulk carriers, it’s found that their calculated buckling capacities according to the current CSR rules might decrease when either increasing plate thickness or reducing the stiffener spacing.

One example, for the side shell plating between hopper and top side tank of a 208k DWT BC, based on the rule requirements in Pt1/Ch8/Sec5/2.3.4 of CSR, stiffener induced buckling (SI) is indicated as their critical mode of failure. With increasing plate thickness from 21.5mm to 26.0mm, as shown in the table below, the buckling utilization factor increases from 1.035 to 1.193, which is physically unreasonable. Similar cases are occasionally found both for some Capesize and Handymax bulk carriers and for some non-CSR ship types assessed using the current CSR buckling formulae.

Preliminary analysis of IACS CSR Maintenance Team indicates that the reason behind the above phenomenon might be caused by the inaccuracy of the current beam-based formulae in CSR for the estimation of the global buckling capacity of stiffened panels under combined stresses. Therefore, related formulae are further investigated based on orthotropic plate theory together with both eigenvalue FE analysis and nonlinear FE analysis, and as a result, the rule changes based on orthotropic plate theory are proposed accordingly.

2. Background

2.1 Introduction In the rule change proposal, the stiffener buckling formulae are updated, and orthotropic plate theory is used to compute the global elastic buckling (GEB) load in order to account for second order effect in the bending moment M0, representing the moment due to lateral displacement of the stiffener:


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𝑀𝑀0 = �𝜋𝜋𝐿𝐿�2𝐸𝐸𝐸𝐸 𝛾𝛾

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺−𝛾𝛾𝑤𝑤0 with 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 − 𝛾𝛾 > 0

where:

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 : Eigenvalue computed as defined in [2.2].

L : Stiffener full length of span.

𝛾𝛾 : Load proportional factor or stress multiplier factor acting on loads.

𝑤𝑤0 : Lateral deflection of the stiffener as further explained in [2.5].

This replaces the current CSR formula:

𝑀𝑀0𝐶𝐶𝐶𝐶𝐶𝐶 = �

𝜋𝜋𝐿𝐿�2𝐸𝐸𝐸𝐸

𝑃𝑃𝑧𝑧𝑐𝑐𝑓𝑓 − 𝑃𝑃𝑧𝑧

𝑤𝑤

The two expressions above are similar and both are using a displacement magnifier to account for the moment due to lateral displacement as illustrated in Figure 1.

Figure 1: The displacement magnifier using beam theory to be replaced by orthotropic plate theory

(Note: 𝐹𝐹𝑥𝑥,𝐹𝐹𝐺𝐺𝑥𝑥 in the formula refer to axial force and elastic buckling capacity of the beam, respectively)

The expression for the moment M0 in the rule change proposal is obtained by assuming out-of-plane sinusoidal displacement due to buckling

𝑤𝑤𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑤𝑤 sin �𝜋𝜋𝐿𝐿

𝑥𝑥� =𝛾𝛾

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 − 𝛾𝛾 𝑤𝑤0 sin �

𝜋𝜋𝐿𝐿

𝑥𝑥�

The moment at mid-span (x = L/2) is found by substituting the eigenvalue 𝛬𝛬𝐺𝐺 and 𝑤𝑤𝑜𝑜𝑜𝑜𝑜𝑜 into M0:

𝑀𝑀0 = −𝐸𝐸𝐸𝐸 𝑑𝑑2𝑤𝑤𝑜𝑜𝑜𝑜𝑜𝑜𝑑𝑑𝑥𝑥2

�𝑥𝑥=𝐿𝐿2

= �𝜋𝜋𝐿𝐿�2𝐸𝐸𝐸𝐸

𝛾𝛾𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 − 𝛾𝛾

𝑤𝑤0

This is similar to the existing formula for M0 in current CSR

𝑀𝑀0𝐶𝐶𝐶𝐶𝐶𝐶 = �

𝜋𝜋𝐿𝐿�2𝐸𝐸𝐸𝐸

𝑃𝑃𝑧𝑧𝑐𝑐𝑓𝑓 − 𝑃𝑃𝑧𝑧

𝑤𝑤

By comparing the two expressions for M0, it can be seen that 𝑐𝑐𝑓𝑓 (elastic support provided by the stiffener) is replaced by 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 and Pz (nominal lateral load acting on the stiffener due to in-plane stresses) is replaced by 𝛾𝛾. The penalty form expression of M0 is to effectively prevent


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overall global elastic buckling (𝑐𝑐𝑓𝑓 − 𝑃𝑃𝑧𝑧 > 0) which is a sound design principle since it may be the most critical limit to govern buckling.

In the formulae for global elastic buckling below, the stress 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 is the average stress for both plate and stiffener, and a Poisson correction should be applied in order to express this in terms of plate stresses 𝜎𝜎𝑥𝑥 and 𝜎𝜎𝑦𝑦:

𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 = 𝜎𝜎𝑥𝑥 − 𝜈𝜈𝜎𝜎𝑦𝑦 𝐴𝐴𝑠𝑠/(𝑜𝑜𝑝𝑝𝑠𝑠 + 𝐴𝐴𝑠𝑠) ≥ 0

where 𝐴𝐴𝑠𝑠 = 𝑜𝑜𝑤𝑤ℎ𝑤𝑤 + 𝑜𝑜𝑓𝑓𝑏𝑏𝑓𝑓 is the stiffener sectional area. In the stiffener capacity formula, the Poisson correction is:

𝝈𝝈𝒙𝒙𝒙𝒙𝒙𝒙𝒙𝒙 = 𝝈𝝈𝒙𝒙 − 𝝂𝝂 𝝈𝝈𝒚𝒚 ≥ 𝟎𝟎

2.2 Global elastic buckling capacity Based on the orthotropic plate theory, the global eigenvalue 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 with corresponding load cases are computed as follows:

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 = 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏+𝜏𝜏 for τ ≠ 0 and (𝜎𝜎𝑥𝑥 > 0 or 𝜎𝜎𝑦𝑦 > 0)

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 = 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 for τ = 0 and (𝜎𝜎𝑥𝑥 > 0 or 𝜎𝜎𝑦𝑦 > 0)

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺 = 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,τ for τ ≠ 0 and (𝜎𝜎𝑥𝑥 ≤ 0 and 𝜎𝜎𝑦𝑦 ≤ 0)

The eigenvalue 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 for an orthotropic plate subjected to biaxial loads is computed as[2,3]:

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 =𝜋𝜋2

𝐿𝐿𝐺𝐺12 𝐿𝐿𝐺𝐺22[𝑚𝑚4𝐷𝐷11𝐿𝐿𝐺𝐺24 + 2(𝐷𝐷12 + 𝐷𝐷33)𝑚𝑚2𝑛𝑛2𝐿𝐿𝐺𝐺12 𝐿𝐿𝐺𝐺22 + 𝑛𝑛4𝐷𝐷22𝐿𝐿𝐺𝐺14 ]

𝑚𝑚2𝐿𝐿𝐺𝐺22 𝑁𝑁𝑥𝑥 + 𝑛𝑛2𝐿𝐿𝐺𝐺12 𝐾𝐾𝑜𝑜𝑡𝑡𝑎𝑎𝑡𝑡𝑁𝑁𝑦𝑦

where 𝑁𝑁𝑦𝑦 = 𝑐𝑐𝜎𝜎𝑦𝑦𝑜𝑜𝑝𝑝 , 𝑁𝑁𝑥𝑥 = 𝜎𝜎𝑥𝑥.𝑎𝑎𝑎𝑎(𝑜𝑜𝑝𝑝𝑠𝑠 + 𝑜𝑜𝑤𝑤ℎ𝑤𝑤 + 𝑜𝑜𝑓𝑓𝑏𝑏𝑓𝑓)/𝑠𝑠 and c is the factor taking into account the edge stress distribution in the attached plating acting perpendicular to the stiffener’s axis taken as c = 0.5(1 + 𝜓𝜓) for 0 ≤ 𝜓𝜓 ≤ 1

c = 12(1−𝜓𝜓) for 𝜓𝜓 < 0 where 𝜓𝜓 is the edge stress ratio.

More details on the factor 𝐾𝐾𝑜𝑜𝑡𝑡𝑎𝑎𝑡𝑡 are presented in [2.3]. The eigenvalue 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 has to be minimized with respect to the wave parameters m and n. This means for given values of a load pair (𝑁𝑁𝑥𝑥 , 𝑁𝑁𝑦𝑦 ), which define a load path line, the buckling points for different combinations of (m, n) need to be tested, from which to identify the combination (m, n) corresponding to the lowest positive buckling point that minimizes the eigenvalue. This is illustrated in Figure 2, where the buckling boundaries (defined by 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 = 1) of three combinations of (m1, n1), (m2, n2) and (m3, n3) are shown, and the load path line is defined by the given actual load pair (𝑁𝑁𝑥𝑥,𝑎𝑎𝑎𝑎𝑜𝑜, 𝑁𝑁𝑦𝑦,𝑎𝑎𝑎𝑎𝑜𝑜). Combination (m2, n2) in the figure gives the lowest buckling point (𝑁𝑁𝑥𝑥𝐺𝐺, 𝑁𝑁𝑦𝑦𝐺𝐺) and thus is the one minimizing the eigenvalue 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏.


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Figure 2: Elastic buckling loads/eigenvalues under combined bi-axial loads, schematically

The eigenvalue 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝜏𝜏 for an orthotropic plate subjected to pure shear load (ref. Bergmann[1]) can be computed as:

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝜏𝜏 =�𝐷𝐷113 𝐷𝐷224

(𝐿𝐿𝐺𝐺1 2)⁄ 2 𝑁𝑁𝑥𝑥𝑦𝑦�8.125 + 5.64�

(𝐷𝐷12 + 𝐷𝐷33)2

𝐷𝐷11𝐷𝐷22− 0.6

(𝐷𝐷12 + 𝐷𝐷33)2

𝐷𝐷11𝐷𝐷22� for 𝐷𝐷11𝐷𝐷22 ≥ (𝐷𝐷12 + 𝐷𝐷33)2

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝜏𝜏 =�2𝐷𝐷11(𝐷𝐷12 + 𝐷𝐷33)

(𝐿𝐿𝐺𝐺1 2)⁄ 2 𝑁𝑁𝑥𝑥𝑦𝑦�8.3 + 1.525

𝐷𝐷11𝐷𝐷22(𝐷𝐷12 + 𝐷𝐷33)2 − 0.493

𝐷𝐷112 𝐷𝐷222

(𝐷𝐷12 + 𝐷𝐷33)4� for 𝐷𝐷11𝐷𝐷22 < (𝐷𝐷12 + 𝐷𝐷33)2

where 𝑁𝑁𝑥𝑥𝑦𝑦 = 𝜏𝜏 × 𝑜𝑜𝑝𝑝. The interaction between bi-axial and shear stresses is accounted for by assuming a parabolic interaction:

𝜸𝜸𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝒃𝒃𝒃𝒃

+ �𝜸𝜸

𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝝉𝝉�𝟐𝟐

= 𝟏𝟏

By solving the quadratic formula for 𝛾𝛾, the interaction between bi-axial and shear stress is accounted for by the following equation:

𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝒃𝒃𝒃𝒃+𝝉𝝉 = 𝜸𝜸 =𝟏𝟏𝟐𝟐𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝝉𝝉𝟐𝟐 �−

𝟏𝟏𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝒃𝒃𝒃𝒃

+ �𝟏𝟏

𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝒃𝒃𝒃𝒃𝟐𝟐 + 𝟒𝟒

𝟏𝟏𝜸𝜸𝑮𝑮𝑮𝑮𝑮𝑮,𝝉𝝉𝟐𝟐 �

This is a parabolic interaction between shear and bi-axial loads which is a common assumption in the literature, rules, etc. This assumption has also been validated both using finite element analysis (FEM) and with Rayleigh-Ritz (RR) method as shown in Figure 3.


PAGE 14 OF 65

Figure 3: Interaction between shear and bi-axial loads

2.3 Introduction of Ktran For globally slender plates (stiffened panels with small size or very long stiffeners) subject to transverse loads, the use of the displacement magnifier with linear theory in the computation of the moment M0 is very conservative. This is schematically illustrated in Figure 4, where the load level for a given displacement is smaller for linear theory compared to nonlinear theory.

For nonlinear theory it is possible to trace the response above elastic buckling, but as a design principle the maximum limit should be global elastic buckling. In order to compensate for the effect that linear theory is conservative for globally slender plates with transverse loads (normal to the stiffeners) a factor Ktran is introduced in the formula for computing eigenvalue 𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 for biaxial loads

𝛾𝛾𝐺𝐺𝐺𝐺𝐺𝐺,𝑏𝑏𝑏𝑏 =𝜋𝜋2

𝐿𝐿𝐺𝐺12 𝐿𝐿𝐺𝐺22[𝑚𝑚4𝐷𝐷11𝐿𝐿𝐺𝐺24 + 2(𝐷𝐷12 + 𝐷𝐷33)𝑚𝑚2𝑛𝑛2𝐿𝐿𝐺𝐺12 𝐿𝐿𝐺𝐺22 + 𝑛𝑛4𝐷𝐷22𝐿𝐿𝐺𝐺14 ]

𝑚𝑚2𝐿𝐿𝐺𝐺22 𝑁𝑁𝑥𝑥 + 𝑛𝑛2𝐿𝐿𝐺𝐺12 𝐾𝐾𝑜𝑜𝑡𝑡𝑎𝑎𝑡𝑡𝑁𝑁𝑦𝑦

where 𝐾𝐾𝑜𝑜𝑡𝑡𝑎𝑎𝑡𝑡 = 0.9. This factor is not introduced for axial loads in stiffeners direction since for this case (column buckling) there is no reserve strength beyond elastic buckling, and stresses cannot be redistributed. For stocky plates, yielding will occur before global elastic buckling as illustrated in Figure 5 and for such cases the influence of the Ktran factor is small.


PAGE 15 OF 65

Figure 4: Illustration of response with linear and non-linear theory for a slender plate

Figure 5: Illustration of response with linear and non-linear theory for a stocky plate

2.4 Poisson correction The stresses 𝜎𝜎𝑥𝑥 and 𝜎𝜎𝑦𝑦 are plate stresses, while the input for global elastic buckling is the average stress 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 of plate and stiffener which is the effective load carrying stress. This average stress is obtained from plate and stiffener stresses, which gives an expression that is dependent on both the Poisson ratio and the relative difference between the plate and stiffener dimensions:

𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 = 𝜎𝜎𝑥𝑥 − 𝜈𝜈𝜎𝜎𝑦𝑦 𝐴𝐴𝑠𝑠/(𝑠𝑠 𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠) ≥ 0 where 𝐴𝐴𝑠𝑠 = 𝑜𝑜𝑤𝑤ℎ𝑤𝑤 + 𝑜𝑜𝑓𝑓𝑏𝑏𝑓𝑓 is the stiffener cross sectional area. In order to check the formula for average stress 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 , three plates with flat bar stiffeners are analysed in FE with the dimensions given in Table 1. Each FE model consists of one flat stiffener and a plate with total width s = 800mm (i.e. s/2 on each side of the stiffener).


PAGE 16 OF 65

Table 1: Dimensions of stiffened plates

L [mm]

s [mm]

𝑜𝑜𝑝𝑝 [mm]

ℎ𝑤𝑤 [mm]

𝑜𝑜𝑤𝑤 [mm]

Panel 1 2000 800 20 200 20 Panel 2 2000 800 20 200 100 Panel 3 2000 800 20 200 40

The applied average stress in x- and y-direction is equal to 100MPa as given in Table 2. In the FE model, the edges are forced to remain straight and the external loads are applied as forces F𝑥𝑥 and F𝑦𝑦 as shown in Figure 6 - Figure 8.

F𝑥𝑥 = σ𝑥𝑥,𝑎𝑎𝑎𝑎 �𝑠𝑠 𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠�

F𝑦𝑦 = σ𝑦𝑦 𝐿𝐿 𝑜𝑜𝑝𝑝

Since the edges are forced to remain straight, exactly the same results can be obtained using displacement control, and this condition is representative for real ship structures and cargo hold models. In the FE model, shell elements are used also for the stiffener, since then it is easy to visualise the stresses graphically. Only the axial stiffness (no bending) are activated since the edges forced to remain straight (i.e. all nodes in stiffener and plate are moving with the same x-displacement). The same results will be obtained if beam (or rod) elements are used since these elements will have the same axial stiffness as the stiffener modelled with shell elements. The computed internal stresses in FE are presented in Table 2 and in Figure 6 - Figure 8. These internal stresses are used as input in the formula for σx,av in order to demonstrate that the same results are achieved for the applied stresses equal to 100 MPa in FEM.

Table 2: Applied stresses and corresponding computed internal stresses in FEM

FEM Applied stress 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎

[MPa]

FEM Applied stress 𝜎𝜎𝑦𝑦

[MPa]

FEM Internal plate stress 𝜎𝜎𝑥𝑥 [MPa]

FEM Internal plate stress 𝜎𝜎𝑦𝑦 [MPa]

FEM Internal stiffener

stress [MPa] Panel 1 100 100 106 100 76 Panel 2 100 100 116.67 100 86.67 Panel 3 100 100 110 100 80

The computed average stresses are compared with FEM results in Table 3 and the same results are obtained by using the formula for 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎.


PAGE 17 OF 65

Table 3: Computed average stress according to formula and comparison with FEM

Formula for 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 computed with internal

stress 𝜎𝜎𝑥𝑥 and 𝜎𝜎𝑦𝑦 from FEM

FEM Applied stress 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 [MPa]

Difference between applied stresses in FEM

and formula for 𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎 Panel 1 100 100 0 Panel 2 100 100 0 Panel 3 100 100 0

The Poisson correction accounts for the effect that the plate stresses are used in the GEB computation, and the stresses need to be converted to applied average stress (𝜎𝜎𝑥𝑥,𝑎𝑎𝑎𝑎) for both stiffener and plate since this is the load carrying stress. The total force acting in x-direction is

F𝑥𝑥 = σ𝑥𝑥,𝑎𝑎𝑎𝑎 �𝑠𝑠 𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠� = [𝜎𝜎𝑥𝑥 − 𝜈𝜈𝜎𝜎𝑦𝑦 𝐴𝐴𝑠𝑠/(𝑠𝑠 𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠)] × (𝑠𝑠 𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠) = 𝜎𝜎𝑥𝑥𝑠𝑠 𝑜𝑜𝑝𝑝 + (𝜎𝜎𝑥𝑥 − 𝜈𝜈𝜎𝜎𝑦𝑦) 𝐴𝐴𝑠𝑠 = 𝜎𝜎𝑥𝑥𝑠𝑠 𝑜𝑜𝑝𝑝 + 𝜎𝜎𝑥𝑥,stiff𝐴𝐴𝑠𝑠 = F𝑥𝑥,plate + F𝑥𝑥,stiff

where Fx,plate is the force in the plate, Fx,stiff is the force in the stiffener, and 𝜎𝜎𝑥𝑥,stiff = 𝜎𝜎𝑥𝑥 − 𝜈𝜈𝜎𝜎𝑦𝑦 is the stiffener stress (equal to σxcor in the rule text). The formula for the stiffener stress is also equal to the FEM results as presented in Table 4

Table 4: Computed stiffener stress according to formula and comparison with FEM

𝜎𝜎𝑥𝑥,stiff = 𝜎𝜎𝑥𝑥 − 𝜈𝜈𝜎𝜎𝑦𝑦 computed with internal

stress 𝜎𝜎𝑥𝑥 and 𝜎𝜎𝑦𝑦 from FEM

FEM Internal stiffener stress [MPa]

Difference between applied stresses in FEM

and formula Panel 1 76 76 0 Panel 2 86.7 86.7 0 Panel 3 80 80 0

Figure 6: Internal plate stresses in FE for Panel 1


PAGE 18 OF 65



2.5 Effect of lateral pressure on second order effects The global elastic buckling is accounted for using plate theory in combination with a displacement amplifier and an initial imperfection. Numerical calculations have been carried out for plates with lateral pressure, which shows that the additional imperfection due to lateral pressure (w1) can give very conservative results. This means that only the geometrical imperfection (w0) should be included in the displacement amplifier. This can also be explained physically, since the global mode for a stiffened plate is an up-and-down local wave buckling pattern as shown in Figure 9. For such cases, the lateral pressure will act both against and with the displacements, giving both negative and positive contribution, and then the overall effect will be small. For a single beam it is physically correct to include an additional imperfection due to lateral pressure since the displacement will be in the same direction as the pressure, but for a stiffened plate it is very conservative. However, the effect of the lateral pressure will still be accounted for in the bending moment M1 in the stiffener capacity formula. The effect of the imperfection w0 and w1 on the computed ultimate capacity for both CSR and the proposed formula has been studied by varying the lateral pressure and results are presented in Figure 10.


PAGE 19 OF 65

It shows that the imperfection w1 has a destabilizing effect for pressure on stiffeners side (negative pressure) and a stabilizing effect for pressure on plate side (positive pressure). However, this effect is very large compared to FEA and formulae, which indicates that only w0 should be used in the formula.

Figure 9: Global buckling mode for a stiffened plate

Figure 10: Effect of varying the lateral pressure

2.6 Boundary conditions for vertically stiffened side shell of single side skin bulk carrier For continuous stiffened plates, the contribution due to lateral pressure (M1) is computed for a clamped plate, while the contribution due to in-plane load (M0) is calculated by assuming harmonic up-and-down buckling pattern (i.e. simply supported stiffener; Euler buckling) as illustrated in Figure 11.


PAGE 20 OF 65

Figure 11: Deformation from in-plane loads and pressure for a regular stiffened plate

The simply supported condition is conservative, and it is a reasonable assumption for stiffened plates between transverse frames which also normally has a regular spacing. However, for vertically stiffened side shell of single side skin bulk carrier, the boundary conditions at the stiffener ends are actually in between clamped and simply supported. This is investigated by linear elastic buckling analysis (i.e. eigenvalue analysis) for three different FE models:

• Model A-C: Side frame plate with clamp condition at stiffener ends. • Model A-S: Side frame plate with simply supported stiffener ends. • Model B: Cargo hold model

The dimensions and boundary conditions for each FE model are presented in Figure 12. Each model is loaded with axial loads with a magnitude to apply the same reference stress in the longitudinal direction. For model A-C and A-S, the edges are free to move in the in-plane directions, but forced to remain straight. For model B, the transverse section is fixed at one end and forced to remain straight and plane at the loaded edge. Results from the linear elastic buckling analysis are presented in Figure 13, and the computed linear elastic buckling load for the cargo hold is equal to 207 MPa, which is in between the results for the simply supported (144 MPa) and the clamped case (244 MPa).


PAGE 21 OF 65

Figure 12: Dimensions and boundary conditions for three different FE models

Figure 13: Results from linear elastic buckling analysis

In the proposed formula, a reduced buckling length is used in order to account for that the boundary condition for a vertically stiffened side shell of single side skin bulk carrier is in between simply supported and clamped. The actual reduction is found by investigating the side shell plating with various lengths, and by comparing the results for the case with full length for both simply supported and clamped stiffener ends. Since it is difficult to find a well-defined global eigenmode (especially for shorter stiffeners), the FE model is idealised


PAGE 22 OF 65

using orthotropic shell elements with the same stiffness coefficient (D11, D12, etc.) as in the proposal. This idealisation is also verified by comparisons with plate models where stiffeners are modelled with shell elements with a fine mesh. The linear elastic buckling results for plates with various effective lengths are shown in Figure 14, and effective length is equal to total stiffener length times a reduction factor LB1,eff. In the figure, the dashed green line is the eigenvalue for the simply supported plate with full length (i.e. LB1,eff = 1), and the dashed red is the same for the clamped condition. To obtain the same relative difference in the buckling load as for the cargo hold model, the effective length reduction factor is equal to 0.75. In order to be conservative, it is proposed to use an effective length reduction factor:

LB1,eff = 0.8 (conservative estimation)

This effective length reduction factor 0.8 is the appropriately determined numerical value between simply supported (LB1,eff=1) and clamped (LB1,eff approximately equals to 0.7). This reduction value is only applicable to vertically stiffened side shell of single side skin bulk carrier, and for these frames there are specific prescriptive requirements for end connections with brackets, etc. (ref. CSR Pt 2, Ch 1 Sec 3). The reduction factor LB1,eff should only be used in the calculation of the global elastic buckling load and M0. In the calculation for the bending moment M1 at the mid-span, the expression for continuous stiffeners is already for a clamped stiffener and stiffener length should not be reduced:

M1 = Ci |𝑝𝑝| 𝑠𝑠 𝐿𝐿2

24 × 103

where L is full stiffener length.

Figure 14: Linear elastic buckling results for plates with varying effective length


PAGE 23 OF 65

2.7 General theory for global elastic buckling The most general anisotropic plate theory is based on stiffness relations for the forces and moments with all cross couplings included. Such relations are given in the matrix form as[4]

κκκεεε

=

3

2

1

3

2

1

333231132313

232221322212

131211312111

333231333231

232221232221

131211131211

3

2

1

3

2

1

DDDQQQDDDQQQDDDQQQQQQCCCQQQCCCQQQCCC

MMMNNN

;

=

κε

DQQC

MN

T (1)

with a symmetric stiffness matrix which means 21 independent stiffness coefficients. The forces s'Ni and s'Mi are per unit width along the plate edges, C is the extensional

stiffness matrix, D is the bending stiffness matrix and Q is the extension- bending stiffness matrix describing the coupling between in-plane forces and bending moments. Q is generally a non-symmetric matrix. The stiffness matrix as a whole is symmetric.

Figure 15: Eccentric stiffened panel with uni-directional stiffener profiles

A stiffened panel has a physical plane plate made of an isotropic material on which eccentrically placed stiffeners also made of isotropic material are welded. This is the typical configuration for stiffened panels in ships and offshore constructions and the lateral support (support normal to the plate plane) is provided by girders/bulkheads along the edges in the physical plate plane. So the analogy between stiffened panels and anisotropic material is obvious from a stiffness point of view. The anisotropy of stiffened panels, where each


PAGE 24 OF 65

component plate is made of isotropic material, is due to the stiffener arrangement and not the material behaviour. The most intuitive approach is then to select the middle-plane of the plate as the reference

plane and to express the forces 321 N,N,N (per unit width) and moments 321 M,M,M (per

unit width) in terms of the extensional strain 321 ,, εεε (ε3 = γ) and bending strain (curvature)

321 ,, κκκ of this plane.

In order to obtain relatively simple expressions for the ideal elastic buckling stresses of stiffened plates, the theory given in Timoshenko & Gere is used. This orthotropic plate theory is based on orthotropic stiffness relations with no coupling between extension and bending. Moreover, in the expressions for the ideal elastic buckling stresses, only the bending stiffness coefficients D11, D12, D21, D22 and D33 enter, and all other items D13, D23, D31 and D32 are zeros. In the extensional stiffness matrix C, C11, C12, C21, C22 and C33 enter, and all other items C13, C23, C31 and C32 are zeros. This means that for an eccentrically stiffened panel the bending stiffness coefficients have to be modified in such a way that bending and extension are decoupled such that

𝑴𝑴 = 𝑫𝑫𝑫𝑫 and 𝑵𝑵 = 𝑪𝑪𝑪𝑪

The non-zero stiffness coefficients of the matrix items are to be calculated by the following expressions:

=+

=

−=

−==

−+−

=

G tν)2(1

tEC

ν1 t EC

ν1t ECC

ts

A)ν(11

ν1 tEC

33

222

22112

s2211

ν

𝐷𝐷11 = 𝐺𝐺𝑜𝑜𝑝𝑝3

12(1−𝜈𝜈2) �1 + 12(1 − 𝜈𝜈2) 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒104

𝑠𝑠𝑜𝑜𝑝𝑝3�

𝐷𝐷12 = 𝐷𝐷21 = 𝐺𝐺𝑜𝑜𝑝𝑝3𝜈𝜈12(1−𝜈𝜈2)


12(1−𝜈𝜈2)


12(1+𝜈𝜈) ⎭⎪⎪⎬

⎪⎪⎫

Bending stiffness coefficients, where Ieff is the bending stiffness of the stiffener with an effective plate breadth beff

In-plane total stiffness coefficients, stiffened panel


PAGE 25 OF 65

3. Numerical verification and consequence assessment Results computed by the formulae with the rule change proposal are compared with results by CSR and FEA for both the global elastic buckling (GEB) load and ultimate strength calculations.

3.1 Results - Global elastic buckling (GEB) Four different panels have been studied and their properties are given in Table 5.

Table 5: Panel dimensions Panel No.

LB1 LB2 tp s hw tw bf tf

1 2400 2400 20 600 250 12 125 15 2 2400 12000 20 600 250 12 125 15 3 5000 12000 20 600 250 12 125 15 4 5000 12000 20 600 350 15 200 20

Computed results of GEB with the proposal are compared with those of finite element linear elastic buckling analysis using Abaqus. Shell elements are used where the section properties are defined with orthotropic theory with direct input of stiffness coefficients as given in Figure 16. The same properties are used in the proposal for the Closed Form Method (CFM). Computed global eigenvalues for various combinations of biaxial loads using FEM and CFM (both with orthotropic panels) are shown in Figure 17 - Figure 20, and the agreement is very good. The theory is valid for all load combinations including compression and tension.

Figure 16: Stiffness coefficients for panel 1,2, and 3 (left) and for panel 4 (right)


PAGE 26 OF 65

Figure 17: Panel 1 - Global elastic buckling capacities for biaxial loads



-1000

-800

-600

-400

-200

0

200

400

600

800

-1000 0 1000 2000 3000 4000Sy [

MPa

]

Sx [MPa]

CFM

FEM

-1000

-800

-600

-400

-200

0

200

400

600

800

-1000 0 1000 2000 3000 4000Sy [

MPa

]

Sx [MPa]

CFM

FEM

-100

-50

0

50

100

150

-100 100 300 500 700 900

Sy [

MPa

]

Sx [MPa]

CFM

FEM


PAGE 27 OF 65


For pure axial loads, the results are compared with the critical stress from beam theory (Euler buckling), FEM using both orthotropic plate theory and a model with actual stiffened panel. The FE model with a stiffened panel is a model consisting of two full panel lengths and with half panel length at each end (½L+L+L+½L) as in the procedure given in TB Rep_Pt1_Ch08_Sec05 which had been used for validation during the development of CSR BC&OT buckling capacity formulae. The eigenmode for panel 3 with FE model of stiffened panel is shown in Figure 21.

Table 6: Global elastic buckling capacities for pure axial loads xσ [MPa]

Panel No.

CFM orthotropic plate

theory

FEM orthotropic plate

theory

FEM stiffened panel

Beam theory (Euler)

beamEx A

EIL

2

,

=πσ

1 3129 3066 3315* 3089 2 3040 3016 3123* 3089 3 715 711 718 712 4 1771 1739 1887 1769

* beam stiffeners without shear deformations

Figure 21: Global elastic mode for Panel 3 with FE model with stiffened panel (extent ½L+L+L+½L)

-100

-50

0

50

100

150

200

250

-500 0 500 1000 1500 2000

Sy [

MPa

]

Sx [MPa]

CFM

FEM


PAGE 28 OF 65

Computed global elastic buckling capacity for pure shear loads using FEM and CFM (both with orthotropic panels) are shown in Figure 22 and the agreement is very good.

Figure 22: Global elastic buckling capacities for pure shear loads

3.2 Results – Test panels In Figure 23 to Figure 27, the proposal is compared with results computed by CSR for the panels defined in Appendix A. For pure axial load, the agreement between CSR and the proposal is good, and the reason for this is because the stiffener buckling approach in CSR which is based on beam theory is good for pure axial loads. However, for transverse and combined loads, it is more accurate to use orthotropic plate theory as in the proposal. It can be seen that the results for CSR and the proposal are similar for many cases. However, in some cases the new proposal for CFM is more conservative (i.e. larger utilisation), and this is especially the case for plates with very slender stiffeners subjected to transverse or combined loads.

Figure 23: Buckling utilization factors of the panels under axial load


PAGE 29 OF 65

Figure 24: Buckling utilization factors of the panels under transverse loads

Figure 25: Buckling utilization factors of the panels under combined in-plane loads

Figure 26: Buckling utilization factors of the panels under combined in-plane loads with pressure on plate side

Figure 27: Buckling utilization factors of the panels under combined in-plane loads with pressure on stiffener side


PAGE 30 OF 65

3.3 Results – stiffened panels with slender stiffeners Stiffened panels with slender stiffeners subjected to transverse loads are analysed with the buckling procedure for the proposal, CSR and FEM[5], respectively. The plate and load definitions are given in Table 7 and Table 8, respectively. For each panel, the plate thicknesses or stiffener spacing are varied to investigate the effect on the utilisation, and the results are presented in Figure 28.

Table 7: Dimensions of stiffened plate panels

L [mm]

s [mm]


ℎ𝑤𝑤 [mm]


𝑏𝑏𝑓𝑓 [mm]

𝑜𝑜𝑓𝑓 [mm]

Stiff. Type

Yield st. plate

[MPa]

Yield st. stiffener [MPa]

Panel1-s500 4800 542 17 240.5 25 - - F 355 355 Panel1-s1000 4800 1084 17 240.5 25 - - F 355 355 Panel2-T14 8800 930 14 480 16 250 20 T 355 355 Panel2-T18 8800 930 18 480 16 250 20 T 355 355 Panel2-T22 8800 930 22 480 16 250 20 T 355 355 Panel3-T14 9000 870 14 450 15 250 27 T 355 355 Panel3-T18 9000 870 18 450 15 250 27 T 355 355 Panel3-T22 9000 870 22 450 15 250 27 T 355 355 Panel4-T12 10880 870 12 465 12 150 15 T 315 315 Panel4-T14 10880 870 14 465 12 150 15 T 315 315 Panel4-T16 10880 870 16 465 12 150 15 T 315 315 Panel4-T18 10880 870 18 465 12 150 15 T 315 315

Table 8: Loads for stiffened plate panels

Axial stress 𝜎𝜎𝑥𝑥1


Transverse stress 𝜎𝜎𝑦𝑦1


Shear stress 𝜏𝜏

Pressure p (fixed)

Panel1-s500 21.9 21.9 51.9 51.9 0 0

Panel1-s1000 21.9 21.9 51.9 51.9 0 0

Panel2-T14 0 0 50 50 0 0

Panel2-T18 0 0 50 50 0 0

Panel2-T22 0 0 50 50 0 0

Panel3-T14 0 0 50 50 0 0

Panel3-T18 0 0 50 50 0 0

Panel3-T22 0 0 50 50 0 0

Panel4-T12 0 0 25 25 0 0

Panel4-T14 0 0 25 25 0 0

Panel4-T16 0 0 25 25 0 0

Panel4-T18 0 0 25 25 0 0

From the results, it shows the agreement between the proposal and FEM is quite good. However, for CSR there is a large difference in many of the cases, and for some plates the utilization obtained by the other two methods is only half of that obtained by CSR. For these


PAGE 31 OF 65

cases, the failure mode in the proposal is global elastic buckling (GEB), and the same can be found using FE with GEB as a cut-off limit. In addition, for the set of panels, it shows that the utilisation factor of CSR increases with increased thicknesses, which is in this case unphysical, while the utilisation factor decreases with increasing thicknesses for the other two methods.

Figure 28: Plates with slender/long stiffeners subjected to transverse loads

For a selected stiffened panel of a CSR bulk carrier, the effect of varying the aspect ratio is studied and the comparisons are presented in Figure 29. For small aspect ratios, the failure mode is plate buckling and the agreement between all the methods is very good. However, for larger aspect ratios the utilisation for CSR is much larger compared to both the proposal and FEM.

Figure 29: Effect of varying the aspect ratio for a stiffened panel with transverse load


PAGE 32 OF 65

3.4 Results – stiffened side shell panels of single side skin Bulk Carriers Consequence assessment is performed for several CSR bulk carriers, with the dimensions and applied critical stresses of their single side shell plating and frames in Table 9 and Table 10, respectively.

Table 9: Dimensions of stiffened plate panels

Panel ID L [mm]

s [mm]


ℎ𝑤𝑤 [mm]


𝑏𝑏𝑓𝑓 [mm]


Stiff. Type

Yield st.

plate [MPa]


Fr265CH2 8205 980 35 504.25 12 175.5 23.5 T 355 355

Fr223CH3 7370 980 21 504.25 12.5 145.5 21.5 T 355 355

Fr202CH4 7370 980 25 504.25 12 125.5 20.5 T 355 355

Fr172CH5 7370 980 18 504.25 12 125.5 20.5 T 355 355

Fr156CH6 7370 980 22 554.25 12 235.5 20.5 T 355 355

Fr118CH7 7370 980 23.5 504.25 12 145.5 20.5 T 355 355

Fr100CH8 7370 980 22 504.25 12 145.5 20.5 T 355 355

210kCKSS23 7280 980 19 552.25 10.5 147.75 20.5 T 355 355 210kCH4SS41 7370 980 18 475 17 150 25 T 355 355 210kCH8SS70 7370 810 24.5 475 16.5 150 25 T 355 355 210kCH8SS57 7370 980 22 475 16.5 150 25 T 355 355 82kBC-SS 9000 870 15 450 10.5 250 22.5 T 355 355 180kBC-SSt18 8800 930 18 480 11.5 250 15.5 T 355 355 180kBC-SSt24 8800 930 24 480 11.5 250 15.5 T 355 355 No1CH 9034 950 20.05 604.25 12 245.5 20.5 T 355 355

No2CH 8530 950 17.56 524.25 10 195.5 17.5 T 355 355

No3CH 8530 950 18.5 544.25 10 245.5 20.5 T 355 355

No4CH 8530 950 17.5 524.25 10 245.5 20.5 T 355 355

No5CH 8530 950 16 534.25 10 245.5 20.5 T 355 355

No6CH 8530 950 16.5 484.25 10 195.5 17.5 T 355 355

No7CH 5593 950 17.5 304.25 10 120.5 11.5 T 355 355


PAGE 33 OF 65

Table 10: Loads for stiffened plate panels

Panel ID Axial stress 𝜎𝜎𝑥𝑥1





Lateral Pressure p

Fr265CH2 0 0 57.3 57.3 78.9 -0.045 Fr223CH3 0 0 75.4 75.4 62.5 -0.082 Fr202CH4 0 0 79.6 79.6 70.6 -0.047 Fr172CH5 0 0 82.3 82.3 87.9 -0.086 Fr156CH6 0 0 82.0 82.0 73.0 -0.049 Fr118CH7 0 0 75.3 75.3 88.6 -0.090 Fr100CH8 0 0 63.7 63.7 76.0 -0.052 210kCKSS23 33.4 33.4 88.6 88.6 87.7 0.059 210kCH4SS41 32.5 32.5 97.1 97.1 38.2 0.049 210kCH8SS70 5.0 5.0 68.0 68.0 59.2 -0.031 210kCH8SS57 31.0 31.0 91.6 91.6 37.5 0.056 82kBC-SS 0 0 63.7 63.7 0 -0.04 180kBC-SSt18 0 0 59.4 59.4 0 -0.04 180kBC-SSt24 0 0 59.4 59.4 0 -0.04 No1CH 75.3 75.3 122.4 122.4 38.9 0.132

No2CH 27.4 27.4 78.8 78.8 9.6 0.034

No3CH 31.0 31.0 116.1 116.1 15.5 0.036

No4CH 34.0 34.0 100.3 100.3 11.7 0.048

No5CH 35.1 35.1 126.2 126.2 23.7 0.036

No6CH 83.1 83.1 78.1 78.1 6.5 0.161

No7CH 26.4 26.4 92.6 92.6 42.6 0.037

For the single side shell plating and frames of several CSR bulk carriers, the utilisation for both existing CSR and the current Proposal but without considering the L1,eff factor in [2.6] are presented in Figure 30. CA results indicate that: (1) For all except single side shell structures of the bulk carriers, there is no scantling impact; (2) For the stiffened side shell panels with plate buckling as failure mode, no changes in utilisation and consequences are introduced. However, for the stiffened side shell panels with stiffener buckling as failure mode, the increase of buckling utilization factors is generally about 15% and at most 30% than current CSR as shown in Figure 30. Therefore, for these panels there will be considerable scantling increase without considering the L1,eff factor. In addition, it’s also found that for stiffened side shell panels with very thick plating, there is considerable reduction of buckling utilization factors, which allows for potentially reducing plate thicknesses or stiffener sizes.


PAGE 34 OF 65

Figure 30: Consequence for the side shell panels (simply supported) of CSR bulk carriers

Note: SS B.C in the title of this figure means simply supported boundary condition.

However, the above analysis without considering the L1,eff factor actually assumes that both ends of the stiffened side shell panels are simply supported. With eigenvalue buckling analysis of the single side shell structure of a typical Bulk Carrier as demonstrated in [2.6], it’s considered that this simply supported boundary assumption is for some cases too conservative compared with the actual constraints applied to the short side of the stiffened side shell panels by the adjacent side shell structures, which should be more in between simply supported and clamped boundary conditions. Therefore, for the stiffened side shell panels in Table 9 and Table 10, assuming the constraints to the short side edge of the stiffened side shell panel as partially clamped by considering of the L1,eff factor in [2.6], the utilisation factors for the Proposal is presented in Figure 31. Compared with existing CSR results, it shows that for almost all cases there is no or only slight changes in utilisation and consequences. Specifically, for stiffened side shell panels with very thick attached plating, same as above discussion, it also allows for reducing plate thicknesses or stiffener sizes. Regarding this partially clamped boundary condition, consequence assessment results for the single side shell plating and frames of some more CSR Bulk Carriers are shown in Figure 32, which give the same conclusions as for the panels above in Table 9 and Table 10.


PAGE 35 OF 65

Figure 31: Consequence for the side shell panels (partially clamped) of CSR bulk carriers

Note: SC B.C in the title of this figure means short side clamped boundary condition.

Figure 32: Consequence for the side shell panels (partially clamped) of more CSR bulk carriers

3.5 Results – Vertically stiffened plates for transverse bulkheads of Oil Tankers For Oil Tanker, consequence assessments have been carried out for vertically stiffened transverse bulkheads, and the panel dimensions and load cases are defined in Table 11 and Table 12. The utilisation for both existing CSR and the Proposal are presented in Figure 33. For each panel, the failure mode is plate buckling and no changes in utilisation and consequences are introduced for such cases.


PAGE 36 OF 65

Table 11: Dimensions of vertically stiffened panels for transverse bulkheads of Oil Tankers

L [mm]

s [mm]


ℎ𝑤𝑤 [mm]


𝑏𝑏𝑓𝑓 [mm

]


Stiff. Typ

e

Yield st.

plate [MPa]


OT150K-BHD-T19 5845 835 19 525 12 150 26 T 315 315 OT150K-BHD-T17.5 5845 835 17.5 525 12 150 26 T 315 315 OT150K-BHD-TP16 5845 835 16 525 12 150 26 T 315 315 VLCC-BHD-T20 6600 825 20 550 14.5 180 32 T 315 315 VLCC-BHD-T18 6600 825 18 550 14.5 180 32 T 315 315 VLCC-SWBHD-T20 4950 825 20 425 12 150 18 T 315 315 VLCC-SWBHD-T18 4950 825 18 425 12 150 18 T 315 315

Table 12: Loads for vertically stiffened plates for transverse bulkheads for Oil Tankers






Pressure p (fixed)

OT150K-BHD-T19 30 30 114 114 0 -0.1 OT150K-BHD-T17.5 30 30 102 102 0 -0.1 OT150K-BHD-TP16 30 30 90 90 0 -0.1 VLCC-BHD-T20 30 30 120 120 0 -0.15 VLCC-BHD-T18 30 30 103 103 0 -0.15 VLCC-SWBHD-T20 30 30 130 130 0 0 VLCC-SWBHD-T18 30 30 112 112 0 0

Figure 33: Consequence for vertically stiffened transverse bulkheads of Oil Tankers

3.6 Results – Enlarged stiffeners on longitudinal bulkheads of Oil Tankers For Oil Tanker, consequence assessments have been carried out for enlarged stiffeners on longitudinal bulkheads, and the panel dimensions and load cases are defined in Table 13 and Table 14. The utilisation for both existing CSR and the Proposal are presented in Figure 34.


PAGE 37 OF 65

For each panel, the failure mode is stiffener induced (SI) buckling but no changes in utilisation and consequences are introduced for such cases since the structures are subject to pure longitudinal stresses.

Table 13: Dimensions of enlarged stiffeners on longitudinal bulkheads of Oil Tankers

L [mm]

s [mm]


ℎ𝑤𝑤 [mm]


𝑏𝑏𝑓𝑓 [mm

]


Stiff. Typ

e

Yield st.

plate [MPa]


150kOT-LBH1 4720 880 17.5 680 12.5 180 16 T 315 315

150kOT-LBH2 4720 857.5

13 110

0 16 300 14 T 315 235

VLCC-LBH1 5750 930 18 680 12.5 180 12.5 T 315 315

VLCC-LBH2 5750 930 18 110

0 15 400 18 T 315 315

Table 14: Loads for enlarged stiffeners on longitudinal bulkheads of Oil Tankers






Pressure p (fixed)

150kOT-LBH1 267 267 0 0 0 0

150kOT-LBH2 197 197 0 0 0 0

VLCC-LBH1 257 257 0 0 0 0

VLCC-LBH2 281 281 0 0 0 0

Figure 34: Consequence for enlarged stiffeners on longitudinal bulkheads of Oil Tankers

4. Conclusions This report contains the draft rule change proposal and technical background regarding the global elastic buckling item in the WP-B of IACS PT PH43.


PAGE 38 OF 65

For some stiffened panels with thick plating and relatively slender stiffeners, such as some side shell structures of single side bulk carriers, the stiffener induced buckling or global elastic buckling modes might be the critical modes of their buckling failures. To further increase the accuracy for buckling checks of these structures, some improvements are made to the current rule.

Based on general elastic orthotropic plate buckling theory, a set of new formulae regarding global elastic buckling capacity is proposed to replace the corresponding ones based on 1-dimensional beam theory in current CSR. The Poisson effect for stiffened panels is further investigated to give a revised Poisson correction formula, which can give better agreement with FE results. The effect of lateral pressure on ultimate strength is also investigated to further revise the second order effect item of the stiffener bending moment. In addition, typically for globally slender plates subject to transverse loads, a correction factor Ktran is introduced to compensate for that linear elastic response is conservative compared to the actual structural nonlinear behaviour. Generally, based on both theoretical analysis and numerical comparisons with NLFEM results and for the purpose of better rule application, the following rule changes are proposed:

(1) A set of new formulae based on elastic orthotropic plate buckling theory is used to calculate global elastic buckling capacity and for the corresponding global buckling check.

(2) To calculate the second order effect item of the stiffener bending moment using the new global elastic buckling capacity formula and considering the actual effect of lateral pressure.

(3) A correction factor Ktran is introduced to account for the actual structural nonlinear behaviour especially for globally slender plates subject to transverse loads.

(4) The Poisson correction formula is revised to give better agreement with FE results. (5) An effective length factor LB1,eff is introduced to appropriately consider the special

boundary conditions of vertically stiffened side shell of single side skin bulk carriers. For numerical verifications of the proposed rule changes, a series of typical ship stiffened panels and some specially selected panels with slender stiffeners are calculated and compared with corresponding NLFEM results, which gives good agreement and verifies the correctness of the proposal. For consequence assessments, the proposed rule changes give almost the same buckling check results as current rule for typical ship stiffened panels subject to pure axial stresses. However, improved results can be obtained for the panels with slender stiffeners and side shell panels of single side skin Bulk Carriers with stiffener induced buckling or global elastic buckling as critical modes, which demonstrates the effectiveness of the proposed rule changes.

The RCP is included in the modified rule text of CSR BC&OT (Jan 2020) Pt1/Ch8/Sec5/2.1 and 2.3.

References (1) Bergmann. S.G., Behaviour of buckled rectangular plates under the action of shear forces, Victor

Pettersen, 1948. (2) Brush and Almroth, Buckling of bars, plates and shells, 1975. (3) C.S. Smith, Design of marine structures in composite materials, 1990. (4) Timoshenko and Gere, Theory of Elastic Stability, 1961. (5) IACS, TB Report for CSR BC & OT: Validation of Non-linear Buckling Procedure, 2015.


PAGE 39 OF 65

A. Appendix. In Table A 1- Table A 4, the test panels and load cases from CSR-OT App. D are defined.

Table A 1: Flatbar stiffeners – panel definition and load cases

Table A 2: HP-bulb flats – panel definition and load cases


PAGE 40 OF 65

Table A 3: Angle stiffeners – panel definition and load cases

Table A 4: T-bar stiffeners – panel definition and load cases


PAGE 41 OF 65

Pt 1, Ch 8, Sec5, Elastic torsional buckling

1. Reason for the rule change For some stiffeners with high webs of the ship structures, such as side frames of single side bulk carriers and enlarged stiffeners used as Permanent Means of Access (PMA) without stiffeners on its webs, the torsional buckling might be the critical modes of their buckling failures. From recent feedback of the industry and also as stated in IACS KC 1450/1459 etc., it is found that in some cases taking the current torsional buckling formula to calculate σ𝑤𝑤, i.e. normal stress due to torsional deformation, the warping stress values might be negative or exceed the material tensile strength. With some analysis, it is found mainly to be caused by the following three aspects:

(1) There are some cases that the condition of 0.4ET eHRσ > cannot be met with, which leads to

unreasonable negative warping stresses. (2) For some cases, the above problem is caused by using the assumed applied axial stress

0.4 eHR instead of the actual axial stress aσ .

(3) For other cases, this may be due to inaccurate estimation of the elastic buckling stress,

ETσ , using the existing calculation formula.

Therefore, regarding the current torsional buckling formulae in CSR, further investigation is performed in this report based on both theoretical and numerical analysis, and some rule changes are proposed accordingly.

2. Rule changes and technical background

2.1 Rule changes on stiffener torsional buckling 2.1.1 Stiffener warping stress

For a stiffener attached to a plate, its normal stress due to torsional deformation, or warping stress, can be calculated using the following formula:

𝜎𝜎𝑤𝑤 = 𝐸𝐸𝑦𝑦𝑤𝑤 �𝑜𝑜𝑒𝑒2

+ ℎ𝑤𝑤�𝛷𝛷0 �𝜋𝜋𝑙𝑙�2� 11−𝛾𝛾𝜎𝜎𝑎𝑎𝜎𝜎𝐺𝐺𝐸𝐸

− 1� with 𝜎𝜎𝐺𝐺𝐸𝐸 − 𝜎𝜎𝑎𝑎 > 0 for stiffener induced failure (SI);

σ𝑤𝑤 = 0 for plate induced failure (PI).

where:

σa : Effective axial stress, in N/mm2, at mid span of the stiffener, acting on the stiffener with

its attached plating.

𝜎𝜎𝑎𝑎 = 𝜎𝜎𝑥𝑥𝑠𝑠𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠

𝑏𝑏eff1𝑜𝑜𝑝𝑝 + 𝐴𝐴𝑠𝑠

Φ0: Coefficient taken as:

𝜙𝜙0 =𝑙𝑙ℎ𝑤𝑤

10−4

σET: Reference stress for stiffener elastic torsional buckling, in N/mm2, as defined in [2.1.2].


PAGE 42 OF 65

2.1.2 Reference stress for stiffener elastic torsional buckling

The reference stress for stiffener elastic torsional buckling, in N/mm2, can be calculated using the following formula:

𝜎𝜎𝐺𝐺𝐸𝐸 =𝐸𝐸𝐸𝐸𝑝𝑝��𝑚𝑚𝜋𝜋𝑙𝑙�2𝐸𝐸ω ∙ 102 +

12(1 + 𝜈𝜈) 𝐸𝐸𝐸𝐸 + �

𝑙𝑙𝑚𝑚𝜋𝜋

�2

𝜀𝜀 ∙ 10−4�

where:

IP : Net polar moment of inertia of the stiffener, in cm4, about point C(as shown in Figure 1

and Figure 2) , as defined in Table 1.

IT : Net St. Venant’s moment of inertia of the stiffener, in cm4, as defined in Table 1.

Iω : Net sectorial moment of inertia of the stiffener, in cm6, about point C (as shown in

Figure 1 and Figure 2) , as defined in Table 1. m : Number of half waves, taken as a positive integer so as to give smallest reference stress

for torsional buckling.

𝜀𝜀 : Degree of fixation, in mm2, taken as:

ε = �3𝑏𝑏𝑜𝑜𝑝𝑝3

+2ℎ𝑤𝑤𝑜𝑜𝑤𝑤3

�−1

for bulb, angle, L2 , L3 and T profiles;

ε =𝑜𝑜𝑝𝑝3

3𝑏𝑏 for flat bars.

Aw : Net web area, in mm2.

Af : Net flange area, in mm2.

Table 1: Moments of inertia

Flat bars(1) Bulb, angle, L2, L3 and T profiles

IP 3

43 10w wh t⋅

22 4( 0.5 )

103

w f ff f

A e tA e − −

+

IT 3

4 1 0.633 10

w w w

w

h t th

− ⋅

3 3

4 4

( 0.5 )1 0.63 1 0.63

3 10 0.5 3 10f f w f f fw

f f f

e t t b t tte t b

−− + − ⋅ − ⋅

Iω

3 3

636 10w wh t⋅

For bulb, angle, L2 and L3 profiles

Af3 + 𝐴𝐴𝑤𝑤3

36 ∙ 106+

𝑒𝑒𝑓𝑓2

106�𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓2 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤2

3−�𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤�

2

4(𝐴𝐴𝑓𝑓 + 𝐴𝐴𝑤𝑤)�

For T profiles

6

23

1012 ⋅fff etb

(1) tw is the net web thickness, in mm. tw_red as defined in [2.3.2] is not to be used in this table.


PAGE 43 OF 65

Figure 1: Stiffener cross sections

Figure 2: Schematic diagram of stiffened panel and coordinate system

2.2 Fundamental theory for stiffener elastic bending and torsional buckling

Considering a beam located along x-axis is subjected to compressive load P and it bends the stiffener in the direction of both of y-axis and z-axis as well as twists it about x-axis, its deformation can be expressed by the following simultaneous differential equations:

EIz 𝑑𝑑4𝑣𝑣𝑑𝑑𝑥𝑥4

+ 𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧𝑑𝑑4𝑤𝑤𝑑𝑑𝑥𝑥4

= 𝑓𝑓𝑦𝑦(𝑃𝑃) (2.1)

𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧𝑑𝑑4𝑣𝑣𝑑𝑑𝑥𝑥4

+ EIy 𝑑𝑑4𝑤𝑤𝑑𝑑𝑥𝑥4

= 𝑓𝑓𝑧𝑧(𝑃𝑃) (2.2)

E𝛤𝛤𝑑𝑑4Φ𝑑𝑑𝑥𝑥4

− 𝐺𝐺𝐸𝐸𝐸𝐸𝑑𝑑2Φ𝑑𝑑𝑥𝑥2

= 𝑚𝑚𝑥𝑥(𝑃𝑃) (2.3)

where:

𝑣𝑣,𝑤𝑤,Φ : Displacements in y and z directions, and rotation angle, respectively. 𝐸𝐸𝑦𝑦 , 𝐸𝐸𝑧𝑧 , 𝐸𝐸𝑦𝑦𝑧𝑧 : Moment inertia of stiffener about each axis. 𝛤𝛤 : Warping resistance moment of inertia of the stiffener. 𝐸𝐸𝐸𝐸 : St. Venant’s moment of inertia of the stiffener. The resulting internal forces of each arbitrary infinitesimal segment of the stiffener due to the compressive load 𝑃𝑃 are


PAGE 44 OF 65

𝑓𝑓𝑦𝑦(𝑃𝑃) = −𝑃𝑃 �𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

+ 𝑧𝑧𝑠𝑠dΦ𝑑𝑑𝑥𝑥2

� (2.4)

𝑓𝑓𝑧𝑧(𝑃𝑃) = −𝑃𝑃 �𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

− 𝑦𝑦𝑠𝑠dΦ𝑑𝑑𝑥𝑥2

� (2.5)

𝑚𝑚𝑥𝑥(𝑃𝑃) = �−𝜎𝜎𝑜𝑜(𝑧𝑧𝑠𝑠 − 𝑧𝑧) �𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

+ (𝑧𝑧𝑠𝑠 − 𝑧𝑧) 𝑑𝑑2Φ𝑑𝑑𝑥𝑥2

� + 𝜎𝜎𝑜𝑜(𝑦𝑦𝑠𝑠 − 𝑦𝑦) �𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

− (𝑦𝑦𝑠𝑠 − 𝑦𝑦)𝑑𝑑2Φ𝑑𝑑𝑥𝑥2

� 𝑑𝑑𝑠𝑠

= 𝑃𝑃(𝑦𝑦𝑠𝑠𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

− 𝑧𝑧𝑠𝑠𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

) −𝑃𝑃𝐴𝐴�𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧 + 𝐴𝐴(𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑠𝑠2�

𝑑𝑑2Φ𝑑𝑑𝑥𝑥2

(2.6)

where:

𝑦𝑦𝑠𝑠, 𝑧𝑧𝑠𝑠 : Coordinates of the stiffener shear centre. 𝐴𝐴 : Sectional area of the stiffener. 𝜎𝜎 : Axial compressive stress working at each segment of the stiffener. t : Thickness of each segment of the stiffener. Fundamental equations of bending and torsional buckling can be obtained by substituting Eq.(2.4), Eq.(2.5) and Eq.(2.6) into Eq.(2.1), Eq.(2.2) and Eq.(2.3) as below:

EIz 𝑑𝑑4𝑣𝑣𝑑𝑑𝑥𝑥4


+ 𝑃𝑃 �𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

+ 𝑧𝑧𝑠𝑠 d2Φdx2

� = 0 (2.7)


+ EIy 𝑑𝑑4𝑤𝑤𝑑𝑑𝑥𝑥4

+ 𝑃𝑃 �𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

− 𝑦𝑦𝑠𝑠d2Φdx2

� = 0 (2.8)

EΓ𝑑𝑑4Φ𝑑𝑑𝑥𝑥4

− [𝐺𝐺𝐸𝐸𝐸𝐸 −𝑃𝑃𝐴𝐴�𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧 + 𝐴𝐴(𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑠𝑠2)}�


− 𝑃𝑃𝑦𝑦𝑠𝑠𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

+ 𝑃𝑃𝑧𝑧𝑠𝑠𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

= 0 (2.9)

2.3 Bending and torsional buckling of stiffener with attached plate

The following geometric and mechanics conditions need to be established for a stiffener and its attached plate with bending and twisting deformation as illustrated in Figure 3: 1) Both the stiffener and the attached plate are deformed together in the yoz plane. 2) The stiffener is fixed in transverse direction at the intersection point between the stiffener and the attached plate (see Point C in Figure 3) and takes the reaction force 𝑞𝑞0, from the attached plate. 3) The attached plate is bended and induces reaction moment about x-axis when the transverse section of stiffener is rotated. Based on Eqs.(2.7)-(2.9), the three conditions listed above can be formulated as the following three corresponding governing equations:


+ 𝐸𝐸𝐸𝐸′y 𝑑𝑑4𝑤𝑤𝑑𝑑𝑥𝑥4

+ 𝑃𝑃′𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

− 𝑃𝑃𝑦𝑦𝑠𝑠d2Φdx2

= 0 (3.1)

𝑣𝑣 + (𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)Φ = 0 (3.2)

𝐸𝐸𝐸𝐸𝑧𝑧 𝑑𝑑4𝑣𝑣𝑑𝑑𝑥𝑥4


+ 𝑃𝑃 �𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

+ 𝑧𝑧𝑠𝑠 d2Φdx2

� = 𝑞𝑞0 (3.3)


PAGE 45 OF 65

𝐸𝐸𝛤𝛤𝑑𝑑4Φ𝑑𝑑𝑥𝑥4

− [𝐺𝐺𝐸𝐸𝐸𝐸 −𝑃𝑃𝐴𝐴�𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧 + 𝐴𝐴(𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑠𝑠2)}�



+ 𝑃𝑃𝑧𝑧𝑠𝑠𝑑𝑑2𝑣𝑣𝑑𝑑𝑥𝑥2

− 𝑞𝑞0(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑥𝑥) + 𝑘𝑘ΦΦ

= 0 (3.4)

where:

𝐸𝐸′𝑦𝑦: Moment of inertia of stiffener with attached plate. 𝑃𝑃′: Axial compressive load P of stiffener with attached plate.

Figure 3: Schematic diagram of stiffener and its attached plate

Displacement 𝑣𝑣 and force 𝑞𝑞0 can be eliminated by substituting Eq.(3.2) and Eq.(3.3) into Eq.(3.1) and Eq.(3.4), and the following fundamental equations of bending and torsional buckling of the stiffener and its attached plate can be obtained:

𝐸𝐸𝐸𝐸𝑦𝑦′𝑑𝑑4𝑤𝑤𝑑𝑑𝑥𝑥4

+ 𝑃𝑃′𝑑𝑑2𝑤𝑤𝑑𝑑𝑥𝑥2

− 𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎) 𝑑𝑑4Φ𝑑𝑑𝑥𝑥4

− 𝑃𝑃𝑦𝑦𝑠𝑠𝑑𝑑2Φ𝑑𝑑𝑥𝑥2

= 0 (3.5)

{𝐸𝐸𝛤𝛤 + 𝐸𝐸𝐸𝐸𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)2}𝑑𝑑4Φ𝑑𝑑𝑥𝑥4

− [𝐺𝐺𝐸𝐸𝐸𝐸 −𝑃𝑃𝐴𝐴�𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧 + 𝐴𝐴(𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑎𝑎2)}�


− 𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)𝑑𝑑4𝑤𝑤𝑑𝑑𝑥𝑥4


+ 𝑘𝑘ΦΦ = 0 (3.6)

Buckling shape functions which meet simply supported boundary conditions at both ends can be taken as below:

𝑤𝑤 = 𝐴𝐴𝑏𝑏 sin𝜋𝜋𝑥𝑥𝑎𝑎

(3.7)

𝛷𝛷 = 𝐴𝐴𝑜𝑜 sin𝑚𝑚𝜋𝜋𝑥𝑥𝑎𝑎

(3.8)

For torsional buckling, its buckling mode of more than one half waves can possibly take place especially for flat bar type stiffener. By substituting Eq.(3.7) and Eq.(3.8) into Eq.(3.5) and Eq.(3.6), simultaneous linear equations regarding Ab and At can be derived. The determinant of the simultaneous linear equations must be zero in order to obtain non-trivial solutions for Ab and At, which can be formulated as:


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�𝐸𝐸𝐸𝐸𝑦𝑦′ �

𝜋𝜋𝑎𝑎�

2− 𝑃𝑃′ 𝑃𝑃𝑦𝑦𝑠𝑠 − 𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧 �

𝜋𝜋𝑎𝑎�

2(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)

𝑃𝑃𝑦𝑦𝑠𝑠 − 𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧 �𝜋𝜋𝑎𝑎�

2(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎) �

𝑚𝑚𝜋𝜋𝑎𝑎 �

2{𝐸𝐸𝛤𝛤 + 𝐸𝐸𝐸𝐸𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)2} + 𝐺𝐺𝐸𝐸𝐸𝐸 −

𝑃𝑃𝐴𝐴�𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧 + 𝐴𝐴(𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑎𝑎2)� + �

𝑎𝑎𝑚𝑚𝜋𝜋�

2𝑘𝑘Φ�

= 0 (3.9)

� 𝛼𝛼1 − 𝑃𝑃′ 𝜉𝜉𝑃𝑃′𝑦𝑦𝑠𝑠 − 𝛼𝛼2𝜉𝜉𝑃𝑃′𝑦𝑦𝑠𝑠 − 𝛼𝛼2 −𝛼𝛼3𝜉𝜉𝑃𝑃′ + 𝛼𝛼4

� = 0 (3.10)

α1 = 𝐸𝐸𝐸𝐸𝑦𝑦′ �𝜋𝜋𝑎𝑎�2

(3.11)

α2 = 𝐸𝐸𝐸𝐸𝑦𝑦𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎) �𝜋𝜋𝑎𝑎�2

(3.12)

α3 =𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧

A+ 𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑎𝑎2 (3.13)

α4 = {𝐸𝐸𝛤𝛤 + 𝐸𝐸𝐸𝐸𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)2} �𝑚𝑚𝜋𝜋𝑎𝑎�2

+ GIT + �a𝑚𝑚𝜋𝜋

�2𝑘𝑘𝜙𝜙 (3.14)

𝑃𝑃 =𝐴𝐴𝐴𝐴𝑠𝑠𝑝𝑝

𝑃𝑃′ = 𝜉𝜉𝑃𝑃′ (3.15)

where:

𝐴𝐴𝑠𝑠𝑝𝑝: Sectional area of the stiffener including half breadth of attached plating on both sides. A quadratic equation with respect to 𝑃𝑃′ can be derived from Eqs.(3.10)-(3.15) and the solutions of the quadratic equation represent bending and torsional buckling capacities.

β1(𝑃𝑃′)2 + 𝛽𝛽2𝑃𝑃′ + 𝛽𝛽3 = 0 (3.16)

β1 = 𝑦𝑦𝑠𝑠2𝜉𝜉2 − 𝛼𝛼3𝜉𝜉 (3.17)

β2 = 𝛼𝛼4 + 𝛼𝛼1𝛼𝛼3 𝜉𝜉 − 2𝛼𝛼2𝑦𝑦𝑠𝑠𝜉𝜉 (3.18)

β3 = 𝛼𝛼22 − 𝛼𝛼1𝛼𝛼4 (3.19)

𝑃𝑃cr′ =−𝛽𝛽2 + �𝛽𝛽22 − 4𝛽𝛽1𝛽𝛽3

2𝛽𝛽1 (3.20)

Two different modes of buckling can be obtained separately where the interaction effect between bending and torsion buckling is negligible whether the transverse section is symmetrical or not.

Pcrb′ = 𝛼𝛼1 (3.21)

Pcrt′ =𝛼𝛼4𝜉𝜉𝛼𝛼3

(3.22)

Therefore, elastic torsional buckling stress can be obtained from the following formula

σcrt =𝑃𝑃𝑎𝑎𝑡𝑡𝑜𝑜′𝐴𝐴𝑠𝑠𝑝𝑝

=𝐸𝐸{𝛤𝛤 + 𝐸𝐸𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)2} �𝑚𝑚𝜋𝜋𝑎𝑎 �

2+ 𝐺𝐺𝐸𝐸𝐸𝐸 + � 𝑎𝑎𝑚𝑚𝜋𝜋�

2𝑘𝑘𝜙𝜙

𝐸𝐸𝑦𝑦 + 𝐸𝐸𝑧𝑧 + (𝑦𝑦𝑠𝑠2 + 𝑧𝑧𝑎𝑎2)𝐴𝐴𝑠𝑠 (3.23)


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2.4 Sectorial moment of inertia of the stiffener, Iω

To be in accordance with the torsional buckling formula defined in CSR, compared with Eq.(3.23), the definition of sectorial moment of inertia of the stiffener, 𝐸𝐸𝜔𝜔 should be:

𝐸𝐸ω = 𝛤𝛤 + 𝐸𝐸𝑧𝑧(𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎)2 (4.1)

where:

𝛤𝛤 : Warping resistance moment of inertia of the stiffener, as defined in Table 2.

𝐸𝐸𝑧𝑧 : Moment of inertia of the stiffener about z-axis, as defined in Table 2.

Therefore, the sectorial moment of inertia for each stiffener type can be derived theoretically as shown in Table 2.

Figure 4: Schematic diagram of three types of stiffener

Table 2: Theoretically derived formulae of 𝑰𝑰𝝎𝝎

Stiffener type

𝛤𝛤 𝐸𝐸𝑧𝑧 𝑧𝑧𝑠𝑠 − 𝑧𝑧𝑎𝑎 𝐸𝐸𝜔𝜔

Flat bar hw3 𝑜𝑜𝑤𝑤3

144

ℎ𝑤𝑤𝑜𝑜𝑤𝑤3

12

ℎ𝑤𝑤2

hw3 𝑜𝑜𝑤𝑤3

144+ℎ𝑤𝑤3 𝑜𝑜𝑤𝑤3

48=ℎ𝑤𝑤3 𝑜𝑜𝑤𝑤3

36

T-bar Af3

144+𝐴𝐴𝑤𝑤3

36

𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓2

12+

Aw𝑜𝑜𝑤𝑤2

12 ℎ𝑤𝑤 +

𝑜𝑜𝑓𝑓2

= 𝑒𝑒𝑓𝑓 Af3


36+ 𝑒𝑒𝑓𝑓2 �

𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓2

12+

Aw𝑜𝑜𝑤𝑤2

12�

Bulb, angle

bar, L2 and L3

Af3


36

𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓2 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤2

3− �𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤�

2

/4𝐴𝐴𝑠𝑠

ℎ𝑤𝑤 +𝑜𝑜𝑓𝑓2

= 𝑒𝑒𝑓𝑓

Af3


36+ 𝑒𝑒𝑓𝑓2 �

𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓2 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤2

3

− �𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤�2

/4�𝐴𝐴𝑓𝑓 + 𝐴𝐴𝑤𝑤��

The formulae of 𝐸𝐸𝜔𝜔 defined in UR S11 and CSR are shown in Table 3. There can be some differences between the proposed formulae and those in existing rules. For flat bar, the formulae of 𝐸𝐸𝜔𝜔 in both UR S11 and CSR are identical to the proposed formula.

For T bar, the form of formulae is different as shown in Table 3. However, their calculated values are almost identical. The formulae in UR S11 and CSR are approximate expressions


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for simplicity.

Table 3: Comparisons of formulae for 𝑰𝑰𝝎𝝎 in UR S11 and CSR

Rules UR S11 CSR


36

hw3 𝑜𝑜𝑤𝑤3

36

T-bar bf3𝑜𝑜𝑓𝑓ℎ𝑤𝑤2

12

bf3𝑜𝑜𝑓𝑓𝑒𝑒𝑓𝑓2

12

Bulb, angle bar, L2 and L3

𝑏𝑏𝑓𝑓3ℎ𝑤𝑤2

12�𝑏𝑏𝑓𝑓 + ℎ𝑤𝑤�2 �𝑜𝑜𝑓𝑓�𝑏𝑏𝑓𝑓

2 + 2𝑏𝑏𝑓𝑓ℎ𝑤𝑤 + 4ℎ𝑤𝑤2 �

+ 3𝑜𝑜𝑤𝑤𝑏𝑏𝑓𝑓ℎ𝑤𝑤�

Af𝑒𝑒𝑓𝑓2𝑏𝑏𝑓𝑓2

12�𝐴𝐴𝑓𝑓 + 2.6𝐴𝐴𝑤𝑤𝐴𝐴𝑓𝑓 + 𝐴𝐴𝑤𝑤

�

With regard to stiffeners of unsymmetrical profiles, the formulae of 𝐸𝐸𝜔𝜔 are different between UR S11 and CSR. In addition, their calculated values are also different among UR S11, CSR and the proposed formula. It can be regarded that the existing formulae are actually some approximate expressions and their accuracies can be further improved. Therefore, a modification of the formulae of 𝐸𝐸𝜔𝜔 is proposed as shown in Table 4.

As for other moments of inertia such as 𝐸𝐸𝐸𝐸 and 𝐸𝐸𝑃𝑃, existing formulae need not to be modified

Table 4: Modification of 𝑰𝑰𝝎𝝎 formulae

Current Proposal


36 Keep as it is

T-bar bf3𝑜𝑜𝑓𝑓𝑒𝑒𝑓𝑓2

12 Keep as it is

Bulb, angle bar, L2 and L3

Af𝑒𝑒𝑓𝑓2𝑏𝑏𝑓𝑓2

12�𝐴𝐴𝑓𝑓 + 2.6𝐴𝐴𝑤𝑤𝐴𝐴𝑓𝑓 + 𝐴𝐴𝑤𝑤

� Af3 + 𝐴𝐴𝑤𝑤3

36 ∙ 106+𝑒𝑒𝑓𝑓2

106�𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓2 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤2

3−�𝐴𝐴𝑓𝑓𝑏𝑏𝑓𝑓 + 𝐴𝐴𝑤𝑤𝑜𝑜𝑤𝑤�

2

4(𝐴𝐴𝑓𝑓 + 𝐴𝐴𝑤𝑤)�

2.5 Effect of attached plate bending on torsional buckling

When the transverse section of a stiffener is rotated, bending of the attached plate yields a reaction moment 𝑀𝑀p in reaction to its rotation. This effect can be considered as similar with a linear rotation spring. Considering a simply supported beam subjected to bending moment 𝑀𝑀 at both ends, the rotation angle at the ends can be calculated as 𝑀𝑀𝑙𝑙/ 4𝐸𝐸𝐸𝐸, and the rotation spring coefficient, kΦ can be obtained as below:

kϕ =𝑀𝑀𝑃𝑃

𝜃𝜃1=𝐸𝐸𝑜𝑜𝑝𝑝3

3𝑏𝑏 (5.1)

where θ1 represents the rotation angle of plate at the intersection point. Meanwhile, the reaction moment, 𝑀𝑀𝑃𝑃 also bends the stiffener web and deforms the transverse section of stiffener. To consider this effect in a simple manner, we assume that the


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transverse section is virtually rotated by θ2 due to stiffener web deformation as shown in Figure 5. The rotation angle θ′2 in Figure 6 can be obtained from the rotation angle of simply supported beam applying a bending moment at one end. A coefficient 𝛼𝛼 is introduced to give the relationship between θ2 and θ2′ .

θ2 = 𝛼𝛼𝜃𝜃2′ = 𝛼𝛼4𝑀𝑀𝑝𝑝ℎ𝑤𝑤𝐸𝐸𝑜𝑜𝑤𝑤3

(5.2)

Therefore, the rotational spring coefficient due to the attached plate bending is taken as:

kϕ′ =𝑀𝑀𝑝𝑝

𝜃𝜃1 + 𝜃𝜃2=

𝐸𝐸3𝑏𝑏𝑜𝑜𝑝𝑝3

+ 𝛼𝛼 4ℎ𝑤𝑤𝑜𝑜𝑤𝑤3

(5.3)

The value of coefficient 𝛼𝛼 is to be determined so as to make the estimated torsional buckling strength to be comparable with corresponding eigenvalue buckling analysis results.

Figure 5: Effect of plate bending and stiffener web bending

Figure 6: Idealized model of stiffener bending


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2.6 Revisions of the formula for warping stress calculation

For torsional buckling requirements in CSR, it is implicitly considered in the lateral buckling formula by incorporating torsion with an additional warping stress, i.e. to combine both lateral and torsional buckling into one formula as below:

𝜎𝜎𝑎𝑎 + 𝜎𝜎𝑏𝑏 + 𝜎𝜎𝑤𝑤𝑅𝑅𝑒𝑒𝑒𝑒

< 1.0

where the warping stress is:

𝜎𝜎𝑤𝑤 = 𝐸𝐸𝑦𝑦′ �hw +𝑜𝑜𝑓𝑓2�𝛷𝛷0 �

𝜋𝜋𝑙𝑙�2�

1

1 − 0.4𝑅𝑅𝑒𝑒𝑒𝑒𝜎𝜎𝐺𝐺𝐸𝐸

− 1� for stiffener induced failure (SI)

𝜎𝜎𝑤𝑤 = 0 for plate induced failure (PI).

Φ0: Coefficient taken as:

Φ0 = 𝑙𝑙ℎ𝑤𝑤

10−3

However, as shown in [1], for some stiffened panels especially those with high web stiffeners, it is found that this formula may not give reasonable or sufficiently accurate results. With some analysis, it is found mainly to be caused by the following three aspects:

(1) There are some cases that the condition of 0.4ET eHRσ > cannot be met with, which leads

to physically unreasonable negative warping stresses. (2) For some cases, the above problem is caused by using the assumed applied axial stress

0.4 eHR instead of the actual axial stress aσ .

(3) For other cases, this may due to inaccurate estimation of the elastic buckling stress, ETσ ,

using the existing calculation formula. Therefore, corresponding to the above three aspects, the rule changes in [2.1] are proposed and explained in detail as below:

(1) A pre-condition of 0ET aσ σ− > is required before calculating the warping stresses for SI

buckling mode.

(2) To use the actually applied axial stress aσ instead of the currently assumed 0.4 eHR

for warping stress calculation; (3) The formula to calculate the reference stress for stiffener elastic torsional buckling is

revised based on a comprehensive theoretical analysis. (4) In addition, based on calibration with series of practical ship panels, the parameter 𝛷𝛷0,

which can be regarded as maximum initial torsional deformation angle, is tuned to a more proper value.

For the above first two rule changes, no further explanation is considered necessary because the meaning is explicitly described above. For the third rule change, comprehensive theoretical background explanations are given in [2.2]-[2.5]. Series of typical ship panels from TB of CSR-OT and practical ship panels are also calculated using rule methods, eigenvalue FE analysis and NLFE analysis, respectively and results compared as shown in [3.1]-[3.3],


PAGE 51 OF 65

which validates all the rule changes as a whole. However, for the last rule change, i.e. the tuning of maximum initial torsional deformation angle 𝛷𝛷0 , although in general it is also mainly validated by numerical calculation and comparisons regarding practical ship panels as shown in [3.1]-[3.3], some specific explanations are also given below. Figure 7-1 and Figure 7-2 show comparisons of the estimated ultimate capacities between NLFEM (curve in green) and proposed formula (curve in red) but with considering existing

initial torsional angle, 𝛷𝛷0 = 𝑙𝑙ℎ𝑤𝑤

10−3. The dimensions of the two calculated typical stiffened

panel models with varying thicknesses are shown in Table 5.

Table 5：Dimensions of two typical stiffened panels

No. Type a s hw bf tp tw tf 1 T-bar 4000 800 550 150 12-25 12.0 25.0 2 T-bar 4000 800 650 150 12-25 13.5 25.0

From Figure 7-1 and Figure 7-2, it is found that the proposed formula together with the existing angle(curve in red) gives too conservative ultimate capacity compared with NLFEM results (curve in green). For the reason of this capacity underestimation, it is found based on NLFE analysis that high web stiffened panels actually does not collapse immediately with initial yielding caused by warping stress at the end tips of stiffener flange. In Figure 8, which gives the distribution of Von-Mises stress and deformation of a stiffened panel at collapse stage based on NLFE analysis, it shows that before reaching material yielding strength, full width of the stiffener flange can bear extra loads. With the above analysis, to make the buckling formula more reasonable and more agreeable

with NLFEM results, the initial torsional angle is finally tuned to 𝛷𝛷0 = 𝑙𝑙ℎ𝑤𝑤

10−4 based on

series of iterative analyses. Figure 7-1 and Figure 7-2 show that the calculated results with this tuning (curve in purple) become close to NLFEA results and keep on safe side.


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Figure 7-1: Comparison on ultimate capacity for No.1 panel in Table 5

Figure 7-2: Comparison on ultimate capacity for No.2 panel in Table 5


PAGE 53 OF 65

Figure 8: Distribution of Von-Mises stress and deformation

3. Numerical verification and consequence assessment

3.1 Results – Elastic torsional buckling (ETB) capacity

To verify the accuracy of the proposed torsional buckling capacity formulae defined in [2.1.2], comparison studies have been carried out with FE eigenvalue buckling analysis method. Several types of stiffeners with different profiles and scantlings as defined in Table 6 are calculated with both methods. For comparison purpose, in the torsional buckling capacity formulae defined in [2.1.2], also as below for the prototype formula, three cases with α = 0.0, 0.5 and 1.0 are considered respectively.

ε = �3𝑏𝑏𝑜𝑜𝑝𝑝3

+ α4ℎ𝑤𝑤𝑜𝑜𝑤𝑤3

�−1

for bulb, angle, L2 , L3 and T profiles;

With comparison of the estimated values based on the formulae and calculated FE eigenvalues, it is found that the case α = 0.5 for T and angle stiffeners and α = 0.0 for flat type stiffeners give results closest to corresponding FE results. This is also the reason why in the final proposed definition of rotation spring coefficient due to the attached plate bending is as follows:

𝜀𝜀 = �3𝑏𝑏𝑜𝑜𝑝𝑝3

+2ℎ𝑤𝑤𝑜𝑜𝑤𝑤3

�−1

for T and angle stiffeners

𝜀𝜀 =𝑜𝑜𝑝𝑝3

3𝑏𝑏 for flat bar stiffeners

Irrelevant of stiffener types and scantlings, the elastic torsional buckling capacity equations can give accurate results which coincide with eigenvalue analysis results with using ABAQUS and MARC as shown in Figure 9.1a to Figure 9.6a. The buckling modes of each stiffener type obtained from eigenvalue analysis are shown in Figure 9.1b to Figure 9.6b.


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Table 6：Dimensions of series stiffened panels

ID Type a s hw bf tp tw tf

Panel 1 T-bar 5000 600 400～1100 250 22 20 22

Panel 2 T-bar 2400 600 400～750 150 20 12 20

Panel 3 T-bar 3200 800 400～1100 200 22 20 22

Panel 4 Flat 3200 700 260～400 22 20

Panel 5 Angle 5000 600 400～1100 200 22 20 22

Panel 6 T-bar 3200 800 600 200 22 12～30 22

Figure 9.1a: Comparison result (Panel 1) Figure 9.1b: FEA results (Panel 1)



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Figure 9.3a: Comparison result (Panel 3) Figure 9.3b：FEA results (Panel 3)

Figure 9.4a：Comparison result (Panel 4) Figure 9.4b: FEA results (Panel 4)

Figure 9.5a：Comparison result (Panel 5) Figure 9.5b：FEA results (Panel 5)


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3.2 Results – CSR-OT App. D panels

3.2.1 Utilisation for CSR-OT App. D panels. In Figure 10.1 to Figure 10.5, the proposal is verified with comparing the stiffener buckling utilization factors obtained from the proposal with ones computed by CSR for the panels defined in CSR-OT App. D. For all load cases, Case 1 to Case 4, the proposal can give almost same results as current CSR. This is because CSR-OT App. D panels are not so high in web height that torsional buckling would take place. Thus additional verifications with using more high-web stiffeners shown in [3.3] have to be done.

Figure 10.1: Axial load

Figure 10.2: Transverse loads


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Figure 10.3: Combined in-plane loads

Figure 10.4: Combined in-plane loads with pressure on plate side

Figure 10.5: Combined in-plane loads with pressure on stiffener side

3.3 Results –Consequence assessment of actual ship types and stiffened panels with high-web stiffeners

Regarding some CSR Bulk Carriers and Oil Tankers, some typical panels from single side shell, longitudinal and transverse bulkheads, hopper transverse web, and outer and inner side shells are chosen for consequence assessment of the proposal rule change. It shows that the calculated buckling results using current CSR and the proposal in [2.1] are almost the same, giving either no or slight scantling impact, as shown in Figure 11.


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Figure 11: Consequence assessment for some typical panels of CSR ships

However, for some PMA structures of different sizes from several CSR ships, the buckling utilization factors calculated using the proposed formulas either increase or decrease a little compared with using current CSR. Especially for some PMA structures that cannot be reasonably calculated for buckling check using the current CSR formula, with the proposal in [2.1], the calculated results agree with NLFE results very well. For the PMA structures and some specially selected stiffened panels with high-web stiffeners as listed in Table 7, FE analyses are also carried out to further verify the proposal in [2.1]. For the verification, both elastic torsional buckling capacities and ultimate capacities are calculated and compared among the proposal, CSR procedure and linear and non-linear FEM as shown in Figure 12.1 to Figure 12.3 and Figure 13.1 to Figure 13.3. With regard to elastic torsional buckling capacities as shown in Figure 12.1a to Figure 12.3a, the results for the proposal are more close to FE results compared to CSR procedure regardless of stiffener types. CSR procedure gives non-conservative results for most cases. The CSR procedure adopts an approximate approach to calculate warping stresses by using 0.4ReH as compressive axial loads instead of actual one. This approximation works well in most cases but in the case where torsional elastic buckling capacities are smaller than 0.4ReH, the ultimate capacities can’t be obtained as it can be seen for the panel, PMA2tp8(T). On the other hand, the proposal gives results closer to non-linear FE, even for the case of PMA2tp8(T).

Table 7: Model of stiffened panel

Plate Stiffener Flat400tp12 3200x700x12mm(HT32) 400x20mm(HT32) Flat500tp12 3200x700x12mm(HT32) 500x20mm(HT32) Flat600tp12 3200x700x12mm(HT32) 600x20mm(HT32) Flat400tp22 3200x700x22mm(HT32) 400x20mm(HT32) Flat500tp22 3200x700x22mm(HT32) 500x20mm(HT32) Flat600tp22 3200x700x22mm(HT32) 600x20mm(HT32)


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PMA1tp8(T) 7000x800x8mm(HT36) 1000x16.5+250x12mm(HT36) PMA1tp20(T) 7000x800x20mm(HT36) 1000x16.5+250x12mm(HT36) PMA2tp8(T) 9000x900x8mm(HT36) 1100x18+275x13mm(HT36)

PMA2tp20(T) 9000x900x20mm(HT36) 1100x18+275x13mm(HT36) PMA3tp16(L2) 5500x950x16mm(HT36) 1000x18+180x20mm(HT36) PMA3tp18(L2) 5500x950x18mm(HT36) 1000x18+180x20mm(HT36) PMA3tp20(L2) 5500x950x20mm(HT36) 1000x18+180x20mm(HT36)

Figure 12.1a: ETB capacity (flat bar) Figure 12.1b: Ultimate capacity (flat bar)

Figure 12.2a: ETB capacity (T bar) Figure 12.2b: Ultimate capacity (T bar)

Figure 12.3a: ETB capacity (L2) Figure 12.3b: Ultimate capacity (L2)


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Figure 13.1: Deformation and Mises stress at collapse stage (Flat600tp12)

Figure 13.2: Deformation and Mises stress at collapse stage (PMA2tp8(T))

Figure 13.3: Deformation and Mises stress at collapse stage (PMA3tp16(T))

4. Conclusions

This report contains the draft rule change proposal and technical background regarding the stiffener torsional buckling item in the WP-B of IACS PT PH43.

For stiffeners with high webs, such as side frames of single side bulk carriers and enlarged stiffeners used as Permanent Means of Access (PMA) without stiffeners on its webs, the torsional buckling might be the critical modes of their buckling failures. To further increase the accuracy for buckling checks of these structures, some improvements are made to current rule.

Based on the fundamental theory of stiffener elastic bending and torsional buckling analysis, a set of governing equations is formulated systematically with respect to torsional buckling of stiffener with attached plate. Compared with the newly obtained full set of formulae, the


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corresponding formulae in current CSR are thoroughly checked and improvements on the calculation formulae of elastic buckling capacity and sectorial moment of inertia of stiffeners are proposed. The effect of attached plate bending on torsional buckling is also investigated to give a set of revised calculation formulae of rotation spring coefficients which are tuned as appropriate for different stiffener types. Based on both theoretical analysis and numerical comparisons with NLFEM results and for the purpose of better rule application, the following rule changes are proposed:

(1) A pre-condition of 0ET aσ σ− > is supplemented to the calculation of warping stresses

for SI buckling mode check.

(2) To use the actually applied axial stress aσ instead of the currently assumed 0.4 eHR for

warping stress calculation; (3) The formula to calculate the reference stress for stiffener elastic torsional buckling is

revised based on a comprehensive theoretical analysis. (4) In addition, based on calibration with series of practical ship panels, the parameter 𝛷𝛷0,

which is maximum initial torsional deformation angle, is tuned to a more proper value. For numerical verifications of the proposed rule changes, a series of typical ship stiffened panels and some specially selected panels with high-web stiffeners are calculated and compared with corresponding NLFEM results, which gives good agreement and verifies the correctness of the proposal. For consequence assessments, the proposed rule changes give almost the same buckling check results as current rule for typical ship stiffened panels while giving improved results for the high-web stiffeners with torsional buckling as critical modes, which demonstrates the effectiveness of the proposed rule changes.

The RCP is included in the modified rule text of CSR BC&OT (Jan 2020) Pt1/Ch8/Sec5/2.3.

References

(6) Timoshenko and Gere, Theory of Elastic Stability, 1961.


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Pt 1, Ch 9, Sec 3, [6.1.1] 1. Reason for the Rule Change The proposal is to clarify the application of rule. 2. Background This RCP is based on feedback from the IMO audit team, See 2015 GBS audit “IACS/2015/FR1-8/OB/17 (Post weld treatment)”. Auditors are primarily concerned that a “corrosion free condition” as a prerequisite for the effectiveness of post weld treatment cannot be ensured in practice. TB report, IACS/2015/FR1-8/OB/17 (Post Weld Treatment) Rev 0-2019-02-08, was made and approved to address this observation. In addition to TB report, the rule text is proposed to be amended to alleviate this concern. 3. Impact in Scantlings No scantling impact is expected. Pt 1, Ch 12, Sec 3, [2.4.6] 1. Reason for the Rule Change For clarity, the Rule change is to make the application of partial penetration welding for the end connection of backing bracket and buttress structure, where applicable. 2. Background Relating to the partial penetration welding application to the backing bracket for the PSMs, it is identified that the current rule text could be interpreted as the subjective welding is only required to be applied to the end connection of PSM on longitudinal/transverse bullhead to the double bottom, which isan inconsistent approach to the typical 300 mm partial penetration welding application marks in figure 3. Furthermore, it is noted that the current rule text does not explicitly require the partial penetration welding for the backing bracket, which is normally arranged in conventional oil tanker design, and this could make potential argument between industries. In principal, the partial penetration welding should be applied to the areas, where high tensile stresses are expected. As long as the end connection of the PSM to the double bottom is considered as the high tensile stressed area, it is obvious that the same or comparable magnitude of high tensile stresses would be identified on both ends of backing bracket of the PSM, where applicable. Hence, the same principal for the partial penetration welding application should be applied to the end connections of backing bracket for the consistent approach. Furthermore, the same principle should be also applicable to the end connection of buttress structure to the double bottom. The Rule changes for this clarification also achieves the consistency with the Figure 3 appended to the [2.4.6] of Ch 12, Sec 3. 3. Impact in Scantlings There is no impact on scantlings due to this change.


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Pt 1, Ch 12, Sec 3, [2.5.2] 1. Reason for the Rule Change This proposal is made to correct the welding factor fweld for cleats and fittings on hatch cover and hatch coaming. This factor is set quite big in spite of the thin thickness of hatch cover skin plate. This big fweld leads to abnormally large leg length of fillet welding compare with hull construction and it is the cause of large deformation of skin plate due to large heat effect during the welding. 2. Background According to the requirement, the size of fillet welding is determined based on the as-built thickness of the member being joined and weld leg length for cleats and fittings becomes extremely big in case of thicker plate. The leg length is depends on the strength of cleats / fittings instead of steel plate thickness.

Figure 1: Example of leg length in case of thicker plate Thickness (HT36) Welds

50.0mm 27.0mm

70.0mm 37.0mm

Example of hatch cover 27mm leg length for 50mm fitting

Comparing with the fweld for hull construction as well as the other fittings and equipment, 0.43 instead of 0.6 is proposed for the cleats and fittings to mitigate the unexpected initial deformation due to the heat effect. 3. Impact in Scantlings There is no impact in scantling of hatch cover and hatch coaming.


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Pt 2, Ch 1, Sec 2, [3.3.6], new 1. Reason for the Rule Change The proposal is to clarify the application of rule. 2. Background In URCN to Jan 2014, Modifying the surface finishing factors provided that adequate protective measures are taken as well as refined stress range calculation are carried out. Auditors requires further clarification of “adequate protective measurement”. To address this observation, it is considered necessary to clarify the Rule requirement in respect of protection of the hatch corner from mechanical damage e.g. grooving by grab wire. For this purpose, the rule text is proposed to be amended to alleviate this concern. Protective measure, i.e. fitting a round bar diagonally across the corner slightly above or below the upper deck level, is to be provided by designers for approval. Pt 2 Ch 1 Sec 5 [2.1.1] and [2.1.2] 1. Reason for the Rule Change

See reason for Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 2. Background The requirements in [2.1.1] and [2.1.2] are in accordance with ICLL, MSC Res. 143(77), Reg. 14-1 (1) and (2)). With regard to coaming height reduction or coaming omitting in [2.1.2], the condition is that the Administration is satisfied. So it is decided that they are to be removed because the height requirement in [2.1.1] is the same as ICLL, and the condition that coaming height may be reduced or the coamings may be omitted entirely in [2.1.2] is to be that the Administration is satisfied. The rule text and corresponding TB Reference are removed. 3. Impact in Scantlings There is no impact in scantling Pt 2 Ch 2 Sec 1 [2.1.1] and [2.2.2] 1. Reason for the Rule Change

See reason for Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 2. Background Regarding separation of cargo tanks, the requirements in [2.1.1] and [2.1.2] are the same as some provisions in SOLAS, Ch II-2, Reg. 4.5.1 (as amended). However, the provisions, which are to be complied with in SOLAS, are not introduced in CSR BC & OT completely. The requirements duplicate those in SOLAS but are not complete compared with SOLAS. These requirements are to be removed from CSR BC & OT and Rule Text updates are proposed for [2.1.1].


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3. Impact in Scantlings There is no impact in scantling Pt 2 Ch 2 Sec 4 [1.1.1] and [1.1.2] 1. Reason for the Rule Change

See reason for Pt 1 Ch 2 Sec 2 [1.2.1] and [1.2.2] 2. Background The application of emergency towing arrangements is described as the SOLAS, Ch II-1, Reg. 3-4 (as amended) and MSC Res. 35(63). The designer’s attention is to be drawn on these requirements. Meanwhile, the Rule requirements from [1.2] to [1.6] assume that the emergency towing arrangements comply with the SOLAS regulation. The rule text and TB Rule Reference updates are proposed. 3. Impact in Scantlings There is no impact in scantling.

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